Home

Sysquake Remote User Manual

1. besself n w0 high besself n wl wh stop besself 5 Description besself calculates a Bessel filter The result is given as zeros poles and gain if there are three output arguments or as numerator and denominator coefficient vectors if there are two output arguments besself n w where w0 is a scalar gives a nth order digital low pass filter with a cutoff frequency of w0 relatively to half the sampling frequency besself n wl wh where the second input argument is a vector of two numbers gives a 2nth order digital bandpass filter with pass band between wl and wh relatively to half the sampling frequency besself n w0 high gives a nth order digital highpass filter with a cutoff frequency of w0 relatively to half the sampling frequency besself n wl wh stop where the second input argument is a vector of two numbers gives a 2nth order digital bandstop filter with stopband between wl and wh relatively to half the sampling frequency With an additional input argument which is the string s besself gives an analog Bessel filter Frequencies are given in rad s See also bilinear Analog to digital conversion with bilinear transformation Syntax zd pd kd numd dend bilinear zc pc kc fs bilinear numc denc fs Description bilinear zc pc kc fs converts the analog continuous time transfer function given by its zeros zc poles pc and gain kc
2. 002000 330 6 24 Structure Functions 0 0 020005 335 6 25 Object Functions 000000002 eee 340 6 26 Logical Functions e a eee 342 6 27 Dynamical System Functions 354 6 28 Input Output Functions 2 005 361 6 29 File System Functions 00020005 377 6 30 Time Functions 0 000000 0 eee ee 378 631 MATHlES ia ga eee eo ea es Be a ae Rae as 380 6 32 Shell ope sath ca OGG eed de EE a AE ieas ed ee 381 6 33 Graphics 0000002 ea ia eee 388 6 34 Remarks on graphics 00020005 390 6 35 Base Graphical Functions 40 4 393 6 36 3D Graphics s sate e g socera e aiae a es 417 aa Gad sha ad Pee aay E ee A ee a 418 6 38 Surface shading 0 00002 419 6 39 Functions 2 0000 0c ee ee 419 yun dienes Bebe 433 6 41 Sysquake for ATEX Functions 0 469 Contents 5 471 ad StdliD i sie Gk de ara by at BERE i babes dem KURESE Se mnie ek Ge 472 l Stall ea a a i e a ee Qa wi a ee Bo ea 489 7 3 ClASSES e aos aa ob OG a a Bank oa ee ee RS 498 TA FARON i cas Bho Stak Gh hh eee hw ee eh dem ae bok Be 507 45 bitfield e a ccd ae le Se Ge Ae beck aw we Aw Rack A a 509 TO MEE ge did ges pe das eo haat Aas a Ya esa E 515 TI 2 MEIN seine fee aah aired aa Hasek Gaal ek fon Ge Rerun sees sere ea 525 7 8 SIGEME o sesh a BS RO ERM a Bika Be Bat Bee ec en ks 548 sia Bade
3. LME Reference 55 Source code not found The source code of a function is not avail able File not found fopen does not find the file specified Bad file ID _1 O function with a file descriptor which neither is stan dard nor corresponds to an open file or device Cannot write to file Attempt to write to a read only file Bad seek Seek out of range or attempted on a stream file Too many open files Attempt to open too many files End of file Attempt to read data past the end of a file Timeout Input or output did not succeed before a too large amount of time elapsed No more OS memory The operating system cannot allocate more memory Bad context Call of a function when it should not application dependent Not supported The feature is not supported at least in the current version 6 7 Character Set There exist different standards to represent characters In LME char acters are stored as 16 bit unsigned integer numbers The mapping between these codes and the actual characters they represent de pends on the application and the operating system Currently on Mac OS X and Linux Sysquake uses the UCS 2 character encoding i e plain 16 bit Unicode characters and on Windows the current system character encoding To make the exchange of files possible without manual conversion all text files used by LME applications can have their character set specified explicitly In Sysquake this includes library files Iml SQ
4. operator lt operator lt operator gt Operator lt Less or equal to Syntax LME Reference operators 143 Description x lt y is true if x is less than or equal to y and false otherwise Comparing NaN not a number to any number yields false including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size le x y is equivalent to x lt y It can be used to redefine this op erator for objects Example 2 3 4 lt 2 4 2 TTF See also operator operator operator lt operator gt operator gt Operator gt Greater or equal to Syntax x gt y ge x y Description x gt y is true if x is greater than or equal to y and false otherwise Comparing NaN not a number to any number yields false including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size ge x y is equivalent to x gt y It can be used to redefine this op erator for objects Example 2 3 4 gt 2 4 2 TFT See also operator operator operator lt operator lt operator gt 144 Sysquake for TEX Operator Not Syntax b not b Description b is false logical 0 if b is different from 0 or false and true otherwise If b is an array the operation is performed on each element not b is equivalent to b It can be used to redefine this operator
5. 1 for el 1 abc 2 5 el end el 1 el abc el 2 5 See also variable assignment function endfunction Definition of a function operator or method LME Reference programming constructs 83 Syntax function f statements function f x1 x2 statements function f xl x2 expr2 statements function y f x1 x2 statements function yl y2 x1 x2 statements function class method statements function statements endfunction Description New functions can be written to extend the capabilities of LME They begin with a line containing the keyword function followed by the list of output arguments if any the function name and the list of input arguments between parenthesis if any The output arguments must be enclosed between parenthesis or square brackets if they are several One or more variable can be shared in the list of input and output arguments When the execution of the function terminates ei ther after the last statement or because of the command return the current value of the output arguments as set by the function s state ments is given back to the caller All variables used in the function s statements are local their value is undefined before the first assign ment and it is illegal to use them in an expression and is not shared with variables in other functions or with recursive calls of the same function D
6. 7 b 17 4 7 Loops and Conditional Execution To repeat the execution of some commands you can use either a for end block or a while end block With for you use a variable as a counter gt for i 1 3 i end With while the commands are repeated as long as some expression is true gt i 1 while i lt 10 i 2 x i end i You can choose to execute some commands only if a condition holds true gt if 2 lt 3 ok else amazing end ans ok 4 8 Functions LME permits you to extend its set of functions with your own This is convenient not only when you want to perform the same computation on different values but also to make you code clearer by dividing the whole task in smaller blocks and giving names to them To define a 28 Sysquake for ATEX new function you have to write its code in a file you cannot do it from the command line In Sysquake put them in a function block Functions begin with a header which specifies its name its input arguments parameters which are provided by the calling expression and its output arguments result of the function The input and out put arguments are optional The function header is followed by the code which is executed when the function is called This code can use arguments like any other variables We will first define a function without any argument which just displays a magic square the sum of each line and the sum of each column function m
7. A 1 2 3 4 subsasgn A struct type subs 1 999 999 999 3 4 subsasgn A struct type subs 1 2 4 See also Operart subsref Reference to a part of an array list or structure Syntax B subsref A s Description When an object variable is subscripted in an expression like A s1 s2 LME evaluates subsref A s where subsref is a method of the class of variable A and s is a structure with two fields s type which is and s subs which is the list of subscripts sl1 s2 If a Subscript is the colon character which stands for all elements along the corresponding dimensions it is represented with the string in s subs When an object variable is subscripted in an expression like A s LME evaluates subsref A s where subsref is a method of the class of variable A and s is a structure with two fields s type which is and s subs which is the list containing the single subscript s When the field of an object variable is retrieved in an expression like A f LME executes subsref A s where s is a structure with two LME Reference programming constructs 79 LA fields s type which is and s subs which is the name of the field f in this case While the primary purpose of subsref is to permit the use of sub scripts with objects a built in implementation of subsref is provided for arrays when s type i
8. Description The expression of the switch statement is evaluated If it yields a number it is compared successively to the result of the expressions of the case statements until it matches one then the statements which follow the case are executed until the next case otherwise or end If the case expression yields a vector or a list a match occurs if the switch expression is equal to any of the elements of the case expres sion If no match is found but otherwise is present the statements following otherwise are executed If the switch expression yields a string a match occurs only in case of equality with a case string ex pression or any element of a case list expression LME Reference programming constructs Example Switch option case arithmetic m mean data case geometric m prod data 1 length data otherwise error unknown option end See also try Error recovery Syntax try end try catch end Description 91 The statements after try are executed If an error occurs execution is switched to the statements following try if any or to the statements following end The error message can be retrieved with lasterr or lasterror If no error occurs the statements between try and end are ignored try ignores two errors the interrupt key Control Break on Windows Command on Mac OS X Control C on other operating systems with a keyboard timeout in Sysquake Remote
9. See also disp fprintf which Library where a function is defined Syntax fullname which name 118 Sysquake for ATEX Description which name returns an indication of where function name is defined If name is a user function or a method prefixed with its class and two colons the result is name prefixed with the library name and a slash If name is a built in function it is prefixed with builtin If itis a variable it is prefixed with var If name is neither a function nor a variable which returns the empty string Examples which logspace stdlib logspace which polynom plus classes polynom plus which sin builtin sin x 2 which x var x See also info 6 13 Sandbox Function sandbox Execute untrusted code in a secure environment Syntax sandbox str sandbox str varin varout sandbox str varout sandbox str varin Description sandbox str executes the statements in string str Functions which might do harm if used improperly are disabled they include those related to the file system to devices and to the network Global and persistent variables are forbidden as well but local variables can be created The same restrictions apply to functions called directly or indirectly by statements in str The purpose of sandbox is to permit LME Reference sandbox function 119 the evaluation of code which comes from untrusted sources such as the Internet sandbox str varin evaluat
10. and respectively and work only element wise with scalar arguments Mathematical functions below accept quaternions as arguments with arrays of quaternions they are applied to each element sepa rately 322 Sysquake for ATEX Function Purpose abs absolute value conj conjugate cos cosine exp exponential log natural logarithm real real part sign quaternion sign normalization sin sine sqrt square root Functions below performs computations on arrays of quaternions Function Purpose cumsum cumulative sum diff differences double conversion to array of double mean arithmetic mean sum sum Functions below are related to array size Function Purpose beginning first subscript cat array concatenation end last subscript flipdim flip array fliplr flip left right flipud flip upside down ipermute dimension inverse permutation isempty test for empty array length length of vector ndims number of dimensions numel number of elements permute dimension permutation repmat array replication reshape array reshaping rot90 array rotation size array size squeeze remove singleton dimensions Finally functions below are related to output and assignment LME Reference quaternions 323 Function Purpose disp display dumpvar conversion to string subsasgn assignment to subarrays or to quaternion parts subsref reference to subarrays or to quaternion parts Function imag is replaced with qimag which g
11. r r r roots pol roots M roots M dim Description roots pol calculates the roots of the polynomial pol The polyno mial is given by the vector of its coefficients highest power first while the result is a column vector With a matrix as argument roots M calculates the roots of the polynomials corresponding to each column of M An optional second argument is used to specify in which dimension roots operates 1 for columns 2 for rows The roots of the i th polynomial are in the i th column of the result whatever the value of dim is Examples roots 1 0 1 1 1 roots 1 0 1 242 1 1 roots 1 1 0 5 1 6 1 2 1 3 roots 1 0 1 2 See also schur Schur factorization Syntax U T schur A T schur A U T schur A c T schur A c Description Sysquake for ATEX schur A computes the Schur factorization of square matrix A i e a unitary matrix U and a square matrix T the Schur matrix such that A U T U If Ais complex the Schur matrix is upper triangular and its diagonal contains the eigenvalues of A if A is real the Schur matrix is real upper triangular except that there may be 2 by 2 blocks on the main diagonal which correspond to the complex eigenvalues of A To force a complex Schur factorization with an upper triangular matrix T schur is given a second input argument c or complex Example U T schur 1
12. sort def abcd abc abc abcd def To sort the rows of an array after the first column one can obtain the permutation vector by sorting the first column and use it as subscripts on the array rows Algorithm Shell sort See also squeeze Suppression of singleton dimensions of an array 276 Sysquake for TEX Syntax B squeeze A Description squeeze A returns an array with the same elements as A but where dimensions equal to 1 are removed The result has at least 2 dimen sions row and column vectors keep their dimensions Examples size squeeze rand 1 2 3 1 4 234 size squeeze 1 5 15 See also sub2ind Conversion from row column subscripts to single index Syntax ind sub2ind size i j Description sub2ind size i j gives the single index which can be used to re trieve the element corresponding to the i th row and the j th column of an array whose size is specified by size size must be either a scalar for square matrices or a vector of two elements or more for other arrays If i and j are arrays they must have the same size the result is calculated separately for each element and has the same size Example M 3 6 8 9 M 2 1 8 sub2ind size M 2 1 7 LME Reference arrays M 3 8 See also tril Extraction of the lower triangular part of a matrix Syntax tril M L L tril M k Description 277 tril M
13. 0 78 le 3 sd discrete time transfer function Ts 1le 3 yl ul 0 1 s 0 8 y2 ul 0 15 s 0 78 See also SS Iti append Append the inputs and outputs of systems Syntax b append al a2 Description append al a2 builds a system with inputs ul u2 and outputs yl y2 where ul and u2 are the inputs of al and yl and y2 their outputs respectively append accepts any number of input arguments Libraries lti 529 See also connect ss augstate Extend the output of a system with its states Syntax b augstate a Description augstate a extends the ss object a by adding its states to its out puts The new output yext is y x where y is the output of a and x is its states Iti beginning First index Syntax var beginning Description In an expression used as an index between parenthesis beginning a gives the first valid value for an index It is always 1 See also Iti c2d Conversion from continuous time to discrete time Syntax b c2d a Ts b c2d a Ts method 530 Sysquake for TEX Description c2d a Ts converts the continuous time system a to a discrete time system with sampling period Ts c2d a Ts method uses the specified conversion method method is one of the methods supported by c2dm See also Iti connect Arbitrary feedback connections Syntax b connect a links in out Description connect a Links in out modifies l
14. 2 6x 9 See also polynom polynom polynom feval polynom diff Polynom derivative Syntax diff a Description diff a differentiates polynomial a Example polynom 3 0 1 4 2 diff p p q 12x 3 2x 4 See also polynom polynom polyder 502 Sysquake for ATEX polynom int Polynom integral Syntax int a Description int a integrates polynomial a Example polynom 3 0 1 4 2 int p 0 6x 5 0 3333x 3 2x 2 2x p q Q Il il See also polynom polynom polynom diff polynom inline Conversion from polynom object to inline function Syntax fun inline a Description inline a converts polynomial a to an inline function which can then be used with functions such as feval and ode45 Example p polynom 3 0 1 4 2 fun inline p fun lt inline function gt dumpvar fun fun fun inline function y f x y polyval 3 0 1 4 2 x Libraries classes 503 See also polynom polynom polynom feval polynom feval Evaluate a polynom object Syntax y feval a x Description feval a x evaluates polynomial a for the value of x If x is a vector or a matrix the evaluation is performed separately on each element and the result has the same size as x Example p polynom 3 0 1 4 2 y feval p 1 5 y 2 46 242 770 1882 See also polynom polynom polynom inline feval ratfun ratfun Ratf
15. Description whitecm n creates a color map with n identical entries corresponding to plain white The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 568 Sysquake for ATEX See also colormap cyan2magentacm graycn green2yellowcm interprgbcm magenta2yellowcm red2yel Lowcm 7 13 polyhedra Library polyhedra implements functions which create solid shapes with polygonal facesin 3D Solids are displayed with plotpoly They are defined by the coordinates of their vertices and by the list of vertex indices for each face Other solids such as cylinder and sphere are generated with parametric equations and displayed with surf Some solids have parameters e g for the number of discrete values used for parameters When called without output argument with an op tional trailing string argument for the edge style the solid is displayed with the current scaling and color map With output arguments ar rays X Y Z expected by surf mesh and plotpoly and index array expected by plotpoly are produced They can be modified to move scale or stretch the solids The following statement makes available functions defined in polyhedra use polyhedra cube Create a cube Syntax cube cube style X Y Z ind cube Description Without output argument cube displays a cube i e a co
16. If the function where the error occurs is called itself by another func tion the whole chain of calls is displayed here primes was called by factor at line 174 in library stdlib Here is the list of errors which can occur For some of them LME attempts to solve the problem itself e g by allocating more memory for the task Stack overflow Too complex expression or too many nested func tion calls Data stack overflow Too large objects on the stack in expres sions or in nested function calls Variable overflow Not enough space to store the contents of a variable LME Reference 51 Code overflow Not enough memory for compiling the program Not enough memory Not enough memory for an operation out side the LME core Algorithm does not converge A numerical algorithm does not converge to a solution or does not converge quickly enough This usually means that the input arguments have invalid values or are ill conditioned Incompatible size Size of the arguments of an operator or a func tion do not agree together Bad size Size of the arguments of a function are invalid Non vector array A row or column vector was expected but a more general array was found Not a column vector A column vector was expected but a more general array was found Not a row vector A row vector was expected but a more general array was found Non matrix array A matrix was expected but an array with more than 2 dimensions was foun
17. See also setdiff zeros Null array Syntax zeros n zeros nl n2 zeros n1 n2 A A A A zeros type Description zeros builds an array whose elements are 0 The size of the array is specified by one integer for a square matrix or several integers either as separate arguments or in a vector for an array of any size An additional input argument can be used to specify the type of the result It must be the string double single int8 int16 LME Reference triangulation 281 int32 int64 uint8 uint16 uint32 or uint64 64 bit arrays are not supported on all platforms Examples zeros 2 3 000 000 zeros 2 0 0 0 0 zeros 1 5 uint16 1x5 uintl6 array 0000 0 See also 6 18 Triangulation Functions delaunay 2 d Delaunay triangulation Syntax t delaunay x y t e delaunay x y Description delaunay x y calculates the Delaunay triangulation of 2 d points given by arrays x and y Both arrays must have the same number of values m The result is an array of three columns Each row corre sponds to a triangle values are indices in x and y The second output argument if requested is a logical vector of size m by 1 elements are true if the corresponding point in x and y belongs to the convex hull of the set of points The Delaunay triangulation is a net of triangles which link all the star
18. Syntax wordlength length a Description length a gives the number of bits of bitfield a Example a bitfield 123 16 length a 16 See also bitfield bitfield length bitfield sign Get the sign of a bitfield Syntax s sign a Description Sign a gets the sign of bitfield a The result is 1 if the most significant bit of a is 1 0 if all bits of a are 0 or 1 otherwise Libraries filter 515 Example a bitfield 5 3 a 0b101 sign a 1 See also bitfield uint8 bitfield uint16 bitfield uint32 Convert a bitfield object to an unsigned integer number Syntax n n n uint8 a uint16 a uint32 a Description uint8 a uint16 a and uint32 a convert bitfield a to a uint8 uint16 or uint32 number respectively If a has more bits than the target integer most significant bits are ignored Example a bitfield 1234 16 uint8 a 210 See also t uint32 bitfield int8 bitfield int16 bitfield int32 bitfield double bitfield bitfield H ee 7 6 filter filter is a library which adds to LME functions for designing analog continuous time and digital discrete time linear filters The following statement makes available functions defined in filter 516 Sysquake for ATEX use filter This library provides three kinds of functions besselap buttap cheblap cheb2ap and ellipap which com pute the zeros poles and gain
19. The following statement makes available functions defined in solids use solids cone Cone Syntax cone cone cap cone cap cone cap X Y Z X Y Z style cone cone n ras 574 Sysquake for ATEX Description Without output argument cone draws a cone approximated by a poly hedron The optional first input argument a logical value which is true by default specifies if the cap is included The optional second input argument an integer specifies the number of discrete values for the parameter which describes its surface By default edges are not drawn An optional third input argument a string specifies the edge style it corresponds to the style argument of surf With three output arguments cone produces the X Y and Z arrays expected by surf or mesh and it does not display anything See also crosscap Cross cap Syntax crosscap crosscap n crosscap n style crosscap crosscap n X Y Z X Y Z i ie Description Without output argument crosscap draws a cross cap a self intersecting surface approximated by a polyhedron With an input argument crosscap n draws a cross cap where the two parameters which describe its surface are sampled with n discrete values By default edges are not drawn An optional second input argu ment a string specifies the edge style it corresponds to the style argument of surf With three output arguments crosscap pro
20. axisbounds Description axisbounds bnds sets the bounds of displayed box along each axis Argument bnds is an array of 2 4 or 6 elements With 2 or 4 elements axisbounds has the same effect as scale Merge with scale See also camorbit Camera orbit around target Syntax camorbit dphi dtheta LME Reference 3D graphics 421 Description camorbit dphi dtheta rotates the camera around the target point by angle dphi around the up vector and by angle dtheta around the vector pointing to the right of the projection plane Both angles are given in radians A positive value of dphi makes the camera move to the right and a positive value of dtheta makes the camera move down See also candotly camzoom campan Tilt and pan camera Syntax campan dphi dtheta Description campan dphi dtheta pans the camera by angle dphi and tilts it by angle dtheta Both angles are in radians More precisely the target point is changed so that the vector from view point to target is rotated by angle dphi around the up vector then by angle dtheta around a right vector a vector which is horizontal in view coordinates Camera position Syntax campos p campos auto campos manual p campos 422 Sysquake for TEX Description campos p sets the view position to p p is a 3D vector campos auto sets the view position to automatic mode so that it follows the target campos manual sets the view position to
21. is the transpose of the real matrix M i e columns and rows are per muted If Mis complex the result is the complex conjugate transpose of M If Mis an array with multiple dimensions the first two dimensions are permuted ctranspose M is equivalent to M It can be used to redefine this operator for objects Examples 1 2 3 4 1 3 24 1 2j 3 4j 1 2j 3 4j See also operator conj Operator Transpose Syntax M transpose M Description M is the transpose of the matrix M i e columns and rows are per muted M can be real or complex If Mis an array with multiple dimen sions the first two dimensions are permuted 138 Sysquake for ATEX transpose M is equivalent to M It can be used to redefine this operator for objects Example 1 2 3 4 13 24 1 2j 3 4j 1 2j 3 4j See also Operator Equality Syntax x y eq x y Description x y is true if x is equal to y and false otherwise Comparing NaN not a number to any number yields false including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size eq x y is equivalent to x y It can be used to redefine this oper ator for objects Example 1 true 1 1 eps false 1 1 eps 2 true inf inf true nan nan false LME Reference operators 139 1 2 3 1 3 3 TFT See also operator
22. tion header Bad function header Syntax error in a function header Missing expression Statement such as if or while without ex pression Unexpected expression Statement such as end or else followed by an expression Incomplete expression Additional elements were expected dur ing the compilation of an expression such as right parenthesis or a sub expression at the right of an operator for not followed by a single assignment for is followed by an expression or an assignment with multiple variables Bad variable name The left hand part of an assignment is not a valid variable name e g 2 3 String without right quote The left quote of a string was found but the right quote is missing Unknown escape character sequence Ina string the backslash character is not followed by a valid escape sequence Unexpected right parenthesis Right parenthesis which does not match a left parenthesis Unexpected right bracket Right bracket which does not match a left bracket LME Reference 53 Unrecognized or unexpected token An unexpected character was found during compilation such as 1 end not in an index expression end was used outside of any index sub expression in an expression beginning not in an index expression beginning was used outside of any index sub expression in an expression matrixcol not in an index expression matrixcol was used outside of any index sub expression in an expression matrixrow not
23. 0 9 e or are ignored Decoding stops at the end of the string or when is reached See also base64encode base64encode Encode data using base64 Syntax strt base64encode strb 308 Sysquake for ATEX Description base64encode strb encodes the contents of string strb which rep resents binary data The result contains only characters A Z a z 0 9 and it is suitable for transmission or storage on media which accept only text Each character of encoded data represents 6 bits of binary data i e one needs four characters for three bytes The six bits represent 64 different values encoded with the characters A to Z a to z 0 to 9 and in this order When the binary data have a length which is not a multiple of 3 encoded data are padded with one or two characters to have a multiple of 4 Base64 encoding is an Internet standard described in RFC 1521 Example s base64encode char 10 S AAECAWQFBgcICQo double base64decode s 1 2 3 4 5 6 7 8 9 10 See also base64decode char Convert an array to a character array string Syntax char A char sl s2 S S Description char A converts the elements of matrix A to characters resulting in a string of the same size Characters are stored in unsigned 16 bit words The shape of A is preserved Even if most functions ignore
24. 72 Sysquake for TEX Sysquake Remote O o htmlspecialchars nttp pvars 6 10 Variable Assignment and Subscript ing Variable assignment Assignment to a variable or to some elements of a matrix variable Syntax var expr varl var2 function Description var expr assigns the result of the expression expr to the variable var When the expression is a naked function call varl var2 function assigns the value of the output arguments of the function to the different variables Usually providing less variables than the function can provide just discards the superfluous output arguments however the function can also choose to perform in a different way an example of such a function is size which returns the number of rows and the number of columns of a matrix either as two numbers if there are two output arguments or as a 1 by 2 vector if there is a single output argument Providing more variables than what the function can provide is an error Variables can store any kind of contents dynamically the size and type can change from assignment to assignment A subpart of a matrix variable can be replaced with the use of In this case the size of the variable is expanded when required padding elements are 0 for numeric arrays and empty ar rays for cell arrays and lists See also LME Reference variables 73 beginning First index of an array Syntax v beginning A
25. In all indexing operations several indices can be specified simulta neously to extract more than one element along a dimension A single colon means all the elements along the corresponding dimension Instead of indices the elements to be extracted can be selected by the true values in a logical array of the same size as the variable the result is a column vector or in a logical vector of the same size as the corresponding dimension Calculating a boolean expression based on the variable itself used as a whole is the easiest way to get a logical array Variable indexing can be used in an expression or in the left hand side of an assignment In this latter case the right hand size can be one of the following An array of the same size as the extracted elements A scalar which is assigned to each selected element of the vari able An empty matrix which means that the selected elements should be deleted Only whole rows or columns or hyper planes for arrays of more dimensions can be deleted i e a 2 5 and b 3 6 8 if b is a row or column vector are legal while c 2 3 is not When indices are larger than the dimensions of the variable the vari able is expanded new elements are set to 0 for numeric arrays false for logical arrays the nul character for character array and the empty array for cell arrays Lists In LME lists have one dimension thus a single index is re quired Be it with
26. M2 Description Between brackets the comma is used to separate elements on the same row in a matrix Elements can be scalars vector or matrices their number of rows must be the same unless one of them is an empty matrix Outside brackets or between parenthesis the comma is used to separate statements or the arguments of functions 148 Sysquake for ATEX horzcat M1 M2 is equivalent to M1 M2 It can be used to rede fine this operator for objects Between braces the comma separates cells on the same row Examples 1 2 12 3 5 ones 2 311 5 Ll abc def abcdef See also Operator Vertical matrix concatenation Syntax M1 M2 vertcat M1 M2 Description Between brackets the semicolon is used to separate rows in a matrix Rows can be scalars vector or matrices their number of columns must be the same unless one of them is an empty matrix Outside brackets the comma is used to separate statements they loose any meaning between parenthesis and give a syntax error vertcat M1 M2 is equivalent to M1 M2 It can be used to rede fine this operator for objects Between braces the semicolon separates rows of cells Examples 1 2 1 2 1 5 3 2 4 5 1 12345 32451 LME Reference operators 149 abc def abc def See also operator operator Operator Range Syntax x1 x2 x1l step x2 colon x1 x2 colon x1 step x2 Description
27. See also asech Inverse hyperbolic secant Syntax y asech x Description asech x gives the inverse hyperbolic secant of x which is complex if x is complex or strictly negative See also sech asec acosh asin Arc sine Syntax y asin x Description asin x gives the arc sine of x which is complex if x is complex or if abs x gt 1 Examples asin 5 0 5236 asin 2 1 5708 1 317j See also LME Reference mathematical functions 157 asinh Inverse hyperbolic sine Syntax y asinh x Description asinh x gives the inverse hyperbolic sine of x which is complex if x is complex Examples asinh 2 1 4436 asinh 0 1 2j 0 1 8055 1 7359j See also Sinh lacosh asin atan Arc tangent Syntax y atan x Description atan x gives the arc tangent of x which is complex if x is complex Example atan 1 0 7854 See also 158 Sysquake for TEX atan2 Direction of a point given by its Cartesian coordinates Syntax phi atan2 y x Description atan2 y x gives the direction of a point given by its Cartesian coor dinates x and y Imaginary component of complex numbers is ignored atan2 y x is equivalent to atan y x if x gt 0 Examples atan2 1 1 0 7854 atan2 1 1 2 3562 atan2 0 0 0 See also atanh Inverse hyperbolic tangent Syntax y atanh x Description atan x gives the inverse hyperbolic tangent of x which is co
28. Syntax A2 flipud A1 Description flipud A1 gives an array A2 which has the same size as Al but where all lines are placed in reverse order Example flipud 1 2 3 4 34 12 See also ind2sub Conversion from single index to row column subscripts Syntax i j ind2sub size ind Description ind2sub size ind gives the subscripts of the element which would be retrieved from an array whose size is specified by size by the single index ind size must be either a scalar for square matrices or a vector of two elements or more for arrays ind can be an array the result is calculated separately for each element and has the same size Example M 3 6 8 9 M 3 8 i j ind2sub size M 3 p Fi N ie M i j 8 256 Sysquake for ATEX See also jsub2ind size interpn Interpolation Syntax Vi interpn xl xn V xil Xin Vi interpn xl xn V xil xin method Description interpn x1 xn V xil xin interpolates data in a space of n dimensions Input data are defined by array V where element V i j Corresponds to coordinates x1 i x2 j etc Interpolation is performed for each coordinates defined by arrays xil xi2 etc which must all have the same size the result is an array of the same size Length of vectors x1 x2 must match the size of V along the corresponding dimension Vectors x1 x2 must be sorted mono tonically increasing or de
29. barh 2 4 3 6 3 5 4 1 Stacked horizontal bar plot barh 1 4 magic 4 stacked Horizontal interval plot barh 1 4 2 4 3 1 5 6 4 6 interval See also circle Add circles to the figure Syntax circle x y r circle x y r style circle x y r style id 398 Sysquake for TEX Description circle x y r draws a circle of radius r centered at x y The argu ments can be vectors to display several circles Their dimensions must match scalar numbers are repeated if necessary The optional fourth and fifth arguments are the style and object ID cf their above In mouse handlers _x0 and _y correspond to the projection of the mouse click onto the circle _nb is the index of the circle in x y and r and _ix is empty Circles are displayed as circles only if the scales along the x and y axis are the same and linear With different linear scales circles are displayed as ellipses With logarithmic scales they are not displayed Examples circle 1 2 5 r 1 circle zeros 10 1 zeros 10 1 1 10 See also colormap Current colormap from scalar to RGB Syntax colormap clut clut colormap Description Command colormap clut changes the color mapping from scalar values to RGB values used by commands such as pcolor image and surf Colormaps are arrays of size n by 3 Each row corresponds to a color the first column is the intensity of red from 0 no red component to 1 maxi
30. beginning function e C beginning obj i n Description In an expression used as an index to access some elements of an array beginning gives the index of the first element line or column depending of the context It is always 1 for native arrays beginning can be overloaded for objects of used defined classes Its definition should be have a header equivalent to function e C beginning obj i n where C is the name of the class obj is the object to be indexed i is the position of the index expression where beginning is used and n is the total number of index expressions See also Operator 0 operator 1 end Last index of an array Syntax v end A end function e I C end obj i n Description In an expression used as an index to access some elements of an ar ray end gives the index of the last element line or column depending of the context end can be overloaded for objects of used defined classes Its definition should be have a header equivalent to function e C end obj i n where C is the name of the class obj is the object to be indexed i is the position of the index expression where end is used n is the total number of index expressions 74 Sysquake for ATEX Examples Last 2 elements of a vector a 1 5 a end 1 end 45 Assignment to the last element of a vector a end 99 a 123499 Extension of a vector a end 1 100 a 123 4 99 100 See
31. coth x Description coth x gives the hyperbolic cotangent of x which is complex if x is See also tanh csc Cosecant Syntax y csc x 166 Sysquake for TEX Description csc x gives the cosecant of x which is complex if x is See also acsc csch Hyperbolic cosecant Syntax y csch x Description csch x gives the hyperbolic cosecant of x which is complex if x is See also diln Dilogarithm Syntax y diln x Description diln x gives the dilogarithm or Spence s integral of x Argument and result are real imaginary part is discarded The dilogarithm is defined as x log t dinco gl Hi 1 t 1 Example diln 0 2 0 7 10 1 0748 0 3261 3 9507 LME Reference mathematical functions 167 double Conversion to double precision numbers Syntax B double A Description double A converts number or array A to double precision A can be any kind of numeric value real complex or integer or a character or logical array To keep the integer type of logical and character arrays the unitary operator should be used instead Examples double uint8 3 3 double AB 65 66 islogical double 1 gt 2 false See also and related functions SingTel cast operator setstr char logical ellipam Jacobi elliptic amplitude Syntax phi phi ellipam u m ellipam u m tol Description ellipam u m gives the Jacobi ellip
32. extracts the lower triangular part of a matrix the result is a matrix of the same size where all the elements above the main di agonal are set to zero A second argument can be used to specify another diagonal O is the main diagonal positive values are above and negative values below Examples magic 3 BWOrFRWOrRWO ll I ct ct 3 5 OUrPRAOUG OUP N OO N UND m NN O See also triu diag 278 Sysquake for ATEX triu Extraction of the upper triangular part of a matrix Description tril M extracts the upper triangular part of a matrix the result is a matrix of the same size where all the elements below the main di agonal are set to zero A second argument can be used to specify another diagonal 0 is the main diagonal positive values are above and negative values below Examples M M magic 3 ct s QOooroooar AWO I il D Ee ee ae NN OS NNO rt rp m ONO See also tril diag union Set union Syntax c union a b c ia ib union a b LME Reference arrays 279 Description union a b gives the union of sets a and b i e it gives the set of members of sets a or b or both Sets are any type of numerical char acter or logical arrays or lists or cell arrays of character strings Mul tiple elements of input arguments are considered as single members the result is always sorted and has unique elements The optional second an
33. false false true true false true false true FTTF xor pi 8 false See also 6 27 Dynamical System Functions This section describes functions related to linear time invariant dy namical systems c2dm Continuous to discrete time conversion Syntax numd dend c2dm numc denc Ts dend c2dm numc denc Ts numd dend c2dm numc denc Ts method dend c2dm numc denc Ts method Ad Bd Cd Dd c2dm Ac Bc Cc Dc Ts method Description numd dend c2dm numc denc Ts converts the continuous time transfer function numc denc to a discrete time transfer function numd dend with sampling period Ts The continuous time transfer function is given by two polynomials in s and the discrete time transfer function is given by two polynomials in z all as vectors of coefficients with highest powers first c2dm numc denc Ts method uses the specified conversion method method is one of LME Reference dynamical system functions 355 zoh or z zero order hold default foh or f first order hold tustin or t Tustin bilinear transformation matched or m Matched zeros poles and gain The input and output arguments numc denc numd and dend can also be matrices in that case the conversion is applied separately on each row with the same sampling period Ts c2dm Ac Bc Cc Dc Ts method performs the conversion from continuous time state space model Ac Bc Cc Dc to discrete time state
34. hist Y N X hist Y m N X hist Y m dim N hist Y X N hist Y X dim Description hist Y gives the number of elements of vector Y in 10 equally spaced intervals A second input argument may be used to specify the number of intervals The center of the intervals may be obtained in a second output argument If Y is an array histograms are computed along the dimension spec ified by a third argument or the first non singleton dimension the re sult N has the same size except along that dimension When the second argument is a vector it specifies the centers of the intervals 480 Sysquake for TEX Example N X hist logspace 1 5 N 45 21 14 11 9 X 1 9 3 7 5 5 7 3 9 1 ifftshift Shift DC frequency of FFT from center to beginning of spectrum Syntax Y ifftshift X Description ifftshift X shifts halves of vector 1 d or matrix 2 d X to move the DC component from the center It should be used before ifft or ifft2 It reverses the effect of fftshift See also isprime Prime number test Syntax b isprime n Description isprime n returns true if nis a prime number or false otherwise If n is a matrix the test is applied to each element and the result is a matrix the same size Examples isprime 7 true isprime 0 2 10 FTF Libraries stdlib 481 See also primes isreal Test for a real number Syntax b isreal x Description isr
35. if both operands are false or 0 and true logical 1 otherwise or b1 b2 is equivalent to b1 b2 It can be used to redefine this operator for objects Example false false true true false true false true FTTT See also Operator Or with lazy evaluation Syntax b1 b2 Description b1 b2 is b1 if b1 is true and b2 otherwise Like with if and while statements b1 is true if it is a nonempty array with only non zero elements b2 is evaluated only if b1 is false b1 b2 bn returns the last operand which is true remaining operands are not evaluated or the last one Example Boolean value which is true if the vector v is empty or if its first ele ment is NaN isempty v isnan v 1 See also operator amp amp LME Reference operators 147 Operator Alternative with lazy evaluation Syntax b x y Description b x y is x if b is true and y otherwise Like with if and while state ments b is true if it is a nonempty array with only non zero elements Only one of x and y is evaluated depending on b Operators and have the same priority parenthesis or brackets should be used if e g x or y is a range Example Element of a vector v or default value 5 if the index ind is out of range ind lt 1 ind gt length v 5 v ind See also operator amp amp operator Operator Horizontal matrix concatenation Syntax M1 M2 M1 M2 horzcat M1
36. is replaced with its elements v may be a list variable or the result of an expres sion Examples 1 abc 3 5 7 1 x 1 string real 2x2 xX 128 Sysquake for ATEX x 3 35 7 1 x 2 2 3j x 1 2 3j real 2x2 x 3 2 x 1 2 3j list x end 1 123 x 1 24 3j list 123 C 1 false ab magic 3 2x2 cell array C 2 1 ab a 1 3 5 fprintf sd a 99 1345 99 See also operator TT operator U Operator dot Structure field access Syntax v field v field expr Description A dot is used to access a field in a structure In v field v is the name of a variable which contains a structure and field is the name of the field In expressions v field gives the value of the field it is an error if it does not exist As the target of an assignment the value of the field is replaced if it exists or a new field is added otherwise if v itself is not defined a structure is created from scratch v itself may be an element or a field in a larger variable provided it is a structure or does not exists in an assignment i e complicated assignments like a 2 f 3 2 5 3 are accepted The syntax v expr permits to specify the field name dynamically at run time as the result of evaluating expression expr LME Reference operators 129 v f is equivalent to v f This syntax is more elegant than functions getfield and setfield Examples N
37. it no need to copy and format numerical results or generate EPS file by hand the package takes care of cumbersome tasks letting you focus on the creative work Sysquake for LaTeX uses the same language and libraries as Sysquake and other products of Calerga Chapter 2 Installing Sysquake for IATEX Installing Sysquake for LaTeX consists in three steps Copy all Sysquake for LaTeX files to your hard disk The default location is opt SysquakeLx on unix systems and C Program Files SysquakeForLaTexX on Windows Update environment variables so that sysquakelatextool is found by the operating system and finds library files This consists in adding the path of the directory containing sysquakelatextool to PATH and setting a new environment variable SYSQUAKELXPATH to the semicolon separated list of paths to library directories Make your TeX distribution able to load the sysquake sty pack age In TDS based distributions TeX Live teTex MiKTeX etc this means adding the path of TexFiles to the list of TDS roots LaTeX must also be able to execute external programs with write18 Depending on the platform this may require to add an option argu ment to pdflatex such as shell escape or enable write18 2 1 Windows Install the latest release of MiKTeX e g MiKTeX 2 6 2742 Then run Sysquake for LaTeX installer and follow the instructions The last step as explained in the installer is manual make your TeX distribu
38. program the default level of details basic pLlotoption is used In the example below the grid for the complex plane of the z transform is displayed with full details Once the figure is displayed the user is free to reduce the level of details with the Grid menu scale equal 2 2 2 2 zgrid plotoption fullgrid plotroots 1 1 5 0 8 6 35 Base Graphical Functions activeregion Region associated with an ID Syntax activeregion xmin xmax ymin ymax id activeregion X Y id 394 Sysquake for ATEX Description The command activeregion defines invisible regions with an ID for interactive manipulations in Sysquake Contrary to most other graph ical objects a hit is detected when the mouse is inside the region not close like with points and lines activeregion xmin xmax ymin ymax id defines a rectangular shape activeregion X Y id defines a polygonal shape The start and end points do not have to be the same the shape is closed automati cally Example Rectangular button If an ID was given to plot without activeregion a hit would be detected when the mouse is close to any of the four cor ners with activeregion a hit is detected when the mouse is inside the rectangle plot 50 70 70 50 50 10 10 30 30 10 activeregion 50 70 10 30 1 See also area Area plot Syntax area y area x y area x y y0 area style area style id Description With
39. read write text mode seek to end See also 6 30 Time Functions clock Current date and time Syntax t clock Description clock returns a 1x6 row vector containing the year four digits the month the day the hour the minute and the second of the current date and time All numbers are integers except for the seconds which are fractional The absolute precision is plus or minus one second with respect to the computer s clock the relative precision is plus or minus 1 microsecond on a Macintosh and plus or minus 1 millisecond on Windows Example clock 1999 3 11 15 37 34 9167 LME Reference time 379 See also tic Start stopwatch Syntax tic Description tic resets the stopwatch Typically tic is used once at the beginning of the block to be timed See also toc Elapsed time of stopwatch Syntax elapsed_time toc Description toc gets the time elapsed since the last execution of tic Typically toc is used at the end of the block of statements to be timed On multi tasking operating systems like Windows Mac OS X and Unix toc measures only the time spent in the LME application Other processes do not have a large impact For instance typing tic at the command line prompt waiting 5 seconds and typing toc will show a value much smaller than 5 Example tic x eig rand 200 toc 0 3046 380 Sysquake for ATEX See also 6 31 MAT files matfiledecode Decode the co
40. sponds to samples and columns to channels nbits can be 8 or 16 With 2 or 3 input arguments wavwrite returns the contents of the WAV file as a vector of class uint8 The default word size is 16 bits per sample and channel Example sr 44100 t O sr sr s Sin 2 pi 740 t wavwrite map2int s 1 1 int16 sr 16 beep wav See also 7 10 date date is a library which adds to LME functions to convert date and time between numbers and strings The following statement makes available functions defined in date use date datestr Date to string conversion Syntax str datestr datetime str datestr date format Description datestr datetime converts the date and time to a string The input argument can be a vector of 3 to 6 elements for the year month day hour minute and second a julian date as a scalar number or a string which is converted by datevec The result has the following format jj mmm yyyy HH MM SS Libraries date 557 where jj is the two digit day mmm the beginning of the month name yyyy the four digit year HH the two digit hour MM the two digit minute and SS the two digit second The format can be specified with a second input argument When datestr scans the format string it replaces the following sequences of characters and keeps the other ones unchanged Sequence Replaced with dd day 2 digits ddd day of week 3 char HH hour 2 digits 01 12 or 00
41. starts a new fragment of code which corresponds to command sqgexpr or environment sysquake in LaTeX Argument fragid is a string made only of letters which is used as fragment identifier it is typically the fragment index as a lowercase roman number starting at 1 Arguments width and height are the dimensions of the figure reserved in LaTeX or negative if the result is text without graphics sqlxstartfragment is used by the Sysquake package it should not be called from tex files See also sqlxendf ragment beginfigure Chapter 7 Libraries Libraries are collections of functions which complement the set of built in functions and operators of LME the programming language of Sysquake To use them type or add in the functions block of the SQ files which rely on them a use command such as use stdlib bench bench implements a benchmark which can be used to com pare the performance of LME on different platforms bitfield bitfield implements constructors and methods for bit fields binary numbers Standard operators are redefined to en able the use of amp and for bitwise operations and subscripts for bit extraction and assignment classes classes implements constructors and methods for poly nomial and rational functions With them you can use standard operator notations such as or colormaps colormaps defines functions which create color maps for command colormap constants constants defines physical constant
42. when ais a transfer function or a matrix of transfer func tions creates a new system described by a matrix of transfer functions where a is repeated n times horizontally and vertically If a is a state space system matrices B C and D are replicated to obtain the same effect repmat a m n or repmat a m n repeats matrix a m times ver tically and n times horizontally See also Libraries lti 541 Iti size Number of outputs and inputs Syntax s size a nout nin size a n size a dim Description With one output argument size a gives the row vector nout nin where nout is the number of outputs of system a and nin its number of inputs With two output arguments size a returns these results separately as scalars size a 1 gives only the number of outputs and size a 2 only the number of inputs See also Iti ssdata Get state space matrices Syntax A B C D ssdata a A B C D Ts ssdata a Description ssdata a where a is any kind of LTI object gives the four matrices of the state space model and optionally the sampling period or the empty array for continuous time systems See also Iti subsasgn Assignment to a part of an LTI system 542 Sysquake for ATEX Syntax var i j a var ix a var select a var field value a subsasgn a s b Description The method subsasgn a permits the use of all kinds of assignments to a part of an LTI system If
43. 0 0 0 1 0 4 0 4 1 0 8 0 8 0 8 i e black dark blue light blue and light gray Warning depending on the size of the figure in pixels and the speed of the computer the computation may be slow several sec onds The number of sizes does not have a big impact Example Roots of the polynomial z 0 8 z 0 7 0 6 z 0 7 0 6 where the coefficients in R have an uncertainty bounded by a unit sphere see Fig 6 14 scale equal 2 2 2 2 erlocus poly 0 8 0 7 0 6j 0 7 0 6j eye 3 zgrid 448 Sysquake for ATEX erlocus Figure 6 14 erlocus poly 8 7 6j 7 6j eye 3 See also plotroots hgrid Hall chart grid Syntax hgrid hgrid style Description hgrid plots a Hall chart in the complex plane of the Nyquist diagram The Hall chart represents circles which correspond to the same mag nitude or phase of the closed loop frequency response The optional argument specifies the style The whole grid is displayed only if the user selects it in the Grid menu or after the command plotoption fullgrid By default only the unit circle and the real axis are displayed The whole grid is made of the circles corresponding to a closed loop magnitude of 0 2 0 5 0 8 1 1 0 8 2 and 5 and to a closed loop phase of plus or minus O 10 20 30 45 60 and 75 degrees LME Reference graphics for dynamical systems 449 Figure 6 15 scale equal 2 2 2 2 hgrid
44. 0 and 1 a and b ceil x gives the smallest integer larger than or equal to x If the argument is a complex number the real and imaginary parts are han dled separately Examples ceil 2 3 3 ceil 2 3 2 ceil 2 3 4 5j 3 4j LME Reference mathematical functions 163 See also complex Make a complex number Syntax z complex x y Description complex x y makes a complex number from its real part x and imag inary part y The imaginary part of its input arguments is ignored Examples complex 2 3 2 3j complex 1 5 2 14 2j 2 2j 3 2j 44 2j 5 2j See also conj Complex conjugate value Syntax w conj z Description conj z changes the sign of the imaginary part of the complex num ber z Example conj 1 2j 3 5j 4 0 1 2 3 5j 40 164 Sysquake for ATEX See also cos Cosine Syntax y cos x Description cos x gives the cosine of x which is complex if x is complex Example cos 0 1 2j 1 2 0327 3 0519j See also cosh Hyperbolic cosine Syntax y cosh x Description cos x gives the hyperbolic cosine of x which is complex if x is com plex Example cosh 0 1 2j 1 0 6421 1 0686j LME Reference mathematical functions 165 See also cos cot Cotangent Syntax y cot x Description cot x gives the cotangent of x which is complex if x is See also acot coth Hyperbolic cotangent Syntax y
45. 2 3 4 U 0 8246 0 5658 0 5658 0 8246 0 T 0 3723 1 0 5 3723 eig 1 2 3 4 ans 0 3723 5 3723 LME Reference linear algebra 243 T schur 1 0 0 0 1 2 0 3 1 T 1 0 0 0 1 2 0 3 1 T schur 1 0 0 0 1 2 0 3 1 c T 1 0 0 0 1 2 4495j 1 0 0 1 2 4495j See also skewness Skewness of a set of values Syntax skewness A skewness A dim S S Description skewness A gives the skewness of the columns of array A or of the row vector A The dimension along which skewness proceeds may be specified with a second argument The skewness measures how asymmetric a distribution is It is 0 for a symmetric distribution and positive for a distribution which has more values much larger than the mean Example skewness randn 1 10000 2 2 6833 See also sqrtm Matrix square root 244 Sysquake for TEX Syntax Y sqrtm X Y err sqrtm X Description sqrtm X returns the matrix square root of X such that sqrtm X 2 X X must be square The matrix square root does not always exist With a second output argument err sqrtm also returns an estimate of the relative error norm sqrtm X 2 X norm X Example Y sqrtm 1 2 3 4 Y 0 5537 0 4644 0 807 0 2124 1 2104 0 3186j 1 7641 0 1458 Y 2 1 2 3 4 See also std Standard deviation Syntax lt lt lt XK xX nou ot ou wn ct a s3s3 lt lt Description std v gives the standard
46. 23 MM minute 2 digits mm month 2 digits mmm month 3 char PM AM or PM QQ quarter Q1 to Q4 SS second 2 digits yy year 2 digits yyyy year 4 digits If the sequence PM is found the hour is between 1 and 12 other wise between 0 and 23 Examples datestr clock 18 Apr 2005 16 21 55 datestr clock ddd mm dd yyyy HH MM PM Mon 04 18 2005 04 23 PM See also julian2cal datevec String to date and time conversion Syntax datetime datevec str Description datevec str converts the string str representing the date and or the time to a row vector of 6 elements for the year month day hour minute and second The following formats are recognized 558 Sysquake for ATEX Example Value 20050418T162603 ISO 8601 date and time 2005 04 18 year month and day 2005 Apr 18 year month and day 18 Apr 2005 day month and year 04 18 2005 month day and year 04 18 00 month day and year 18 04 2005 day month and year 18 04 05 day month and year 16 26 03 hour minute and second 16 26 hour and minute PM afternoon Unrecognized characters are ignored If the year is given as two digits it is assumed to be between 1951 and 2050 Examples datevec Date and time 20050418T162603 2005 4 18 16 26 3 datevec 03 57 PM 0 0 0 15 57 0 datevec 1 Aug 1291 1291 8 1 0 0 0 datevec At 16 30 on 11 04 07 2007 11 4 16 30 0 See also weekday Week day of a given date Syn
47. 3 true isvector 1 2 true isvector 7 true isvector 1 2 3 4 false See also tength Icm Least common multiple Syntax q lcm a b Description lcm a b gives the least common multiple of integer numbers a and b Example lcm 72 56 504 See also gcd log Natural base e logarithm Syntax y log x 188 Sysquake for ATEX Description log x gives the natural logarithm of x It is the inverse of exp The result is complex if x is not real positive Example log 1 0 1 10 1 2j 0 3 1416j inf 0 2 3026 0 8047 1 1071j See also log10 Decimal logarithm Syntax y logl10 x Description log10 x gives the decimal logarithm of x defined by log10 x log x log 10 The result is complex if x is not real positive Example log10 1 0 1 10 1 2j 0 1 3644j inf 0 1 0 3495 0 4808 See also loglp Logarithm of x plus one Syntax y loglp x LME Reference mathematical functions Description logip x is log 1 x with improved precision for small x Example loglp le 15 le 15 log 1 le 15 1 1102e 15 See also log2 Base 2 logarithm Syntax y log2 x Description log2 x gives the base 2 logarithm of x defined 189 as log2 x log x log 2 The result is complex if x is not real positive Example log2 1 2 1024 2000 5 1 10 10 9658 2 3219 4 5324j See also mod Modulo Syntax m mod x y 190 Sysquake fo
48. 3 octal or 2 hex adecimal digits are decoded the first non octal or non hexadecimal digit marks the end of the sequence The null character can conve niently be encoded with its octal representation 0 provided it is not followed by octal digits it should be written 000 in that case It is an error when another character is found after the backslash Single ticks can be represented either by a backslash followed by a single tick or by two single ticks Depending on the application and the operating system strings can contain directly Unicode characters encoded as UTF 8 or MBCS multi byte character sequences 16 bit characters encoded with uhhhh escape sequences are always accepted and handled correctly by all built in LME functions low level input output to files and devices which are byte oriented is an exception explicit UTF 8 conversion should be performed if necessary Lists and cell arrays Lists are ordered sets of other elements They may be made of any type including lists Literal lists are enclosed in braces elements are separated with commas 1 3 6 2 9 abc 1 xx Lists can be empty List s purpose is to collect any kind of data which can be assigned to variables or passed as arguments to functions Cell arrays are arrays whose elements or cells contain data of any type They differ from lists only by having more than one dimension LME Reference 47 Most functions which expect lis
49. 5 3 9 a 1 3 9 abc 8 1 6 3 5 7 4 9 2 3 9 Removing elements in a list and have the same effect here a 4 a 1 3 9 abc 3 9 a 1 3 a 3 91 13 9 Replacing NaN with empty arrays in a cell array abc nan 2 false C cellfun x any isnan x C See also Operator endl Operator Brackets Syntax matrix_elements 126 Sysquake for ATEX Description A pair of brackets is used to define a 2 d array given by its elements or by submatrices The operator or spaces is used to separate elements on the same row and the operator or newline is used to separate rows Since the space is considered as a separator when it is in the direct scope of brackets it should not be used at the top level of expressions as long as this rule is observed each element can be given by an expression Inside brackets commas and semicolons are interpreted as calls to horzcat and vertcat Brackets themselves have no other effect than changing the meaning of commas semicolons spaces and new lines the expression 1 for instance is strictly equivalent to 1 The empty array is a special case Since horzcat and vertcat also accept cell arrays brackets can be used to concatenate cell arrays too Examples 1 2 3 5 C1 1 2 C2 ab false C1 C2 2x2 cell array Compare this with the effect of braces where e
50. 9 Local and Global Variables 30 33 5 1 Package options 00000002 eee 33 5 2 Other packages 0000 eee ee 34 5 3 COMMANAS of e Be a BR eo ae ew 35 5 4 Environment 0 0 0 000000000 ce eee 36 4 Sysquake for ATEX 39 6 1 Program format 2 000002 eee eee 39 6 2 Function Call s ea ce ee ee ee ee ee ee eee 40 6 3 LIDMAneS i s s A ws soshal oh SE ea he ae es a AE 42 Cech ak ee bea Pa we oh eG SO A ak 42 6 5 Input and Output 20000 0008 49 6 6 ErrorMessages 0000 eee es 50 6 7 Character Set e ei asi ees Eek ey aie bea each es 55 6 8 Formatted text 0 000000 002 eee 56 6 9 List of Commands Functions and Operators 62 6 10 Variable Assignment and Subscripting 72 6 11 Programming Constructs 2 004 79 6 12 Miscellaneous Functions 00 94 Sie th As ie Be meat ace aaa Ge ee de 118 Les ERR ES EP oe ME Re Ow ee ee eee 120 6 15 Mathematical Functions 4 152 ap ais e e AOE eee aah eae a ee ae 203 6 17 Array FUNCIONS s s sos a eee ee ee ee ee eS 248 6 18 Triangulation Functions 004 281 6 19 Integer Functions 00 0000 eee 287 6 20 Non Linear Numerical Functions 291 6 21 String Functions 0000002 ee eee 307 6 22 Quaternions 0 0000 eee ee ee ee 320 6 23 List Functions
51. 97 98 99 100 count 4 data count sread abcd inf 3 data 97 98 99 count 3 See also sscanf Decoding of formatted numbers LME Reference input output 375 Syntax r sscanf str format r count scanf str format r count nchar scanf str format Description Numbers characters and strings are extracted from the first argu ment Exactly what is extracted is controlled by the second argument which can contain the following elements Substring in format Meaning single character string integer number in decimal unsigned integer number in hexadecimal unsigned integer number in octal integer number floating point number Oo K aN H X X X X XXXL e floating point number g floating point number other character exact match i recognizes an optional sign followed by a decimal number an hexadecimal number prefixed with 0x or 0X a binary number prefixed with Ob or OB or an octal number prefixed with 0 The decoded elements are accumulated in the output argument either as a column vector if the format string contains d 0 x i f e or g or a string if the format string contains only c s or literal values If a star is inserted after the percent character the value is decoded and discarded A width as one or more decimal characters can be inserted before s d x o i f e or g it limits the number of characters to be decoded In the input string sp
52. Example Hall chart grid with a Nyquist diagram see Fig 6 15 scale equal 1 5 1 5 1 5 1 5 hgrid nyquist 20 poly 1 2 1j 2 1j See also plotoption hstep Step response plot of a discrete time transfer function followed by a continuous time transfer function Syntax hstep numd dend Ts numc denc hstep numd dend Ts numc denc style hstep numd dend Ts numc denc style id 450 Sysquake for TEX scale 0 5 hstep 1 0 5 1 1 1 5 Figure 6 16 scale 0 5 hstep 1 1 0 5 1 1 1 5 Description A step is filtered first by numd dend a discrete time transfer function with sampling period Ts the resulting signal is converted to continuous time with a zero order hold and filtered by the continuous time transfer function numc denc Most discrete time controllers are used with a zero order hold and a continuous time system hstep can be used to display the simu lated output of the system when a step is applied somewhere in the loop e g as a reference signal or a disturbance The transfer func tion numd dend should correspond to the transfer function between the step and the system input the transfer function numc denc should be the model of the system Note that the simulation is performed in open loop If an unsta ble system is stabilized with a discrete time feedback controller all closed loop transfer functions are stable however the simulation with hstep which uses t
53. M1 M2 is equivalent to M1 inv M2 mrdivide x y is equivalent to x y It can be used to redefine this operator for objects LME Reference operators Example 9 3 3 2 6 1 2 3 4 5 1 4 10 2 25 See also Operator Scalar right division Syntax x y M1 M2 M X x JM rdivide x y Description 133 The first operand is divided by the second operand If both operands are matrices with a size different from 1 by 1 their size must be equal the division is performed element wise If one operand is a scalar it is repeated to match the size of the other operand rdivide x y is equivalent to x y It can be used to redefine this operator for objects Examples 3 10 3 5 12 See also 134 Sysquake for ATEX Operator Matrix left division Syntax x y M1 M2 x M mldivide x y Description x y divides the second operand by the first operand If the first operand is a scalar it divides each element of the second operand Otherwise it must be a square matrix M1 M2 is equivalent to inv M1 M2 mldivide x y is equivalent to x y It can be used to redefine this operator for objects Examples 3 9 3 1 2 3 4 2 6 2 4 10 NZ ON 5 See also Operator Scalar left division Syntax M1 M2 M1 x ldivide x y LME Reference operators 135 Description The second operand is divided by the first operand If both ope
54. M2 dim convolves along the dimension dim 1 for columns and 2 for rows If one of the matrices has only one column or one row it is repeated to match the size of the other argument Example conv 1 2 1 2 3 1476 conv 1 2 1 2 3 4 5 6 2 147 6 4 13 16 12 208 Sysquake for ATEX See also conv2 Two dimensions convolution of matrices Syntax M M conv2 M1 M2 conv2 M1 M2 kind Description conv2 M1 M2 convolves the matrices M1 and M2 along both directions The optional third argument specifies how to crop the result Let nl1 ncl size M1 and nl2 nc2 size M2 With kind full default value the result M has nl1 nl2 1 lines and ncl nc2 1 columns With kind same the result M has nl1 lines and ncl columns this options is very useful if M1 represents equidistant samples in a plane e g pixels to be filtered with the finite impulse response 2 d filter M2 With kind valid the result M has nl1 nl2 1 lines and ncl nc2 1 columns or is the empty matrix if M1 represents data filtered by M2 the borders where the convolution sum is not totally included in M1 are removed Examples conv2 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 13 65 3 5 12 21 16 9 12 27 45 33 18 11 24 39 28 15 7 15 24 17 9 conv2 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 same 12 21 16 27 45 33 24 39 28 conv2 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 valid 45 See also conv LME Reference line
55. Refine set to 1 and 4 Events EventTime InitialStep MaxStep MinStep NormControl false OnEvent OutputFecn PreArg Refine RelTol le 3 Stats false Option refine 301 ode45 is typically able to use large time steps to achieve the requested tolerance When plotting the output however interpolating it with straight lines produces visual artifacts This is why ode45 inserts 3 interpolated points for each calculated point based on the fifth order approximation calculated for the integration Refine is 4 by default In the following code curves with and without interpolation are com pared see Fig 6 2 Note that the numbers of evaluations of the func tion being integrated are the same mu 1 fun t y y 2 mux 1 y 1 2 y 2 y 1 302 Sysquake for ATEX t y ode45 fun 0 5 2 0 odeset Refine 1 Stats true Number of function evaluations 289 Successful steps 42 Failed steps error too large 6 size y 43 2 ti yi ode45 fun 0 5 2 0 odeset Stats true Number of function evaluations 289 Successful steps 42 Failed steps error too large 6 size yi 169 2 plot ti yi g plot t y State based events For simulating a ball bouncing on the ground an event is generated every time the ball hits the ground and its speed is changed instan taneously Let y 1 be the height of the ball above the ground and
56. See also 7 2 stat stat is a library which adds to LME advanced statistical functions The following statement makes available functions defined in stat use stat bootstrp Bootstrap estimate Syntax stats samples bootstrp n fun D1 Description bootstrp n fun D picks random observations from the rows of ma trix or column vector D to form n sets which have all the same size as D then it applies function fun a function name or reference or an inline function to each set and returns the results in the columns of stats Up to three different set of data can be provided bootstrp gives an idea of the robustness of the estimate with re spect to the choice of the observations 490 Sysquake for TEX Example D rand 1000 1 bootstrp 5 std D 0 2938 0 2878 0 2793 0 2859 0 2844 geomean Geometric mean of a set of values Syntax m geomean A m geomean A dim Description geomean A gives the geometric mean of the columns of array A or of the row vector A The dimension along which geomean proceeds may be specified with a second argument The geometric mean of vector v of length n is defined as v Example geomean 1 10 4 5287 mean 1 10 5 5 exp mean log 1 10 4 5287 See also harmmean Harmonic mean of a set of values Syntax m harmmean A m harmmean A dim Libraries stat 491 Description harmmean A gives the harmonic mean of the columns of arr
57. a vector as second argu ment the array can be replicated along more than two dimensions Examples repmat 1 2 3 4 1 2 1212 3434 repmat abc 3 abcabcabc abcabcabc abcabcabc See also reshape Rearrange the elements of an array to change its shape 270 Sysquake for TEX Syntax A2 reshape A1 A2 reshape Al nl n2 A2 reshape Al nl n2 Description reshape Al1 gives a column vector with all the elements of array Al which is read row wise If Al is a variable reshape Al1 is the same as Al reshape Al n1 n2 or reshape Al nl n2 changes the dimensions of array A1 so that the result has m rows and n columns Al must have nl n2 elements read line wise both Al and the result have the same elements When dimensions are given as separate elements one of them can be replaced with the empty array it is replaced by the value such that the number of elements of the result matches the size of input array Example reshape 1 2 3 10 20 30 3 2 1 2 3 10 20 30 reshape 1 12 3 123 4 5 6 7 8 9 10 11 12 See also operator rot90 Array rotation Syntax A2 rot90 A1 A2 rot90 Al k LME Reference arrays 271 Description rot90 A1 rotates array Al 90 degrees counter clockwise the top left element of Al becomes the bottom left element of A2 If Al is an array with more than two dimensions each plane corresponding to the first two dimensions is
58. also Operator operator length global persistent Declaration of global or persistent variables Syntax global x y persistent xy Description By default all variables are local and created the first time they are assigned to Local variables can be accessed only from the body of the function where they are defined but not by any other function even the ones they call They are deleted when the function exits If the function is called recursively i e if it calls itself directly or indirectly distinct variables are defined for each call Similarly local variables defined in the workspace using the command line interface cannot be referred to in functions On the other hand global variables can be accessed by multiple functions and continue to exist even after the function which cre ated them exits Global variables must be declared with global in each functions which uses them They can also be declared in the workspace There exists only a single variable for each different name Declaring a global variable has the following result LME Reference variables 75 If a previous local variable with the same name exists it is deleted If the global variable does not exist it is created and initialized with the empty array Every access which follows the declaration in the same function or workspace uses the global variable Like global variables persistent variables are preserv
59. an attempt to execute an untrusted function in a sandbox The error can be handled only outside the sandbox 92 Sysquake for TEX Examples a l a 2 555 Index out of range a try a 2 end 555 555 try a 2 catch 333 end 555 333 555 try a catch 333 end 555 a ur il 55 See also lasterr lLasterror jerror until End of repeat until loop See also use Import libraries Syntax use lib use libl lib2 Description Functions may be defined in separate files called ibraries use makes them available in the current context so that they may be called by the functions or statements which follow Using a library does not make available functions defined in its sublibraries however libraries may be used multiple times in each context where their functions are referenced All use statements are parsed before execution begins They may be placed anywhere in the code typically before the first function LME Reference programming constructs 93 They cannot be skipped by placing them after an if statement Like wise try catch cannot be used to catch errors useifexists should be used if the absence of the library is to be ignored See also useifexists Import libraries if they exist Syntax useifexists lib useifexists libl lib2 Description useifexists has the same syntax and effect as use except that li braries which are not found are ignored without error See als
60. chebyl n w0 high chebyl n wl wh stop chebyl s Description chebyl calculates a Chebyshev type 1 filter The result is given as zeros poles and gain if there are three output arguments or as nu merator and denominator coefficient vectors if there are two output arguments cheby1 n w0 where w0 is a scalar gives a nth order digital low pass filter with a cutoff frequency of w0 relatively to half the sampling frequency cheby1 n wl wh where the second input argument is a vector of two numbers gives a 2nth order digital bandpass filter with pass band between wl and wh relatively to half the sampling frequency cheby1 n w0 high gives a nth order digital highpass filter with a cutoff frequency of w0 relatively to half the sampling frequency Libraries filter 521 cheby1 n wl wh stop where the second input argument is a vector of two numbers gives a 2nth order digital bandstop filter with stopband between wl and wh relatively to half the sampling frequency With an additional input argument which is the string s cheby1 gives an analog Chebyshev type 1 filter Frequencies are given in rad s See also cheby2 Chebyshev type 2 filter P k cheby2 n w0 um den cheby2 n w0 cheby2 n wl wh cheby2 n w0 high cheby2 n wl wh stop cheby2 s Description cheby2 calculates a Chebyshev t
61. converts its arguments to a string and writes it to the standard output fprintf fd format a b spec ifies the output file descriptor See sprintf for a description of the conversion process LME Reference input output 367 Example fprintf d 2f 3E g n 1 3 pi 1 2 00 3 000E0 3 1416 22 Caveat Same limitations as sprintf See also fread Raw input Syntax a count fread fd a count fread fd size a count fread fd size precision Description fread fd reads signed bytes from the file descriptor fd until it reaches the end of file It returns a column vector whose elements are signed bytes between 128 and 127 and optionally in the second output argument the number of bytes it has read fread fd size reads the number of bytes specified by size If size is a scalar that many bytes are read and result in a column vector If size is a vector of two elements m n mxn elements are read row by row and stored in an m by n matrix If the end of the file is reached before the specified number of elements have been read the number of rows is reduced without throwing an error The optional second output argument always gives the number of elements in the result With a third argument fread fd size precision reads inte ger words of 1 2 or 4 bytes or IEEE floating point numbers of 4 bytes single precision or 8 bytes double precision The meaning of the string precision is described in
62. copy copies output to both fd and fdTarget if copy is true instead of redirecting it only to fdTarget If copy is false the result is the same as with two input arguments With zero or one input argument and without output argument redirect displays the current redirection for the specified file descrip tor 1 or 2 or for both of them Note that the redirection itself may alter where the result is displayed With an output argument redirect returns a 1 by 2 row vector if the file descriptor is specified or a 2 by 2 matrix otherwise The first column contains the target file descriptor and the second column 1 for copy mode and 0 for pure redirection mode Examples Create a new file diary txt and copy to it both standard output and error fd fopen diary txt redirect 1 fd true redirect 2 fd true w Stop copying standard output and error and close file redirect 1 1 redirect 2 2 fclose fd Redirect standard error to standard output and get the redirection state redirect 2 1 redirect stdout fd 1 gt fd 1 stderr fd 2 gt fd 1 redirect 2 372 Sysquake for ATEX stderr fd 2 gt fd 1 redirect sprintf Formatted conversion of objects into a string Syntax s sprintf format a b Description sprintf converts its arguments to a string The first parameter is the format string All the characters are copied verbatim to the out put string except for
63. cumprod M1 dim operates along the dimension dim column wise if dim is 1 row wise if dim is 2 Examples cumprod 1 2 3 4 5 6 1 2 3 4 10 18 cumprod 1 2 3 4 5 6 2 12 6 4 20 120 See also prod cumsum LME Reference linear algebra 211 cumsum Cumulative sums Syntax M2 cumsum M1 M2 cumsum M1 dim Description cumsum M1 returns a matrix M2 of the same size as M1 whose ele ments M2 i j are the sum of all the elements M1 k j with k lt i cumsum M1 dim operates along the dimension dim column wise if dim is 1 row wise if dim is 2 Examples cumsum 1 2 3 4 5 6 123 579 cumsum 1 2 3 4 5 6 2 13 6 49 15 See also dare Discrete time algebraic Riccati equation Syntax X L K dare A B Q X L K dare A B Q R Description dare A B Q calculates the stable solution X of the following discrete time algebraic Riccati equation X A XA A XB B XB 1B XA Q All matrices are real Q and X are symmetric 212 Sysquake for TEX With four input arguments dare A B Q R with R real symmetric solves the following Riccati equation X A XA A XB B XB R 1B XA Q With two or three output arguments X L K dare also re turns the gain matrix K defined as K B XB R B XA and the column vector of closed loop eigenvalues L eig A BK Example 4 2 1 2 0 1 2 1 ee DOOWD gt Y x lt X oul uo ou tl C
64. depends only on the default character size neither on the line length nor on the plot scale Its color and thickness are the same as the line s Many graphical commands accept data for more than one line If the style string contains several sequences of styles the first line bor rows its style from the first sequence the second line from the second sequence and so on If there are not enough styles they are recycled A sequence is one or two style specifications one of them for the color and the other one for the dash pattern or the symbol shape in any or der Sequences of two specifications are used if possible Commas may be used to remove ambiguity Here are some examples plot 0 1 0 1 0 1 1 1 2 2 3 3 k r The first line from 0 1 to 1 1 is black and dashed the second line from 0 2 to 1 2 is red and dash dot and the third line from 0 3 to 1 3 is black and dashed again plot 0 1 0 1 0 1 1 1 2 2 3 3 rbk The first line is red the second line is blue and the third line is black plot 0 1 0 1 0 1 1 1 2 2 3 3 br The first and third lines are blue and dashed and the second line is red and solid plot 0 1 0 1 0 1 1 1 2 2 3 3 H cccccc The first and third lines are dotted and the second line is gray solid and thick 392 Sysquake for ATEX Graphic ID The second optional argument is the graphic ID It has two purposes First it specifies that the graphic
65. e it repeats n times the same operation Up to a scalar factor the result is an ap proximation of the n th order derivative based on equidistant samples diff A n dim operates along dimension dim If the second argu ment n is the empty matrix the default value of 1 is assumed Examples diff 1 3 5 4 8 2 2 14 diff 1 3 5 4 8 2 0 35 diff 1 3 5 4 8 2 3 9 8 1 2 HONA i 2 4 6 See also cumsum dlyap Discrete time Lyapunov equation Syntax X dlyap A C LME Reference linear algebra 215 Description dlyap A C calculates the solution X of the following discrete time Lyapunov equation AXA X C 0 All matrices are real 3 1 2 7 1 2 dlyap KL 4 A XI ol N 1 0505 3 2222 1 2117 3 2317 11 213 4 8234 1 4199 5 184 2 7424 See also dot Scalar product Syntax v3 dot v1 v2 v3 dot v1 v2 dim Description dot v1 v2 gives the scalar products of vectors v1 and v2 v1 and v2 must be row or columns vectors of same length or arrays of the same size then the scalar product is performed along the first dimension not equal to 1 A third argument dim may be used to specify the dimension the scalar product is performed along Examples dot 1 2 3 0 0 1 3 dot 1 2 3 7 1 3 4 0 0 0 2 0 2 4 2 216 Sysquake for ATEX See also eig Eigenvalues and eigenvectors of a matrix Syntax e eig M V D eig M Descript
66. element can be manipulated by the user When the user clicks in a figure Sysquake scans all the curves which have a non negative graphic ID the default value of all com mands is 1 making the graphical object impossible to grasp and sets z0 _x0 _y0 _id and _ix such that they correspond to the nearest element if it is close enough to the mouse coordinates Second the ar gument _id is set to the ID value so that the mousedown mousedrag and mouseup handlers can identify the different objects the user can manipulate In applications without live interactivity such as Sysquake Remote the graphic ID argument is accepted for compatibility reasons but ignored Scale Before any figure can be drawn on the screen the scale or equiv alently the portion of the plane which is represented on the screen must be determined The scale depends on the kind of graphics and consequently is specified in the draw handler but can be changed by the user with the zoom and shift commands What the user specifies has always the priority If he or she has not specified a new scale the scale command found in the draw handler is used scale xMin xMax yMin yMax If scale is not used or if some of the limits are NaN not an number a default scale is given by the plot commands themselves If used the scale command should always be executed before any plot command because several of them use the scale to calculate traces only over the visibl
67. f With two input arguments fftn f size resizes f by cropping or by padding f with zeros See also filter Digital filtering of data Syntax filter filter filter y xf f X0 X0 dim y b a u y b a u y b a u ilter Description filter b a u filters vector u with the digital filter whose coefficients are given by polynomials b and a The filtered data can also be an array filtered along the first non singleton dimension or along the dimension specified with a fifth input argument The fourth argu ment if provided and different than the empty matrix is a ma trix whose columns contain the initial state of the filter and have max length a length b 1 element Each column correspond to a signal along the dimension of filtering The result y which has the same size as the input can be computed with the following code if u is a vector a a a l if length a gt length b b b zeros 1 length a length b else a a zeros 1 length b length a end length x 220 Sysquake for ATEX for i 1 length u y i b 1 u i x 1 for j 1 n 1 x j b j 1 u i x j 1 a j 1 y i end x n b n 1 u i a n 1 y i end The optional second output argument is set to the final state of the filter Examples filter 1 2 1 2 3 ones 1 10 1 1 2 41 11 22 8 47 121 5 6 5 6 5 6 5 p 0 8 filter 1 p 1 p u p u 1 low
68. files sq and SQ data files sqd Versions of Sysquake using Uni code currently Mac OS X and Linux convert automatically files with a charset specification The character set specification is a comment line with the following format charset charsetname or charset charsetname 56 Sysquake for ATEX Spaces between the comment mark and the keyword charset are ignored The comment line must be the first or the second line of the text file The character set charsetname must be one of the following ascii or uSascii ASCII utf 8 or utf8 UTF 8 unicode iso 8859 1oriso latin 1 ISO Latin 1 Windows 1252 macintosh or macosroman Mac OS Classic Here are advices about the use of character set specifications both for the current transition phase where Sysquake for Windows does not use Unicode and for the future If you need only ASCII typically because you work in English or for files without text or where unaccented letters are acceptable do not add any character set specification ASCII is a subset of all supported character sets or add charset ascii as an indication that the file should contain only 7 bit characters If you need accented characters found in western European lan guages use ISO 8859 1 with an explicit character set specification on Windows and other platforms if you need cross platform compatibility or any character set with a character set specification otherwise If you need another native character set
69. first cell is empty First column Second column First row Cell 1 1 Cell 1 2 J Second row Cell 2 1 Cell 2 2 Preformatted Preformatted text is a block of text displayed literally including line feeds Preformatted text is preceded with a line containing and is 60 Sysquake for ATEX followed by a line containing Example This is some C code double f double x return 2 x x f x 2x HH In preformatted text lines which begin with with leading spaces or not must have an additional space which is discarded in the output Horizontal rules Horizontal rules are single lines containing four hyphens Example Paragraph Paragraph following a horizontal rule Extensions Sequences lt lt and lt lt lt are reserved for extensions Character style Inside paragraphs indented paragraphs headings list elements and table cells the following two character sequences toggle on or off the corresponding style It is not mandatory to nest spans of styled char acters between matching style sequences Style is reset at the end of text block Markup Style Bold Italic Monospace two commas Subscript Superscript __ two underscores Underlined Double stars and double sharps are interpreted as the beginning of list items when they are the first characters of a line in the context of LME Reference 61 lists To be interpreted as style markup they can be pre
70. flips a matrix left right gt flipud 1 2 3 4 ans 3 4 12 gt fliptr 1 2 3 4 ans 21 4 3 To sort the elements of each column of a matrix or the elements of a row vector use the sort function Tutorial 25 gt sort 2 4 8 7 1 3 ans 123478 To get the size of a matrix you can use the size function which gives you both the number of rows and the number of columns unless you specify which of them you want in the optional second argument gt size rand 13 17 ans 13 17 gt size rand 13 17 1 ans 13 gt size rand 13 17 2 ans 17 4 4 Polynomials LME handles only numerical values Therefore it cannot differenti ate functions like f x sin e However a class of functions has a paramount importance in numerical computing the polynomials Poly nomials are weighted sums of powers of a variable such as 2x 3x 5 LME which handles only matrices stores the coefficients of poly nomials in row vectors i e 2x 3x 5 is represented as 2 3 5 and 2x 3x as 2 0 0 0 3 0 Adding two polynomials would be like adding the coefficient vectors if they had the same size in the general case however you had better use the function addpol which can also be used for subtraction gt addpol 1 2 3 7 ans 49 gt addpol 1 2 2 4 5 ans 257 gt addpol 1 2 2 4 5 ans 2 3 3 Multiplication of polynomials corresponds to convolution no need to understa
71. for TEX See also asec cos sech Hyperbolic secant Syntax y sech x Description acot x gives the hyperbolic secant of x which is complex if x is See also jasech seci cosh sin Sine Syntax y sin x Description sin x gives the sine of x which is complex if x is complex Example sin 2 0 9093 See also sinc Sinc LME Reference mathematical functions 199 Syntax y sinc x Description Sinc x gives the sinc of x i e Sin pi x pixx if x 0 or 1 if x 0 The result is complex if x is complex Example sinc 1 5 0 2122 See also single Conversion to single precision numbers Syntax B single A Description single A converts number or array A to single precision A can be any kind of numeric value real complex or integer or a character or logical array Single literal numbers can be entered as a floating point number with the single suffix Examples single pi 3 1416single Single AB 1x2 single array 65 66 3 7e4single 37000single 200 Sysquake for TEX See also and related functions Logical sinh Hyperbolic sine Syntax y sinh x Description sinh x gives the hyperbolic sine of x which is complex if x is com plex Example sinh 2 3 6269 See also sqrt Square root Syntax r sqrt z Description sqrt z gives the square root of z which is complex if z is not real positive Exam
72. g cos 9 and z rsin 9 474 Sysquake for ATEX Example phi theta r cart2sph 1 2 3 phi 1 1071 theta 0 9303 r 3 7417 See also circshift Shift the elements of a matrix in a circular way Syntax B circshift A shift_vert B circshift A shift_vert shift_hor Description circshift A sv shifts the rows of matrix A downward by sv rows The sv bottom rows of the input matrix become the sv top rows of the output matrix sv may be negative to go the other way around circshift A sv sh shifts the rows of matrix A downward by sv rows and its columns to the right by sh columns The sv bottom rows of the input matrix become the sv top rows of the output matrix and the sh rightmost columns become the sh leftmost columns See also Tliptr flipud blkdiag Block diagonal matrix Syntax X blkdiag Bl B2 Libraries stdlib 475 Description blkdiag B1 B2 creates a block diagonal matrix with matrix blocks B1 B2 etc Its input arguments do not need to be square Example blkdiag 1 2 3 4 5 120 340 0 0 blkdia 12 0 0 0 0 1 2 3 4 See also diag compan Companion matrix Syntax X compan pol Description compan pol gives the companion matrix of polynomial pol a square matrix whose eigenvalues are the roots of pol Example compan 2 3 4 5 1 5 2 0 2 5 1 0 0 0 0 0 0 0 1 0 0 0 See also 476 Sysquake for TEX corrcoef Corr
73. gives the median of vector v i e the value x such that half of the elements of v are smaller and half of the elements are larger NaN values are ignored nanmedian A gives a row vector which contains the median of the columns of A With a second argument nanmedian A dim operates along dimension dim See also nansum Sum after discarding NaNs Syntax nansum A nansum A dim y y Libraries stat 495 Description nansum v returns the sum of the elements of vector v NaN values are ignored nansum A returns a row vector whose elements are the sums of the corresponding columns of array A nansum A dim returns the sum of array A along dimension dim the result is a row vector if dim is 1 or a column vector if dim is 2 See also pdist Pairwise distance between observations Syntax d pdist M d pdist M metric d pdist M metric p Description pdist calculates the distance between pairs of rows of the observation matrix M The result is a column vector which contains the distances between rows i and j with i lt j It can be resized to a square matrix with squareform By default the metric used to calculate the distance is the eu clidean distance but it can be specified with a second argument euclid euclidean distance seuclid standardized euclidean distance mahal Mahalanobis distance cityblock sum of absolute values minkowski Minkowski metric with parameter p The s
74. i n where C is the name of the class obj is the object to be indexed i is the position of the index expression where matrixcol is used and n is the total number of index expressions Example Set to 0 the NaN values which are not in the first column A 1 nan 5 nan 7 2 3 1 2 A matrixcol gt 1 amp isnan A 0 A 1 0 5 nan 7 2 3 1 2 See also matrixrow beginning jend matrixrow First index in a subscript expression Syntax A matrixrow function e C matrixrow obj i n LME Reference variables 77 Description In an expression used as a single subscript to access some elements of an array A expr matrixrow gives an array of the same size as A where each element is the row index For instance for a 2 by 3 matrix matrixrow gives the 2 by 3 matrix 1 2 3 1 2 3 In an expression used as the first of multiple subscripts to access some elements of an array A expr matrixrow gives a row vec tor of length size A 1 whose elements are the indices of each row It is equivalent to the range beginning end matrixrow is useful in boolean expressions to select some ele ments of an array matrixrow can be overloaded for objects of used defined classes Its definition should be have a header equivalent to function e C matrixrow obj i n where C is the name of the class obj is the object to be indexed i is the position of the index expression where matrixrow is used and n is the total
75. initial state y0 It may stop before reaching tend if the integration step cannot be reduced enough to obtain the required tolerance If the function is continuous you can try to reduce MinStep in the options argument see below The optional fourth argument may contain options It is either the empty array for the default options or the result of odeset the use of a vector of option values is deprecated Additional input arguments of ode45 are given as additional input arguments to the function specified by fun They permit to parame terize the ODE Example Let us integrate the following ordinary differential equation Van Der Pol equation parameterized by p x u 1 x x x Let y x and y2 x their derivatives are y2 play ye yi y5 and may be computed by the following function function yp f t y mu yp y 2 mu 1 y 1 2 y 2 y 1 The following ode45 call integrates the Van Der Pol equation from 0 to 10 with the default options starting from x 0 2 and x 0 0 with u 1 see Fig 6 1 LME Reference non linear numerical functions 297 Van der Pol equation mu 1 Figure 6 1 Van der Pol equation with u 1 integrated with ode45 t y ode45 f 0 10 2 0 1 The plot command expects traces along the second dimension con sequently the result of ode45 should be transposed plot t y See also odeset Options for ordinary differential equation inte
76. list listin LME Reference structures 335 Example replist 1 abc 3 1 abc 1 abc 1 abc See also 6 24 Structure Functions fieldnames List of fields of a structure Syntax fields fieldnames strct Description fieldnames strct returns the field names of structure strct as a list of strings Example fieldnames struct a 1 b 1 5 a b See also orderfields getfield Value of a field in a structure Syntax value getfield strct name 336 Sysquake for TEX Description getfield strct name gets the value of field name in structure strct It is an error if the field does not exist getfield s f gives the same value as s f getfield is especially useful when the field name is not fixed but is stored in a variable or is the result of an expression See also isfield Test for the existence of a field in a structure Syntax b isfield strct name Description isfield strct name is true if the structure strct has a field whose name is the string name false otherwise Examples isfield struct a 1 3 x abc x true isfield struct a 1 3 x abc X false See also jusstruct struct isstruct Test for a structure object Syntax b isstruct obj LME Reference structures 337 Description isstruct obj is true if the object obj is a struct
77. logarithm does not always exist With a second output argument err Logm also returns an estimate of the relative error norm expm lLogm X X norm X Example logm 1 2 3 4 lt lt ll 0 3504 2 3911j 0 9294 1 0938j 1 394 1 6406j 1 0436 0 7505j expm Y 1 5 5511e 16j 2 7 7716e 16j 3 8 3267e 16j 4 See also lu LU decomposition 228 Sysquake for ATEX Syntax L U P lu A L2 U lu A Y lu A Description With three output arguments Lu A computes the LU decomposition of matrix A with partial pivoting i e a lower triangular matrix L an up per triangular matrix U and a permutation matrix P such that P A L U If Ain an m by n mytrix L is m by min m n U is min m n by n and P is m by m A can be rank deficient With two output arguments Lu A permutes the lower triangular matrix and gives L2 P L such that A L2x U With a single output argument lu gives Y L U eye n Example X 1 2 3 4 5 6 7 8 8 L U P lu X L 1 0 0 0 143 1 0 0 571 0 5 1 U ree 7 8 8 0 0 857 1 857 0 0 0 5 P 001 10 0 010 P X L U ans 000 000 000 See also lyap Continuous time Lyapunov equation LME Reference linear algebra 229 Syntax lyap A B C lyap A C Description lyap A B C calculates the solution X of the following continuous time Lyapunov equation AX XB C 0 All matrices are real With two input arguments lyap A C solves the following Lya pu
78. manual mode With an output argument campos gives the current view position See also canorbit CamvVa CamZOOm camproj Projection kind Syntax camproj str str camproj Description camproj str sets the projection mode string str can be either orthographic or o for a parallel projection or perspective or p for a projection with a view point at a finite distance With an output argument camproj gives the current projection mode See also camorbit CamvVa CamZoom camroll Camera roll around view direction Syntax camroll dalpha LME Reference 3D graphics 423 Description camroll dalpha rotates the up vector by angle dalpha around the vector from view position to target dalpha is given in radians A positive value makes the scene rotate counterclockwise See also camdolly camorbit campan Ccamzoom camtarget Target position Syntax camtarget p camtarget auto camtarget manual p camtarget Description camtarget p sets the target to p p is a 3D vector camtarget auto sets the target to automatic mode so that it fol lows the center of the objects which are drawn camtarget manual sets the target to manual mode With an output argument camtarget gives the current target See also candoTly camorbit camzoom camup Up vector Syntax camup p camup auto camup manual p camup 424 Sysquake for ATEX Description camu
79. mesh x y Z mesh z mesh x y z color mesh z color mesh kind mesh kind style mesh kind style id Description mesh x y z plots a mesh defined by 2 D arrays x y and z Ar guments x and y must have the same size as z or be vectors of size z 2 and size z 1 elements respectively If x and y are miss ing their default values are coordinates from 1 to size z 2 along x axis and from 1 to size z 1 along y axis Color is obtained by mapping the full range of z values to the color map mesh x y z color maps values of array color to the color map color must have the same size as z and contain values between 0 and 1 which are mapped to the color map mesh kKind specifies which side of the mesh is visible kind is a string of 1 or 2 characters f if the front side is visible the side where increasing y are on the left of increasing x coordinates and b if the back side is visible Default is equivalent to fb mesh style specifies the line or symbol style of the mesh The default is to map z or color values to the color map mesh 1d specifies the ID used for interactivity in Sysquake LME Reference 3D graphics 429 See also surf plotpoly plot3 Generic 3D plot Syntax plot3 x y z plot3 x y z style plot3 x y z style id Description The command plot3 displays 3D graphical data in the current figure The data are given as three vectors o
80. model Ac Bc Cc Dc defined by x kK 1 Agx k Bgu k y k Cgx k Dgu k and dx ae A x t B u t y t C x t D u t Example numd dend c2dm 1 1 1 5 t numd 0 7143 0 7143 dend 1 0 4286 LME Reference dynamical system functions 357 numc denc d2cm numd dend numc 3 8858e 17 1 denc 11 See also dmargin Robustness margins of a discrete time system Syntax gm psi wc wx gm psi wc wx dmargin num den Ts dmargin num den Description The open loop discrete time transfer function is given by the two poly nomials num and den with sampling period Ts default value is 1 If the closed loop system with negative feedback is unstable all out put arguments are set to an empty matrix Otherwise dmargin calcu lates the gain margins gm which give the interval of gain for which the closed loop system remains stable the phase margin psi al ways positive if it exists which defines the symmetric range of phases which can be added to the open loop system while keeping the closed loop system stable the critical frequency associated to the gain mar gins where the open loop frequency response intersects the real axis around 1 and the cross over frequency associated to the phase mar gin where the open loop frequency response has a unit magnitude If the Nyquist diagram does not cross the unit circle psi and wx are empty Examples Stable closed loop Nyquist i
81. number into a boolean Syntax B logical A Description logical x converts array or number A to logical boolean type All nonzero elements of A are converted to true and zero elements to false Logical values are stored as 0 for false or 1 for true in unsigned 8 bit integers They differ from the uint8 type when they are used to select the elements of an array or list Examples a 1 3 a 1 0 1 Index out of range a 1 3 a logical 1 0 1 13 See also LME Reference logical functions 353 true Boolean constant true Syntax b true B true n B true nl1 n2 B true nl n2 Description The boolean constant true can be used to set the value of a variable It is equivalent to Logical 1 The constant 1 is equivalent in many cases indices to get or set the elements of an array are an important exception With input arguments true builds a logical array whose elements are true The size of the array is specified by one integer for a square matrix or several integers either as separate arguments or in a vec tor for an array of any size Examples true true islogical true true true 2 TT TT See also xor Exclusive or Syntax b3 xor b1 b2 354 Sysquake for ATEX Description xor b1 b2 performs the exclusive or operation between the corre sponding elements of b1 and b2 b1 and b2 must have the same size or one of them must be a scalar Examples xor
82. number of index expressions See also beginning subsasgn Assignment to a part of an array list or structure Syntax A subsasgn A s B Description When an assignment is made to a subscripted part of an object in a statement like A s1 s2 B LME executes A subsasgn A s B where subsasgn is a method of the class of variable A and s is a struc ture with two fields s type which is and s subs which is the list of subscripts sl s2 If a subscript is the colon character which stands for all elements along the corresponding dimensions it is represented with the string in s subs When an assignment is made to a subscripted part of an object in a statement like A s B LME executes A subsasgn A s B where subsasgn is a method of the class of variable A and s is a structure with two fields s type which is and s subs which is the list containing the single subscript s 78 Sysquake for ATEX When an assignment is made to the field of an object in a statement like A f B LME executes A subsasgn A s B where s is a structure with two fields s type which is and s subs which is the name of the field f in this case While the primary purpose of subsasgn is to permit the use of sub scripts with objects a built in implementation of subsasgn is provided for arrays when s type is for lists when s type is a list and for structures when s typeis Examples
83. of Sysquake for LaTeX for your platform to a suitable location such as opt SysquakeLx Have bin i386 linux sysquakelatextool in your PATH There are two ways to do it put a symbolic link to bin i386 linux sysquakelatextool in one of your existing program directories such as usr bin or usr local bin sudo ln s opt SysquakeLx bin i386 linux sysquakelatextool usr bin Install 11 or add opt SysquakeLx SysquakeLx bin i386 linux to your environment variable PATH Create a new environment variable named SYSQUAKELXPATH which contains a list of semicolon separated paths for the directories containing Iml libraries The default value is Lib opt SysquakeLx Lib which finds standard libraries if Sysquake for LaTeX is installed in opt SysquakeLx Edit file texmf cnf type kpsewhich texmf cnf in a terminal to find it and append the full path of directory TexFiles to TEXM FLOCAL separated from previous path s with a semicolon For example TEXMFLOCAL SELFAUTOPARENT texmf local opt SysquakeLx TexFiles Chapter 3 Sysquake for IATEX Tutorial Once installed Sysquake for LaTeX is a LaTeX package which you can use as easily as any other package In this tutorial we will create a technical report which analyzes some of the properties of a rational function Sysquake for LaTeX works at the level of the TeX program which with LaTeX definitions is usually called Latex for creating DVI files or pdflatex for cre
84. of the prototype of analog low pass filter with a cutoff frequency of 1 rad s They correspond respec tively to Bessel Butterworth Chebyshev type 1 Chebyshev type 2 and elliptic filters besself butter chebyl cheby2 and ellip which provide a higher level interface to design filters of these different types In addition to the filter parameters degree bandpass and band stop ripples one can specify the kind of filter lowpass highpass bandpass or bandstop and the cutoff frequency or frequencies The result can be an analog or a digital filter given as a rational transfer function or as zeros poles and gain lp2lp lp2hp lp2bp and lp2bs which convert analog lowpass filters respectively to lowpass highpass bandpass and bandstop with specified cutoff frequency or frequencies Transfer functions are expressed as the coefficient vectors of their nu merator num and denominator den in decreasing powers of s Laplace transform for analog filters or z z transform for digital filters or as the zeros z poles p and gain k besselap Bessel analog filter prototype Syntax z p k besselap n Description besselap n calculates the zeros the poles and the gain of a Bessel analog filter of degree n with a cutoff angular frequency of 1 rad s See also besself Bessel filter Libraries filter 517 Syntax z p k besself n w0 num den besself n w0 besself n wl wh
85. of those of matrix A Q has rank A columns which are the first left singular vectors of A that is those corresponding to the largest singular values Example orth 1 2 3 1 2 4 1 2 5 0 4609 0 788 0 5704 8 9369e 2 0 6798 0 6092 See also LME Reference linear algebra 235 pinv Pseudo inverse of a matrix Syntax M2 M2 inv M1 inv M1 e TT Description pinv M1 returns the pseudo inverse of matrix M For a nonsingular square matrix the pseudo inverse is the same as the inverse For an arbitrary matrix possibly nonsquare the pseudo inverse M2 has the following properties size M2 size M1 M1 M2 M1 M1 M2 M1 M2 M2 and the norm of M2 is minimum To pseudo inverse is based on the singular value decomposition where only the singular values larger than some small threshold are considered This threshold can be specified with an optional second argument If M1 is a full rank matrix with more rows than columns pinv returns the least square solution pinv M1 xy M1 M1 M1 xy of the over determined system M1 x y Examples 0 2 0 4 1 2 M2 x 1 2 1 2 M2 x 1 2 M2 0 2 0 4 See also poly Characteristic polynomial of a square matrix or polynomial coefficients based on its roots 236 Sysquake for TEX Description With a matrix argument poly M returns the characteristic polyno mial det x eye size M M of the square matrix M The roots of the characteristic p
86. on Windows do not add any character set specification and switch to UTF 8 as soon as a uni code version of Sysquake becomes available 6 8 Formatted text Like most text based programming languages LME primary text for mat is plain text without any character or paragraph formatting To improve readability it also supports formatted text Formatting in formation used to change the character font and size and how para graphs are aligned is embedded in the text itself as markup i e spe cial sequences of characters unlikely to occur in normal text This is similar to HTML or LaTeX but with a simpler syntax similar to what is used in wikis and blogs The markup which has been chosen for LME is Creole a collaborative effort to create a common markup language to be used across different wikis and more precisely the open source NME implementation Formatted text output channel In addition to the standard output channel file descriptor 1 and the standard error channel file descriptor 2 LME has a special channel for formatted output file descriptor 4 Anything written to that chan nel is parsed so that markup constructs are interpreted the result is LME Reference 57 displayed in the command window Each write command should con tain a whole block of text with markup there is no buffering across output commands Not all versions of LME support formatted output and how format ted output is displayed depends on the applicat
87. option Latexingraphics is used are not modified The main purpose of this option is to let an author make minor modifications on a platform where sysquakelatextool is not sup ported noseparatelogfile By default standard error channel used when evaluating Sysquake code fragments is redirected to a file whose base name is the same as the main tex file and the suffix is Lmelog This file can be opened after typesetting to check that all code fragments were executed correctly This is especially im portant when an error occurs when a code fragment is replaced with sysquake in the typeset result The noseparatelogfile option merges the Sysquake standard er ror channel with other LaTeX messages 5 2 Other packages Package sysquake sty depends on other packages for some tasks You should also import them with usepackage if necessary graphicx Used to include EPS graphics generated by Sysquake for LaTex epstopdf Used by pdflatex to convert EPS graphics to PDF pdflatex cannot include directly EPS files hyperref Used for hypertext links created by writing text markup to file descriptor 4 The preamble of an article which needs all these packages could be as follows There is no harm in importing superfluous packages documentclass article usepackage latexingraphics noseparatelogfile sysquake usepackage graphicx usepackage epstopdf usepackage hyperref TeX Package Reference 35 begin document e
88. option argument used by functions which display frequency and time responses such as nyquist and step Options are specified with name value pairs where the name is a string which must match exactly the names in the table below Case is significant Options which are not specified have a de fault value The result is a structure whose fields correspond to each option Without any input argument responseset creates a structure with all the default options Note that functions such as nyquist and step also interpret the lack of an option argument as a request to use the default values Contrary to other functions which accept options in structures such as ode45 empty array cannot be used it would be interpreted incorrectly as a numerical argument When its first input argument is a structure responseset adds or changes fields which correspond to the name value pairs which follow Here is the list of permissible options Name Default Meaning Delay 0 time delay NegFreq false negative frequencies Range time or frequency range tOnly false samples for specified time only lsim LME Reference graphics for dynamical systems 461 Option Delay is used only by continuous time frequency response and time response functions for frequency responses it subtracts a phase of delay w where w is the angular frequency Option NegFreq is used in Nyquist and Nichols diagrams continuous time or discrete time when true the response is
89. optional second output is set to what follows immediately the token including separators If no token is found it is the same as str An optional second input argument contains the separators in a string Examples Strings are displayed with quotes to show clearly the separators strtok ab cde ab t r strtok ab cde r strtok 2 5 3 318 Sysquake for TEX See also strtrim Remove leading and trailing blank characters from a string Syntax s2 strtrim s1 Description strtrim s1 removes the leading and trailing blank characters from string s1 Blank characters are spaces code 32 tabulators code 9 carriage returns code 13 line feeds code 10 and null characters code 0 Example double tAB CD r n 0 32 9 65 66 32 32 67 68 13 10 0 double strtrim tAB CD n r 0 65 66 32 32 67 68 See also deblank upper Convert all lowercase letters to lowercase Syntax s2 upper s1 LME Reference strings 319 Description upper s1 converts all the lowercase letters of string s1 to uppercase Currently only ASCII letters without diacritic are converted Example upper abcABC123 ABCABC123 See also utf8decode Decode Unicode characters encoded with UTF 8 Syntax str utf8decode b Description utf8decode b decodes the contents of uint8 or int8 array b which represents Unicode characters encoded with UTF 8 Each Unicode
90. pass with matching initial state 5 5 2 5 16 5 328 5 2624 5 4099 5 3279 u See also funm Matrix function Syntax Y funm X fun Y err funm X fun Description funm X fun returns the matrix function of square matrix X specified by function fun fun takes a scalar input argument and gives a scalar output It is either specified by its name or given as an inline function or a function reference With a second output argument err funm also returns an estimate of the relative error LME Reference linear algebra 221 Examples funm 1 2 3 4 sin 0 4656 0 1484 0 2226 0 6882 X 1 2 3 4 funm X inline 1 x 2 x 0 25 0 75 1 125 1 375 eye 2 X 2 eye 2 X 0 25 0 75 1 125 1 375 See also ifft Inverse Fast Fourier Transform Syntax f ifft F f ifft F n f ifft F n dim Description ifft returns the inverse Discrete Fourier Transform inverse DFT Up to the sign and a scaling factor the inverse DFT and the DFT are the same operation for a vector ifft d conj fft d length d ifft has the same syntax as fft Examples ft 1 2 3 4 See also 222 Sysquake for TEX ifft2 Inverse 2 d Fast Fourier Transform Syntax f ifft2 F f ifft2 F size f ifft2 F nr nc f ifft2 F n Description ifft2 returns the inverse 2 d Discrete Fourier Transform inverse DFT along dimensions 1 and 2 With two or three input arguments ifft2 resiz
91. plain white The following statement makes available functions defined in colormaps use colormaps Functions are typically used directly as the argument of colormap colormap blue2yellow2red black2orangecm Colormap with shades from black to orange Syntax cm black2orangecm cm black2orangecm n Description black2orangecm n creates a color map with n entries corresponding to color shades from black to orange The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also colormap interprgbcm cyan2magentacm graycm green2yellowcm huecm 562 Sysquake for TEX black2red2whitecm Colormap with shades from black to red and white Syntax black2red2whitecm black2red2whitecm n cm cm Description black2red2whitecm n creates a color map with n entries corresponding to color shades from black to red and white The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also blue2greencm Colormap with shades from blue to green Syntax cm cm blue2greencm blue2greencm n Description blue2greencm n creates a color map with n entries corresponding to color shades from blue to green The color map i
92. plotoption fullgrid By default only the unit circle the stability limit is displayed Example Typical use for poles or zeros displayed in the z plane scale equal 1 2 1 2 1 2 1 2 zgrid LME Reference Sysquake for IATgX functions 469 plotroots pol See also 6 41 Sysquake for IATEX Functions Functions described in this section are specific to Sysquake for LaTeX Some of them are documented because they appear in Ime files gen erated when tex files which include file sysquake sty are typeset but they should not be called directly sqixendfragment End fragment of code Syntax sqlxendf ragment f ragid Description sqlxendfragment fragid ends a fragment of code beginning with beginlmefragment Argument fragid a string must match the first argument of the previous beginlmefragment sqlxendfragment is used by the Sysquake package it should not be called from tex files See also beginlmef ragment sqlxvalue Format value for an equation Syntax sqlxvalue v str sqlxvalue v 470 Sysquake for TEX Description sqlxvalue v inserts value v in the LaTeX output enclosed in ensuremath so that it can be used in math mode as well as in text mode With an output argument sqlxvalue v returns its result in a string sqixstartfragment Start a new fragment of code Syntax sqlxstartfragment fragid width height Description sqlxstartfragment fragid width height
93. qe SR sey a a sie ae tans a ait its wee ie Ain fa ate a ae 554 bk td ih ele aie ote kok an ee Bs we oe ed ca a 556 LL CONStANES is 4 e a Oe aio oe a ee we Eo Aa 559 7 12 COlOMMAPS 2 ec oe ee ee eee ee 560 ws ont aa dh dh E Rate he Pode eats hho oe Be 8 568 7 14 solids 0 000 a a a a a a a G a 573 578 Chapter 1 Introduction LaTeX is a typesetting environment for generating high quality docu ments for reading on screen typically as PDF or for printing Devel oped by Leslie Lamport with the support of many other contributors and the American Mathematics Society AMS LaTeX is extremely pop ular for scientific publications LaTeX is a set of macros for TeX From TeX it inherits the high quality of text layout and equation formatting and implementations on most computer platforms it adds facilities for sections page layout table of contents index etc Like TeX LaTeX takes as input a text file which contains the text to typeset and commands introduced with an escape character To create a document the author typically writes this file in a text editor processes it with latex a command line tool displays the result in a viewer and continues with more iterations until a satisfactory result is obtained Numerical results and graphics come usually from other applications such as Sysquake Sysquake for LaTeX is a LaTeX package and a command tool which let you embed Sysquake commands directly in LaTeX source files With
94. response of the discrete time transfer function numd dend with sampling period Ts Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning dimpulse Ad Bd Cd Dd Ts plots the impulse response of the discrete time state space model Ad Bd Cd Dd defined as x k 1 Agx k Bgu t y k Cgx k Dau k where u k is a unit discrete impulse The state space model must have a scalar input and may have a scalar or vector output With output arguments dimpulse gives the output and the time as column vectors No display is produced Example dimpulse 1 poly 0 9 0 7 0 6j 0 7 0 6j 1 See also dinitial Time response plot of a discrete time linear state space model with initial conditions Syntax dinitial Ad Bd Cd Dd Ts x0 dinitial Ad Bd Cd Dd Ts x0 opt 440 Sysquake for TEX dinitial style dinitial style id y t dinitial Description dinitial Ad Bd Cd Dd Ts x0 plots the output s of the discrete time state space model Ad Bd Cd Dd with null input and initial state x0 The model is defined as x k 1 Adx k Bgu t y k Cax k Dau k where u k is null Sampling period is Ts The state space model may have a scalar or vector output The simulation time range can be provided in a structure opt cre ated with responseset It is a vector of two elements the start
95. result is a row vector if X is a scalar or a row vector or a column vector otherwise The result depends on the computer architecture LME Reference linear algebra 203 Example typecast luint32 uint8 1x4 uint8 array 0 0 Oo 1 typecast pi uint8 1x8 uint8 array 64 9 33 251 84 68 45 24 See also swapbytes 6 16 Linear Algebra addpol Addition of two polynomials Syntax p addpol p1 p2 Description addpol p1 p2 adds two polynomials p1 and p2 Each polynomial is given as a vector of coefficients with the highest power first e g x 2x 3 is represented by 1 2 3 Row vectors and column vectors are accepted as well as matrices made of row vectors or col umn vectors provided one matrix is not larger in one dimension and smaller in the other one addpol is equivalent to the plain addition when both arguments have the same size Examples addpol 1 2 3 2 5 148 addpol 10 addpol 124 457 1 2 3 2 5 subtraction 2 1 2 3 4 5 6 1 1 See also 204 Sysquake for ATEX balance Diagonal similarity transform for balancing a matrix Syntax B balance A T B balance A Description balance A applies a diagonal similarity transform to the square ma trix A to make the rows and columns as close in norm as possible Balancing may reduce the 1 norm of the matrix and improves the accuracy of the computed eigenvalues and or eigenvectors To avoid round of
96. spaces when they are en closed in brackets Parenthesis and square brackets are equivalent as far as LME is concerned parenthesis are preferred in LME code but square brackets are available for compatibility with third party appli cations In some cases a simpler syntax can be used when the function has only literal character strings as input arguments The following conditions must be satisfied No output argument Each input argument must be a literal string without any space tabulator comma or semicolon beginning with a letter a digit or one of minus slash dot colon or star containing at least one letter or digit In that case the following syntax is accepted left and right columns are equivalent fun strl fun str1 fun strl str2 fun stri str2 fun abc def fun abc def Arguments can also be quoted strings in that case they may con tain spaces tabulators commas semicolons and escape sequences beginning with a backslash see below for a description of the string data type Quoted and unquoted arguments can be mixed fun a bc n fun a bc n fun strl str 2 fun strl str 2 42 Sysquake for ATEX This command syntax is especially useful for functions which accept well known options represented as strings such as format loose 6 3 Libraries Libraries are collections of user functions identified in LME by a name Typical
97. such as M placed after their unique operand In Sysquake their arguments are always evaluated from left to right Since they do not require parenthesis or comma their priority matters Priority specifies when subexpressions are considered as a whole as the argu ment of some operator For instance in the expression at b c where x denotes the multiplication the evaluation could result in a b c or at b c however since operator s priority is higher than operator s the expression yields a b c without ambiguity Here is the list of operators from higher to lower priority LA r A A unary J f AN lt gt lt gt Most operators have also a functional syntax for instance a b can also be written plus a b This enables their overriding with new definitions and their use in functions such as feval which take the name of a function as an argument Here is the correspondence between operators and functions 122 Sysquake for TEX a b vertcat a b a b minus a b a b horzcat a b axb mtimes a b a b colon a b a b mrdivide a b a b c colon a b c a b mldivide a b alb or a b a b times a b a amp b and a b a b rdivide a b a lt b le a b a b ldivide a b a lt b lt a b a b mpower a b a gt b ge a b a b power a b a gt b gt a b a not a b eq a b a uminus a a b ne a b a uplus a a b same a b a ctranspose a a unsame a b a trans
98. system a is continuous time or static and false oth erwise See also Iti isdt Test for a discrete time LTI Syntax b isdt a Description isdt a is true if system a is discrete time or static and false other wise Libraries lti 535 See also isct Iti isempty Test for an LTI without input output Syntax b isempty a Description isempty a is true if system a has no input and or no output and false otherwise See also Iti isproper Test for a proper causal LTI Syntax b isproper a Description isproper a is true if lti object a is causal or false otherwise An ss object is always causal A tf object is causal if all the transfer functions are proper i e if the degrees of the denominators are at least as large as the degrees of the numerators Iti issiso Test for a single input single output LTI Syntax b issiso a 536 Sysquake for ATEX Description issiso a is true if lti object a has one input and one output single input single output system or SISO or false otherwise Iti minreal Minimum realization Syntax b b minreal a minreal a tol Description minreal a modifies lti object a in order to remove states which are not controllable and or not observable For tf objects identical zeros and poles are canceled out minreal a tol uses tolerance tol to decide whether to discard a state or a pair of pole zero Iti minus S
99. the list of all persistent variables With an output argument info persistent gives the list of the persis tent variable names info libraries displays the list of all loaded libraries With an output argument info libraries gives the list of the library names info usedlibraries displays the list of libraries available in the current context With an output argument info usedlibraries gives the list of the names of these libraries info threads displays the ID of all threads With an output argu ment info threads gives a list of structures which describe each thread with the following fields id thread ID totaltime execution time in seconds Only the first character of the argument is meaningful info b is equivalent to info builtin Examples info LME 4 5 info b abs acos acosh etc LME Reference miscellaneous functions 105 info v ans 1x1 complex vars info v dumpvar vars struct name ans defined true struct name vars defined false See also inline Creation of inline function Syntax fun inline funstr fun inline expr fun inline expr argl fun inline funstr param fun inline expr argl paramstruct Description Inline function are LME objects which can be evaluated to give a result as a function of their input arguments Contrary to functions declared with the function keyword inline funct
100. the phase margin where the open loop frequency response LME Reference dynamical system functions 359 has a unit magnitude If the Nyquist diagram does not cross the unit circle psi and wx are empty Examples Stable closed loop Nyquist inside unit circle gm psi wc wx margin 0 5 poly 1 1 1 gm 2 16 psi wc 0 1 7321 wx Stable closed loop Nyquist crosses unit circle gm psi wc wx margin 4 poly 1 1 1 gm 0 25 2 psi 0 4737 we 0 1 7321 wx 1 2328 Unstable closed loop gm psi wc wx margin 10 poly 1 1 1 gm psi we wx Caveats Contrary to many functions margin cannot be used with several trans fer functions simultaneously because not all of them may correspond simultaneously to either stable or unstable closed loop systems See also ss2tf Conversion from state space to transfer function Syntax num den ss2tf A B C D den ss2tf A B C D num den ss2tf A B C D iu den ss2tf A B C D iu 360 Sysquake for ATEX Description A continuous time linear time invariant system can be represented by the state space model Ax t Bu t Cx t Du t dx a y t where x t is the state u t the input y t the output and ABCD four constant matrices which characterize the model If it is a single input single output system it can also be represented by its trans fer function num den num den s
101. the table below 368 Sysquake for ATEX precision meaning int8 signed 8 bit integer 128 lt x lt 127 char signed 8 bit integer 128 lt x lt 127 int16 signed 16 bit integer 32768 lt x lt 32767 int32 signed 32 bit integer 2147483648 lt x lt 2147483647 int64 signed 64 bit integer 9 223372e18 lt x lt 9 223372e18 uint8 unsigned 8 bit integer 0 lt x lt 255 uchar unsigned 8 bit integer 0 lt x lt 255 uint16 unsigned 16 bit integer 0 lt x lt 65535 uint32 unsigned 32 bit integer 0 lt x lt 4294967295 uint64 unsigned 64 bit integer 0 lt x lt 18 446744e18 single 32 bit IEEE floating point double 64 bit IEEE floating point By default multibyte words are encoded with the least significant byte first little endian The characters b can be appended to spec ify that they are encoded with the most significant byte first big en dian for symmetry U is accepted and ignored By default the output is a double array To get an output which has the same type as what is specified by precision the character can be inserted at the beginning For instance uint8 reads bytes and stores them in an array of class uint8 int32 b reads signed 32 bit words and stores them in an array of class int32 after performing byte swapping if necessary and char reads bytes and stores them in a character row vector i e a plain string See also fwrite s read
102. the terminating string end sysquake Standard output produced by functions like disp and fprintf is inserted in the output where it is processed by LaTeX Remark the sysquake environment uses the verbatim package In the beamer class the fragile option should be added to frames where it is used See below for an example Example In the fragment below the eigenvalues of a magic square are com puted and displayed in a loop Function fprintf is used to display symbols with and sqlxvalue to format the numbers R s eigenvalues lambda_i are begin sysquake R magic 3 lambda eig R for i 1 length lambda fprintf lambda_ d s n i sqlxvalue lambda i end end sysquake begin sysquake w h end sysquake begin sysquake w h starts an environment where sysquake code is evaluated like begin sysquake In addition an EPS file with width w and height h is created with all the output produced by graphical commands the code fragment contains This EPS is inserted in the LaTeX document in a picture environment with the includegraphics command of package graphicx Example The fragment below defines a floating figure with the step response of the transfer function 1 s 3 2s 2 3s 4 We assume the TeX Package Reference 37 latexingraphics option of the sysquake package so that text is processed by LaTeX see above begin figure begin center begin sysquake 300 200 ste
103. the third dimension Example sys tf 1 1 2 3 tf 2 1 2 3 4 evalfr sys 0 1j 3j ans 1x2x4 array 1 0 3333 0 5 2 0 25 0 25j 0 5 0 5j 3 5 8824e 2 0 2353j 0 4 0 2j 4 8 3333e 2 8 3333e 2j 5 3846e 2 6 9231e 2 See also ss ctrb Controllability matrix Syntax C crtb a Description ctrb a gives the controllability matrix of system a which is full rank if and only if a is controllable Libraries lti 533 See also Iti dcgain Steady state gain Syntax g dcgain a Description dcgain a gives the steady state gain of system a See also norm Iti feedback Feedback connection Syntax c feedback a b c feedback a b sign c feedback a b ina outa c feedback a b ina outa sign Description feedback a b connects all the outputs of lti object a to all its inputs via the negative feedback lti object b feedback a b sign applies positive feedback with weight sign the default value of sign is 1 feedback a b ina outa specifies which inputs and outputs of a to use for feedback The inputs and outputs of the result always cor respond to the ones of a See also connect 534 Sysquake for ATEX Iti inv System inverse Syntax b inv a Description inv a gives the inverse of system a See also Iti isct Test for a continous time LTI Syntax b isct a Description isct a is true if
104. the transfer function evaluated at e where w is a real number be tween 0 and 7 inclusive other definitions include the range between m and 27 which gives a symmetric diagram with respect to the real axis The range of frequencies is selected automatically between 0 and t or can be specified in an optional argument w a vector of normalized frequencies 444 Sysquake for ATEX dnyquist 40P 20 Figure 6 12 dnyquist 3 poly 0 9 0 7 0 6j 0 7 0 6j Further options can be provided in a structure opt created with responseset fields NegFreq and Range are utilized The optional ar guments style and id have their usual meaning With output arguments dnichols gives the real and imaginary parts of the frequency response and the frequency as column vectors No display is produced In Sysquake when the mouse is over a Nyquist diagram in addition to the complex value which can be retrieved with _z0 or _x0 and _y0 the normalized frequency is obtained in _q Example Nyquist diagram with the same scale along both x and y axis and a Hall chart grid reduced to a horizontal see Fig 6 12 scale equal hgrid dnyquist 3 poly 0 9 0 7 0 6j 0 7 0 6j See also LME Reference graphics for dynamical systems 445 dsigma Singular value plot for a discrete time state space model Syntax dsigma Ad Bd Cd Dd Ts dsigma Ad Bd Cd Dd Ts w dsigma Ad Bd Cd Dd Ts opt dsigma Ad Bd Cd Dd Ts w opt dsig
105. the variable is a matrix of transfer func tions subsasgn produces the expected result converting the right hand side of the assignment to a matrix of transfer function if required If the variable is a state space model the result is equivalent the re sult remains a state space model For state space models changing all the inputs or all the outputs with the syntax var expr sys or var expr sys is much more efficient than specifying both sub scripts or a single index The syntax for field assignment var field value is defined for the following fields for state space models A B C and D matrices of the state space model for transfer functions num and den cell arrays of coefficients for both var string and Ts scalar or empty array for continuous time systems Field assignment must preserve the size of matrices and arrays The syntax with braces var i vaLlue is not supported See also operator O Iti subsref Extraction of a part of an LTI system Syntax var i j var ix var select var field b subsref a s Libraries lti 543 Description The method subsref a permits the use of all kinds of extraction of a part of an LTI system If the variable is a matrix of transfer func tions subsref produces the expected result If the variable is a state space model the result is equivalent the result remains a state space model For state space models extracting all the inputs or all the
106. time and the end time Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time The optional arguments style and id have theirjusual meaning With output arguments dinitial gives the output and the time as column vectors No display is produced See also disim Time response plot of a discrete time linear system with arbitrary in put Syntax dlsim numd dend u Ts dlsim Ad Bd Cd Dd u Ts dlsim Ad Bd Cd Dd u Ts x0 dlsim t dlsim opt style dlsim style id dlsim opt style je opt style id dlsim LME Reference graphics for dynamical systems 441 Description dlsim numd dend u Ts plots the time response of the discrete time transfer function numd dend with sampling period Ts The input is given in real vector u where the element i corresponds to time i 1 Ts Input samples before 0 and after Length u 1 are 0 dlsim Ad Bd Cd Dd u Ts plots the time response of the discrete time state space model Ad Bd Cd Dd defined as x kK 1 Agx k Bgu t y k Cax k Dqu k where the system input at time sample k is u k For single input systems u can also be a row vector dlsim Ad Bd Cd Dd u Ts x0 starts with initial state x0 at time t 0 The length of x0 must match the number of states The default initial state is the zero vector The simulation time range can be prov
107. value to multiple variables to swap the contents of two variables or to assign the elements of a list to different variables Examples Swap variable a and b 2 abc b deal b a a b a o Il bc o NI O Copy the same random matrix to variables x y and z x y z deal rand 5 98 Sysquake for TEX Assign the elements of list l to variables v1 v2 and v3 l 1 abc 3 5 v1 v2 v3 deal l vl See also varargin varargout operator dumpvar Dump the value of an expression as an assignment to a variable Syntax dumpvar value dumpvar name value dumpvar fd name value str dumpvar value str dumpvar name value Description dumpvar fd name value writes to the channel fd the standard out put by default a string which would set the variable name to value if it was evaluated by LME If name is omitted only the textual represen tation of value is written With an output argument dumpvar stores result into a string and produces no output Examples dumpvar 2 3 5 a 6 dumpvar a a 6 abc dumpvar string s string abc a S LME Reference miscellaneous functions 99 See also error Display an error message and abort the current computation Syntax error str error format argl arg2 error identifier format argl arg2 Description Outside a try block error str display
108. values far away from the mean Example kurtosis rand 1 10000 1 8055 See also linprog Linear programming 226 Sysquake for ATEX Syntax x linprog c A b x linprog c A b xlb xub Description Linprog c A b solves the following linear programming problem mincx s t Ax lt b The optimum x is either finite infinite if there is no bounded solution or not a number if there is no feasible solution Additional arguments may be used to constrain x between lower and upper bounds linprog c A b xlb xub solves the following lin ear programming problem mincx s t Ax lt b X 2 Xib X lt Xub If xub is missing there is no upper bound xlb and xub may have less elements than x or contain inf or inf corresponding elements have no lower and or upper bounds Examples Maximize 3x 2y subject to x y lt 9 3x y lt 18 x lt 7 and y lt 6 c 3 2 A 1 1 3 1 1 0 0 1 b 9 18 7 6 x linprog c A b x 4 5 4 5 A more efficient way to solve the problem with bounds on variables c 3 2 A 1 1 3 1 b 9 18 xlb LME Reference linear algebra 227 inprog c A b xlb xub Check that the solution is feasible and bounded all isfinite x true logm Matrix logarithm Syntax Y logm X Y err Logm X Description logm X returns the matrix logarithm of X the inverse of the matrix exponential X must be square The matrix
109. vectors With a 6th or 4th string argument quiver style uses the specified style to draw the arrows Example Force field complex numbers are used to simplify computation scale equal z fevalx plus 5 0 5 5 1lj 5 0 5 5 z0 0 2 0 3j 1 20 sign z z0 max abs z z0 2 3 real z imag z real f imag f uiver x y u v f x y u v q See also scale Set the scale Syntax scale xmin xmax ymin ymax scale xmin xmax scale xmin xmax ymin ymax zmin zmax LME Reference base graphics 413 scale features scale features usersettablefeatures scale features xmin xmax ymin ymax scale features usersettablefeatures xmin xmax ymin ymax sc scale Description Without output argument the scale command which should be placed before any other graphical command sets the scale and scale options The last parameter contains the limits of the plot either for both x and y axis or only for the x axis in 2D graphics or for x y and z axis for 3D graphics The limits are used only if the user has not changed them by zooming The first parameter s specify some properties of the scale and which one can be changed by the user There are two ways to specify them with a string or with one or two integer numbers The recom mended way is with a string The list below enumerates the possible values equal Same linear scale for x and y axis Typically us
110. x1 x2 gives a row vector with the elements x1 x1 1 x1 2 etc until x2 The last element is equal to x2 only if x2 x1 is an integer and smaller otherwise If x2 lt x1 the result is an empty matrix x1 step x2 gives a row vector with the elements x1 x1 step x1 2 step etc until x2 The last element is equal to x2 only if x2 x1 step is an integer With fractional numbers rounding er rors may cause x2 to be discarded even if x2 x1 step is almost an integer If x2 sign step lt x1l sign step the result is an empty matrix If x1 or step is complex a complex vector is produced with the expected contents The following algorithm is used to generate each element z xl while real z x1 x2 x1 lt 1 add z to the vector z Z step end This algorithm is robust enough to stop even if x2 is not on the complex straight line defined by x1 and step If x2 x1 and step are orthogonal it is attempted to produce an infinite number of elements which will obviously trigger an out of memory error This is similar to having a null step in the real case Note that the default step value is always 1 for consistency with real values Choosing for instance sign x2 x1 would have made the 150 Sysquake for TEX generation of lists of indices more difficult Hence for a vector of purely imaginary numbers always specify a step colon x1 x2 is equivalent to x1 x2 and colon x1 step x2 to x1 step x2 It can be used to redefine this o
111. y 2 its speed SI units are used The state space model is y y 2 9 81 An event occurs when the ball hits the ground value y 1 isterminal false direction 1 When the event occurs a new state is computed yn 0 damping xy 2 To integrate this the following functions are defined function yp ballfun t y damping yp y 2 9 81 function v te d ballevents t y damping v y 1 event when the height becomes negative te false do not terminate d 1 only for negative speeds function yn ballonevent t y i damping yn 0 damping y 2 Ball state is integrated during 5 s see Fig 6 3 with opt odeset Events ballevents OnEvent ballonevent t y ode45 ballfun 0 5 2 0 opt 1 plot t y LME Reference non linear numerical functions 303 Bouncing ball integrated with events Figure 6 3 Bouncing ball integrated with events Time events with discontinuous function If the function being integrated has discontinuities at known time in stants option EventTime can be used to insure an accurate switching time Consider a first order filter with input u t where u t O for t lt 1 and u t 1 fort gt 1 The following function is defined for the state derivative function yp filterfun t y yp y t lt 1 0 1 A single time event is generated at t 1 opt odeset EventTime inf 1 t y
112. 0 For positive integer values n n 1 LME Reference mathematical functions 177 Examples gamma 5 24 gamma 3 inf gamma 3 5 0 2701 See also gammainc Incomplete gamma function Syntax y gammainc x a Description gammainc x a gives the incomplete gamma function of x and a Ar guments and result are real imaginary part is discarded x must be nonnegative The incomplete gamma function is defined as 1 x gammainc x a t 16 dt r a Jo Example gammainc 2 1 5 0 7385 See also gammalin Logarithm of gamma function Syntax y gammaln x 178 Description Sysquake for ATEX gammaln x gives the logarithm of the gamma function of x Argument and result are real imaginary part is discarded gammaln does not rely on the computation of the gamma function to avoid overflows for large numbers Examples gammaln 1000 5905 2204 gamma 1000 inf See also gammainc betaln gcd Greatest common divisor Syntax q gcd a b Description gcd a b gives the greatest common divisor of integer numbers a and b Example gcd 72 56 8 See also goldenratio Golden ratio constant LME Reference mathematical functions 179 Syntax x goldenratio Description goldenratio is the golden ration v5 1 2 up to the precision of its floating point representation Example goldenratio 1 6180 See also hypot Hypotenuse Sy
113. 123 See also fieldnames rmfield Deletion of a field in a structure Syntax strctout rmfield strctin name Description strctout rmfield strctin name makes a structure strctout with the same fields as strctin except for field named name which is re moved If field name does not exist strctout is the same as strctin Example x rmfield struct a 1 3 b abc a fieldnames x b LME Reference structures 339 See also struct set field get field orderfields setfield Assignment to a field in a structure Syntax strctout setfield strctin name value Description strctout setfield strctin name value makes a structure strctout with the same fields as strctin except that field named name is added if it does not exist yet and is set to value s setfield s f v has the same effect as s f v setfield is especially useful when the field name is not fixed but is stored in a variable or is the result of an expression See also struct Creation of a structure Syntax strct struct fieldnamel valuel fieldname2 value2 Description struct builds a new structure Input arguments are used by pairs to create the fields for each pair the first argument is the field name provided as a string and the second one is the field value Example X X struct a 1 b 2 5 a 1 x b 2345 340 Sysquake for TEX See also 6 25 Object Func
114. 2 hello 2 string 1 s f tow e wma real 1x2 string Q See also Operator getfieldjsetfietd Operator Addition Syntax xX y M1 M2 M x plus x y Description With two operands both operands are added together If both operands are matrices with a size different from 1 by 1 their size must be equal the addition is performed element wise If one operand is a scalar it is added to each element of the other operand With one operand no operation is performed except that the result is converted to a number if it was a string or a logical value like with all mathematical operators and functions For strings each character is replaced with its numerical encoding The prefix is actually a synonym of double 130 Sysquake for ATEX plus x y is equivalent to x y and uplus x to x They can be used to redefine these operators for objects Example 2 3 5 1 2 3 5 47 3 4 2 5 6 See also Operator Subtraction or negation Syntax X y M1 M2 M x minus x y X M uminus x Description With two operands the second operand is subtracted from the first operand If both operands are matrices with a size different from 1 by 1 their size must be equal the subtraction is performed element wise If one operand is a scalar it is repeated to match the size of the other operand With one operand the sign of each element is changed minus x y is equivalent to
115. 2 out 0 5 0 5 2 0 5 0 5 dist 0 259 See also Lloyds dpcmenco ulawcompress mu law compressor 554 Sysquake for TEX Syntax output ulawcompress input output ulawcompress input mu Description ulawcompress input a compresses signal input with mu law method using parameter mu input can be a real array of any size and dimension The default value of mu is 255 The compressor and its inverse the expander are static nonlin ear filters used to improve the signal noise ratio of quantized signals The compressor should be used before quantization or on a signal represented with a higher precision See also ulLawexpand jalawcompress ulawexpand mu law expander Syntax output ulawexpand input output ulawexpand input mu Description ulawexpand input a expands signal input with mu law method us ing parameter a input can be a real array of any size and dimension The default value of mu is 255 See also ulawcompressS ja lLawexpand 7 9 wav wav is a library which adds to LME functions for encoding and decoding WAV files WAV files contain digital sound The wav library supports un compressed 8 bit and 16 bit monophonic and polyphonic WAV files It can also encode and decode WAV data in memory without files The following statement makes available functions defined in wav Libraries wav 555 use wav wavread WAV decoding Syntax samples samplerate nbits wavread fi
116. 21 The partition is optimized with Lloyds and the same signal is quan tized again The distortion is reduced partition_opt codebook_opt Lloyds in codebook partition_opt 2 9 0 5 1 3 codebook_opt 4 1 1 7 0 4 2 2 i out dist quantiz in partition_opt codebook_opt dist 1 0543 See also Libraries sigenc 553 quantiz Table based signal quantization Syntax i quantiz input partition i output distortion quantiz input partition codebook Description quantiz input partition quantizes signal input using partition as boundaries between different ranges Range from to partition 1 corresponds to code 0 range from partition 1 to partition 2 corresponds to code 1 and so on input may be a real array of any size and dimension partition must be a sorted vector The output i is an array of codes with the same size and dimension as input quantiz input partition codebook uses codebook as a look up table to convert back from codes to signal It should be a vector with one more element than partition With a second output argu ment quantiz gives codebook i With a third output argument quantiz computes the distortion be tween input and codebook i i e the mean of the squared error Example partition 1 0 1 codebook 2 0 5 0 5 2 in randn 1 5 in 0 1799 9 7676e 2 1 1431 0 4986 1 0445 i out dist quantiz in partition codebook l 2 1 0 1
117. 5 L K dare A B Q R 2327 9552 1047 113 1047 113 496 0624 L 0 2315 0 431 K 9 3492 2 1995 X A X A A X B B X xB R B X A Q 1 0332e 9 4 6384e 10 4 8931le 10 2 2101e 10 See also Care deconv Deconvolution or polynomial division Syntax q deconv a b q r deconv a b LME Reference linear algebra 213 Description q r deconv a b divides the polynomial a by the polynomial b re sulting in the quotient q and the remainder r All polynomials are given as vectors of coefficients highest power first The degree of the remainder is strictly smaller than the degree of b deconv is the inverse of conv a addpol conv b q r Examples q r deconv 1 2 3 4 5 1 3 2 q 1 14 r 6 3 addpol conv q 1 3 2 r 12345 See also det Determinant of a square matrix Syntax d det M Description det M is the determinant of the square matrix M which is O up to the rounding errors if M is singular The function rank is a numerically more robust test for singularity Examples det 1 2 3 4 2 det 1 2 1 2 0 See also poly 214 Sysquake for ATEX diff Differences Syntax dm diff A dm diff A n dm diff A n dim dm diff A dim Description diff A calculates the differences between each elements of the columns of matrix A or between each elements of A if it is a row vector diff A n calculates the n th order differences i
118. 5 7 See also 478 Sysquake for TEX fftshift Shift DC frequency of FFT from beginning to center of spectrum Syntax Y fftshift XxX Description fftshift X shifts halves of vector 1 d or matrix 2 d X to move the DC component to the center It should be used after fft or fft2 See also ifftshitt filter2 Digital 2 d filtering of data Syntax Y Y filter2 F X filter2 F X shape Description filter2 F X filters matrix X with kernel F with a 2 d correlation The result has the same size as X An optional third argument is passed to conv2 to specify another method to handle the borders filter2 and conv2 have three differences arguments F and X are permuted filtering is performed with a correlation instead of a con volution i e the kernel is rotated by 180 degrees and the default method for handling the borders is same instead of full See also hankel Hankel matrix Libraries stdlib 479 Syntax X hankel c r Description hankel c r creates a Hankel matrix whose first column contains the elements of vector c and whose last row contains the elements of vector r A Hankel matrix is a matrix whose antidiagonals have the same value In case of conflict the first element of r is ignored The default value of r is a zero vector the same length as c Example hankel 1 3 3 8 1 2 3 4 5 6 23 4 5 6 7 3 4 5 67 8 See also toeplitz diag hist Histogram Syntax N X
119. 72 text 416 tf2ss 360 thick line 390 thin line 390 tic 379 times 12 title 417 toc 379 toeplitz torus 5 8 trace transpose 122 trapzZ 488 LET trimmean 497 triu 278 true 353 try PI tsearch 284 tsearchn 285 typecast uint16 287 uint32 uint64 287 uint8 287 NI ulawcompress 553 ulawexpand uminus 122 union 278 unique ne E unsetenv B87 until 92 uplus upper use useifexists utf8decode utf8encode 319 var varargin varargout 115 variables vertcat voronoi 286 voronoin 287 wa rand E wavread wavwrite 555 weekday which while whitecm 567 xor 353 zeros 280 zgrid zscore 498 587
120. 8 B13 uint16 B15 uint32 515 uints 515 class bitfield beginning ar sic disp double end 512 find 512 length sign class lti append beginning bodemag bodephase c2d 529 connect 530 d2c 530 Sysquake for ATEX end evalf r 532 feedback 533 impulse 545 inv 534 isct 534 isdt 534 isempty isproper issiso 535 lsim 546 minreal 536 minus gt 36 mldivide 537 mrdivide 537 mtimes 537 nichols 546 norm 538 nyquist parallel plus repmat 540 series 540 size 541 ssdata 541 step 547 ee subsref tfdata 543 uminus 547 uplus 548 class polynom diff 501 Toop EDD double feval 503 inline 502 int 502 polynom subst class ratfun den 505 diff 505 disp feval Index inline 506 num 504 ratfun class ratio char disp double ratio 50 class ss augstate 529 ctrb 532 initial 545 obsv ss 526 class tf tf 627 543 clc 362 clear 96 clock 378 colon 122 color 390 colormap compan ee cond 206 cone aE conj B BO contour B99 contour3 425 conv 207 conv2 208 corrcoef 476 cos 164 cosh cot 165 coth 165 cov 209 eee Se cross 209 eae 574 CSC csch 166 TES ctranspose 122 cube 568 cumprod 210 581 cumsum 211 cumtrapz cyan2magentacm cylinder d2cm dare dash
121. Integers whose absolute value is smaller than 2 52 about 4 5e15 have an exact representation though Literal double numbers constant numbers given by their numerical value have an optional sign an integer part an optional fractional part following a dot and an optional exponent The exponent is the power of ten which multiplies the number it is made of the letter e or E followed by an optional sign and an integer number Numbers too large to be represented by the floating point format are changed to plus or minus infinity too small numbers are changed to 0 Here are some examples numbers on the same line are equivalent 123 123 123 123 00 12300e 2 2 5 25e 1 0 25e1 0 25e 1 0 0 0 0 le 99999 inf 1e999999 inf 1e999999 Literal integer numbers may also be expressed in hexadecimal with prefix Ox in octal with prefix 0 or in binary with prefix Ob The four literals below all represent 11 stored as double Oxb Literal integer numbers stored as integers and literal single numbers are followed by a suffix to specify their type such as 2int16 for the number 2 stored as a two byte signed number or 0x300uint32 for the LME Reference 45 number whose decimal representation is 768 stored as a four byte un signed number All the integer types are valid as well as single This syntax gives the same result as the call to the corresponding function e g 2int16 is the same as int16 2 except when the integer num b
122. Minimum of a function in R n Syntax x fminsearch fun x0 x fminsearch options x fminsearch options x y didConverge fminsearch Description fminsearch fun x0 finds numerically a local minimum of func tion fun fun is either specified by its name or given as an anonymous or inline function or a function reference It has at least one input argument x a real scalar vector or array and it returns one output LME Reference non linear numerical functions 293 argument a scalar real number fminsearch finds the value x such that fun x is minimized starting from point x0 The optional third input argument may contain options It is either the empty array for default options or the result of optimset Remaining input arguments of fminsearch if any are given as ad ditional input arguments to function fun They permit to parameterize the function For example fminsearch fun x0 2 5 calls fun as fun x 2 5 and minimizes its value with respect to x The first output argument of fminsearch is the value of x at opti mum The second output argument if it exists is the value of fun x at optimum The third output argument if it exists is set to true if fminsearch has converged to an optimum or to false if it has not in that case other output arguments are set to the best value obtained With one or two output arguments fminsearch throws an error if it does not converge Al
123. O and the number of out put arguments listed in the function definition The function can use nargout to check whether some output arguments are not used so that it can avoid computing them or do something else With one argument nargout fun returns the maximum number of output arguments a function can provide fun may be the name of a built in or user function a function reference or an inline function Functions with a variable number of output arguments such as feval give 1 112 Sysquake for ATEX Example A function which prints nicely its result when it is not assigned or used in an expression function y multiplyByTwo x if nargout gt 0 y 2 X else fprintf The double of f is f n x 2 x end Maximum number of output arguments of svd nargout svd 3 See also nargin varargout function rethrow Throw an error described by a structure Syntax rethrow s Description rethrow s throws an error described by structure s which contains the same fields as the output of Lasterror rethrow is typically used in the catch part of a try catch construct to propagate further an error but it can also be used to initiate an error like error Example The error whose identifier is LME indexOutOfRange is handled by catch other errors are not try catch err lasterror if err identifier LME indexOutOfRange else LME Reference miscellaneous functions rethr
124. Q fe Sysquake for IATEX User Manual Embedding numerical results and graphics in ATEX November 2007 2 Sysquake for ATEX Copyright 1999 2007 Calerga Sarl No part of this publication may be reproduced transmitted or stored in any form or by any means including electronic mechanical recording or oth erwise without the express written permission of Calerga Sarl The information provided in this manual is for reference and information use only and Calerga assumes no responsibility or liability for any inaccura cies or errors that may appear in this documentation Sysquake LME Calerga the Calerga logo and icons are copyrighted and are protected under the Swiss and international laws Copying this software for any reason beyond archival purposes is a violation of copyright and viola tors may be subject to civil and criminal penalties Sysquake LME and Calerga are trademarks of Calerga Sarl All other trademarks are the property of their respective owners Sysquake for IATEX User Manual August 2007 Yves Piguet Calerga Sarl Lausanne Switzerland Most of the material in LyME User Manual has first been written as a set of XHTML files with lots of cross reference links Since X HTML is not very well suited for printing it has been converted to ATX with the help of a home made conversion utility Additional XML tags have been used to benefit from TEx features e g raster images have been replaced with EPS images eq
125. Re e qr A false Qe Re qr A false 240 Sysquake for ATEX Description With three output arguments qr A computes the QR decomposition of matrix A with column pivoting i e a square unitary matrix Q and an upper triangular matrix R such that A E Q R With two output argu ments qr A computes the QR decomposition without pivoting such that A Q R With a second input argument with the value false if A has m rows and n columns with m gt n qr produces an m by n Q and an n by n R Bottom rows of zeros of R and the corresponding columns of Q are discarded With column pivoting the third output argument e is a permutation vector A e Q R Example Q R qr 1 2 3 4 5 6 0 169 0 8971 0 4082 0 5071 0 276 0 8165 0 8452 0 345 0 4082 R 5 9161 7 4374 0 0 8281 0 0 Q R qr 1 2 3 4 5 6 false Q 0 169 0 8971 0 5071 0 276 0 8452 0 345 R 5 9161 7 4374 0 0 8281 See also rank Rank of a matrix Syntax x rank M x rank M e LME Reference linear algebra 241 Description rank M returns the rank of matrix M i e the number of lines or columns linearly independent To obtain it the singular values are computed and the number of values significantly larger than O is counted The value below which they are considered to be 0 can be specified with the optional second argument Examples rank 1 1 0 0 1 rank 1 1 0 1j 2 See also roots Roots of a polynomial Syntax
126. Real square root Syntax y realsqrt x Description realsqrt x gives the real square root of x The imaginary part of x is ignored The result is NaN if x is negative Example realsqrt 1 0 1 10 1 2j nan 0 1 3 1623 1 See also rem Remainder of a real division 196 Sysquake for TEX Syntax r rem x y Description rem x y gives the remainder of x divided by y i e a number r be tween 0 and sign x abs y such that x q y r with integer q Imaginary parts if they exist are ignored Examples rem 10 7 3 rem 10 7 3 rem 10 7 3 rem 10 7 3 See also round Rounding to the nearest integer Syntax y round x Description round x gives the integer nearest to x If the argument is a complex number the real and imaginary parts are handled separately Examples round 2 3 2 round 2 6 3 round 2 3 2 LME Reference mathematical functions 197 See also Floor ceil sign Sign of a real number or direction of a complex number Syntax s sign x z2 sign z1 Description With a real argument sign x is 1 if x gt 0 0 if x 0 or 1 if x lt 0 With a complex argument sign z1 is a complex value with the same phase as Z1 and whose magnitude is 1 Examples sign 2 1 sign 1 1j 0 7071 0 7071j sign 0 5 01 See also sec Secant Syntax y sec x Description sec x gives the secant of x which is complex if x is 198 Sysquake
127. T file With one argument matfileencode s writes to the standard output which should be un common since MAT files contain non printable bytes Only arrays are supported scalar matrices arrays of more than two dimensions real or complex numerical logical or char Examples S a 123 b abc S fd fopen data mat wb matfileencode fd s fclose fd Function variables can be used to save all variables v variables fd fopen var mat wb matfileencode fd v fclose fd See also matf ledecode variables 6 32 Shell This section describes functions related to the Unix or Windows shell They are available only on Windows and on Unix or Unix like systems such as Mac OS X The versions for Unix and Windows have significant differences 382 Most functions described here are defined on both Unix and Win dows to avoid errors when loading functions which contain con ditional code for Unix and Windows Functions with an empty im plementation return the error Not supported Table below gives the status of all commands On Windows some of the functionality of unix is provided by dos The main difference is that dos does not give any output except cd Set or get the current directory Command cd cputime dir dos getenv ispc pwd setenv sleep unix unsetenv Unix supported supported supported not supported supported false supported s
128. TI transfer function constructor Syntax a H a tf num den a oe denlist a tf Ts a tf TS var a tf b a tf gain n a tf b Description tf num den creates an LTI object which represents the continuous time transfer function specified by descending power coefficient vec tors num and den tf num den Ts creates an LTI object which repre sents a discrete time transfer function with sampling period Ts In both cases num and den may be replaced with cell arrays of coef ficients whose elements are the descending power coefficient vectors 528 Sysquake for TEX The number of rows is the number of system outputs and the number of columns is the number of system inputs An additional argument var may be used to specify the variable of the Laplace s default or p or z transform z default or q tf b6 where b is an LTI object creates a transfer function of the same kind continuous discrete time sampling time and variable as b tf b converts the LTI object b to a transfer function tf gain where gain is a matrix creates a matrix of gains Examples Simple continuous time system with variable p p is used only for dis play sc tf 1 1 2 3 4 p SC continuous time transfer function 1 p 3 2p 2 3p 4 Matrix of discrete time transfer functions for one input and two out puts with a sampling period of 1ms sd tf 0 1 0 15 1 0 8 1
129. _mass 1 6749286e 27 kg plank_constant 6 6260755e 34 Js plank_constant_reduced 1 0545727e 34 Js plank_mass 2 17671e 8 kg proton_mass 1 6726231e 27 kg solar_radius 6 9599e8 m speed_of_light 299792458 m s speed_of_sound 340 29205 m s stefan_boltzmann_constant 5 67051le 8 W m 2 K 4 vacuum_permittivity 8 854187817e 12 As Vm 7 12 colormaps colormaps is a library containing functions related to color maps Color maps are tables of colors which can be used with the colormap function they are used by functions such as image and surf to map values to colors All functions accept at least the number of colors n as input ar gument and produce an n by 3 real double array which can be used directly as the argument of colormap The default value of n is 256 colormaps defines the following functions Libraries colormaps 561 Function Description black2orangecm color shades from black to orange black2red2whitecm color shades from black to red and white blue2greencm color shades from blue to green blue2yellow2redcm color shades from blue to yellow and red cyan2magentacm color shades from cyan to magenta graycm gray shades from black to white green2yellowcm color shades from green to yellow huecm color shades with hue from red to red throw green and blu interprgbcm colormap created with linear interpolation magenta2yellowcm color shades from magenta to yellow red2yellowcm color shades from red to yellow sepiacm sepia shades whitecm
130. a single index or a vector of indices indexed el ements are grouped in a list New elements also provided in a list can be assigned to indexed elements if the list to be assigned has a single element the element is assigned to every indexed element of the variable Cell arrays cell arrays are subscripted like other arrays The re sult or the right hand side of an assignment is also a cell array or a list for the syntax v select lists are to cell arrays what column vectors are to non cell arrays To create a single logical array for se lecting some elements function cellfun may be useful To remove 124 Sysquake for ATEX cells the right hand side of the assignment can be the empty list or the empty array Examples Ordering evaluation 1 2 3 9 Extracting a single element a row and a column 1 2 3 4 5 6 a a 2 3 Extracting a sub array with contiguous rows and non contiguous columns a 1 2 1 3 1 2 13 46 Array elements as a vector a 3 5 o mBW Ww OUBWNE Selections of elements where a logical expression is true a a gt 5 5 6 a sum a 1 gt 6 23 5 6 LME Reference operators 125 Assignment a 1 5 99 a 9 erel 2309 5600 Extraction and assignment of elements in a list 1 2 7 3 abc magic 3 x 5 2 7 3 x 5 ab cde a 2 I a 2 a 1 ab abc 8 1 6 3 5 7 4 9 2 cde a 2
131. aces and tabulators are skipped before decoding s d x 0 i f e or g The format string is recycled as many times as necessary to de code the whole input string The decoding is interrupted if a mismatch occurs The optional second output argument is set to the number of ele ments decoded successfully may be different than the length of the first argument if decoding strings The optional third output argument is set to the number of characters which were consumed in the input string 376 Sysquake for ATEX Examples sscanf f 2 3 f f 2 3 sscanf 12a34x778 d c 12 97 34 120 778 sscanf abc def abcdef sscanf abc def abc def sscanf 12 34 xd d 34 sscanf 0275a0ff 2x 2 117 160 255 S c See also swrite Store data in a string Syntax s swrite data s swrite data precision Description swrite fd data stores the contents of the matrix data in a string The third parameter is the precision whose meaning is the same as for fread Its default value is uint8 Examples swrite 65 68 ABCD double swrite 1 2 int16 1020 LME Reference file system 377 double swrite 1 2 int16 b 0102 See also 6 29 File System Functions Access to any kind of file can be useful to analyze data which come from other applications such as experimental data and to generate results in a f
132. agicsum3 magic_3 magic 3 sum_of_each_line sum magic_3 2 sum_of_each_column sum magic_3 1 You can call the function just by typing its name in the command line gt magicsum3 magic_3 816 357 492 sum_of_each_line sum_of_each_column 15 15 15 This function is limited to a single size For more generality let us add an input argument function magicsum n magc magic n sum_of_each_line sum magc 2 sum_of_each_column sum magc 1 When you call this function add an argument gt magicsum 2 magc 13 42 sum_of_each_line 4 6 sum_of_each_column 55 Tutorial 29 Note that since there is no 2 by 2 magic square magic 2 gives some thing else Finally let us define a function which returns the sum of each line and the sum of each column function sum_of_each_line sum_of_each_column magicSum n magc magic n sum_of_each_line sum magc 2 sum_of_each_column sum magc 1 Since we can obtain the result by other means we have added semi colons after each statement to suppress any output Note the upper case S in the function name for LME this function is different from the previous one To retrieve the results use the same syntax gt sl sc magicSum 3 15 15 15 You do not have to retrieve all the output arguments To get only the first one just type gt sl magicSum 3 Sl 15 15 15 When you retrieve only one output argument you can use it direct
133. al 10 Figure 6 17 Example of initial Example Response of a continuous time system whose initial state is 5 3 see Fig 6 17 initial 0 3 0 1 0 8 0 4 2 3 1 3 2 1 2 1 5 3 See also Isim Time response plot of a continuous time linear system with piece wise linear input Syntax lsim numc denc u t lsim numc denc u t opt lsim Ac Bc Cc Dc u t lsim Ac Bc Cc Dc u t opt lsim Ac Bc Cc Dc u t x0 lsim Ac Bc Cc Dc u t x0 opt lsim style 454 Sysquake for TEX lsim style id y t lsim Description lsim numc denc u t plots the time response of the continuous time transfer function numd dend The input is piece wise linear it is de fined by points in real vectors t and u which must have the same length Input before t 1 and after t end is 0 The input used for the simulation is interpolated to have a smooth response lsim Ac Bc Cc Dc u t plots the time response of the continuous time state space model Ac Bc Cc Dc defined as dx Ae Acx t B u t y t C x t Dcu t where the system input at time sample t i is u i For single input systems u can also be a row vector Lsim Ac Bc Cc Dc u t x starts with initial state x0 at time t 0 The length of x0 must match the number of states The default initial state is the zero vector Options can be provided in a structure opt created with responseset Range The range is a
134. al numbers between 0 to 1 maximum intensity The default value of n is 256 See also black2orangecm black2red2whitecm blue2greencm blue2yellow2redcm cyan2magentacm graycm green2yellowcm magenta2yellowcm interprgbcm Colormap with entries obtained by linear interpolation Syntax cm cm interprgbcm i r g b interprgbcm i r g b n Description interprgbcm i r b g n creates a color map with n entries Color shades are interpolated between colors defined in RGB color space by corresponding elements of r g and b defined for input in i These four arguments must be vectors of the same length larger or equal to 2 with elements between 0 and 1 Argument i must have monotonous entries with i 1 0 and i end 1 The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of nis 256 566 Sysquake for ATEX See also colormap black2orangecm black2red2whitecm blue2greencm blue2yellow2redcm cyan2magentacm graycm green2yellowcm magenta2yellowcm red2yel lLowcm magenta2yellowcm Colormap with shades from black to orange Syntax cm magenta2yellowcm cm magenta2yellowcm n Description magenta2yellowcm n creates a color map with n entries correspond ing to color shades from magenta to yellow The color map is an n by 3 array with one color per row columns corresp
135. ally the magnitude is displayed with a logarithmic or dB scale use scale Uindb Jor scale Uinlog Uindb before dnichols The range of frequencies is selected automatically between 0 and t or can be specified in an optional argument w a vector of normalized frequencies Further options can be provided in a structure opt created with responseset fields NegFreq and Range are utilized The optional ar LME Reference graphics for dynamical systems 443 guments style and id have their usual meaning With output arguments dnichols gives the magnitude and phase of the frequency response and the frequency as column vectors No display is produced In Sysquake when the mouse is over a Nichols diagram in addition to the magnitude and phase which can be retrieved with _y0 and _x0 the normalized frequency is obtained in _q Example scale lindb ngrid dnichols 3 poly 0 9 0 7 0 6j 0 7 0 6j See also dnyquist Nyquist diagram of a discrete time system Syntax dnyquist numd dend dnyquist numd dend w dnyquist numd dend opt lt dnyquist numd dend w opt dnyquist style dnyquist style id w dnyquist re im dnyquist re im w dnyquist Description The Nyquist diagram of the discrete time transfer function given by polynomials numd and dend is displayed in the complex plane In dis crete time the Nyquist diagram is the locus of the complex values of
136. andboxtrust fun Description sandboxtrust fun sets a flag associated with function fun so that fun is executed without restriction even when called from a sandbox All functions called directly or indirectly from a trusted function are executed without restriction except if a nested call to sandbox is per formed Argument fun can be a function reference or the name of a function as a string the function must be a user function not a built in one The purpose of sandboxtrust is to give back some of the capabili ties of unrestricted code to code executed in a sandbox For instance if unsecure code must be able to read the contents of a specific file a trusted function should be written for that It is very important for the trusted function to check carefully its arguments such as file paths or URL Example Function which reads the contents of file data txt function data readFile fd fopen data txt data fread fd inf xchar fclose fd Execution of unsecure code which may read this file sandboxtrust readFile sandbox d readFile See also sandbox 6 14 Operators Operators are special functions with a syntax which mimics mathe matical arithmetic operations like the addition and the multiplication LME Reference operators 121 They can be infix Such as x y separating their two arguments called operands prefix such as x placed before their unique operand or postfix
137. ands for invoking Sysquake 3 3 Your first Sysquake computation Since we are anxious to get a result let us begin with something re ally simple We will insert the result of 5 2 as an equation Add the following line between maketitle and end document We all know that 5 2 sqexpr 5 2 don t we Running LaTeX twice will produce the result you expect Why twice Like other features of LaTeX the Sysquake package needs two phases to do its magic During the first phase commands create a file named sq tutorial lme the name is built after the tex file which contains Sysquake code or more exactly code written in the programming language of Sysquake LME During the second phase loading the sysquake package runs the command line tool sysquakeLlatextool to interpret this file The result is written to a file named sq tutorial lmeout and if there are graphics to EPS files This file contains the result as LaTeX definitions for instance sqexpr 5 2 above creates the following definition def lmef ragi 7 Sysquake for ATEX Tutorial 15 Here the name is lmefrag followed by the expression number in lowercase roman notation this might change in future versions Sysquake commands in addition to producing sq tutorial lme also invoke the corresponding definition to insert the result in the final typeset document Note that typesetting documents is often already performed by run ning Latex twice or even more times to re
138. ap fun listinl Description map fun listinl evaluates function fun successively for each corresponding elements of the remaining arguments which must be lists or cell arrays It returns the result s of the evaluation as list s or cell array s with the same size as inputs Input lists which contain a single element are repeated to match other arguments if necessary fun is the name of a function as a string a function reference or an inline function Examples map max 2 6 4 7 1 1 100 6 7 100 map inline x 10 3 7 16 13 17 26 nr nc map size 1 abc 4 7 3 4 nr 1 1 2 nc 1 3 2 map size 1 abc 4 7 3 4 S 1 1 1 3 2 2 map disp hello lme hello lme map atan2 1 2 3 0 4636 0 3218 S 334 Sysquake for ATEX See also num2list Conversion from array to list Syntax list num2list A list num2list A dim Description num2list A creates a list with the elements of non cell array A num2list A dim cuts array A along dimension dim and creates a list with the result Examples num2list 1 5 1 2 3 4 5 num2list 1 2 3 4 1 2 3 4 num2list 1 2 3 4 1 1 2 3 4 num2list 1 2 3 4 2 1 3 2 4 See also replist Replicate a list Syntax listout replist listin n Description replist listin n makes a new list by concatenating n copies of
139. aph 58 Sysquake for ATEX Indenting Indented paragraphs are paragraphs which begin with a colon Mul tiple colons define the level of indenting Indented paragraphs can contain styled text links spans of verbatim text and inline exten sions Example This is a normal paragraph This is an indented paragraph in two lines This is more indented Headings Headings are made of a single line beginning with an equal character The number of equal characters defines the level of title from main title to sub sub subtitle Headings may end with a sequence of equal characters Examples Level 1 heading Paragraph Another level 1 heading without trailing equal character Level 2 heading Level 3 heading Lists Lists are collections of items There are three kinds of lists unnum bered lists numbered lists and definition lists Lists can be nested they end with the next heading indented paragraph table or blank line Unnumbered lists are represented as indented paragraphs with a bullet Each item begins with a star character it can span multiple lines Numbered lists are represented as indented paragraphs with a number Items are numbered automatically with consecutive integers starting at 1 Each item begins with a sharp character it can span multiple lines Definition lists are made of two kinds of items title typically dis played in bold font and definition typically displ
140. ar algebra 209 cov Covariance Syntax M M M cov data cov data 0 cov data 1 Description cov data returns the best unbiased estimate m by m covariance ma trix of the n by m matrix data for a normal distribution Each row of data is an observation where n quantities were measured The co variance matrix is real and symmetric even if data is complex The diagonal is the variance of each column of data cov data Q is the same as cov data cov data 1 returns the m by m covariance matrix of the n by m matrix data which contains the whole population Example cov 1 2 2 4 3 5 11 5 1 5 2 3333 See also cross Cross product Syntax v3 v3 cross vl v2 cross vl v2 dim Description cross vl v2 gives the cross products of vectors v1 and v2 v1 and v2 must be row or columns vectors of three components or arrays of 210 Sysquake for ATEX the same size containing several such vectors When there is ambi guity a third argument dim may be used to specify the dimension of vectors 1 for column vectors 2 for row vectors and so on Examples cross 1 2 3 0 0 1 2 1 0 cross 1 2 3 7 1 3 4 0 0 0 2 0 2 0 12 8 6 0 14 See also cumprod Cumulative products Syntax M2 cumprod M1 M2 cumprod M1 dim Description cumprod M1 returns a matrix M2 of the same size as M1 whose ele ments M2 i j are the product of all the elements M1 k j with k lt i
141. as a string uint8 uint16 uint32 uint64 int8 int16 int32 or int64 64 bit integers are not supported on all platforms The input interval is mapped to its full range In all cases input values outside the interval are clipped to the minimum or maximum values Examples map2int 0 2 0 2 1 2 1x5 uint8 array 0 51 102 153 204 255 255 map2int 1 3 7 10 uint16 1x3 uint16 array 6553 19660 45875 map2int 1 3 7 0 10 int16 1x3 int16 array 26214 13107 13107 See also and related functions LME Reference non linear numerical functions 291 6 20 Non Linear Numerical Functions fminbnd Minimum of a function Syntax x y fminbnd fun x0 x y fminbnd fun xlow xhigh x y fminbnd options x y fminbnd options x y didConverge fminbnd Description fminbnd fun finds numerically a local minimum of function fun fun is either specified by its name or given as an anonymous or inline function or a function reference It has at least one input argument x and it returns one output argument also a real number fminbnd finds the value x such that fun x is minimized Second argument tells where to search it can be either a starting point or a pair of values which must bracket the minimum The optional third argument may contain options It is either the empty array for default options or the result of optim
142. ase equal to m 1 2k with integer k are displayed The whole grid is made of the lines corresponding to a closed loop magnitude of 12 6 3 0 3 6 and 12 dB 456 scale lindb 4 pi 0 0 01 100 ngrid Sysquake for ATEX Figure 6 19 scale lindb Example ngrid nichols 7 1 3 See also Agra nichols Nichols diagram of a continuous time system Syntax nichols numc nichols numc denc nichols numc denc denc nichols numc denc nichols style nichols style w nichols mag phase mag phase w opt w opt id nichols nichols 4 pi 0 0 01 100 ngrid LME Reference graphics for dynamical systems 457 scale lindb nichols 1 1 4 Figure 6 20 scale lindb nichols 1 1 4 Description nichols numc denc displays the Nichols diagram of the continuous time transfer function given by polynomials numc and denc In con tinuous time the Nichols diagram is the locus of the complex val ues of the transfer function evaluated at jw where w is real posi tive displayed in the phase magnitude plane Usually the magnitude is displayed with a logarithmic or dB scale use scale lindb or scale tinlog Tindb before nichols The range of frequencies is selected automatically or can be speci fied in an optional argument w a vector of frequencies Further options can be provided in a structu
143. ating PDF files You can also choose among different applications which hide the details of running Latex and displaying the result such as emacs on unix platforms TeXShop on Mac OS X or Texmaker on Windows and other platforms Sysquake for LaTeX is perfectly compatible with them actually it just ignores them So you can keep your favorite editing environment And now let us begin Remark the complete source file of another example comes with Sysquake for LaTex 3 1 Creating a LaTeX file Use your usual method to create a LaTeX file and write a few things to check that everything runs correctly For this tutorial we will make a report The LaTeX file named for example sq tutorial tex contains something like documentclass 1llpt report begin document title Sysquake for LaTex tutorial author Calerga 14 Sysquake for ATEX date 13 August 2007 maketitle end document 3 2 Add the Sysquake package Packages are sets of definitions which add new commands to LaTeX They are listed in the preamble of LaTeX files between documentclass and begin document The Sysquake package is added like any other package documentclass 1l1lpt report usepackage sysquake begin document If there is an error when you typeset this file the file sysquake sty is probably not found Please refer to section Sysquake for LaTeX Instal lation for more informations Assuming there is no error we can now use LaTeX comm
144. ation matrix Syntax R q2mat q Description R q2mat q gives the 3x3 orthogonal matrix R corresponding to the rotation given by scalar quaternion q For a vector a x y z and its representation as a pure quaternion aq quaternion x y z the rotation can be performed with quaternion multiplication bq q aq q or matrix multiplication b Rxa Input argument q does not have to be normalized a quaternion cor responding to a given rotation is defined up to a multiplicative factor Example rpy2q 0 1 0 3 0 2 q2mat q R 0 9363 0 1688 0 3080 0 1898 0 9810 0 0954 0 2955 0 0954 0 9506 aq quaternion 1 2 3 qxaq q 5228i 2 0336j 2 7469k 1 2 3 a q R a R See also q2rpy rpy2q q2rpy Conversion from quaternion to attitude angles LME Reference quaternions 325 Syntax pitch roll yaw q2rpy q Description q2rpy q gives the pitch roll and yaw angles corresponding to the rotation given by quaternion q It is the inverse of rpy2q All angles are given in radians If the input argument is a quaternion array the results are arrays of the same size conversion from quaternion to angles is performed independently on corresponding elements See also rpy2q q2str Conversion from quaternion to string Syntax str q2str q Description q2str q converts quaternion q to its string representation with the same format as disp See also quaternion format qimag Q
145. ative for b fseek returns 0 if success ful or 1 if the position is outside the limits of the file contents See also ftell 370 Sysquake for TEX ftell Get the current read or write position in a file Syntax position ftell fd Description ftell fd gives the current file position associated with file descriptor fd The file position is the offset with respect to the beginning of the file at which the next input function will read or the next output function will write The offset is expressed in bytes With text files ftell may not always correspond to the number of characters read or written See also fseek feof fwrite Raw output Syntax count count fwrite fd data fwrite fd data precision Description fwrite fd data writes the contents of the matrix data to the out put referenced by the file descriptor fd The third parameter is the precision whose meaning is the same as for fread Its default value is 7uint8 See also redirect Redirect or copy standard output or error to another file descriptor LME Reference input output 371 Syntax redirect fd fdTarget redirect fd fdTarget copy redirect fd R redirect fd redirect R redirect Description redirect fd fdTarget redirects output from file descriptor fd to fdTarget fd must be 1 for standard output or 2 for standard error If fdTarget fd the normal behavior is restored redirect fd fdTarget
146. ay A or of the row vector A The dimension along which harmmean proceeds may be specified with a second argument The inverse of the harmonic mean is the arithmetic mean of the inverse of the observations Example harmmean 1 10 3 4142 mean 1 10 5 5 See also geomean mean iqr Interquartile range Syntax m m iqr A iqr A dim Description iqr A gives the interquartile range of the columns of array A or of the row vector A The dimension along which iqr proceeds may be specified with a second argument The interquartile range is the difference between the 75th per centile and the 25th percentile Example iqr rand 1 1000 0 5158 See also 492 Sysquake for TEX mad Mean absolute deviation Syntax m m mad A mad A dim Description mad A gives the mean absolute deviation of the columns of array A or of the row vector A The dimension along which mad proceeds may be specified with a second argument The mean absolute deviation is the mean of the absolute value of the deviation between each observation and the arithmetic mean Example mad rand 1 1000 0 2446 See also nancorrcoef Correlation coefficients after discarding NaNs Syntax S nancorrcoef X S nancorrcoef X1 X2 Description nancorrcoef X calculates the correlation coefficients of the columns of the m by n matrix X NaN values are ignored The result is a square n by n matrix whose diagonal is 1 na
147. ayed indented Titles begin with a semicolon at the beginning of a line Definitions either follow the title separated with a colon or they begin on a new line beginning with a colon LME Reference 59 List nesting can combine different kinds of lists Sublist items begin with multiple markers the first one s corresponding to the enclosing list s For instance items of an unnumbered list nested in a num bered list should start with at the beginning of the line without any preceding space List markers must be used in a consistent way for example at the beginning of a line in an unnumbered list is not interpreted as a second level numbered list item but as monospace style see below Examples First item of unnumbered list Second item in two lines Third item without space spaces are optional First item of numbered list Second item First item on unnumbered sublist x Second item Thirst item of top level numbered list First title of definition list Definition of first item Second title Second definition beginning on the same line Paragraph separated with a blank line Tables Tables are rectangular array of cells They are made of one line per row Each cell begins with character Heading cells typically dis played in bold face begin with Rows may end with a trailing Example In the table below the first row and the first column contain headings The very
148. bar in a group columns of y and w2 to the relative width of each group Default values used if w is missing or is the empty matrix is 0 8 for both w1 and w2 bar kind where kind is a string specifies the kind of bar plot The following values are recognized grouped Columns of y are grouped horizontally default stacked Columns of y are stacked vertically interval Bars defined with min and max val 396 bar 2 4 3 6 2 4 bar 1 4 magic 4 stacked ft pn Hoo OHC 2 Sysquake for ATEX bar 2 4 3 6 3 5 4 1 2 4 2 4 3 1 5 6 4 6 interval Jy Figure 6 5 Example of bar with different options With interval intervals are defined by two consecutive rows of y which must have an even number of rows The optional arguments style and id have their usual meaning bar uses default colors when argument style is missing Examples Simple bar plot see Fig 6 5 bar 2 4 3 6 3 5 4 1 Stacked bar plot bar 1 4 magic 4 stacked Interval plot bar 1 4 2 4 3 1 5 6 4 6 See also barh Horizontal bar plot interval LME Reference base graphics 397 Syntax barh x barh y x barh y x w barh kind barh kind style barh id Description barh plots a bar plot with horizontal bars Please see bar for a de scription of its behavior and arguments Examples Simple horizontal bar plot
149. be provided in a structure opt created with responseset fields Delay NegFreq and Range are utilized The op tional arguments style and id have their usual meaning With output arguments nyquist gives the real and imaginary parts of the frequency response and the frequency as column vectors No display is produced In Sysquake when the mouse is over a Nyquist diagram in addition to the complex value which can be retrieved with _z0 or _x0 and _y0 the frequency is obtained in _q LME Reference graphics for dynamical systems scale equal nyquist 1 1 3 Figure 6 21 scale equal nyquist 1 1 2 3 Example Nyquist diagram of a third order system see Fig 6 21 scale equal hgrid nyquist 20 poly 1 2 1j 2 1j See also plotroots Roots plot Syntax plotroots pol plotroots pol style plotroots pol style id Description 459 plotroots pol displays the roots of the polynomial pol in the com plex plane If this argument is a matrix each line corresponds to a 460 Sysquake for TEX different polynomial The default style is crosses it can be changed with a second argument Example scale equal plotroots den x plotroots num o See also responseset Options for frequency responses Syntax options responseset options responseset namel valuel options responseset optionsO namel valuel Description responseset namel1 valuel creates the
150. be scalar arrays of the same size or a mix of both Examples bitshift 1 3 8 bitshift 8 2 2 2 4 8 16 32 LME Reference logical functions 349 bitshift 15 0 3 4 15 14 12 8 See also bitxor Bitwise exclusive OR Syntax c bitxor a b Description The input arguments are converted to 32 bit unsigned integers each bit of the result is the binary exclusive OR of the corresponding bits of the inputs The inputs can be scalar arrays of the same size or a scalar and an array Examples bitxor 1 3 2 bitxor 1 6 1 032547 bitxor 7uint8 1234int16 1237int16 See also false Boolean constant false Syntax b false B false n B false nl n2 B false nl n2 350 Sysquake for TEX Description The boolean constant false can be used to set the value of a variable It is equivalent to Logical 0 The constant 0 is equivalent in many cases indices to getjor set the elements of an array are an important exception With input arguments false builds a logical array whose elements are false The size of the array is specified by one integer for a square matrix or several integers either as separate arguments or in a vec tor for an array of any size Examples false false islogical false true false 2 3 FFF FFF See also graycode Conversion to Gray code Syntax g graycode n Description graycode n converts the integer number n to Gray
151. be written in base 2 8 10 or 16 0b1100 014 12 and Oxc all represent the same number The following operators and functions may be used with bitfield arguments with results analog to the corresponding functions of LME Logical functions operate bitwise plus amp and minus or mtimes i not mrdivide xor mldivide eq uminus ne uplus Indexes into bit fields are non negative integers 0 represents the least significant bit and wordlength 1 the most significant bit Unlike arrays bits are not selected with logical arrays but with other bit fields where ones represent the bits to be selected for example a 0b1011 selects bits 0 1 and 3 This is consistent with the way bitfield find is defined Libraries bitfield 511 Examples a bitfield 123 16 a 0b0000000001111011 b a b 0b1111111110000100 b arx 5 b 0b0000001001100111 See also bitfield disp bitfield double bitfield disp Display a bitfield object Syntax disp a Description disp a displays bitfield a It is also executed implicitly when LME displays the bitfield result of an expression which does not end with a semicolon See also bitfield double Convert a bitfield object to a double number Syntax n double a Description double a converts bitfield a to double number 512 Sysquake for ATEX Example a bitfield 123 16 double a 123 See also bitfield bitfield bitfield e
152. bound Checking upper bound Inverse quadratic interpolation 2 1 5649 1 Inverse quadratic interpolation 1 5649 1 571 2 Inverse quadratic interpolation 1 571 1 5708 1 5649 Inverse quadratic interpolation 1 5708 1 5708 1 571 Inverse quadratic interpolation 1 5708 1 5708 1 571 ans 1 5708 See also quad Numerical integration Syntax y quad fun a b y quad fun a b tol y quad fun a b tol trace y quad fun a b tol trace Description quad fun a b integrates numerically function fun between a and b fun is either specified by its name or given as an anonymous or inline function or a function reference The optional fourth argument is the requested relative tolerance of the result It is either a positive real scalar number or the empty ma trix or missing argument for the default value which is sqrt eps The optional fifth argument if true or nonzero makes quad displays information at each step Additional input arguments of quad are given as additional input arguments to function fun They permit to parameterize the function LME Reference strings 307 Example 2 te dt 0 quad t txexp t 0 2 0 5940 See also 6 21 String Functions base64decode Decode base64 encoded data Syntax strb base64decode strt Description base64decode strt decodes the contents of string strt which rep resents data encoded with base64 Characters which are not A Z a 2
153. buted random number Syntax x randn M randn n M randn nl1 n2 M randn nl1 n2 randn seed s Description randn builds a scalar pseudo random number chosen from a normal distribution with zero mean and unit variance The current imple mentation is based on a scalar 64 bit seed which theoretically al lows 2 64 different numbers This seed can be set with the arguments randn seed s where s is a scalar or a vector of two components The seed is not the same as the seed of rand randn seed s re turns the empty array as output argument To discard it the state ment should be followed by a semicolon randn n randn n1 n2 and randn n1 n2 return an n by n square matrix or an array of arbitrary size whose elements are pseudo random numbers chosen from a normal distribution Examples randn 1 5927 randn 1 3 0 7856 0 6489 0 8141 randn seed Q LME Reference arrays 269 randn 1 5927 See also rand repmat Replicate an array Syntax A2 repmat Al n A2 repmat Al m n A2 repmat Al nl Description repmat creates an array with multiple copies of its first argument It can be seen as an extended version of ones where 1 is replaced by an arbitrary array The number of copies is m in the vertical direction and n in the horizontal direction The type of the first argument number character or logical value is preserved With
154. ceded by a space Example This is italic text bold text and bold italic textx Escape character The tilde character when followed by any character except space tab or line feed is an escape character it is not displayed and the next character loses its special meaning if any Example Two stars or xx or xx tilde is rendered as Two stars or or tilde Verbatim Verbatim text is a sequence of characters enclosed between and After all characters are output verbatim without any markup interpreting until the next or the end of text block Multiple spaces and tabs and single line feeds are still converted to single spaces though Verbatim text is an alternative to the escape character it is more convenient for sequences of characters Example xx is rendered as xx Line break Except in preformatted blocks line breaks are not preserved The sequence forces a line break Example The next line of this paragraph begins here 62 Sysquake for ATEX Links Hypertext links URLs are enclosed between and The text dis played as the link is either the same as the URL itself if there is no character or it is what follows No markup is recognized in the URL part what follows can contain styled text and verbatim text Spaces surrounding are ignored Examples Simple link http www calerga
155. character corresponds to one two or three bytes of UTF 8 code The result is a standard character array with a single row Invalid codes for example when the beginning of the decoded data does not correspond to a character boundary are ignored See also utf8encode Encode a string of Unicode characters using UTF 8 Syntax b utf8encode str 320 Sysquake for ATEX Description utf8encode b encodes the contents of character array str using UTF 8 Each Unicode character in str corresponds to one two or three bytes of UTF 8 code The result is an array of unsigned 8 bit integers If the input string does not contain Unicode characters the output is invalid Example b utf8encode abc 200 2000 20000 b 1x10 uint8 array 97 98 99 195 136 223 144 228 184 160 str utf8decode b str 1x6 uint16 array 97 98 99 200 2000 20000 See also 6 22 Quaternions Quaternion functions support scalar and arrays of quaternions Basic arithmetic operators and functions are overloaded to support expres sions with the same syntax as for numbers and matrices Quaternions are numbers similar to complex numbers but with four components instead of two The unit imaginary parts are named i j and k A quaternion can be written w ix jy kz The following relationships hold j k ijk 1 It follows that the product of two quaternions is not commutative for instance ij k but ji k Quaternions are convenien
156. cify the type of the result double for double precision default or single for single precision Examples 1 inf 0 inf inf See also LME Reference mathematical functions 183 isfinite Test for finiteness Syntax b isfinite x Description isfinite x is true if the input argument is a finite number neither infinite nor nan and false otherwise The result is performed on each element of the input argument and the result has the same size Example isfinite 0 1 nan inf TTFF See also isfloat Test for a floating point object Syntax b isfloat x Description isfloat x is true if the input argument is a floating point type dou ble or single and false otherwise Examples isfloat 2 true isfloat 2int32 false See also isnumeric isinteger 184 Sysquake for ATEX isinf Test for infinity Syntax b isinf x Description isinf x is true if the input argument is infinite neither finite nor nan and false otherwise The result is performed on each element of the input argument and the result has the same size Example isinf 0 1 nan inf FF FT See also isinteger Test for an integer object Syntax b isinteger x Description isinteger x is true if the input argument is an integer type includ ing char and logical and false otherwise Examples isinteger 2int16 true isinteger false true isinteger abc tru
157. clear discards the contents of all the local variables including input arguments With string input arguments clear vl v2 discards the contents of the enumerated variables Note that the variables are specified by strings clear is a normal function which evaluates its arguments if they are enclosed between parenthesis You can also omit parenthesis and quotes and use clear is usually not necessary because local variables are auto matically discarded when the function returns It may be useful if a large variable is used only at the beginning of a function or at the command line interface clear functions or clear f removes the definition of all func tions It can be used only from the command line interface not in a function Examples In the example below clear b evaluates its argument and clears the variable whose name is a clear b without parenthesis and quotes does not evaluate it the argument is the literal string b LME Reference miscellaneous functions 97 a Undefined variable b a clear b b Undefined variable b a See also variable assignment deal Copy input arguments to output arguments Syntax v1 v2 deal e vl v2 deal el e2 Description With a single input argument deal provides a copy of it to all its output arguments With multiple input arguments deal provides them as output arguments in the same order deal can be used to assign a
158. code The argu ment n can be an integer number of class double converted to an unsigned integer or any integer type If it is an array conversion is performed on each element The result has the same type and size as the input Gray code is an encoding which maps each integer of s bits to another integer of s bits such that two consecutive codes i e graycode n and graycode n 1 for any n have only one bit which differs LME Reference logical functions Example graycode Q 7 01326754 See also igraycode Conversion from Gray code Syntax n igraycode g Description 351 igraycode n converts the Gray code g to the corresponding integer It is the inverse of graycode The argument n can be an integer num ber of class double converted to an unsigned integer or any integer type If it is an array conversion is performed on each element The result has the same type and size as the input Example igraycode graycode Q 7 01234567 See also islogical Test for a boolean object Syntax b islogical obj 352 Sysquake for ATEX Description islogical obj is true if obj is a logical value and false otherwise The result is always a scalar even if obj is an array Logical values are obtained with comparison operators logical operators test functions and the function Logical Examples islogical eye 10 false islogical eye 10 true See also logical Transform a
159. code the signal in vector in with differential pulse code modulation The result can be used with dpcmenco to encode signals with similar properties If the second input argument is a vector it is used as the predictor and not optimized further its first element must be zero If the third input argument is a vector it is used as an initial guess for the codebook which has the same length An optional fourth input argument provides the tolerance the default is le 7 If only the predictor is required only the input and the predictor order must be supplied as input arguments See also dpcmenco dpcmdeco lloyds 552 Sysquake for TEX lloyds Optimal quantization Syntax partition codebook partition codebook partition codebook Lloyds input n Lloyds input codebook0 lloyds tol Description Lloyds input n computes the optimal partition and codebook for quantizing signal input with n codes using the Lloyds algorithm If the second input argument is a vector Lloyds input codebook0 uses codebook as an initial guess for the codebook The result has the same length A third argument can be used to specify the tolerance used as the stopping criterion of the optimization loop The default is 1e 7 Example We start from a suboptimal partition and compute the distortion partition 1 0 1 codebook 2 0 5 0 5 2 in 5 0 6 3 i out dist quantiz in partition codebook dist 2 14
160. column vector arguments area x y displays the area between the horizontal axis y 0 and the points given by x and y When the second argument is an array with as many rows as elements in x area x Y displays the contribution of each column of Y summed along each row When both the first and second arguments are ar rays of the same size area X Y displays independent area plots for corresponding columns of X and Y without summation LME Reference base graphics 395 With a single argument area y takes integers 1 2 n for the horizontal coordinates With a third argument area x y y0 displays the area between the horizontal line y y0 and values defined by y The optional arguments style and id have their usual meaning area uses default colors when argument style is missing Example area rand 20 10 See also bar Vertical bar plot Syntax bar y bar x y bar x y w bar kind bar kind style bar id Description bar x y plots the columns of y as vertical bars centered around the corresponding value in x If x is not specified its default value is l size y 2 bar x y w where w is scalar specifies the relative width of each bar with respect to the horizontal distance between the bars with values smaller than 1 bars are separated with a gap while with values larger than 1 bars overlap If wis a vector of two components w1 w2 wl corresponds to the relative width of each
161. com x Link with link text http www calerga com Calerga x Link with styled link text http ww calerga com Calerga 6 9 List of Commands Functions and Op erators Programming keywords LME Reference Programming operators and functions nargout sandbdpox strzTun 63 64 Sysquake for TEX Strings base64decode Structures setfield LME Reference 65 Logical operators Bitwise functions bitall 66 Sysquake for ATEX Set functions intersect setdiff union Ismene setxor _unaquel Constants Arithmetic functions LME Reference Other scalar math functions 67 68 Sysquake for ATEX Geometry functions cross dot cross dotl Statistic functions Triangulation functions delaunay griddatan voronoi searc voronoin griddata Quaternion operators LME Reference 69 Quaternion math functions quaternion Quaternion other functions fminbnd ode23 loptimset fminsearch ode45 tzero pdeset Dynamical systems functions c2am margin ss2EF azem margint E255 70 Input output matTiledecode ma 1leencode Files Basic graphics Sysquake for ATEX LME Reference 71 Graphics for dynamical systems nichols nyquis Data compression Image input output imageread imageset imagewrite Date and time mene me ey Unix
162. computed for negative frequencies instead of positive frequencies Option Range should take into account the sampling period for discrete time commands where it is specified Examples Default options responseset Delay 0 NegFreq false Nyquist diagram of e 5 s 1 nyquist 1 1 1 responseset Delay 1 Complete Nyquist diagram of 1 s 2s 2s 1 with dashed line for negative frequencies nyquist 2 nyquist 2 responseset NegFreq true See also bodemag bodephase dbodemag dbodephase dnichols nyquist sigma step rlocus Root locus Syntax rlocus num den rlocus num den style rlocus num den style id 462 Sysquake for TEX Description The root locus is the locus of the roots of the denominator of the closed loop transfer function characteristic polynomial of the system whose open loop transfer function is num den when the gain is be tween 0 and inclusive The characteristic polynomial is num k den with k gt 0 rlocus requires a causal system with real coefficients i e Length den gt Length num Note that the rlocus is defined the same way in the domain of the Laplace transform the z transform and the delta transform The root locus is made of Length den 1 branches which start from each pole and end to each zero or to a real or complex point at infinity The locus is symmetric with respect to the real axis because the coefficients of the characterist
163. continue It is an error to execute continue outside any loop See also define Definition of a constant LME Reference programming constructs 81 Syntax define c expr define c expr Description define c expr assign permanently expression expr to c It is equiv alent to function y c y expr Since c does not have any input argument the expression is usually constant A semicolon may follow the definition but it does not have any effect define must be the first element of the line spaces and comments are skipped Examples define e define g define c define G exp 1 9 81 299792458 6 672659e 11 See also unction for Loop controlled by a variable which takes successively the value of the elements of a vector or a list Syntax for v vect s1 end for v list s1 end 82 Sysquake for ATEX Description The statements between the for statement and the corresponding end are executed repeatedly with the control variable v taking succes sively every column of vect or every element of list List Typically vect is a row vector defined with the range operator You can change the value of the control variable in the loop how ever next time the loop is repeated that value is discarded and the next column of vect is fetched Examples for i 1 3 i end i 1 i 2 i 3 for i 1 3 i end i 1 2 3 for i 1 2 5 end i i 5 for i 1 3 break end i i
164. creasing but they do not have to be spaced uniformly Interpolated points outside the input volume are set to nan Input and output data can be complex Imaginary parts of coordinates are ignored The default interpolation method is multilinear An additional input argument can be provided to specify it with a string only the first character is considered Argument Meaning 0 or nearest nearest value r lt lower coordinates gt higher coordinates 1 or linear multilinear Method lt takes the sample where each coordinate has its in dex as large as possible lower or equal to the interpolated value and smaller than the last coordinate Method gt takes the sample where each coordinate has its index greater or equal to the interpo lated value Examples One dimension interpolation LME Reference arrays 257 interpn 1 2 5 8 0 1 0 2 0 5 1 0 2 3 7 nan 0 2000 0 3000 0 8333 interpn 1 2 5 8 0 1 0 2 0 5 1 0 2 3 7 0 nan 0 2000 0 2000 1 0000 Three dimension interpolation D cat 3 0 1 2 3 4 5 6 7 interpn 0 1 0 1 0 1 D 0 2 0 7 0 5 3 1000 Image rotation we define original coordinates between 0 5 and 0 5 in vector c and arrays X and Y and the image as a linear gradient between 0 and 1 0 5 X interpn c c D Xi Yi isnan E 0 5 c 5 0 01 0 5 X repmat c 101 1 Y X phi 0 2 Xi co
165. ctions loaded in memory Syntax inmem list inmem Description inmem displays the list of user defined functions loaded in memory with the library where they are defined With an output argument inmem gives a list of structures which describe each user defined function loaded in memory with the following fields library library name class class name for functions name function name See also isglobal Test for the existence of a global variable Syntax b isglobal str 108 Sysquake for ATEX Description isglobal str returns true if the string str is the name of a global variable defined as such in the current context See also iskeyword Test for a keyword name Syntax b iskeyword str list iskeyword Description iskeyword str returns true if the string str is a reserved keyword which cannot be used as a function or variable name or false other wise Keywords include if and global but not the name of built in functions like sin or i Without input argument iskeyword gives the list of all keywords Examples iskeyword otherwise true iskeyword break case catch continue else elseif end endfunction for function global if otherwise persistent private public repeat return switch try until use useifexists while See also lasterr Las
166. cts a and b in such a way that the inputs of the result is applied to both a and b and the outputs of the result is their sum parallel a b ina inb outa outb specifies which inputs are shared between a and b and which outputs are summed The inputs of the result are partitioned as ua uab ub and the outputs as ya yab yb Inputs uab are fed to inputs ina of a and inb of b inputs ua are fed to the remaining inputs of a and ub to the remaining inputs of b Similarly outputs yab are the sum of outputs outa of a and outputs outb of b and ya and yb are the remaining outputs of a and b respectively See also series Iti plus System sum Syntax a b plus a b c C Description a b computes the system whose inputs are fed to both a and b and whose outputs are the sum of the outputs of a and b If a and b are transfer functions or matrices of transfer functions this is equivalent to a sum of matrices See also parallel 540 Sysquake for TEX Iti series Series connection Syntax 0 Cc series a b series a b outa inb Description series a b connects the outputs of lti object a to the inputs of lti object b series a b outa inb connects outputs outa of a to inputs inb of b Unconnected outputs of a and inputs of b are discarded See also times paratte Iti repmat Replicate a system Syntax b repmat a n b repmat a m n b repmat a m n Description repmat a n
167. cutoff angular frequency w0 lp2hp z0 p0 k0 w0 converts a filter given by its ze ros poles and gain lp2hp num0 den0 w0 converts a filter given by its numerator and denominator coefficients in decreasing powers of s The new filter F s is 1 F s Fo Wos where Fo s is the filter prototype Libraries lti 525 See also Ip2ip Lowpass prototype to lowpass filter conversion Syntax z p k lp2lp z0 pO kO w0 num den lp2lp num0 den0 w0 Description lp2lp convert a lowpass analog filter prototype with unit angular fre quency to a lowpass analog filter with the specified cutoff angular frequency w0 lp2lp z0 p0 k0 w0 converts a filter given by its ze ros poles and gain Lp2lp num0 den0 w0 converts a filter given by its numerator and denominator coefficients in decreasing powers of s The new filter F s is S F s Fo Wo where Fo s is the filter prototype See also 7 7 Iti Library lti defines methods related to objects which represent linear time invariant dynamical systems LTI systems may be used to model many different systems electro mechanical devices robots chemical processes filters etc LTI systems map one or more inputs u to one or more outputs y The mapping is defined as a state space model or as a matrix of transfer functions either in continuous time or in discrete time Methods are provided to create combine and analyze LTI objects Graphical methods ar
168. d Name Arguments Effect line n Set the current line number to n pragma line 120 sets the current line number as reported by er ror messages or used by the debugger or profiler to 120 This can be useful when the LME source code has been generated by process ing another file and line numbers displayed in error messages should refer to the original file 6 2 Function Call Functions are fragments of code which can use input arguments as parameters and produce output arguments as results They can be built in LME built in functions loaded from optional extensions or defined with LME statements user functions A function call is the action of executing a function maybe with input and or output arguments LME supports different syntaxes LME Reference 41 fun fun fun inl fun inl in2 outl fun out1 out2 fun out1 out2 fun out1 out2 fun Input arguments are enclosed between parenthesis They are passed to the called function by value which means that they cannot be mod ified by the called function When a function is called without any input argument parenthesis may be omitted Output arguments are assigned to variables or part of variables structure field list element or array element A single output argu ment is specified on the left on an equal character Several output arguments must be enclosed between parenthesis or square brackets arguments can simply be separated by
169. d Non square matrix A square matrix was expected but a rectan gular matrix was found Index out of range Index negative or larger than the size of the array Wrong type String or complex array instead of real etc Non integer argument An argument has a fractional part while an integer is required Argument out of range An argument is outside the permitted range of values Non scalar argument An argument is an array while a scalar number is required Non object argument An object is required as argument Not a permutation The argument is not a permutation of the integers from 1 to n Bad argument A numerical argument has the wrong site or the wrong value Unknown option A string option has an invalid value 52 Sysquake for TEX Object too large An object has a size larger than some fixed limit Undefined variable Attempt to retrieve the contents of a variable which has not been defined Undefined input argument Attempt to retrieve the contents of an input argument which was neither provided by the caller nor defined in the function Undefined function Attempt to call a function not defined Too few or too many input arguments Less or more arguments in the call than what the function accepts Too few or too many output arguments Less or more left side variables in an assignment than the function can return Syntax error Unspecified compile time error function keyword without function name Incomplete func
170. d bitor Bitwise OR Syntax c bitor a b Description The input arguments are converted to 32 bit unsigned integers each bit of the result is the binary OR of the corresponding bits of the inputs The inputs can be scalar arrays of the same size or a scalar and an array If the input arguments are of type double so is the result and the operation is performed on 32 bits Examples bitor 1 2 3 bitor 1 6 1 133557 bitor 7uint8 1234int16 1239int16 See also bitset Bit assignment Syntax b b bitset a n bitset a n v 348 Sysquake for ATEX Description bitset a n sets the n th bit of integer a to 1 a can be an integer or a double The result has the same type as a n 1 corresponds to the least significant bit With 3 input arguments bitset a n v sets the bit to 1 if v is nonzero or clears it if v is zero The inputs can be scalar arrays of the same size or a mix of them If ais of type double so is the result and n is limited to 32 Examples bitset 123 10 635 bitset 123 1 0 122 bitset 7uint8 1 8 1x8 uint8 array 7 7 7 15 23 39 71 135 See also bitshift Bit shift Syntax b bitshift a shift b bitshift a shift n Description The first input argument is converted to a 32 bit unsigned integer and shifted by shift bits to the left if shift is positive or to the right if it is negative With a third argument n only n bits are retained The inputs can
171. d ix sort A dim dir list_sorted ix sort list list_sorted ix sort list dir Description sort A sorts separately the elements of each column of array A or the elements of A if it is a row vector The result has the same size as A Elements are sorted in ascending order with NaNs at the end For complex arrays numbers are sorted by magnitude The optional second output argument gives the permutation array which transforms A into the sorted array It can be used to reorder elements in another array or to sort the rows of a matrix with respect to one of its columns as shown in the last example below Order of consecutive identical elements is preserved If a second numeric argument dim is provided the sort is performed along dimension dim columns if dim is 1 rows if 2 etc An additional argument can specify the ordering direction It must be the string ascending or a for ascending order or descending or d for descending order In both cases NaNs are moved to the end sort list sorts the elements of a list which must be strings Cell arrays are sorted like lists not column wise like numeric arrays The LME Reference arrays 275 second output argument is a row vector The direction can be specified with a second input argument Examples sort 3 6 2 3 9 1 2 1223369 sort 2 5 3 nan 4 2 6 1 1 2 1 1 6 4 2 nan 5 3 sort 2 5 3 nan 4 2 6 1 1 d 6 5 3 2 4 2 nan 1 1
172. d third output arguments are vectors of in dices such that if c ia ib union a b then elements of c are the elements of a ia or b ib the intersection of a ia and b ib is empty Example a a bc bbb de b z bc aa bbb c ia ib union a b c a aa bbb bc de z ia 1324 ib 31 a ia a bbb bc de b ib aa z Set exclusive or can also be computed as the union of a and b minus the intersection of a and b setdiff union a b intersect a b a aa de z See also Setditt unique Keep unique elements Syntax v2 unique vl list2 unique list1 b ia ib unique a 280 Sysquake for ATEX Description With an array argument unique v1 sorts its elements and removes duplicate elements Unless v1 is a row vector v1 is considered as a column vector With an argument which is a list of strings unique list sorts its elements and removes duplicate elements The optional second output argument is set to a vector of indices such that if b ia unique a then b is a ia The optional third output argument is set to a vector of indices such that if b ia ib unique a then ais b ib Examples b ia ib unique 4 7 3 8 7 1 3 b 13478 ia 63124 ib 3425412 unique def ab def abc ab abc def
173. damping and natural frequency grid for the poles of a discrete time system Syntax zgrid zgrid style zgrid damping freq zgrid damping freq style 468 Sysquake for ATEX scale equal 1 2 1 2 1 2 1 2 zgrid ae cL iy gt ae c gt R HU A we i ry o 1 ere e _ YZ ui U p B 3 eam Rez eras ty Y SS o s QOS A a ime CEE Me Kt Y Figure 6 25 scale equal 1 2 1 2 1 2 1 2 zgrid Description With no or one argument zgrid plots a grid of lines with constant relative damping and natural frequencies in the complex plane of z see Fig 6 25 The style argument has its usual meaning With two or three arguments zgrid plots only the lines for the specified values of damping and natural frequency The damping and the natural frequency w are defined the same way as for the sgrid function with the mapping z e a normalized sampling frequency is assumed With a damping the line z and its complex conjugate Z are generated by z e 14 V1 4 6 u_ with O lt u lt Umax and Umax chosen such that the line has a positive imaginary part With a natural frequency w typically in the range 0 for a null frequency to n for the Nyquist frequency the line is generated by e where v is such that the curve is inside the unit circle The whole grid is displayed only if the user selects it in the Grid menu or after the command
174. denc bodemag numc denc w 434 Sysquake for TEX bodemag numc denc opt bodemag numc denc w opt bodemag Ac Bc Cc Dc bodemag Ac Bc Cc Dc w bodemag Ac Bc Cc Dc opt reo i Bc Cc Dc w opt w bodemag style bodemag style id mag w bodemag Description bodemag numc denc plots the magnitude of the frequency response of the continuous time transfer function numc denc The range of fre quencies is selected automatically or can be specified in an optional argument w a vector of frequencies Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning bodemag Ac Bc Cc Dc plots the magnitude of the frequency response Y jw U jw of the continuous time state space model Ac Bc Cc Dc defined as joXGw A XYw B UUw Yw C X jw DcUljw With output arguments bodemag gives the magnitude and the fre quency as column vectors No display is produced Examples Green plot for 1 s 2s 35 4 with s jw see Fig 6 9 bodemag 1 1 2 3 4 g The same plot between w 0 and w 10 scale 0 10 bodemag 1 1 2 3 4 g See also bodephase dbodemag bodephase Phase Bode diagram for a continuous time system LME Reference graphics for dynamical systems 435 scale logdb bodemag 1 1 2 3 4 Figure 6 9 scale logdb bod
175. denominator of a as a row vector of descending power coefficients See also ratfun ratfun ratfun diff Ratfun derivative Syntax diff a Description diff a differentiates ratfun a Example r ratfun 1 3 0 1 2 5 q diff r q 4x 3 21x 2 30x 2 4x 2 20x 25 506 Sysquake for ATEX See also ratfun ratfun ratfun inline Conversion from ratfun to inline function Syntax fun inline a Description inline a converts ratfun a to an inline function which can then be used with functions such as feval and ode45 See also ratfun ratfun ratfun feval ratfun feval Evaluate a ratfun object Syntax y feval a x Description feval a x evaluates ratfun a for the value of x If x is a vector ora matrix the evaluation is performed separately on each element and the result has the same size as x Example r ratfun 1 3 0 1 2 5 y feval r 1 5 y 0 7143 2 3333 5 0000 8 6923 13 4000 See also ratfun ratfun Teval Libraries ratio 507 7 4 ratio Library ratio implements the constructors and methods of class ratio for rational numbers It is based on long integers so that the precision is limited only by available memory Basic arithmetic operators and functions are overloaded to support expressions with the same syntax as for numbers The following statement makes available functions defined in ratio use ratio ratio ratio Ratio object const
176. deviation of vector v normalized by length v 1 With a second argument std v p normalizes by length v 1 if pis true or by Length v if p is false std M gives a row vector which contains the standard deviation of the columns of M With a third argument median M p dim operates along dimension dim LME Reference linear algebra Example std 1 2 5 6 10 12 4 3359 See also sum Sum of the elements of a vector Syntax x V V sum v sum M sum M dim Description 245 sum v returns the sum of the elements of vector v sum M returns a row vector whose elements are the sums of the corresponding columns of matrix M sum M dim returns the sum of matrix M along dimension dim the result is a row vector if dim is 1 or a column vector if dim is 2 Examples sum 1 5 15 sum 1 5 15 sum 1 2 3 5 6 7 6 8 10 sum 1 2 3 5 6 7 1 6 8 10 sum 1 2 3 5 6 7 2 6 18 See also prod 246 Sysquake for ATEX svd Singular value decomposition Syntax s svd M U S V svd M U S V svd M false Description The singular value decomposition U S V svd M decomposes the m by n matrix M such that M UxS V where S is an m by n diagonal matrix with decreasing positive diagonal elements the singular values of M Uis an m by m unitary matrix and V is an n by n unitary matrix The number of non zero diagonal elements of S up to rounding errors gives the rank of M When M is
177. does not display anything 570 Sysquake for ATEX See also dodecahedron greatstellateddodecahedron greaticosahedron plotpoly greaticosahedron Create a great dodecahedron Syntax greaticosahedron greaticosahedron style X Y Z ind greaticosahedron Description Without output argument greaticosahedron displays a great icosa hedron i e a regular nonconvex solid whose twenty faces are equi lateral triangles By default edges are not drawn An optional string input argument specifies the edge style With four output arguments greaticosahedron produces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also icosahedron greatdodecahedron plotpoly greatstellateddodecahedron Create a great stellated dodecahedron Syntax greatstellateddodecahedron greatstellateddodecahedron style X Y Z ind greatstellateddodecahedron Description Without output argument greatstellateddodecahedron displays a great stellated dodecahedron i e a regular nonconvex solid whose twelve faces are regular star pentagons and where each vertex is com mon to three faces By default edges are not drawn An optional string input argument specifies the edge style Libraries polyhedra 571 With four output arguments greatstellateddodecahedron pro duces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also dodecahedron greatd
178. duces the X Y and Z arrays expected by surf or mesh and it does not display anything See also Libraries solids 575 cylinder Cylinder Syntax cylinder cylinder cap cylinder cap n cylinder cap n style X Y Z cylinder X Y Z cylinder n Description Without output argument cylinder draws a cylinder approximated by a polyhedron The optional first input argument a logical value which is true by default specifies if caps are included The optional second input argument an integer specifies the number of discrete values for the parameter which describes its surface By default edges are not drawn An optional third input argument a string specifies the edge style it corresponds to the style argument of surf With three output arguments cylinder produces the X Y and Z arrays expected by surf or mesh and it does not display anything See also klein Klein bottle Syntax klein klein p klein p n klein p n style X Y Z 576 Sysquake for TEX Description Without output argument klein draws a Klein bottle approximated by a polyhedron With an input argument klein p uses parameters stored in structure p The following fields are used Field Description Default value rO average tube radius 0 7 d tube variation h half height 3 With two input arguments klein p n draws a Klein bottle where the two parameters which describe its surface are sampled with n dis cr
179. e isinteger 3 false LME Reference mathematical functions See also 1snumeric isfloat isnan Test for Not a Number Syntax b isnan x Description 185 isnan x is true if the input argument is nan not a number and false otherwise The result is performed on each element of the input argument and the result has the same size Example isnan 0 1 nan inf FFTF See also isnumeric Test for a numeric object Syntax b isnumeric x Description isnumeric x is true if the input argument is numeric real or complex scalar vector or array and false otherwise Examples isnumeric pi true isnumeric abc false 186 Sysquake for ATEX See also isscalar Test for a scalar number Syntax b isscalar x Description isscalar x is true if the input argument is scalar real or complex number of any floating point or integer type character or logical value and false otherwise Examples isscalar 2 true isscalar 1 2 5 false See also isnumeric isvector size isvector Test for a vector Syntax b isvector x Description isvector x is true if the input argument is a row or column vec tor real or complex 2 dimension array of any floating point or inte ger type character or logical value with one dimension equal to 1 or empty array and false otherwise LME Reference mathematical functions 187 Examples isvector 1 2
180. e body of the function varargout is a normal variable Its value can be set globally with the brace notation or element by element with varargout i nargout may be used to know how many output arguments to produce Example Here is a function which differentiates a vector of values as many times as there are output arguments function varargout multidiff v for i 1l nargout v diff v varargout i v end In the call below 1 3 7 2 5 3 1 8 is differentiated four times v1 v2 v3 v4 multidiff 1 3 7 2 5 3 1 8 vl 2 4 5 3 2 2 7 v2 2 9 8 5 0 9 v3 11 17 13 5 9 v4 28 30 18 4 See also nargout varargin function variables Contents of the variables as a structure Syntax v variables Description variables returns a structure whose fields contain the variables de fined in the current context LME Reference miscellaneous functions 117 Example 3 1 ow 5 iabl varia a b real 1x5 See also warning Write a warning to the standard error channel Syntax warning msg warning format argl arg2 Description warning msg displays the string msg It should be used to notify the user about potential problems not as a general purpose display func tion With more arguments warning uses the first argument as a format string and displays remaining arguments accordingly like fprintf Example warning Doesn t converge
181. e based on the corresponding graphical func tions the numerator and denominator coefficient vectors or the state space matrices are replaced with an LTI object They accept the same optional arguments such as a character string for the style The following statement makes available functions defined in lti 526 Sysquake for ATEX use lti Ss SS LTI state space constructor Syntax v wv v Ww voovoogvogo tod it wd oueou Description ss A B C D creates an LTI object which represents the continuous time state space model t Bu D u t ss A B C D Ts creates an LTI object which represents the discrete time state space model with sampling period Ts xX k 1 y k In both cases if D is 0 it is resized to match the size of B and C if neces sary An additional argument var may be used to specify the variable of the Laplace s default or p or z transform z default or q ss A B C D b where b is an LTI object creates a state space model of the same kind continuous discrete time sampling time and variable as b ss b converts the LTI object b to a state space model Examples sc ss 1 1 2 2 5 0 sc continuous time LTI state space system A 1 B 1 2 Libraries lti 527 2 5 D 0 0 0 0 sd ss tf 1 1 2 3 4 0 1 sd discrete time LTI state space system Ts 0 1 A 2 3 4 1 0 0 0 1 0 B 1 0 0 C 0 0 1 D 0 See also tf tf L
182. e character f all levels are filled with the default colors Regions with a value of z smaller than the lowest level are left trans parent an explicit lower level should be specified to fill the whole rectangle Examples A function is evaluated over a grid of two variables x and y and is drawn with contour see Fig 6 6 meshgrid 2 0 40 10 xp x 0 2 2 y 0 3 2 exp x 0 5 2 y 0 1 2 0 1 x scale equal contour z 1 1 1 1 x y z e 400 contour Figure 6 6 Example of contour Filled contours 2 0 80 20 repmat u 81 1 x exp x 0 2 2 y 0 3 2 exp x 0 5 2 y 1 2 Q0 1 xX 0 5 sin y levels 1 0 2 1 scale equal contour z 1 1 1 1 levels f NX KC See also fontset Options for fonts Syntax options fontset options fontset namel valuel options fontset options0 namel valuel Sysquake for ATEX LME Reference base graphics 401 Description fontset namel valuel creates the font description used by text Options are specified with name value pairs where the name is a string which must match exactly the names in the table below Case is significant Options which are not specified have a default value The result is a structure whose fields correspond to each option Without any input argument fontset creates a structure with all the default options When its
183. e for TEX Description unsetenv name removes the definition of the environment variable of the specified name Argument is a string If no such environment variable exists unsetenv does nothing Environment variables are defined in the context of the application they can be accessed in the application or in processes it launches Environment variables of the calling process command shell for in stance are not changed unsetenv is not defined for Windows Example unsetenv CONTROLDEBUG See also 6 33 Graphics LME provides low level commands for basic shapes as well as high level commands for more specialized plots Low level commands Low level commands add simple shapes such as lines marks polygons circles and images With them you can display virtually everything you want Arguments of these commands are such that it is very easy to work globally with ma trices without computing each value sequentially in a loop High level commands High level commands perform some com putation of their own to process their arguments before displaying the result This has two benefits first the code is simpler more compact and faster to develop Second command execution is faster because the additional processing is not interpreted by LME but implemented as native machine code The information related to interactive manipulation is often easier to use too Most of these functions are related to automatic control and
184. e is utilized The optional arguments style and id have their usual meaning Sigma is the equivalent of bodemag for multiple input systems For single input systems it produces the same plot With output arguments sigma gives the singular values and the frequency as column vectors No display is produced See also step Step response plot of a continuous time linear system Syntax step numc denc step numc denc opt step Ac Bc Cc Dc step Ac Bc Cc Dc opt step style step style id y t step Description step numc denc plots the step response of the continuous time transfer function numc denc Further options can be provided in a structure opt created with responseset fields Delay and Range are utilized The optional argu ments style and id have their usual meaning step Ac Bc Cc Dc plots the step response of the continuous time state space model Ac Bc Cc Dc defined as dx a Acx t B u t y t C _x t D u t where u is a unit step The state space model must have a scalar input and may have a scalar or vector output With output arguments step gives the output and the time as col umn vectors No display is produced LME Reference graphics for dynamical systems 467 step 1 1 4 0 2 Figure 6 24 step 1 1 2 3 4 Example Step response of the continuous time system 1 s 2s 35 4 see Fig 6 24 step 1 1 4 b See also zgrid Relative
185. e largest integer of the type specified by string type which can be uint8 uint16 uint32 uint64 int8 int16 int32 or int64 64 bit integers are not sup ported on all platforms The result has the corresponding integer type Examples intmax 2147483647int32 intmax uint16 65535uint16 See also uint8 and related functions intmin Smallest integer Syntax i intmin i intmin type Description Without input argument intmin gives the smallest signed 32 bit inte ger intmin type gives the largest integer of the type specified by string type which can be uint8 uint16 uint32 uint64 int8 int16 int32 or int64 64 bit integers are not sup ported on all platforms The result has the corresponding integer type Examples intmin 2147483648int32 intmin uint16 Ouint16 290 Sysquake for TEX See also uint8 and related functions map2int Mapping of a real interval to an integer type Syntax B B B map2int A map2int A vmin vmax map2int A vmin vmax type Description map2int A vmin vmax converts number or array A to 8 bit unsigned integers Values between vmin and vmax in A are mapped linearly to values 0 to 255 With a single input argument the default input interval is O to 1 map2int A vmin vmax type converts A to the specified type which can be any integer type given
186. e matrix argument Example polyvalm 1 2 8 2 1 0 1 16 5 0 11 See also primes List of primes Syntax v primes n Description primes n gives a row vector which contains the primes up to n 486 Sysquake for ATEX Example primes 20 2 3 5 7 11 13 17 19 See also sortrows Sort matrix rows Syntax S index sortrows M S index sortrows M sel S index sortrows M sel dim Description sortrows M sort the rows of matrix M The sort order is based on the first column of M then on the second one for rows with the same value in the first column and so on sortrows M sel use the columns specified in sel for comparing the rows of M A third argument dim can be used to specify the dimen sion of the sort 1 for sorting the rows or 2 for sorting the columns The second output argument of sortrows gives the new order of the rows or columns as a vector of indices Example sortrows 3 1 2 2 2 1 2 1 2 NOrRN 0 2 1 2 2 3 1 See also sort sph2cart Spherical to Cartesian coordinates transform Libraries stdlib 487 Syntax x y Z sph2cart phi theta r Description x y z sph2cart phi theta r transforms polar coordinates phi theta and r to Cartesian coordinates x y and z such that x rcos g cos 9 y rsin g cos 9 and z rsin 9 Example x y Z sph2cart 1 2 3 x 0 6745 See also subspace Angle between two subspace
187. e range or to adjust the density of the calculated points of the traces If you need to know the limits of the displayed area in your draw handler use scale to get them right after setting the default scale so that you take into account the zoom and shift specified by the user scale optString defXMin defXMax defYMin defYMax sc scale xMin sc 1 xMax sc 2 yMin sc 3 yMax sc 4 LME Reference base graphics 393 Grids In addition to the scale ticks displayed along the bounding frame grids can be added to give visual clues and make easier the interpretation of graphics X and Y grids are vertical or horizontal lines displayed in the figure background They can be switched on and off by the user in the Grid menu or switched on by programs with the plotoption command they are set off by default In the example below both X and Y grids are switched on plotoption xgrid plotoption ygrid plot rand 1 10 Commands which display grids for special kind of graphics are also available Command Intended use hgrid nyquist dnyquist ngrid nichols dnichols sgrid plotroots rlocus continuous time zgrid plotroots rlocus discrete time They can be used without argument to let the user choose the level of details none means the command does not produce any out put basic is the default value and gives a simple non obstructive hint a single line or a circle and full gives more details To change by
188. e value of x such that cdf distribution x is p The distribution is specified with the first argument a string case is ignored t and T are equivalent Additional arguments must be provided for some distributions The distributions are given in the table below Default values for the parameters when mentioned mean that the parameter may be omitted Distribution Name Parameters Beta beta aandb x chi2 deg of freedom v Y gamma shape q and scale A F f deg of freedom v and v2 lognormal logn mean 0 and st dev 1 normal norm mean 0 and st dev 1 Student s T t deg of freedom v uniform unif limits of the range 0 and 1 See also cdf pd imag Imaginary part of a complex number Syntax im imag z Description imag z is the imaginary part of the complex number z or 0 if z is real 182 Sysquake for ATEX Examples imag 1 2 2 imag 1 0 See also inf Infinity Syntax x inf x Inf x inf n x inf nl1 n2 x inf nl1 n2 x inf type Description inf is the number which represents infinity Most mathematical func tions accept infinity as input argument and yield an infinite result if appropriate Infinity and minus infinity are two different quantities With integer non negative arguments inf creates arrays whose elements are infinity Arguments are interpreted the same way as zeros and ones The last argument of inf can be a string to spe
189. eal x is true if x is a real scalar or a matrix whose entries are all real Examples isreal 2 5 true isreal 2 3 2j false isreal exp pi 1j true See also isnumeric isfloat 1sscalar linspace Sequence of linearly spaced elements Syntax V V linspace x1 x2 linspace x1 x2 n Description linspace x1 x2 produces a row vector of 100 values spaced linearly from x1 and x2 inclusive With a third argument linspace x1 x2 n produces a row vector of n values 482 Sysquake for ATEX Examples Linspace 1 10 1 0000 1 0909 1 1818 9 9091 10 0000 linspace 1 2 6 1 0 1 2 1 4 1 6 1 8 2 0 See also logspace Sequence of logarithmically spaced elements Syntax V V logspace x1 x2 logspace x1 x2 n Description logspace x1 x2 produces a row vector of 100 values spaced log arithmically from 10 x1 and 10 x2 inclusive With a third argument logspace x1 x2 n produces a row vector of n values Example logspace 0 1 1 0000 1 0235 1 0476 9 5455 9 7701 10 0000 See also inspace operator median Median Syntax X Vv Vv median v median M median M dim Libraries stdlib 483 Description median v gives the median of vector v i e the value x such that half of the elements of v are smaller and half of the elements are larger median M gives a row vector which contains the median of the columns of M With a second argument median M dim operates alon
190. ear interpolation between quaternions ql and q2 The result is on the smallest great circle arc defined by normalized q1 and q2 for values of real number t between 0 and 1 If q1 or q2 is O the result is NaN If they are opposite the great circle arc going through 1 or 1i is picked If input arguments are arrays of compatible size same size or scalar the result is a quaternion array of the same size conversion 328 Sysquake for TEX from angles to quaternion is performed independently on corresponding elements Example q qslerp 1 rpy2q 0 1 1 5 0 0 33 0 66 1 roll pitch yaw q2rpy q roll 0 0000 0 1843 0 2272 0 0000 pitch 0 0000 0 3081 0 6636 1 0000 yaw 0 0000 0 4261 0 8605 1 5000 See also rpy2q q2rpy quaternion Quaternion creation Syntax quaternion quaternion w quaternion c quaternion x y Z quaternion w x Yy Z quaternion w v OGO DDD tod woud uu Description With a real argument quaternion x creates a quaternion object whose real part is w and imaginary parts are 0 With a complex argument quaternion c creates the quaternion object real c iximag c With four real arguments quaternion w x y z creates the quaternion object w ix x j y k z With three real arguments quaternion x y z creates the pure quaternion object i x j y k z In all these cases the arguments may be scalars or arrays of the same size With two arguments quaternion w v cr
191. eates a quaternion object whose real part is w and imaginary parts is array v v must have one more dimension than w for the three imaginary parts LME Reference quaternions 329 Without argument quaternion returns the zero quaternion object The real or imaginary parts of a quaternion can be accessed with field access such as g w q x q y q Z and q v Examples q quaternion 1 2 3 4 q 1 21 3j 4k q 5 6 21 3j 4k q q 28 41 6 8k Q q 2 2 q 5 2x2 quaternion array 0 0 quaternion 1 5 3 7 a2alaw lt 1 51 3j 7k NWwWuc See also rpy2q Conversion from attitude angles to quaternion Syntax q rpy2q pitch roll yaw Description rpy2q pitch roll yaw gives the quaternion corresponding to a rotation of angle yaw around the z axis followed by a rotation of angle pitch around the y axis followed by a rotation of angle roll round the x axis All angles are given in radians The result is a normalized quaternion whose real part is cos 9 2 and imaginary part sin 9 2 Vxi Vyj vzk for a rotation of 9 around unit vector 330 Sysquake for ATEX Vx Vy ve The rotation is applied to a point xyz given as a pure quaternion a xi yj zk giving point a also as a pure quaternion then b q a q and a q b q The rotation can also be seen as changing coordinates from body to absolute where the body s attitude is given by pitch roll and yaw In order to have the usual meaning of pitch ro
192. ed as equal with IEEE numbers nan nan is false and 0 0 is true Contrary to the inequality operator returns a single boolean even if its operands are arrays unsame a b is equivalent to a b Example 1 5 1 5 false 1 5 1 5 FFFFF 1 2 3 4 5 true 1 2 3 4 5 Incompatible size nan nan false nan nan true See also operator lt joperator lt oper Operator lt Less than Syntax x lt y lt x y Description x lt y is true if x is less than y and false otherwise Comparing NaN not a number to any number yields false including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size lt x y is equivalent to x lt y It can be used to redefine this oper ator for objects 142 Sysquake for ATEX Example 2 3 4 lt 2 4 2 FTF See also operator operator operator lt operator gt operator gt Operator gt Greater than Syntax x gt y gt x y Description x gt y is true if x is greater than y and false otherwise Comparing NaN not a number to any number yields false including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size gt x y is equivalent to x gt y It can be used to redefine this oper ator for objects Example 2 3 4 gt 2 4 2 FFT See also operator operator
193. ed at the top If style is e the raster image is scaled down such that each pixel has the same size otherwise the specified position is filled with the raster image You should use e when you want a better quality but do not add other elements in the figure Such as marks or lines and do not have interaction with the mouse Pixels on the screen are interpolated using the bilinear method if style is 1 and the bicubic method if style is 3 Examples Two ways to display a table of 10 by 10 random color cells see Fig 6 7 image rand 10 rand 10 rand 10 image rand 10 10 3 A ramp of gray shades image uint8 0 255 Operator and function meshgrid can be used to create the x and y coordinates used to display a function z x y as an image X Y meshgrid pi 0 1 pi Z cos X 2 Y 2 2 image Z 1 1 1 1 3 See also 404 Sysquake for ATEX Figure 6 7 Example of image label Plot labels Syntax label label_x label label_x label_y Description label label_x label_y displays labels for the X and Y axis Its arguments are strings The label for the Y axis may be omitted Examples step 1 1 2 3 4 label t s y m With literal strings the may be more convenient label Re Im See also text legend LME Reference base graphics 405 Uniform random X Normal random Figure 6 8 Example of legend legend Plot le
194. ed between func tion calls but they cannot be shared between different functions They are declared with persistent They cannot be declared outside a function Different persistent functions can have the same name in different functions Examples Functions to reset and increment a counter function reset global counter counter 0 function value increment global counter counter counter 1 value counter Here is how the counter can be used reset i increment i 1 increment u Il 2 See also unction matrixcol First index in a subscript expression Syntax A matrixcol function e C matrixcol obj i n 76 Sysquake for ATEX Description In an expression used as a single subscript to access some elements of an array A expr matrixcol gives an array of the same size as A where each element is the column index For instance for a 2 by 3 matrix matrixcol gives the 2 by 3 matrix 1 1 1 2 2 2 In an expression used as the second of multiple subscripts to access some elements of an array A expr or A expr matrixcol gives a row vector of length size A 2 whose elements are the indices of each column It is equivalent to the range beginning end matrixcol is useful in boolean expressions to select some ele ments of an array matrixcol can be overloaded for objects of used defined classes Its definition should have a header equivalent to function e C matrixcol obj
195. ed for representation of the complex plane such as the roots of a poly nomial or a Nyquist diagram For 3D graphics same effect as daspect 1 1 1 lLock See below Linlin Linear scale for both axis lLinlog Linear scale for the x axis and logarithmic scale for the y axis loglin Logarithmic scale for the x axis and linear scale for the y axis Loglog Logarithmic scale for both axis lindb Linear scale for the x axis and dB scale for the y axis logdb Logarithmic scale for the x axis and dB scale for the y axis Lindb logdb Linear scale for the x axis and dB scale for the y axis The user can choose a logarithmic scale for the x axis anda logarithmic or linear scale for the y axis This last setting shows how to enable the options the user can choose in Sysquake The setting and the enabled options are separated by a dash if a simple setting is specified the enabled options are assumed 414 Sysquake for TEX to be the same Enabling dB always permits the user to choose a loga rithmic or linear scale and enabling a logarithmic scale always permits to choose a linear scale The equal option cannot be combined with anything else When the properties are specified with one or two integer numbers each bit corresponds to a property Only the properties in bold in the table below can be set by the user whatever the setting is Bit Meaning 0 log x 2 tick on x axis 3 grid for x ax
196. ed order of the columns of array A or of the row vector A The dimension along which moment proceeds may be specified with a third argument Example moment randn 1 10000 3 3 011 See also LME Reference linear algebra 233 norm Norm of a vector or matrix Syntax x norm v x norm v kind x norm M x norm M kind Description With one argument norm calculates the 2 norm of a vector or the induced 2 norm of a matrix The optional second argument specifies the kind of norm Kind Vector Matrix none or2 sqrt sum abs v 2 largest singular value induced 2 norm 1 sum abs V largest column sum of abs inf or inf max abs v largest row sum of abs inf min abs v largest row sum of abs p sum abs V p 1 p invalid fro sqrt sum abs v 2 sqrt sum diag M M Examples norm 3 4 5 norm 2 5 9 3 10 2194 norm 2 5 9 3 1 11 See also null Null space Syntax Z null A 234 Sysquake for ATEX Description null A returns a matrix Z whose columns are an orthonormal basis for the null space of m by n matrix A Z has n rank A columns which are the last right singular values of A that is those corresponding to the negligible singular values Example null 1 2 3 1 2 4 1 2 5 0 8944 0 4472 8 0581le 17 See also orth Orthogonalization Syntax Q orth A Description orth A returns a matrix Q whose columns are an orthonormal basis for the range
197. elation coefficients Syntax S corrcoef X S corrcoef X1 X2 Description corrcoef X calculates the correlation coefficients of the columns of the m by n matrix X The result is a square n by n matrix whose diag onal is 1 corrcoef X1 X2 calculates the correlation coefficients of X1 and X2 and returns a 2 by 2 matrix It is equivalent to corrcoef X1 X2 Example corrcoef 1 3 2 5 4 4 7 10 1 0 8915 0 8915 1 corrcoef 1 5 5 1 1 1 1 1 1 See also cumtrapz Cumulative numerical integration with trapezoidal approximation Syntax S cumtrapz Y S cumtrapz X Y S cumtrapz X Y dim Libraries stdlib 477 Description cumtrapz Y calculates an approximation of the cumulative integral of a function given by the samples in Y with unit intervals The trape zoidal approximation is used If Y is neither a row nor a column vector integration is performed along its columns The result has the same size as Y The first value s is are 0 cumtrapz X Y specifies the location of the samples A third argu ment may be used to specify along which dimension the integration is performed Example cumtrapz 2 3 0 2 5 cumtrapz 1 2 0 2 5 1 2 3 5 See also factor Prime factors Syntax v factor n Description factor n gives a row vector which contains the prime factors of n in ascending order Multiple prime factors are repeated Example factor 350 2 5
198. emag 1 1 2 3 4 Syntax bodephase numc denc bodephase numc denc w bodephase numc denc opt bodephase numc denc w opt bodephase Ac Bc Cc Dc bodephase Ac Bc Cc Dc w bodephase Ac Bc Cc Dc opt ee a Bc Cc Dc w opt style style id ya bodephase x bodephase bodephase phase w Description bodephase numc denc plots the phase of the frequency response of the continuous time transfer function numc denc The range of fre quencies is selected automatically or can be specified in an optional argument w a vector of frequencies Further options such as time delay can be provided in a structure opt created with responseset fields Delay and Range are utilized The optional arguments style and id have theirjusual meaning bodemag Ac Bc Cc Dc plots the phase of the frequency response YQw J U jw of the continuous time state space model Ac Bc Cc Dc defined as 436 Sysquake for ATEX scale loglin bodephase 1 1 2 3 4 Figure 6 10 scale loglin bodephase 1 1 2 3 4 jwX jw AcX jw BcU ljw YGw C XUw D UGw With output arguments bodephase gives the phase and the frequency as column vectors No display is produced Example Green plot for arg 1 s 2s 3s 4 with s jw see Fig 6 10 bodephase 1 1 2 3 4 9 See also bodemag dbodephase responseset dbodemag Magnitude Bode diagram for a discrete time system LME Re
199. en legend Legend in top right corner default Legend in top left corner Legend in bottom right corner Legend in bottom left corner Margin space for title and labels No margin xgrid Grid of vertical lines for the x axis noxgrid No grid for the x axis ygrid Grid of horizontal lines for the y axis noygrid xygrid noxygrid No grid for the y axis Grid of vertical and horizontal lines for the x and y axis No grid for the x and y axis grid Normal details for grids displayed by sgrid zgrid etc nogrid fullgrid fill3d figure Example Removal of grids displayed by sgrid zgrid etc More details for grids displayed by sgrid zgrid etc In 3D graphics zoom in so that the bounding box fills the Display of a photographic image without frame plotoption noframe image photo See also legend LME Reference base graphics 411 polar Polar plot Syntax polar theta rho polar style polar style id Description Command polar displays graphical data in the current figure with po lar coordinates The data are given as two vectors of coordinates theta in radians and rho Depending on the style the points are displayed as individual marks or are linked with lines If x and y are matrices each row is considered as a separate line or set of marks if only one of them is a matrix the other one a vector is reused f
200. ents otherwise These arguments are the same as those expected by function error When the intermediate code is optimized assert can be ignored It should be used only to produce errors at an early stage or as a de bugging aid not to trigger the try catch mechanism The expression should not have side effects The most common use of assert is to check the validity of input arguments Example function y fact n assert length n 1 amp amp isreal n amp amp n round n LME nonIntArg y prod 1 n See also error warning try builtin Built in function evaluation Syntax argoutl builtin fun arginl Description yl y2 builtin fun x1 x2 evaluates the built in func tion fun with input arguments x1 x2 etc Output arguments are as signed to yl y2 etc Function fun is specified by its name as a String builtin is useful to execute a built in function which has been re defined Example Here is the definition of operator plus so that it can be used with character strings to concatenate them function r plus a b if ischar a amp amp ischar b r a b else r builtin plus a b end 96 Sysquake for ATEX The original meaning of plus for numbers is preserved 1 2 3 ab cdef abcdef See also feval clear Discard the contents of a variable Syntax clear clear vl v2 clear functions Description Without argument
201. eof Check end of file status Syntax b feof fd Description feof fd is false if more data can be read from file descriptor fd and true if the end of the file has been reached Example Count the number of lines and characters in a file fopen and fclose are not available in all LME applications fd fopen data txt lines 0 characters 0 while feof fd str fgets fd lines lines 1 characters characters length str end fclose fd 364 Sysquake for TEX See also ftell fflush Flush the input and output buffers Syntax fflush fd Description fflush fd discards all the data in the input buffer and forces data out of the output buffer when the device and their driver permits it fflush can be useful to recover from errors fgetl Reading of a single line Syntax line fgetl fd line fgetl fd n Description A single line of at most n characters is read from a text file The end of line character is discarded See also fgets Reading of a single line Syntax line fgets fd line fgets fd n LME Reference input output 365 Description A single line of at most n characters is read from a text file Unless the end of file is encountered before the end of line always a single line feed is preserved See also fgetl fscanf format Default output format Syntax format format format format format format format format
202. equence of private function definitions in a library Syntax private Description In a library functions which are defined after the private keyword are private private may not be placed in the same line of source code as any other command comments are possible though In a library functions are either public or private Private functions can only be called from the same library while public functions can also be called from contexts where the library has been imported with a use command Functions are public by default Example Here is a library for computing the roots of a second order polynomial Only function roots2 may be called from the outside of the library private function d discr a b c d b2 4xaxcC public function r roots2 p a p 1 b p 2 c p 3 d discr a b c 5 b sqrt d b sqrt d 2 a See also 88 Sysquake for TEX public Mark the beginning of a sequence of public function definitions in a library Syntax public Description In a library functions which are defined after the public keyword are public public may not be placed in the same line of source code as any other command comments are possible though In a library functions are either public or private Private functions can only be called from the same library while public functions can also be called from contexts where the library has been imported with a use command Functions are publ
203. er They can be used to limit the computation of data displayed by plot to the visible area LME Reference base graphics 415 Examples Here are some suggestions for the most usual graphics Time response default linlin is fine Bode mag scale logdb Bode phase scale loglin D bode mag scale lindb logdb 0 pi Ts D bode phase scale Linlin loglin 0 pi Ts Poles scale equal D poles scale equal 1 1 1 1 Nyquist scale equal 1 5 1 5 1 5 1 5 Nichols scale lindb Use of scale to display a sine in the visible x range scale 0 10 sc scale xmin sc 1 xmax sc 2 x xmin xmax xmin 0 0 01 1 101 values between xmin and xmax default x range between 0 and 10 maybe changed by the user 1x2 or 1x4 y sin x plot x y See also plotoption scalefactor Change the scale displayed in axis ticks and labels Syntax scalefactor f f scalefactor Description scalefactor f sets the factor used to display the ticks and the la bels Its argument f can be a vector of two real positive numbers to set separately the x axis and the y axis or a real positive scalar to set the same factor for both axis The normal factor value is 1 so that the ticks correspond to the graphical contents With a different factor the contents are displayed with the same scaling but the ticks and labels are changed as if the graphical data had been scaled by the factor F
204. er cannot be represented with a double then the number is rounded to the nearest value which can be represented with a double Compare the expressions below Expression Value uint64 123456789012345678 123456789012345696 123456789012345678uint64 123456789012345678 Literal complex numbers are written as the sum or difference of a real number and an imaginary number Literal imaginary numbers are written as double numbers with an i or j suffix like 2i 3 7e5j or Oxffj Functions i and j can also be used when there are no variables of the same name but should be avoided for safety reasons The suffices for single and imaginary can be combined as isingle or jsingLe in this order only 2jsingle 3single 4isingle Command format is used to specify how numbers are displayed Strings Strings are stored as arrays usually row vectors of 16 bit unsigned numbers Literal strings are enclosed in single quotes Example of string The second string is empty For special characters the following es cape sequences are recognized 46 Sysquake for ATEX Character Escape seq Character code Null 0 0 Bell a 7 Backspace b 8 Horizontal tab t 9 Line feed n 10 Vertical tab v 11 Form feed f 12 Carriage return r 13 Single tick V 39 Single tick two 39 Backslash 92 Hexadecimal number xhh hh Octal number ooo 000 16 bit UTF 16 uhhhh unicode UTF 16 code For octal and hexadecimal representations up to
205. er respect it behaves like strcmp Examples strcmpi abc aBc true strcmpi Abc abd 2 true See also strmatch String match Syntax i i i strmatch str strMatrix strmatch str strList strmatch exact Description strmatch str strMatrix compares string str with each row of the character matrix strMatrix it returns the index of the first row whose beginning is equal to str or 0 if no match is found Case is significant strmatch str strList compares string str with each element of list strList which must be strings With a third argument which must be the string exact str must match the complete row or element of the second argument not only the beginning LME Reference strings 317 Examples strmatch abc axyz uabc abcd efgh E E ee uabc abcd efgh exact strmatch abe ABC any abcdefg ab abcd See also strtok Token search in string Syntax strtok str strtok str separators token remainder token remainder Description strtok str gives the first token in string str A token is defined as a substring delimited by separators or by the beginning or end of the string by default separators are spaces tabulators carriage returns and line feeds If no token is found i e if str is empty or contains only separator characters the result is the empty string The
206. eric plot Syntax plot y plot x y plot style plot style id Description The command plot displays graphical data in the current figure The data are given as two vectors of coordinates x and y If x is omitted its default value is 1 size y 2 Depending on the style the points are displayed as individual marks or are linked with lines The stairs style s can be used to link two successive points with a horizontal line followed by a vertical line If x and y are matrices each row is considered as a separate line or set of marks if only one of them is a matrix the other one a row or column vector is replicated to match the size of the other argument The optional fourth argument is an identification number which is used for interactive manipulation It should be equal or larger than 1 If present and a mousedown mousedrag and or mouseup handler exists the position of the mouse where the click occurs is mapped to the closest graphical element which has been displayed with an ID for the command plot the closest point is considered lines linking the points are ignored If such a point is found at a small distance the built in variables _x0 _y0 and _z0 are set to the position of the point before it is moved the variable _id is set to the ID as defined by the command plot the variable _nb is set to the number of the row and the variable _ix is set to the index of the column of the matrix x and y Exampl
207. erse hyperbolic cosine of x which is complex if x is complex or if x lt 1 Examples acosh 2 1 3170 acosh 0 1 2j 0 1 5708j 1 5286 1 1437j See also cosh jasinh acos acot Inverse cotangent Syntax y acot x Description acot x gives the inverse cotangent of x which is complex if x is 154 Sysquake for TEX See also acoth Inverse hyperbolic cotangent Syntax y acoth x Description acoth x gives the inverse hyperbolic cotangent of x which is complex if x is complex or is in the range 1 1 See also coth acot acsc Inverse cosecant Syntax y acsc x Description acsc x gives the inverse cosecant of x which is complex if x is com plex or is in the range 1 1 See also acsch Inverse hyperbolic cosecant Syntax y acsch x LME Reference mathematical functions 155 Description acsch x gives the inverse hyperbolic cosecant of x which is complex if x is See also angle Phase angle of a complex number Syntax phi angle z Description angle z gives the phase of the complex number z i e the angle be tween the positive real axis and a line joining the origin to z angle 0 is O Examples angle 1 3j 1 2490 angle 0 1 1 0 3 1416 See also asec Inverse secant Syntax y asec x Description asec x gives the inverse secant of x which is complex if x is complex or is in the range 1 1 156 Sysquake for ATEX
208. erson Here is what it means when it relates to LME Data encapsulation Objects contain data but the data cannot be accessed directly from the outside All accesses are performed via special functions called methods What links a particular method to a particular object is a class Class are identified with a name When an object is created its class name is specified The names of methods able to act on objects of a particular class are prefixed with the class name followed with two colons Objects are special structures whose contents are accessible only to its methods Function and operator overloading Methods may have the same name as regular functions When LME has to call a function it first checks the type of the input arguments If one of them is an object the corresponding method is called rather than the function defined for non object arguments A method which has the same name as a function or another method is said to overload it User functions as well as built in ones can be LME Reference 49 overloaded Operators which have a function name for instance x y can also be written plus x y can also be overloaded Special functions called object constructors have the same name as the class and create new objects They are also methods of the class even if their input arguments are not necessarily objects Inheritance A class subclass may extend the data and methods of another class base class or parent It is said to
209. es Sine between 0 and 27 xX 2 x pi gt x 0 100 0 01 y sin x plot x y Ten random crosses plot rand 1 10 rand 1 10 x LME Reference base graphics 409 A complete SQ file for displaying a red triangle whose corners can be moved interactively on Sysquake variables x y x and y are 1 by 3 vectors init x y init init handler figure Triangle draw drawTri x y mousedrag x y dragTri x y ix _xl _yl functions function x y init x 1 1 0 y 1 1 2 subplots Triangle function drawTri x y scale equal 3 3 3 3 plot x y rf 1 filled red triangle with ID 1 function x y dragTri x y ix x1 yl if isempty ix cancel not a click over a point end x ix xl y ix yl See also plotoption Set plot options Syntax plotoption str Description plotoption sets the initial value of one of the plot options the user can change Its single argument a character string can take one of the following values frame Rectangular frame with tick marks and a white background around the plot noframe No frame no tickmarks no white background 410 Sysquake for ATEX label Subplot name above the frame nolabel legend nolegend trlegend tllegend brlegend bllegend margin nomargin No subplot name Legend if it has been set with Legend Hidd
210. es 42 variable assignment 72 logspace ame lp2bp lp2bs lp2hp lt 122 aa lu jae mad 492 magenta2yellowcm 566 magic map 333 map2int 290 margin 358 markup Sysquake for TEX output channel 56 reference material 427 mat filedecode 380 matfileencode 381 matrixcol 75 matrixrow max 229 md5 mean 230 median 482 mesh 428 meshgrid methods min 231 minus 122 mldivide 122 mod 189 moment 232 mpower 122 mrdivide mtimes 122 nan 190 nancorrcoef 492 nancov 493 nanmean 493 nanmedian 494 nansum 494 nargin nargout 111 nchoosek ndgrid ndims ne 122 ng Pe nichols nnz 264 norm 233 not 122 null 233 num2cel1 264 num2 list 334 number 44 numel nyquist 458 Index object 48 octahedron 571 ode23 295 ode45 295 odeset 297 ones 266 operator amp 144 14 Qa g U1 Ses as Ge E ay o el P oe Cres S NI Gal WI AI Gal Go BS Bye CO EB ILNI WT LIL NN WwW N Ul SILOS AAN Il gt a i ee cll a E oS el A i i Oj gt D NJ Jj 00 2 I ma X Bie an EAEN En EES EN R x Iyon m e i pe uoj amp kan fa A 00 NI m E optimset 305 or 122 orderfields 337 orth 234 otherwise p s NJI 585 plotoption 409 plotpoly plotroots plus 122 polar polyder polyfit poly
211. es the statements in string str ina context with local variables equal to the fields of structure varin With an output argument sandbox collects the contents of all vari ables in the fields of a single structure An error is thrown when the argument of sandbox attempts to exe cute one of the functions which are disabled This error can be caught by a try catch construct outside sandbox but not inside its argument so that unsuccessful attempts to circumvent the sandbox are always reported to the appropriate level Examples Evaluation of two assignments the second value is displayed and the variables are discarded at the end of the evaluation sandbox a 2 b 3 5 b 345 Evaluation of two assignments the contents of the variables are stored in structure result result sandbox a 2 b 3 5 result a 2 b real 1x3 Evaluation with local variables x and y initialized with the field of a structure Variable z is local to the sandbox in x 12 in y 1 10 sandbox z x y in Z 13 14 15 16 17 18 19 20 21 22 Attempt to execute the untrusted function fopen and to hide it from the outside Both attempts fail fopen is trapped and the security violation error is propagated outside the sandbox sandbox try fd fopen etc passwd end Security violation fopen See also sandboxtrust eval 120 Sysquake for ATEX sandboxtrust Escape the sandbox restrictions Syntax s
212. es the two first di mensions by cropping or by padding with zeros ifft2 F nr nc re sizes first dimension to nr rows and second dimension to nc columns In ifft2 F size the new size is given as a two element vector nr nc ifft2 F n is equivalent to ifft2 F n n If the first argument is an array with more than two dimensions ifft2 performs the inverse 2 d DFT along dimensions 1 and 2 sepa rately for each plane along remaining dimensions ifftn performs an inverse DFT along each dimension Up to the sign and a scaling factor the inverse 2 d DFT and the 2 d DFT are the same operation ifft2 has the same syntax as See also ifftn Inverse n dimension Fast Fourier Transform Syntax f ifftn F f ifftn F size Description ifftn F returns the inverse n dimension Discrete Fourier Transform of array F inverse DFT along each dimension of F With two input arguments ifftn F size resizes F by cropping or by padding F with zeros LME Reference linear algebra 223 Up to the sign and a scaling factor the inverse n dimension DFT and the n dimension DFT are the same operation ifftn has the same syntax as fftn See also hess Hessenberg reduction Syntax P H hess A H hess A Description hess A reduces the square matrix A A to the upper Hessenberg form H using an orthogonal similarity transformation P H P A The result H is zero below the first subdiagonal and has the same eigenvalues as A E
213. esents bi nary data strb can be a string only the least significant byte of each character is considered or an array of bytes of class uint8 or int8 The result is a string of 40 hexadecimal digits It is believed to be hard to create the input to get a given digest or to create two inputs with the same digest LME Reference strings 315 shal fd calculates the SHA1 digest of the bytes read from file descriptor fd until the end of the file The file is left open SHA1 digest is an Internet standard described in RFC 3174 Example SHA1 digest of the three characters a b and c shal abc a9993e364706816aba3e25717850c26c9cd0d89d See also strcmp String comparison Syntax b strcmp sl1 s2 b strcmp sl1 s2 n Description strcmp s1 s2 is true if the strings s1 and s2 are equal i e same length and corresponding characters are equal strcmp s1 s2 n compares the strings up to the n th character Note that this function does not return the same result as the strcmp function of the standard C library Examples strcmp abc abc true strcmp abc def false strcmp abc abd 2 true strcemp abc abc 5 false See also 316 Sysquake for TEX strcmpi String comparison with ignoring letter case Syntax b strcmpi sl s2 b strcmpi sl1 s2 n Description strcmpi compares strings for equality ignoring letter case In every oth
214. ete values By default edges are not drawn An optional third input argument a string specifies the edge style it corresponds to the style argument of surf With three output arguments klein produces the X Y and Z arrays expected by surf or mesh and it does not display anything See also crosscap klein8 Figure 8 Klein bottle immersion Syntax klein8 klein8 r klein8 r klein8 r X Y Z lt los style Description Without output argument klein8 draws a figure 8 Klein bottle immer sion a closed self intersecting surface with one face approximated by a polyhedron With an input argument klein8 r draws the sur face with a main radius of r the default value is 1 With two input arguments klein8 r n samples the two parame ters which describe its surface with n discrete values By default edges are not drawn An optional third input argument a string specifies the edge style it corresponds to the style argument of surf Libraries solids 577 With three output arguments klein8 produces the X Y and Z ar rays expected by surf or mesh and it does not display anything See also klein crosscap surf sphere Sphere Syntax sphere sphere n sphere n style X Y Z sphere X Y Z sphere n Description Without output argument sphere draws a sphere approximated by a polyhedron With an input argument sphere n draws a sphere where the two parameters wh
215. f coordinates x y and z De pending on the style the points are displayed as individual marks or are linked with lines If x y and z are matrices each row is considered as a separate line or set of marks row or column vectors are replicated to match the size of matrix arguments if required plot3 id specifies the D used for interactivity in Sysquake See also plotpoly Plot polygons in 3D space Syntax plotpoly x y z ind plotpoly x y z strip plotpoly x y z fan plotpoly x y z color ind plotpoly x y z color strip plotpoly x y z color fan plotpoly vis plotpoly vis style plotpoly vis style id 430 Sysquake for ATEX Description plotpoly x y z ind plots polygons whose vertices are given by vectors x y and z Rows of argument ind contain the indices of each polygon in arrays x y and z Vertices can be shared by several poly gons Color of each polygon is mapped linearly from the z coordinate of the center of gravity of its vertices to the color map Each poly gon can be concave but must be planar and must not self intersect different polygons may intersect plotpoly x y z strip plots a strip of triangles Triangles are made of three consecutive vertices their indices could be defined by the following array ind_strip in Ordering is such that triangles on the same side of the strip have the same orientation plotpoly x y z
216. f errors balance scales A with powers of 2 balance returns the balanced matrix B which has the same eigen values and singular values as A and optionally the diagonal scaling matrix T such that T A T A Example A 1 2e6 3e 6 4 T B balance A T 16384 0 0 3 125e 2 B 1 3 8147 1 5729 4 See also care Continuous time algebraic Riccati equation Syntax X L K care A B Q X L K care A B Q R X L K care A B Q R S LME Reference linear algebra 205 Description care A B Q calculates the stable solution X of the following continuous time algebraic Riccati equation A X XA XBB X Q 0 All matrices are real Q and X are symmetric With four input arguments care A B Q R with R real symmetric solves the following Riccati equation A X XA XBR 1B X Q 0 With five input arguments care A B Q R S solves the following equation A X XA S XB R7H S B X Q 0 With two or three output arguments X L K care also re turns the gain matrix K defined as K R 1B x and the column vector of closed loop eigenvalues L eig A BK Example A 4 2 1 2 B 0 1 C 2 1 Q C C R 5 X L K care A B Q R lt lt Il 1 07 3 5169 3 5169 23 2415 L 4 3488 2 2995 K 0 7034 4 6483 A x X X A XxB R B X Q 1 7319e 14 1 1369e 13 8 5265e 14 6 2528e 13 See also dare 206 Sysquake for TEX chol Cholesky decomposit
217. f values Syntax m trimmean A prc m trimmean A prc dim 498 Sysquake for ATEX Description trimmean A prc gives the arithmetic mean of the columns of array A or of the row vector A once prc 2 percent of the values have been re moved from each end The dimension along which trimmean proceeds may be specified with a third argument trimmean is less sensitive to outliers than the regular arithmetic mean See also zscore Z score normalized deviation Syntax Y zscore X Y zscore X dim Description zscore X normalizes the columns of array X or the row vector X by subtracting their mean and dividing by their standard deviation The dimension along which zscore proceeds may be specified with a sec ond argument 7 3 classes Library classes implements the constructors and methods of two classes polynom for polynomials and ratfun for rational functions Basic arithmetic operators and functions are overloaded to support expressions with the same syntax as for numbers and matrices The following statement makes available functions defined in classes use classes polynom polynom Polynom object constructor Libraries classes 499 Syntax a polynom a polynom coef Description polynom coef creates a polynom object initialized with the coeffi cients in vector coef given in descending powers of the variable Without argument polynom returns a polynom object initialized to 0 The foll
218. fan plots triangles which share the first ver tex and form a fan Their indices could be defined by the following array ind_fan a in eere UB WSd af 2 3 4 etc l plotpoly x y z color uses color instead of z to set the filling color of each polygon color is always a real double array or scalar whose elements are between 0 and 1 How it is interpreted depends on its size A scalar defines the color of all polygons it is mapped to the color map A vector of three elements defines the RGB color of all polygons row vector if there are 3 vertices to avoid ambiguity A vector with as many elements as x y and z defines the color of each vertex column vector if there are 3 vertices to avoid ambiguity Polygons have the mean value of all their vertices which is mapped to the color map LME Reference 3D graphics 431 An array with as many columns as elements in x y and z defines the RGB color of each vertex Polygons have the mean value of all their vertices plotpoly vis uses string vis to specify which side of the sur face is visible f for front only b for back only or fb or bf for both sides The front side is defined as the one where vertices have an anticlockwise orientation The default is f plotpoly vis style uses string style to specify the style of edges plotpoly id specifies the used for interactivity in Sysquake See also sens
219. ference graphics for dynamical systems 437 Syntax dbodemag numd dend Ts dbodemag numd dend Ts w dbodemag numd dend Ts opt dbodemag numd dend Ts w opt dbodemag Ad Bd Cd Dd Ts dbodemag Ad Bd Cd Dd Ts w dbodemag Ad Bd Cd Dd Ts opt dbodemag Ad Bd Cd Dd Ts w opt dbodemag style dbodemag style id mag w dbodemag Description dbodemag numd dend Ts plots the magnitude of the frequency re sponse of the discrete time transfer function numd dend with sampling period Ts The range of frequencies is selected automatically or can be specified in an optional argument w a vector of frequencies Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning dbodemag Ad Bd Cd Dd Ts plots the magnitude of the frequency response Y jw UGw of the discrete time state space model Ad Bd Cd Dd defined as zX Z Y z AgX z BgU z CgX z DgU z where z e 7s With output arguments dbodemag gives the magnitude and the fre quency as column vectors No display is produced Example dbodemag 1 poly 0 9 0 7 0 6 0 7 0 6j 1 See also dbodephase dbodephase Phase Bode diagram for a discrete time system 438 Sysquake for TEX Syntax dbodephase numd dend Ts dbodephase numd dend Ts w dbodephase numd dend Ts opt dbodephase numd dend Ts w opt db
220. first input argument is a structure fontset adds or changes fields which correspond to the name value pairs which follow Here is the list of permissible options empty arrays mean auto matic Name Default Meaning Font fad font name Size 10 character size in points Bold false true for bold font Italic false true for italic font Underline false true for underline characters Color 0 0 0 text color The default font is used if the font name is not recognized The color is specified as an empty array black a scalar gray or a 3 element vector RGB of class double O black 1 maximum brightness or uint8 O black 255 maximum brightness Examples Default font fontset Font Size 10 Bold false Italic false Underline false Color real 1x3 See also text fplot Function plot 402 Sysquake for TEX Syntax fplot fun fplot fun limits fplot fun limits style fplot fun limits style id fplot fun limits style id pl p2 Description Command fplot fun limits plots function fun specified by its name as a string a function reference or an inline function The function is plotted for x between Limit 1 and Limit 2 the default limits are 5 5 The optional third and fourth arguments are the same as for all graphical commands Remaining input arguments of fplot if any are given as additional input arguments to function fun They permit to parameterize the functio
221. for objects Examples true false 1 0 3 false FTFT See also pitomp Operator amp And Syntax b1 amp b2 and b1 b2 Description b1 amp b2 performs the logical AND operation between the corresponding elements of b1 and b2 the result is true logical 1 if both operands are different from false or 0 and false logical 0 otherwise and b1 b2 is equivalent to b1 amp b2 It can be used to redefine this operator for objects LME Reference operators 145 Example false false true true amp false true false true FFFT See also operator 6 Operator amp And with lazy evaluation Syntax b1 amp amp b2 Description b1 amp amp b2 is b1 if b1 is false and b2 otherwise Like with if and while statements b1 is true if it is a nonempty array with only non zero elements b2 is evaluated only if b1 is true b1 amp amp b2 amp amp amp amp bn returns the last operand which is false remain ing operands are not evaluated or the last one Example Boolean value which is true if the vector v is made of pairs of equal values mod length v 2 0 amp amp v 1 2 end v 2 2 end The second operand of amp amp is evaluated only if the length is even See also Operator Or Syntax b1 b2 or bl1 b2 146 Sysquake for ATEX Description b1 b2 performs the logical OR operation between the corresponding elements of b1 and b2 the result is false logical 0
222. for the remaining arguments of cellfun 250 Sysquake for TEX cellfun differs from map in two ways the result is a non cell array and remaining arguments of cellfun are provided directly to fun Examples cellfun isempty 1 ones 5 FT TF map isempty 1 ones 5 2x2 cell array cellfun size 1 ones 5 2 10 05 See also diag Creation of a diagonal matrix or extraction of the diagonal elements of a matrix Syntax M diag v M diag v k v diag M v diag M k Description With a vector input argument diag v creates a square diagonal ma trix whose main diagonal is given by v With a second argument the diagonal is moved by that amount in the upper right direction for pos itive values and in the lower left direction for negative values With a matrix input argument the main diagonal is extracted and returned as a column vector A second argument can be used to spec ify another diagonal Examples diag 1 3 10 0 2 00 Woo LME Reference arrays 3 D NNO Q OONO H A OWOo He UJ lt Q H Qa H yeo NUOV AUI looodoe g M 1 See also eye Identity matrix Syntax M eye n M eye m n M eye m n M eye type Description 251 eye builds a matrix whose diagonal elements are 1 and other elements 0 The size of the matrix is specified by one integer for a square ma trix or two integers either as t
223. format format format format format short short e short g long long e long g bank i j loose compact Description format changes the format used by command disp and for output produced with expressions which do not end with a semicolon The following arguments are recognized 366 Sysquake for ATEX Arguments Meaning none fixed format with 0 or 4 digits loose spacing short fixed format with 0 or 4 digits short e exponential format with 4 digits short g general format with up to 4 digits long fixed format with 0 or 15 digits long e exponential format with 15 digits long g general format with up to 15 digits bank fixed format with 2 digits for currencies orl for nonzero space for zero i symbol i to represent the imaginary unit j symbol j to represent the imaginary unit loose empty lines to improve readability compact no empty line Format for numbers for imaginary unit symbol and for spacing is set separately Format displays compactly numeric and boolean arrays positive numbers and complex numbers with a positive real part are displayed as negative numbers or complex numbers with a negative real part as pure imaginary nonzero numbers as I and zeros as spaces The default format is format short g format j and format compact See also fprintf Formatted output Syntax n n fprintf fd format a b fprintf format a b Description fprintf format a b
224. fscanf Reading of formatted numbers Syntax r fscanf fd format r count fscanf fd format Description A single line is read from a text file and numbers characters and strings are decoded according to the format string The format string follows the same rules as sscanf LME Reference input output 369 The optional second output argument is set to the number of ele ments decoded successfully may be different than the length of the first argument if decoding strings Example Read a number from a file fopen and fclose are not available in all LME applications fd fopen test txt rt fscanf fd f 2 3 fclose fd See also sscanf fseek Change the current read or write position in a file Syntax status fseek fd position status fseek fd position mode Description fseek fd position mode changes the position in an open file where the next input output commands will read or write data The first ar gument fd is the file descriptor returned by fopen or similar functions fopen is not available in all LME applications The second argument is the new position The third argument mode specifies how the posi tion is used b absolute position from the beginning of the file c relative position from the current position e offset from the end of the file must be lt 0 The default value is b Only the first character is checked so beginning is a valid altern
225. g dimension dim Example median 1 2 5 6 inf 5 See also perms Array of permutations Syntax M perms v Description perm v gives an array whose rows are all the possible permutations of vector v Example perms 1 PNR N WW 2 1 3 3 1 2 See also sort 484 Sysquake for ATEX pol2cart Polar to Cartesian coordinates transform Syntax x y pol2cart phi r x y Z pol2cart phi r z Description x y pol2cart phi r transforms polar coordinates phi and r to Cartesian coordinates x and y such that x rcos and x rsin x y z pol2cart phi r z transforms cylindrical coordinates to Cartesian coordinates leaving z unchanged Example x X pol2cart 1 2 0806 y elel 6829 See also polyfit Polynomial fit Syntax pol polyfit x y n Description polyfit x y n calculates the polynomial given as a vector of de scending power coefficients of order n which best fits the points given by vectors x and y The least square algorithm is used Libraries stdlib 485 Example pol pol 0 6667 5 5714 12 7619 9 8000 polyval pol 1 5 1 9429 1 2286 3 6571 5 2286 1 9429 polyfit 1 5 2 1 4 5 2 3 polyvalm Value of a polynomial with square matrix argument Syntax Y polyvalm pol X Description polyvalm pol X evaluates the polynomial given by the coefficients pol in descending power order with a squar
226. gend Syntax legend str legend str style Description legend str style displays legends for styles defined in string style In string str legends are separated by linefeed characters n Legends are displayed at the top right corner of the figure in a frame All styles are permitted symbols lines and filling They are recycled if more legends are defined in str With a single input argument Legend str uses the default style k Example Legend for two traces see Fig 6 8 plot 1 20 rand 1 20 randn 1 20 _x legend Uniform random nNormal random _x 406 Sysquake for ATEX See also Label line Plot lines Syntax line A b line A b style line A b style id Description line displays one or several straight line s Each line is defined by an equation of the form a x azy b The first argument of Line is a matrix which contains the coefficients az in the first column and a2 in the second column each row corresponds to a different line The second argument is a column vector which contains the coefficients b If one of these arguments has one row and the other has several or none the same row is duplicated to match the other size In figures with a logarithmic scale only horizontal and vertical lines are allowed The optional third and fourth arguments are the same as for all graphical commands In mouse handlers _x0 and _y correspond to the projection of
227. gorithm fminsearch implements the Nelder Mead simplex method It starts from a polyhedron centered around x0 the simplex Then at each it eration either vertex x_i with the maximum value fun x_i is moved to decrease it with a reflexion expansion a reflexion or a contraction or the simplex is shrinked around the vertex with minimum value It erations stop when the simplex is smaller than the tolerance or when the maximum number of iterations or function evaluations is reached then an error is thrown Examples Minimum of a sine near 2 displayed with 15 digits fprintf 15g n fminsearch sin 2 4 712388977408411 Maximum of xey xy 0 1x The function if defined as an anony mous function stored in variable fun fun x y x exp xX y 2 x y O 1 x 2 In Sysquake the contour plot can be displayed with the following com mands X Y meshgrid 0 0 02 3 0 0 02 3 contour feval fun X Y 0 3 0 3 0 1 0 05 0 5 The maximum is obtained by minimizing the opposite of the function rewritten to use as input a single variable in R 294 Sysquake for TEX mfun X X 1 exp X 1 X 2 2 X 1 X 2 1 X 1 2 fminsearch mfun 1 2 2 1444 0 3297 For the same function with a constraint x lt 1 the objective function can be modified to return co for inputs outside the feasible region note that we can start on the constraint boundary but starting from the infeasible regi
228. gration Syntax options odeset options odeset namel valuel options odeset optionsO namel valuel Description odeset namel valuel creates the option argument used by ode23 and ode45 Options are specified with name value pairs where 298 Sysquake for TEX the name is a string which must match exactly the names in the table below Case is significant Options which are not specified have a de fault value The result is a structure whose fields correspond to each option Without any input argument odeset creates a structure with all the default options Note that ode23 and ode45 also interpret the lack of an option argument or the empty array as a request to use the default values When its first input argument is a structure odeset adds or changes fields which correspond to the name value pairs which follow Here is the list of permissible options empty arrays mean auto matic Name Default Meaning AbsTol le 6 maximum absolute error Events none state based event function EventTime none time based event function InitialStep 10 MinStep initial time step MaxStep time range 10 maximum time step MinStep time range le6 minimum time step NormControl false error control on state norm OnEvent none event function OutputFcn none output function PreArg list of prepended input arguments Refine 1 4 for ode45 refinement factor RelTol le 3 maximum relative erro
229. grid plots only the lines for the specified values of damping and natural frequency Let p and p be the complex conjugate roots of the polynomial s 2wZs w where w is the natural frequency and lt 1 the damping The locus of roots LME Reference graphics for dynamical systems 465 with a constant damping is generated by Im p 1 Rep with Rep lt 0 The locus of roots with a constant natural frequency w is a circle of radius w The whole grid is displayed only if the user selects it in the Grid menu or after the command plotoption fullgrid By default only the imaginary axis the stability limit is displayed Example Typical use for poles or zeros displayed in the s plane scale equal sgrid plotroots pol See also sigma Singular value plot for a continuous time state space model Syntax sigma Ac Bc Cc Dc sigma Ac Bc Cc Dc w sigma Ac Bc Cc Dc opt sigma Ac Bc Cc Dc w opt sigma style sigma style id sv w sigma Description Sigma Ac Bc Cc Dc plots the singular values of the frequency re sponse of the continuous time state space model Ac Bc Cc Dc de fined as jwX jw A X jw B UGw Yw CcX jw DCUGw The range of frequencies is selected automatically or can be specified in an optional argument w a vector of frequencies 466 Sysquake for TEX Further options can be provided in a structure opt created with responseset field Rang
230. gt operator gt opera Operator Object equality Syntax a same a b Description a bis true if ais the same as b and false otherwise a and b must have exactly the same internal representation to be considered as equal with IEEE floating point numbers nan nan is true and 0 0 is false Contrary to the equality operator returns a single boolean even if its operands are arrays same a b is equivalent to a b Example 1 5 1 5 true 1 5 1 5 TTTTT 1 2 3 4 5 false 1 2 3 4 5 Incompatible size nan nan true nan nan false See also 140 Sysquake for TEX Operator Inequality Syntax xX y ne x y Description x y is true if x is not equal to y and false otherwise Comparing NaN not a number to any number yields true including to NaN If x and or y is an array the comparison is performed element wise and the result has the same size ne x y is equivalent to x y It can be used to redefine this oper ator for objects Example 17 1 false inf inf false nan nan true 1 2 3 1 3 3 FTF See also operator gt operator gt joper Operator Object inequality Syntax a unsame a b LME Reference operators 141 Description a b is true if a is not the same as b and false otherwise a and b must have exactly the same internal representation to be consider
231. he current directory It has the same effect as cd without input argument See also setenv Set the value of an environment variable Syntax setenv name value setenv name Description setenv name value sets the value of the environment variable of the specified name Both arguments are strings If no such environ ment variable exists it is created With a single input argument setenv creates an empty environ ment variable or remove the value of an exisisting environment vari able Environment variables are defined in the context of the application they can be accessed in the application or in processes it launches Environment variables of the calling process command shell for in stance are not changed setenv is not defined for Windows Example setenv CONTROLDEBUG 1 See also getenv junsetenv sleep Suspend execution for a specified amount of time LME Reference shell 387 Syntax sleep t Description sleep t suspend execution during t seconds with a resolution of a microsecond Example sleep le 3 unix Execute a Unix command Syntax unix str Description unix str executes a command with the default shell No input can be provided and output is directed to the standard output of LME Examples unix ls unix cc o calc calc c calc See also dos unsetenv Remove an environment variable Syntax unsetenv name 388 Sysquak
232. he integral of the polynomial pol1 whose zero order coefficient is 0 Both polynomi als are given as vectors of their coefficients highest power first The result is a row vector A second input argument can be used to specify the integration constant 238 Sysquake for ATEX Example Y polyint 1 2 3 4 5 Y 0 2 0 5 1 2 5 0 y polyder Y 1 2 3 4 5 Y polyint 1 2 3 4 5 10 Y 0 2 0 5 1 2 5 10 See also polyval Numerical value of a polynomial evaluated at some point Syntax y polyval pol x Description polyval pol x evaluates the polynomial pol at x which can be a scalar or a matrix of arbitrary size The result has the same size as x Examples polyval 1 3 8 2 18 polyval 1 2 1 5 34567 See also prod Product of the elements of a vector LME Reference linear algebra Syntax x prod v v prod M v prod M dim Description prod v returns the product of the elements of vector v 239 prod M returns a row vector whose elements are the products of the corre sponding columns of matrix M prod M dim returns the product of matrix M along dimension dim the result is a row vector if dim is 1 or a column vector if dim is 2 Examples prod 1 5 120 prod 1 5 120 prod 1 2 3 5 6 7 5 12 21 prod 1 2 3 5 6 7 1 5 12 21 prod 1 2 3 5 6 7 2 6 210 See also qr QR decomposition Syntax Q R E qr A Q R qr A Qe
233. he object obj is a character string false other wise Strings can have more than one line Examples ischar abc true ischar 0 false ischar false ischar true LME Reference strings 311 ischar abc def true See also isdigit Test for digits Syntax b isdigit s Description For each character of string s isdigit s is true if it is a digit 0 to 9 and false otherwise Examples isdigit al23bAB12 FTTTFFFTTFF See also isletter Test for letters Syntax b isletter s Description For each character of string s isletter s is true if it is a letter and false otherwise Letters with diacritical signs are not considered as letters 312 Sysquake for TEX Examples isletter a TTTTF See also isspace Test for spaces Syntax b isspace s Description For each character of string s isspace s is true if it is a space a tabulator a carriage return or a line feed and false otherwise Example isspace a tb c nd 01031010 See also lower Convert all uppercase letters to lowercase Syntax s2 lower s1 Description lower s1 converts all the uppercase letters of string s1 to lowercase Currently only ASCII letters without diacritic are converted LME Reference strings 313 Example Lower abcABC123 abcabc123 See also md5 Calculate MD5 digest Syntax digest md5 s
234. he projection rotate around the image of the target View angle The view angle defines the part of the 3D space which is projected onto the image window in perspective projections It is zero in orthographic mode All of these parameters can be set automatically Here is how the whole projection and scaling process is performed Scale data separately along each direction according to daspect Find bounding box of all displayed data or use limits set with scale Find radius of circumscribed sphere of bounding box If the target point is automatic set it to the center of the bound ing box otherwise use position set with camtarget If the view point is automatic set it to direction 3 2 1 at infinity in orthographic mode or in that direction with respect to the target point at a distance such that the view angle of the circumscribed sphere is about 6 degrees otherwise use position set with campos If the up vector is automatic set it to 0 0 1 vertical pointing upward otherwise use position set with camup Compute the corresponding homogeneous matrix transform LME Reference 3D graphics 419 Set the base scaling factor so that the circumscribed sphere fits the display area Apply an additional zoom factor which depends on camva and camzoom 6 38 Surface shading Surface and mesh colors add information to the image helping the viewer in interpreting it Colors specified by the style argument al
235. he unstable model of the system may diverge if it is run over a long enough time period because of round off errors But in most cases this is not a problem Example Exact simulation of the output of a continuous time system whose in put comes from a zero order hold converter see Fig 6 16 amp unstable system continuous time transfer function LME Reference graphics for dynamical systems 451 num 1 den 1 1 sampling at Ts 1 too slow only for illustration Ts 1 numd dend c2dm num den Ts stabilizing proportional controller My 1 5 transfer function between ref signal and input conv kp dend addpol conv kp numd dend continuous time output for a ref signal step scale 0 10 hstep b a Ts num den discrete time output exact dstep conv b numd conv a dend Ts b a 0 See also impulse Impulse response plot of a continuous time linear system Syntax impulse numc denc impulse numc denc opt impulse Ac Bc Cc Dc ee a Bc Cc Dc opt impulse style impulse i le id y t imputse Description impulse numc denc plots the impulse response of the continuous time transfer function numc denc Further options can be provided in a structure opt created with responseset fields Delay and Range are utilized The optional argu ments style and id have their usual meaning impulse Ac Bc Cc Dc plots the impulse res
236. hen possible conversion to integer types discards most significant bytes If X is an array conversion is performed on each element the result has the same size The imaginary part if any is discarded only with conversions to integer types Example cast pi int8 3int8 See also uint8 and related functions cdf Cumulative distribution function Syntax y cdf distribution x y cdf distribution x al y cdf distribution x al a2 Description cdf distribution x calculates the integral of a probability density function from oo to x The distribution is specified with the first argu ment a string case is ignored t and T are equivalent Additional arguments must be provided for some distributions The distributions are given in the table below Default values for the parameters when mentioned mean that the parameter may be omitted 162 Distribution Name Beta beta Cauchy cauchy X chi x chi2 Y gamma exponential exp F f half normal half normal Laplace laplace lognormal logn normal norm Rayleigh rayl Student s T t uniform unif Weibull weib See also ceil Rounding towards infinity Syntax y ceil x Description Sysquake for ATEX Parameters a and b a and b deg of freedom v deg of freedom v shape and A mean deg of freedom v and v2 9 mean and scale mean 0 and st dev 1 mean 0 and st dev 1 b deg of freedom v limits of the range
237. i in A becomes dimension i in the result Examples size permute rand 3 4 5 2 3 1 534 LME Reference arrays 259 See also isempty Test for empty matrices or empty lists Syntax isempty M isempty list b b Description isempty obj gives true if obj is the empty array the empty string or the empty list and false otherwise Examples isempty T true isempty 0 false isempty true isempty true isempty false See also iscell Test for cell arrays Syntax b iscell X Description iscell X gives true if X is a cell array or a list and false otherwise 260 Sysquake for ATEX Examples iscell 1 2 true iscell 1 2 true islist 1 2 false See also ismember Test for set membership Syntax b ismember m s Description ismember m s tests if elements of array m are members of set s The result is a logical array the same size as m each element is true if the corresponding element of m is a member of s or false otherwise m must be a numerical array or a cell array matching type of set s Example S m b a bc bbb de d a x de a z ismember m s on i i 4m il TF TF See also length Length of a vector or a list LME Reference arrays 261 Syntax n n length v length list Description length v gives the length of
238. ic by default the public keyword is not required at the beginning of the library See also private function repeat Loop controlled by a boolean expression Syntax repeat sl until expr Description The statements between the repeat statement and the corresponding until are executed repeatedly at least once until the expression of the until statement yields true nonempty and all elements different from 0 and false LME Reference programming constructs Example v repeat v v sum v 1 until v end gt 100 V 1 2 4 8 16 32 64 128 See also return Early return from a function Syntax return Description 89 return stops the execution of the current function and returns to the calling function The current value of the output arguments if any is returned return can be used in any control structure such as if or try or at the top level Example function dispFactTable n display the table of factorials from 1 ton if n return nothing to display end fwrite i i n for i 1 n fwrite 2d 3d n i prod 1 i end See also function 90 Sysquake for ATEX switch Conditional execution of statements depending on a number or a string Syntax switch expr case el sl case e2 e3 s23 case e4 e5 s45 otherwise so end switch string case strl sl case str2 s2 case str3 str4 s34 otherwise so end
239. ic polynomial are real By definition closed loop poles for the current gain i e the roots of num den are on the root locus and move on it when the gain change rlocus plots only the root locus not the particular val ues of the roots for the current gain a null gain or an infinite gain If necessary these values should be plotted with plot roots The part of the root locus which is calculated and drawn depends on the scale If no scale has been set before explicitly with scale or implicitly with plotroots or pTot the default scale is set such that the zeros of num and den are visible As with other plots the id is used for interactive manipulation Manipulating a root locus means changing the gain of the controller which keeps the locus at the same place but makes the closed loop poles move on it Other changes are done by dragging the open loop poles and zeros which are plotted by plotroots To change the gain you must also plot the current closed loop poles with the plotroots function and use the same ID so that the initial click identifies the nearest closed loop pole and the mouse drag makes Sysquake use the root locus to calculate the change of gain which can be retrieved in _q see the example below Examples Root locus of Ss 3s 2 s 2s 35 4 with open loop poles and zeros added with plotroots see Fig 6 22 num 1 3 2 den 1 2 3 4 scale equal 4 1 2 2 sgrid rlocus num de
240. ich describe its surface are sampled with n discrete values By default edges are not drawn An optional second input argu ment a string specifies the edge style it corresponds to the style argument of surf With three output arguments sphere produces the X Y and Z ar rays expected by surf or mesh and it does not display anything See also sphericon Sphericon Syntax sphericon sphericon n sphericon n style X Y Z sphericon MG Nyse SPne recon 578 Sysquake for TEX Description Without output argument sphericon draws a sphericon a 3D shape made from a bicone with a 90 degree apex cut by a plane containing both apices where one half is rotated by 90 degrees approximated by a polyhedron With an input argument sphericon n draws a spheri con where the two parameters which describe its surface are sampled with n discrete values By default edges are not drawn An optional second input argu ment a string specifies the edge style it corresponds to the style argument of surf With three output arguments sphericon produces the X Y and Z arrays expected by surf or mesh and it does not display anything See also sphere crosscap surf torus Torus Syntax torus torus torus r n torus r n style X Y Z r r r Description Without output argument torus draws a torus approximated by a polyhedron with a main radius of 1 and a tube radius of 0 5 With an i
241. ided in a structure opt cre ated with responseset It is a vector of two elements the start time and the end time Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time The optional arguments style and id have their usual meaning With output arguments dlsim gives the output and the time as column vectors or an array for the output of a multiple output state space model where each row represents a sample No display is produced Example Simulation of a third order system with a rectangular input see Fig 6 11 u repmat ones 1 10 zeros 1 10 1 3 dlsim 1 poly 0 9 0 7 0 6j 0 7 0 6j u 1 rs See also dnichols Nichols diagram of a discrete time system 442 Sysquake for ATEX disim 0 Figure 6 11 dlsim 1 poly 0 9 0 7 0 6j 0 7 0 6j u Syntax dnichols numd dend dnichols numd dend w dnichols numd dend opt dnichols numd dend w opt dnichols style dnichols style id w dnichols mag phase dnichols mag phase w dnichols Description dnichols numd dend displays the Nichols diagram of the discrete time transfer function given by polynomials numd and dend In dis crete time the Nichols diagram is the locus of the complex values of the transfer function evaluated at e where w is a real number between 0 and 7 inclusive displayed in the phase magnitude plane Usu
242. ifferent kinds of variables can be declared explicitly with global and persistent When multiple functions are defined in the same code source for instance in a library the body of a function spans from its header to the next function or until the endfunction keyword whichever comes first Function definitions cannot be nested endfunction is required only when the function definition is followed by code to be executed outside the scope of any function This includes mixed code and function definitions entered in one large entry in a command 84 Sysquake for ATEX line interface or applications where code is mainly provided as state ments but where function definitions can help and separate libraries are not wished note that libraries cannot contain code outside func tion definitions they do never require endfunction Like function endfunction must be the first element of a line Not all of the input and output arguments are necessarily used The caller fixes the number of input and output arguments which can be retrieved by the called function with nargin and nargout respec tively The unused input arguments from nargin 1 to the last one are undefined unless a default value is provided in the function defi nition with the definition function f x y 2 y is 2 when f is called with a single input argument The unused output arguments from nargout 1 to the last one do not have to be set but may be To redefine an operator
243. in gives a list of structures which describe each built in function with the following fields name function name module module name trusted true if the function is trusted info operators displays the list of operators With an output argument info operators gives a list of structures like info builtin info functions displays the list of user defined functions with the library where they are defined Parenthesis denote functions known by LME but not loaded they also indicate spelling errors in function or variable names With an output argument info functions gives a list of structures which describe each user defined function with the following fields 104 Sysquake for ATEX library library name name function name loaded true if loaded info methods displays the list of methods With an output argu ment info methods gives a list of structures which describe each method with the following fields library library name class class name name function name loaded true if loaded info variables displays the list of variables with their type and size With an output argument info methods gives a list of struc tures which describe each variable with the following fields name function name defined true if defined info global displays the list of all global variables With an output argument info global gives the list of the global variable names info persistent displays
244. in an index expression matrixrow was used outside of any index sub expression in an expression matrixrow or matrixcol used in the wrong index matrixrow was used in an index which was not the first one or matrixcol was used in an index which was not the only one or the second one Compilation overflow Not enough memory during compilation Too many nested subexpressions The number of nested of subexpressions is too high Variable table overflow A single statement attempts to define too many new variables at once Expression too large Not enough memory to compile a large ex pression Too many nested and The maximum depth of nested subexpressions function argument lists arrays and lists is reached Too many nested programming structures Not enough mem ory to compile that many nested programming structures such as if while switch etc Wrong number of input arguments Too few or too many input arguments for a built in function during compilation Wrong number of output arguments Too few or too many out put arguments for a built in function during compilation Too many indices More than two indices for a variable Variable not found A variable is referenced but appears neither in the arguments of the function nor in the left part of an assign ment 54 Sysquake for ATEX Unbounded language construct if while for switch or try without end Unexpected end The end statement does not match an if Swi
245. inf 0 2 3026 0 194 Sysquake for ATEX See also realmax realmin Largest and smallest real numbers Syntax realmax realmax n realmax n1 n2 realmax nl1 n2 realmax type realmin realmin x x KK KK XK tou ud id ud oueou Description realmax gives the largest positive real number in double precision realmin gives the smallest positive real number in double precision which can be represented in normalized form i e with full mantissa precision With integer non negative arguments realmax and realmin create arrays whose elements are all set to the respective value Arguments are interpreted the same way as zeros and ones The last argument of realmax and realmin can be a string to spec ify the type of the result double for double precision default or single for single precision Examples realmin 2 2251e 308 realmin single 1 1755e 38 realmax 1 7977e308 realmax single 3 4028e38single realmax eps realmax inf See also LME Reference mathematical functions 195 realpow Real power Syntax z realpow x y Description realpow x y gives the real value of x to the power y The imaginary parts of x and y are ignored The result is NaN if it is not defined for the input arguments If the arguments are arrays their size must match or one of them must be a scalar number the power is performed element wise See also realsqrt
246. inherit from the parent In LME objects from a subclass contain in a special field an object of the parent class the field name has the same name as the parent class If LME does not find a method for an object it tries to find one for its parent great parent etc if any An object can also inherit from several parents Here is an example of the use of polynom objects which as can be guessed from their name contain polynomials Statement use classes imports the definitions of methods for class polynom and oth ers use classes p polynom 1 5 0 1 p x 3 5x 241 p2 3 p polynom 1 0 a l X 6 10x 54 25x 44 2x 34 13xX 2415x4 1 6 5 Input and Output LME identifies channels for input and output with non negative integer numbers called file descriptors File descriptors correspond to files devices such as serial port network connections etc They are used as input argument by most functions related to input and output such as fprintf for formatted data output or fgets for reading a line of text Note that the description below applies to most LME applications For some of them files command prompts or standard input are ir relevant or disabled and standard output does not always correspond to the screen At least four file descriptors are predefined Value Input Output Purpose 0 Input Standard input from keyboard 1 Output Standard output to screen 2 Output Standard error to screen 3 Output Prom
247. input simple output SISO system or lists of sublists of zeros poles and gains for multiple input multiple output MIMO systems Examples sd zpk 0 3 0 8 0 5j 0 8 0 5j 10 0 1 sd discrete time transfer function Ts 0 1 10z 3 z 2 1 6z 0 89 See also Iti bodemag Magnitude of the Bode plot Syntax bodemag a bodemag a Description bodemag a plots the magnitude of the Bode diagram of system a See also podephase Libraries lti 545 Iti bodephase Phase of the Bode plot Syntax bodephase a bodephase a Description bodephase a plots the magnitude of the Bode diagram of system a See also Iti impulse Impulse response Syntax impulse a impulse a Description impulse a plots the impulse response of system a See also Iti initial Time response with initial conditions Syntax initial a x0 initial a x0 546 Sysquake for TEX Description initial a x0 plots the time response of state space system a with initial state xO and null input See also Iti lsim Time response Syntax lsim a u t lsim a u t Description lsim a u t plots the time response of system a For continuous time systems The input is piece wise linear it is defined by points in real vectors t and u which must have the same length Input before t 1 and after t end is 0 For discrete time systems u is sampled a
248. int quantiz quaternion Quaternions 320 quiver 411 rand randn 268 586 range 496 rene Baer rdivide 122 real 193 reallog realmax 194 realmin 194 realpow realsqrt red2yellowcm 566 redirect B70 rem 195 repeat replist repmat reshape responseset rethrow 112 return rlocus 46 rmfield 3 roots 24 rot90 round rpy2q 39 B 8 00 NI ml NI e FERI 02 ml zE se ov gt OS o DR an Ww sandbox 118 a rust 120 scale 4 tor Sy scalef r 415 schur paz sec sech 198 ies sepiacm set 48 setdiff 271 setenv 386 setstr setxor 272 sgrid 464 shal scale o sensor3 setfield 339 eric sign Sysquake for ATEX sin 198 sinc single eet size 273 skewness 243 sleep 386 smallstellateddodecahedron sort 274 sortrows 486 Pee sph2cart 486 sqlxendfragment 469 sqlxstartfragment 470 sqlxvalue 469 sqrt 200 sqrtm 243 squareform 497 squeeze sread 3 ss2tf sscanf 3 4 aa 391 std 2 step 466 st org str2obj strcmp 3 D ast ceraetch BTS strtok 317 strtrim 318 struct 339 structure style 390 style parameter 390 sub2ind subsasgn subspace 487 subsref sum 245 surf 432 Index svd 246 swapbytes 201 switch 90 swrite 376 symbol shape 390 tan 201 tanh 202 tetrahedron 5
249. ion Syntax M2 chol M1 Description If a square matrix M1 is symmetric or hermitian and positive definite it can be decomposed into the following product M M7 M2 where M2 is an upper triangular matrix The Cholesky decomposi tion can be seen as a kind of square root The part of M1 below the main diagonal is not used because M1 is assumed to be symmetric or hermitian An error occurs if M1 is not positive definite Example chol 5 3 3 8 Wux On I Il co w 2361 1 3416 2 4900 See also cond Condition number of a matrix Syntax x cond M LME Reference linear algebra 207 Description cond M returns the condition number of matrix M i e the ratio of its largest singular value divided by the smallest one or infinity for singu lar matrices The larger the condition number the more ill conditioned the inversion of the matrix Examples cond 1 0 0 1 1 cond 1 1 1 1 le 3 4002 0008 See also conv Convolution or polynomial multiplication Syntax v conv vl1 v2 M conv M1 M2 M conv M1 M2 dim Description conv vl v2 convolves the vectors v1 and v2 giving a vector whose length is length v1 length v2 1 The result is a row vector if both arguments are row vectors and a column vector if both arguments are column vectors Otherwise arguments are considered as matrices conv M1 M2 convolves the matrices M1 and M2 column by columns conv M1
250. ion eig M returns the vector of eigenvalues of the square matrix M V D eig M returns a diagonal matrix D of eigenvalues and a matrix V whose columns are the corresponding eigenvectors They are such that M V VD Examples eig 1 2 3 4 0 3723 5 3723 V D eig 1 2 2 1 V 0 7071 0 7071 0 7071 0 7071 D 1 0 03 1 2 2 1 V 0 7071 2 1213 0 7071 2 1213 V xD 0 7071 2 1213 0 7071 2 1213 See also expm Exponential of a square matrix LME Reference linear algebra 217 Syntax M2 expm M1 Description expm M is the exponential of the square matrix M which is usually different from the element wise exponential of M given by exp Examples expm 1 1 1 1 4 1945 3 1945 3 1945 4 1945 exp 1 1 1 1 2 7183 2 7183 2 7183 2 7183 See also fft Fast Fourier Transform Syntax F fft f F fft f n F fft f n dim Description fft f returns the discrete Fourier transform DFT of the vector f or the DFT s of each columns of the array f With a second argument n the n first values are used if n is larger than the length of the data zeros are added for padding An optional argument dim gives the dimension along which the DFT is performed it is 1 for calculating the DFT of the columns of f 2 for its rows and so on fft f dim specifies the dimension without resizing the array fft is based on a mixed radix Fast Fourier Transform if the data length is non prime I
251. ion and the platform For instance in shell applications word wrap is applied to paragraphs but bold or italic text is rendered as plain text Example T 27 3 fprintf 4 Report nTemperature is 1f n T The same markup is used in LME applications at other places For example in Sysquake the version and help information can contain markup Markup reference Text with markup is plain text with a few character sequences which are recognized by the markup processor and change the format of the result The result is styled text with titles paragraphs with justifica tion bold and italic faces etc There are two kinds of markup constructs blocks and inline Blocks are paragraphs indented or not headings list items table cells horizontal rules and block extensions Inline constructs are charac ter styles verbatim text and links they are part of blocks Except in preformatted blocks sequences of spaces or tabs are replaced with single spaces Paragraphs Paragraphs are made of lines whose first character is not one of nor sequence or lt lt lt Lines can begin with sequences and Paragraphs end with blank lines or next heading list table in dented paragraph preformatted block of text horizontal rule or block extension They can contain styled text links spans of verbatim text and inline extensions Example This is a paragraph written in two lines This is another paragr
252. ion queueput and retrieved from the end of the vector with function queueget function queueput x global QUEUE QUEUE x QUEUE function x queueget global QUEUE x QUEUE end QUEUE end Both functions must declare QUEUE as global otherwise the variable would be local even if there exists also a global variable defined else where The first time a global variable is defined its value is set to the empty matrix In our case there is no need to initialized it to another value Here is how these functions can be used Tutorial 31 gt queueput 1 gt queueget ans 1 gt queueput 123 gt queueput 2 3j gt queueget ans 123 gt queueget ans 2 3j To observe the value of QUEUE from the command line interface QUEUE must be declared global there If a local variable already exists it is discarded gt global QUEUE gt QUEUE QUEUE gt queueput 25 gt queueput 17 gt QUEUE QUEUE 17 25 Chapter 5 IATeEX Package Reference This section documents the LaTeX package sysquake sty This pack age permits to embed in LaTeX source files tex suffix fragments of code written in LME the programming language of Sysquake All these fragments are evaluated successively in the same context definitions and variables can be reused across multiple fragments 5 1 Package options The package is imported in the main LaTeX source file with usepackage usepackage sys
253. ions can be assigned to vari ables passed as arguments and built dynamically Evaluating them with feval is faster than using eval with a string because they are compiled only once to an intermediate code They can also be used as the argument of functions such as fzero and fmin inline funstr returns an inline function whose source code is funstr Input argument funstr follows the same syntax as a plain function The function name is ignored inline expr returns an inline function with one implicit input ar gument and one result The input argument expr is a string which evaluates to the result The implicit input argument of the inline func tion is a symbol made of a single lower case letter different from i and j such as x or t which is found in expr If several such symbols are found the one closer to x in alphabetical order is picked inline expr argl returns an inline function with one re sult and the specified arguments arg1 etc These arguments are also given as strings 106 Sysquake for TEX Inline functions also accept an additional input argument which cor respond to fixed parameters provided when the function is executed inline funstr param where funstr is a string which contains the source code of a function stores param together with the function When the function is called param is prepended to the list of input arguments inline expr args paramstruct is a simplified way to create an inline function when
254. ires it even if it is real gt sqrt 2 ans 04 1 4142i gt exp 3 2j Tutorial 21 ans 8 3585 18 2637j gt log 8 3585 18 2637j ans 3 2j To get the real or imaginary part of a complex number use the func tions real or imag respectively gt real 2 3j ans 2 gt imag 2 3j ans 3 Complex numbers can be seen as vectors in a plane Then addition and subtraction of complex numbers correspond to the same opera tions applied to the vectors The absolute value of a complex number also called its magnitude is the length of the vector gt abs 3 4j ans 5 gt sqrt 3 2 4 2 ans 5 The argument of a complex number is the angle between the x axis real axis and the vector counterclockwise It is calculated by the angle function gt angle 2 3j ans 0 9828 The last function specific to complex numbers we will mention here is conj which calculates the conjugate of a complex number The con jugate is simply the original number where the sign of the imaginary part is changed gt conj 2 3j ans 2 3j Real numbers are also complex numbers with a null imaginary part hence gt abs 3 ans 22 Sysquake for ATEX gt conj 3 ans gt angle 3 ans gt angle 3 ans 3 1416 4 3 Vectors and Matrices LME manipulates vectors and matrices as easily as scalars To define a matrix enclose its contents in square brackets and use commas to separate eleme
255. is 4 labels on x axis 6 log y 7 dB y 8 tick on y axis 9 grid for y axis 10 labels on y axis 12 same scale on both axis 13 minimum grid 14 maximum grid scale lock locks the scale as if the user had done it by hand It fixes only the initial value the user may change it back afterwards The scale is usually limited to a range of 1e 6 for linear scales and a ratio of 1e 6 for logarithmic scales This avoids numerical problems such as when a logarithmic scale is chosen and the data contain the value 0 In some rare cases a large scale may be required The lock option is used to push the limits from 1e 6 to 1e 24 for both linear and logarithmic scales A second argument must be provided scale lock xmin xmax ymin ymax The command must be used in a draw handler or from the command line interface To add other options use a separate scale command scale logdb scale lock le 5 1e8 le 9 1e9 The scale is locked and the user may not unlock it In the example above note also that a single string argument can be written without quote and parenthesis if it contains only letters and digits With an output argument scale returns the current scale as a vec tor xmin xmax ymin ymax If the scale is not fixed the vector is empty If only the horizontal scale is set the vector is xmin xmax During a mouse drag both the x and y are fixed The values returned by scale reflect the zoom chosen by the us
256. it angular fre quency to a bandpass analog filter with the specified center angular frequency w0 and bandwidth ww lp2bp z0 p0 k0 wc ww converts a filter given by its zeros poles and gain lp2bp num0 den0 wc ww converts a filter given by its numerator and denominator coefficients in decreasing powers of s The new filter F s is S w w 4 F s Fo A Wws where Fo s is the filter prototype The filter order is doubled See also Ip2bs Lowpass prototype to bandstop filter conversion 524 Sysquake for ATEX Syntax z p k lp2bs z0 pO kO wc ww num den lp2bs num0 denO wc ww Description lp2bs convert a lowpass analog filter prototype with unit angular fre quency to a bandstop analog filter with the specified center angular frequency w0 and bandwidth ww lp2bs z0 p0 k0 wc ww converts a filter given by its zeros poles and gain lp2bs num0 den0 wc ww converts a filter given by its numerator and denominator coefficients in decreasing powers of s The new filter F s is s Fo gt F s F S2 w2 w2 4 where Fo s is the filter prototype The filter order is doubled See also Up2Up lp2hp tp2bp Ip2hp Lowpass prototype to highpass filter conversion Syntax z p k lp2hp z0 pO kO w0 num den lp2hp num0 den0 w0 Description lp2hp convert a lowpass analog filter prototype with unit angular fre quency to a highpass analog filter with the specified
257. ives a quaternion with the real part set to zero because there are three imaginary compo nents instead of one with complex numbers Operators and functions which accept multiple arguments convert automatically double arrays to quaternions ignoring the imaginary part of complex numbers Conversion to numeric arrays with double adds a dimension for the real part and the three imaginary parts For example converting a scalar quaternion gives a 4 by 1 double column vector and converting a 2 by 2 quaternion array gives a 2 by 2 by 4 double array Real and imaginary components can be accessed with the field access notation q wis the real part of q q x q y and q z are its imaginary parts and q v is its imaginary parts as an array similar to the result of double but without the real part Compatibility note native functions for quaternions replace library quaternion which defined quaternion scalars and matrices It is much faster and supports arrays of more than two dimensions on the other hand matrix oriented functions are not supported anymore and the result of dumpvar is not directly compatible isquaternion Test for a quaternion Syntax b isquaternion q Description isquaternion q is true if the input argument is a quaternion and false otherwise Examples isquaternion 2 false isquaternion quaternion 2 true 324 Sysquake for ATEX See also quaternion isnumeric q2mat Conversion from quaternion to rot
258. k side is visible Default is equivalent to fb surf style specifies the line or symbol style of the mesh between surface cells or the fill style of the surface The default is to map z or color values to the color map for the surface cells and not to draw cell bounds mesh id specifies the ID used for interactivity in Sysquake See also 6 40 Graphical Functions for Dynamical Systems Graphical commands described in this section are related to automatic control They display the time responses and frequency responses of linear time invariant systems defined by transfer functions or state space models in continuous time Laplace transform or discrete time z transform Some of these functions can return results in output arguments instead of displaying them These values depend not only on the input arguments but also on the current scale of the figure For instance the set of frequencies where the response of the system is evaluated for the Nyquist diagram is optimized in the visible area Option Range of responseset can be used when this behavior is not suitable such as for phase portraits using lsim Output can be used for uncommon display purposes such as special styles labels or export Evaluation or simulation functions not related to graphics like polyval ode45 or filter are better suited to other usages bodemag Magnitude Bode diagram of a continuous time system Syntax bodemag numc
259. lements are not con catenated but used as cells C1 C2 1x2 cell array LME Reference operators 127 See also Operator Braces Syntax list_elements cells v index v indexl index2 v index expr fun v i Description A pair of braces is used to define a list or a cell array given by its elements In a list the operator is used to separate elements In a cell array the operator is used to separate cells on the same row the operator is used to separate rows Braces without semicolons produce a list braces with semicolon s produce a cell array v index is the element of list variable v whose index is given index must be an integer between 1 for the first element and length v for the last element v index may be used in an expression to extract an element or on the left hand side of the equal sign to assign a new value to an element Unless it is the target of an assignment v may also be the result of an expression If v is a cell array v index is the element number index v index1 index2 gives the specified cell of a cell array v itself may be an element or a field in a larger variable provided it is a list i e complicated assignments like a 2 f 3 2 5 3 are accepted In an assignment when the index or indices are larger than the list or cell array size the variable is expanded with empty arrays In the list of the input arguments of a function call v
260. lename samples samplerate nbits wavread filename n filename n1 n2 data wavread wavread samples samplerate nbits samples samplerate nbits YS we ioe jonas jy peat Description wavread filename reads the WAV file filename The result is a 2 d array where each row corresponds to a sample and each column to a channel Its class is the same as the native type of the WAV file i e int8 or int16 wavread filename n where n is a scalar integer reads the first n samples of the file wavread filename n1 n2 where the second input argument is a vector of two integers reads samples from n1 to n2 the first sample corresponds to 1 Instead of a file name string the first input argument can be a vector of bytes of class int8 or uint8 which represents directly the contents of the WAV file In addition to the samples wavread can return the sample rate in Hz such as 8000 for phone quality speech or 44100 for CD quality music and the number of bits per sample and channel See also Wavwrite wavwrite WAV encoding Syntax wavwrite samples samplerate nbits filename data wavwrite samples samplerate nbits data wavwrite samples samplerate 556 Sysquake for TEX Description wavwrite samples samplerate nbits filename writes a WAV file filename with samples in array samples sample rate samplerate in Hz and nbits bits per sample and channel Rows of samples corre
261. ll and yaw the x axis must be aligned with the direction of motion the y axis with the lateral direction and the z axis with the vertical direction with the usual sign conventions for cross products Two common choices are x pointing forward y to the left and z upward or x forward y to the right and z downward If input arguments are arrays of compatible size same size or scalar the result is a quaternion array of the same size conversion from angles to quaternion is performed independently on corresponding elements Example Conversion of two vectors from aircraft coordinates x axis forward y axis to the left z axis upward to earth coordinates x directed to the north y to the west z to the zenith In aircraft coordinates vectors are 2 0 0 propeller position and 0 5 0 left wing tip The air craft attitude has a pitch of 10 degrees upward i e 10 degrees with the choice of axis and null roll and yaw a l rpy2q 0 10 pi 180 0 q 0 9962 0 0872j q x quaternion 2 0 0 q 1 96961 0 3473k q quaternion 0 5 0 q 5j See also q2rpy 6 23 List Functions apply Function evaluation with arguments in lists LME Reference lists 331 Syntax listout apply fun listin listout apply fun listin nargout Description Listout apply fun listin evaluates function fun with input argu ments taken from the elements of list Listin Output arguments are grouped in list Listout Func
262. lled with more argu ments than what can be assigned to the other arguments remaining ones are collected in a list and assigned to varargin In the body of the function varargin is a normal variable Its elements may be LME Reference miscellaneous functions 115 accessed with the brace notation varargin i nargin is always the total number of arguments passed to the function by the caller When the function is called with fewer arguments than what is de clared varargin is set to the empty list Example Here is a function which accepts any number of square matrices and builds a block diagonal matrix function M blockdiag varargin M for block varargin block takes the value of each input argument m n size block M end 1 end m end 1 end n block end In the call below varargin contains the list ones 3 2 ones 2 3 blockdiag ones 3 2 ones 2 3 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0 0 3 See also varargout Remaining output arguments Syntax function varargout fun varargout Description varargout is a special variable which may be used to dispatch output arguments In the function declaration it must be used as the last or unique output argument When the function is called with more 116 Sysquake for ATEX output arguments than what can be obtained from the other argu ments remaining ones are extracted from the list varargout In th
263. lumns of matrix M mean M dim returns the mean of matrix M along dimension dim the result is a row vector if dim is 1 or a column vector if dim is 2 Examples mean 1 5 7 5 mean 1 5 7 5 mean 1 2 3 5 6 7 345 mean 1 2 3 5 6 7 1 345 mean 1 2 3 5 6 7 2 2 6 See also min Minimum value of a vector or of two arguments Syntax x min v v ind min v v fiy dim v ind min M dim M3 SCE Description min v returns the largest number of vector v NaN s are ignored The optional second smallest argument is the index of the minimum in v if several elements have the same minimum value only the first one is obtained The argument type can be double single or integer of any size min M operates on the columns of the matrix M and returns a row vector min M dim operates along dimension dim 1 for columns 2 for rows min M1 M2 returns a matrix whose elements are the minimum be tween the corresponding elements of the matrices M1 and M2 M1 and 232 Sysquake for TEX M2 must have the same size or be a scalar which can be compared against any matrix Examples mx ix min 1 3 2 5 8 7 mx 1 ix 1 min 1 3 5 nan 2 1 5 min 1 3 5 nan 2 12 22 See also max moment Central moment of a set of values Syntax m moment A order m moment A order dim Description moment A order gives the central moment moment about the mean of the specifi
264. ly in an expression gt magicSum 3 3 ans 18 18 18 One of the important benefits of defining function is that the variables have a limited scope Using a variable inside the function does not make it available from the outside thus you can use common names such as x and y without worrying about whether they are used in some other part of your whole program For instance let us use one of the variables of magicSum gt mage 77 magc 77 30 Sysquake for ATEX gt magicSum 3 magc ans 92 92 92 gt magc magc 77 4 9 Local and Global Variables When a value is assigned to a variable which has never been refer enced a new variable is created It is visible only in the current con text the base workspace for assignments made from the command line interface or the current function invocation for functions The variable is discarded when the function returns to its caller Variables can also be declared to be global i e to survive the end of the function and to support sharing among several functions and the base workspace Global variables are declared with keyword global global x global y z A global variable is unique if its name is unique even if it is declared in several functions In the following example we define functions which implement a queue which contains scalar numbers The queue is stored in a global variable named QUEUE Elements are added at the front of the vector with funct
265. ly they are stored in a file whose name is the library name with a Iml suffix for instance library stdlib is stored in file stdlib Iml Before a user function can be called its library must be loaded with the use statement use statements have an effect only in the context where they are placed i e ina library or the command line interface or a Sysquake SQ file this way different libraries may define functions with the same name provided they are not used in the same context In a library functions can be public or private Public functions may be called from any context which use the library while private functions are visible only from the library they are defined in 6 4 Types Numerical logical and character arrays The basic type of LME is the two dimensional array or matrix Scalar numbers and row or column vectors are special kinds of matrices Ar rays with more than two dimensions are also supported All elements have the same type which are described in the table below Two non numerical types exist for character arrays and logical boolean arrays Cell arrays which contain composite types are described in a section below Type Description double 64 bit IEEE number complex double Two 64 bit IEEE numbers single 32 bit IEEE number complex single Two 32 bit IEEE numbers uint32 32 bit unsigned integer int32 32 bit signed integer uint16 16 bit unsigned integer int16 16 bit signed integer uint8 8 bit unsigned in
266. ma style dsigma style id sv w dsigma Description dsigma Ad Bd Cd Dd Ts plots the singular values of the frequency response of the discrete time state space model Ad Bd Cd Dd de fined as zX z Y z AaX z BgU z CgX z DgU zZ where z e 7s and Ts is the sampling period Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning dsigma is the equivalent of dbodemag for multiple input systems For single input systems it produces the same plot The range of frequencies is selected automatically or can be speci fied in an optional argument w a vector of frequencies With output arguments dsigma gives the singular values and the frequency as column vectors No display is produced See also dbodemag dbodephase sigma responseset dstep Step response plot of a discrete time linear system 446 Sysquake for ATEX Syntax dstep numd dend Ts dstep numd dend Ts opt dstep Ad Bd Cd Dd Ts pees ae Bd Cd Dd Ts opt dstep style es si AA id y t dstep Description dstep numd dend Ts plots the step response of the discrete time transfer function numd dend with sampling period Ts Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning d
267. ment Additional parameters can be passed directly in the anony mous function with captured variables instead of being supplied as additional arguments the code becomes clearer Examples Function reference quad sin 0 pi 2 Anonymous function a 2 fun x sin a x fun 3 0 2794 quad fun 0 2 0 8268 Without anonymous function parameter a should be passed as an additional argument after all the input arguments defined for quad including those which are optional when parameters are missing quad inline sin a x x a 0 2 false a 0 8268 Anonymous functions are actually stored as inline functions with all captured variables dumpvar fun inline function y f a x y sin a x 2 152 Sysquake for TEX See also 6 15 Mathematical Functions abs Absolute value Syntax x abs z Description abs takes the absolute value of each element of its argument The result is an array of the same size as the argument each element is non negative Example abs 2 3 0 3 4j 2305 See also acos Arc cosine Syntax y acos x Description acos x gives the arc cosine of x which is complex if x is complex or if abs x gt 1 LME Reference mathematical functions 153 Examples acos 2 0 1 3170j acos 1 2j 1 5708 1 1437 1 5286j See also acosh Inverse hyperbolic cosine Syntax y acosh x Description acosh x gives the inv
268. ment dim is provided the operation is performed along that dimension Examples any 1 2 3 2 true any 1 2 3 gt 5 false See also bitall Check whether all the corresponding bits are true Syntax v bitall A v bitall A dim b bitall v 344 Sysquake for ATEX Description bitall A performs a bitwise AND on the elements of the columns of array A or the elements of a vector If a second argument dim is provided the operation is performed along that dimension A can be a double or an integer array For double arrays bitall uses the 32 least significant bits Examples bitall 5 3 1 bitall 7uint8 6uint8 3uint8 6uint8 2 2x1 uint8 array 6 2 See also bitany jall bitand bitand Bitwise AND Syntax c bitand a b Description Each bit of the result is the binary AND of the corresponding bits of the inputs The inputs can be scalar arrays of the same size or a scalar and an array If the input arguments are of type double so is the result and the operation is performed on 32 bits Examples bitand 1 3 1 bitand 1 6 1 1031010 bitand 7uint8 1234int16 2int16 LME Reference logical functions 345 See also bitany Check whether any of the corresponding bits is true Syntax v bitany A bitany A dim bitany v vV b Description bitany A performs a bitwise OR on the elements of the columns of array A or the elements of a vector If a second a
269. mplex if x is complex or if abs x gt 1 Examples atanh 0 5 0 5493 atanh 2 0 5493 1 5708j LME Reference mathematical functions 159 See also beta Beta function Syntax y beta z w Description beta z w gives the beta function of z and w Arguments and result are real imaginary part is discarded The beta function is defined as 1 B z w i t tdt 0 Example beta 1 2 0 5 See also betainc Incomplete beta function Syntax y betainc x z w Description betainc x z w gives the incomplete beta function of x z and w Arguments and result are real imaginary part is discarded x must be between 0 and 1 The incomplete beta function is defined as 1 B z w x I z w et t tdt 0 160 Sysquake for ATEX Example betainc 0 2 1 2 0 36 See also betaln Logarithm of beta function Syntax y betaln z w Description betaln z w gives the logarithm of the beta function of z and w Ar guments and result are real imaginary part is discarded Example betaln 0 5 2 0 2877 See also ganmatn cast Type conversion Syntax Y cast X type LME Reference mathematical functions 161 Description cast X type converts the numeric array X to the type given by string type which can be double single int8 or any other signed or unsigned integer type char or logical The number value is preserved w
270. mum intensity the second column the intensity of green and the third column the intensity of blue Input values are mapped uniformly to one of the discrete color entries O to the first row and 1 to the last row With an input argument colormap clut sets the colormap to clut With an output argument colormap returns the current col ormap LME Reference base graphics 399 See also contour Level curves Syntax contour z contour z xmin xmax ymin ymax contour z xmin xmax ymin ymax levels contour z xmin xmax ymin ymax levels style Description contour z plots seven contour lines corresponding to the surface whose samples at equidistant points 1 size z 2 in the x direction and 1 size z 1 on the y direction are given by z Contour lines are at equidistant levels With a second non empty argument xmin xmax ymin ymax the samples are at equidistant points between xmin and xmax in the x direction and between ymin and ymax in the y direction The optional third argument levels if non empty gives the num ber of contour lines if it is a scalar or the levels themselves if it is a vector The optional fourth argument is the style of each line from the minimum to the maximum level styles are recycled if necessary The default style is kbrmgcy When the style is f for a filled region the corresponding level is filled on the side with a lower value of z If the style argument is the singl
271. n exp x is the exponential of x i e 2 7182818284590446 x Example exp 0 1 0 5j pil 1 2 7183 1j See also expm1 Exponential minus one Syntax y expm1 x Description expm1 x is exp x 1 with improved precision for small x Example expm1 le 15 le 15 exp le 15 1 1 1102e 15 See also LME Reference mathematical functions 175 factorial Factorial Syntax y factorial n Description factorial n gives the factorial n of nonnegative integer n If the input argument is negative or noninteger the result is NaN The imag inary part is ignored Examples factorial 5 120 factorial nan 1 See also fix Rounding towards 0 Syntax y f1ix x Description fix x truncates the fractional part of x If the argument is a complex number the real and imaginary parts are handled separately Examples fix 2 3 2 fix 2 6 2 176 Sysquake for ATEX See also floor Rounding towards infinity Syntax y floor x Description floor x gives the largest integer smaller than or equal to x If the argument is a complex number the real and imaginary parts are han dled separately Examples floor 2 3 2 floor 2 3 3 See also gamma Gamma function Syntax y gamma x Description gamma x gives the gamma function of x Argument and result are real imaginary part is discarded The gamma function is defined as Cc roo P le t dt
272. n 490 hess hgrid hist horzcat 122 hstep 449 huecm 565 hypot i 180 icdf 180 icosahedron 5 1 if 85 ifft 221 ifft2 222 ifftn 222 ifftshift 480 igraycode 351 impulse include 86 ind2sub 255 inf 182 info 103 initial inline 105 inmem 107 int16 287 int32 28 int64 287 int8 287 interpn 256 eet DET intersect 257 intmax 288 intmin 289 inv 224 ipermute 258 isa iscell 259 ischar 310 isdigit 311 isempty isfield 336 N 583 isfinite 183 isfloat 183 isglobal 107 isinf isinteger iskeyword 108 isletter 311 islist B32 epee eon ismember isnan 185 isnumeric 185 isobject 341 ispc 385 isprime isquaternion isreal 481 isscalar 186 isspace isstruct isunix B85 isvector 186 join 331 klein 575 klein8 576 kron 224 kurtosis 225 Label 404 lasterr 108 lasterror 109 Lem 187 ldivide 122 le 122 legend length library colormaps constants 559 date 556 filter 515 lti ratio 507 Sigenc 548 stat 489 584 stdlib 472 wav 554 Lightangle 426 Line 406 Line3 427 linprog linspace 481 list 46 List2num 332 lloyds LME 39 command syntax comments 40 error messages 50 file descriptor 49 function call 40 input output libraries program format 39 statements 39 typ
273. n plotroots num plotroots den The second example shows how rlocus can be used interactively in Sysquake o x LME Reference graphics for dynamical systems 463 rlocus 2 0 Oo O 2 4 2 0 Figure 6 22 Example of rlocus figure Root Locus draw myPlotRLocus num den mousedrag num myDragRLocus num _q function function myPlotRLocus num den scale equal 3 1 2 2 rlocus num den 1 plotroots addpol num den 1 function num myDragRLocus num q if isempty q cancel else num q num end Caveat The Laguerre algorithm is used for fast evaluation roots and plotroots are based on eig and have a better accuracy but their evaluation for a single polynomial is typically 10 times slower The price to pay is a suboptimal precision for multiple roots and or high order polynomials 464 Sysquake for TEX scale equal 2 2 2 2 sgrid 2 Figure 6 23 scale equal 2 2 2 2 sgrid See also sgrid Relative damping and natural frequency grid for the poles of a continuous time system Syntax sgrid sgrid style sgrid damping freq sgrid damping freq style Description With no or one argument sgrid plots a grid of lines with constant relative damping and natural frequencies in the complex plane of s see Fig 6 23 The style argument has its usual meaning With two or three arguments s
274. n For example fplot fun 0 10 1 2 5 calls fun as y fun x 2 5 and displays its value for x between 0 and 10 Examples Plot a sine fplot sin Plot x 0 3 a exp 3x in red for x 2 3 with a 7 2 and an identifier of 1 fun inline function y f x a y x 0 7 3 2 axexp 3 x 2 fplot fun 2 3 F L 2 See also image Raster RGB or grayscale image Syntax image gray image red green blue image rgb LME Reference base graphics 403 image xmin xmax ymin ymax image style image id Description image displays a raster image an image defined by a rectangular ar ray of patches of colors called pixels The raster image can be either grayscale or color A grayscale image is defined by a double matrix of pixel values in the range O black to 1 white or by a uint8 ma trix in the range 0 black to 255 white A color image is defined by three matrices of equal size corresponding to the red green and blue components or by an array with three planes along the 3rd dimen sion Each component is defined between 0 black to 1 maximum intensity with double values or between 0 black to 255 maximum intensity with uint8 values The position is defined by the the minimum and maximum coordi nates along the horizontal and vertical axis The raster image is scaled to fit The first line of the matrix or matrices is display
275. n array the re sult has the same size each element corresponds to an element of the input eps single gives the smallest single positive number e such that 1single e can be distinguished from lsingle eps double gives the same value as eps without input argument Examples eps 2 2204e 16 1 eps 1 2 2204e 16 eps 2 1 1102e 16 172 Sysquake for ATEX 1 eps 2 1 0 See also erf Error function Syntax y erf x Description erf x gives the error function of x Argument and result are real imaginary part is discarded The error function is defined as 2 fx oe erf0 a e dt 0 Example erf 1 0 8427 See also erfc Complementary error function Syntax y erfc x LME Reference mathematical functions 173 Description erfc x gives the complementary error function of x Argument and result are real imaginary part is discarded The complementary error function is defined as fc x 1 erf x dt ertTc x 1 ern x e VT Jx Example erfc 1 0 1573 See also erfinv Inverse error function Syntax x erfinv y Description erfinv y gives the value x such that y erf x Argument and result are real imaginary part is discarded y must be in the range 1 1 values outside this range give nan Example y erf 0 8 0 7421 erfinv y 0 8 See also 174 Sysquake for TEX exp Exponential Syntax y exp x Descriptio
276. n u m sin cn u m cos dn u m 1 msin 9 ellipj u m tol uses tolerance tol the default tolerance is eps 170 Sysquake for ATEX Examples sn cn dn ellipj 2 7 0 6 sn 0 8773 cn 0 4799 dn 0 7336 sin ellipam 2 7 0 6 0 8773 ellipj 0 5 0 3 0 0000 0 8188 0 9713 0 4114 0 5341 0 9930 See also ellipke Complete elliptic integral Syntax K E ellipke m K E ellipke m tol Description K E ellipke m gives the complete elliptic integrals of the first kind K F m and second kind E E m defined as T 2 dt m SS 0 v1l msin t 1 2 Em V1 msin tdt 0 Parameter m must be in 0 1 ellipke m tol uses tolerance tol the default tolerance is eps LME Reference mathematical functions 171 Example K K E ellipke 0 3 1 7139 E 1 4454 See also ellip eps Difference between 1 and the smallest number x such that x gt 1 Syntax e e e eps eps x eps type Description Because of the floating point encoding of real numbers the absolute precision depends on the magnitude of the numbers The relative precision is characterized by the number given by eps which is the smallest double positive number such that 1 eps can be distinguished from 1 eps x gives the smallest number e such that x e has the same sign as x and can be distinguished from x It takes into account whether x is a double or a single number If x is a
277. ncorrcoef X1 X2 calculates the correlation coefficients of X1 and X2 and returns a 2 by 2 matrix ignoring NaN values It is equiva lent to nancorrcoef X1 X2 Libraries stat 493 See also nancov Covariance after discarding NaNs Syntax M nancov data M nancov data 0 M nancov data 1 Description nancov data returns the best unbiased estimate m by m covariance matrix of the n by m matrix data for a normal distribution NaN values are ignored Each row of data is an observation where n quantities were measured nancov data is the same as nancov data nancov data 1 returns the m by m covariance matrix of the n by m matrix data which contains the whole population NaN values are ignored See also nanmean Mean after discarding NaNs Syntax y nanmean A y nanmean A dim Description nanmean v returns the arithmetic mean of the elements of vector v nanmean A returns a row vector whose elements are the means of the corresponding columns of array A nanmean A dim returns the mean of array A along dimension dim the result is a row vector if dim is 1 or a column vector if dim is 2 In all cases NaN values are ignored 494 Sysquake for ATEX Examples nanmean 1 2 nan nan 6 7 147 nanmean 1 2 nan nan 6 7 2 1 5 6 5 nanmean nan nan nan See also nanmedian Median after discarding NaNs Syntax nanmedian A nanmedian A dim y y Description nanmedian v
278. nd Last bit position in a bitfield Syntax a end Description In the index expression of a bitfield end is the position of the most significant bit i e 1 less than the word length See also bitfield bitfield bitfield beginning bitfield find Find the ones in a bitfield Syntax ix find a Description find a finds the bits equal to 1 in bitfield a The result is a vector of bit positions in ascending order the least significant bit is number 0 Libraries bitfield Example itfield 123 16 See also bitfield int8 bitfield int16 bitfield int32 513 Convert a bitfield object to a signed integer number with sign exten sion Syntax n n n int8 a int16 a int32 a Description int8 a int16 a and int32 a convert bitfield a to an int8 int16 or int32 number respectively If a has less bits than the target integer and the most significant bit of a is 1 sign extension is performed i e the most significant bits of the result are set to 1 so that it is negative If a has more bits than the target integer most significant bits are ignored Example a bitfield 9 4 a 0x1001 i int8 a 1 210 b bitfield i o 0b11111001 514 Sysquake for ATEX See also Cc H ee t uint32 bitfield int8 bitfield int16 bitfield int32 bitfield double bitfield bitfield bitfield length Word length of a bitfield
279. nd document 5 3 Commands Any text output produced by code fragments is inserted in the LaTeX source code It is subject to further evaluation by LaTex sqeval sqeval code evaluates code fragment code which is processed as LaTeX code before being evaluated by Sysquake for LaTeX hence char acters aimed at Sysquake which have a meaning in LaTeX such as and must be escaped Examples Simple value The result is sqeval disp 1 2 Passing a LaTeX counter newcounter n setcounter n 10 The square root of arabic n is sqeval disp sqrt arabic n sgexpr sqexpr expr evaluates expression expr formats it for LaTeX and inserts it in the result The expression should not be terminated by a semicolon or comma Currently the following types are supported real or complex dou ble scalar or 2 dim arrays and strings Options set by format are used scientific notation is displayed with the mathematical notation a multiplication by 10 to some integer power instead of the classic computer notation sqexpr can be used in text as well as in math mode Examples Scalar value in an equation Some small number frac pi e 10 sqexpr pi exp 10 36 Sysquake for ATEX Magic square of size 5 I M_5 sqexpr magic 5 5 4 Environment begin sysquake end sysquake Text in the sysquake environment is interpreted verbatim as LME code All LaTeX constructs are ignored except for
280. nd what it means here of the coefficient vectors gt conv 1 2 2 4 5 ans 2 8 13 10 Hence x 2 2x 4x 5 2x34 8x 13x 10 26 Sysquake for ATEX 4 5 Strings You type strings by bracketing them with single quotes gt Hello World ans Hello World If you want single quotes in a string double them gt Easy isn t it ans Easy isn t it Some control characters have a special representation For example the line feed used in LME as an end of line character is n gt Hello nWorld ans Hello World Strings are actually matrices of characters You can use commas and semicolons to build larger strings gt a bc de f ans abc def 4 6 Variables You can store the result of an expression into what is called a variable You can have as many variables as you want and the memory permits Each variable has a name to retrieve the value it contains You can change the value of a variable as often as you want gt a 3 gt a 5 ans 8 gt a 4 gt a 5 ans 9 Note that a command terminated by a semicolon does not display its result To see the result remove the semicolon or use a comma if you have several commands on the same line Implicit assignment to variable ans is not performed when you assign to another variable or when you just display the contents of a variable Tutorial 27 gt a 3 a 3 gt a 7 b 3 2x gt x a a
281. ndgrid x Description ndgrid x1 xn produces n arrays of n dimensions Array i is obtained by reshaping input argument i as a vector along dimension i and replicating it along all other dimensions to match the length of other input vectors All output arguments have the same size With one input argument ndgrid reuses it to match the number of output arguments Y X ndgrid y x is equivalent to X Y meshgrid x y Example X1 X2 ndgrid 1 3 X1 1 1 1 2 2 2 3 3 3 X2 1 2 3 1 2 3 1 2 3 See also ndims Number of dimensions of an array Syntax n ndims A 264 Sysquake for ATEX Description ndims A returns the number of dimensions of array A which is at least 2 Scalars row and column vectors and matrices have 2 dimen sions Examples ndims magic 3 2 ndims rand 3 4 5 3 See also nnz Number of nonzero elements Syntax n nnz A Description nnz A returns the number of nonzero elements of array A See also find num2cell Conversion from numeric array to cell array Syntax C num2cell A C num2cell A dims LME Reference arrays 265 Description num2cell A creates a cell array the same size as numeric array A The value of each cell is the corresponding elements of A num2cell A dims cuts array A along dimensions dims and creates a cell array with the result Dimensions of cell array are the same as dimensions of A for dimensions not in dims and 1 f
282. ned by x is NaN griddatan is based on Delaunay triangulation The interpolation method used in each simplex is linear by default or can be specified with an additional input argument a string Argument Meaning 0 or nearest nearest value 1 or linear linear See also de launayn tsearcnn griddata jinterpn tsearch Search of points in triangles Syntax ix tsearch x y t xi yi Description tsearch x y t xi yi searches in which triangle is located each point given by the corresponding elements of arrays xi and yi Corresponding elements of arrays x and y represent the vertices of the triangles and rows of array t represent their indices in x and y array t is usually the result of deLaunay Dimensions of x and y and of xi and yi must be equal The result is an array with the same size as Xi and yi where each element is the row index in t of the first triangle which contains the point or NaN if the point is outside all triangles i e outside the convex hull of points defined by x and y if t is a proper triangulation such as the one computed with delaunay LME Reference triangulation 285 Example Search for triangles containing points 0 0 and 0 1 corresponding to Delauny triangulation of 20 random points x randn 20 1 y randn 20 1 t delaunay x y xi 0 0 yi 0 1 ix tsearch x y t xi yi See also tsearchn delaunay vorono1 tsearchn Search
283. ng the drag q contains the spherical coordinates phi and theta such that _pl param R cos q_ 1 cos q_ 2 sin q_ 1 cos q_ 2 sin q_ 2 With five input arguments sensor3 type param typeALt paramALt id specifies an alternative constraint surface used when the modifier key is held down Examples simple XY plane phi theta without modifier R with modifier with plane and ignored 2nd param See also surf Plot a surface defined by a grid in 3D space Syntax surf x y zZ surf z surf x y z color surf z color surf vis surf vis style surf vis style id Description surf x y z plots a surface defined by 2 D arrays x y and z Ar guments x and y must have the same size as z or be vectors of size z 2 and size z 1 elements respectively If x and y are miss ing their default values are coordinates from 1 to size z 2 along x axis and from 1 to size z 1 along y axis Color of each surface cell is obtained by mapping the average z values to the color map LME Reference graphics for dynamical systems 433 surf x y zZ color maps values of array color to the color map color must have the same size as z and contain values between 0 and 1 surf Vis specifies which side of the surface is visible vis is a string of 1 or 2 characters f if the front side is visible the side where increasing y are on the left of increasing x coordinates and b if the bac
284. noi cells The second output argument p is a list of vectors of row indices in v each element describes the Voronoi cell corresponding to a point in x In each cell vertices are sorted counterclockwise Voronoi tessalation is a tessalation a partition of the plane such that each region is the set of points closer to one of the initial point than to any other one Two regions are in contact if and only if their initial points are linked in the corresponding Delaunay triangulation Example Voronoi tessalation of 20 random points x rand 20 1 y rand 20 1 v p voronoi x y These points are displayed as crosses with Sysquake graphical func tions The scale is fixed because Voronoi polygons can have vertices which are far away from the points clf scale equal 0 1 0 1 plot x y x Voronoi polygons are displayed in a loop skipping unbounded poly gons The first vertex is repeated to have closed polygons Since plot expects row vectors vertex coordinates are transposed LME Reference integers 287 for pl p if any pl 1 pl pl p1 1 plot v p1 1 v p1 2 end end See also voronoin delLaunay voronoin N d Voronoi tessalation Syntax v p voronoin x Description voronoin x calculates the Voronoi tessalation of the set of points given by the rows of arrays x in a space of dimension n size x 2 The first output argument v is an array of n columns
285. nov equation AX XA C 0 Example 3 1 2 1 3 5 6 2 1 2 7 8 3 2 1 4 5 8 9 lyap A B C 0 1635 0 1244 0 2628 0 1311 0 7797 0 7645 A B C X XIN See also dlyap max Maximum value of a vector or of two arguments Syntax x max v v ind max v v As dim v ind max M dim M3 kr 1 M2 230 Sysquake for TEX Description max v returns the largest number of vector v NaN s are ignored The optional second output argument is the index of the maximum in v if several elements have the same maximum value only the first one is obtained The argument type can be double single or integer of any size max M operates on the columns of the matrix M and returns a row vector max M dim operates along dimension dim 1 for columns 2 for rows max M1 M2 returns a matrix whose elements are the maximum between the corresponding elements of the matrices M1 and M2 M1 and M2 must have the same size or be a scalar which can be compared against any matrix Examples mx ix max 1 3 2 5 8 7 mx 8 ix 5 max 1 3 5 nan 2 3 5 max 1 3 5 nan 2 23 5 2 See also min mean Arithmetic mean of a vector Syntax x mean v v mean M v mean M dim Description mean v returns the arithmetic mean of the elements of vector v mean M returns a row vector whose elements are the means of the LME Reference linear algebra 231 corresponding co
286. nput argument torus r draws a torus with tube radius r With two input arguments torus r n draws a torus where the two param eters which describe its surface are sampled with n discrete values By default edges are not drawn An optional third input argument a string specifies the edge style it corresponds to the style argument of surf With three output arguments torus produces the X Y and Z arrays expected by surf or mesh and it does not display anything See also abs acos acosh 153 acot 153 acoth acsc 154 acsch 154 activeregion 393 addpol 203 alawcompress alawexpand all 342 and 122 t22 155 asec 55 asech asin 156 asinh eee atan 157 atan2 atanh axisbounds balance 204 bar barh base64decode 307 base64encode 307 beginning besselap besself beta betainc 159 arg Index betaln 160 bilinea oH 517 bitall 3 bitand 344 Ba bitset B47 Ban bitshift bitxor sao black2orangecn 561 black2red2whitecm blkdiag 474 blue2greencm 562 blue2yellow2redcm bodemag bodephase bootstrp break 79 builtin 95 piece Ta butter bwrite 361 c2dm 354 a a camorbit 420 campan campos camproj camroll camtarget 423 camup camva 424 camzoom 424 care 580 cart2pol cart2sph circle 397 circshift 474 class 340 class bitfield int16 513 int32 513 int
287. nside unit circle gm psi wc wx dmargin 0 005 poly 0 9 0 9 gm 2 38 psi wc 0 0 4510 wx Stable closed loop Nyquist crosses unit circle 358 Sysquake for ATEX gm pSi wc wx dmargin 0 05 poly 0 9 0 9 gm 0 2 3 8 psi 0 7105 we 0 0 4510 wx 0 2112 Unstable closed loop gm psi wc wx dmargin 1 poly 0 9 0 9 gm psi we wx Caveats Contrary to many functions dmargin cannot be used with several transfer functions simultaneously because not all of them may cor respond simultaneously to either stable or unstable closed loop sys tems See also margin Robustness margins of a continuous time system Syntax gm psi wc wx margin num den Description The open loop continuous time transfer function is given by the two polynomials num and den If the closed loop system with negative feedback is unstable all output arguments are set to an empty ma trix Otherwise margin calculates the gain margins gm which give the interval of gain for which the closed loop system remains stable the phase margin psi always positive if it exists which defines the sym metric range of phases which can be added to the open loop system while keeping the closed loop system stable the critical frequency as sociated to the gain margins where the open loop frequency response intersects the real axis around 1 and the cross over frequency asso ciated to
288. nt Times Bold true Size 18 Color 1 0 0 text 1 1 4 2 Abc font See also title Subplot title Syntax title string Description title string sets or changes the title of the current subplot See also 6 36 3D Graphics Three dimension graphic commands enable the representation of ob jects defined in three dimensions x y and z on the two dimension screen The transform from the 3D space to the screen is performed as if there were a virtual camera in the 3D space with a given posi tion orientation and angle of view related to the focal length in a real camera 418 Sysquake for ATEX 6 37 Projection The projection is defined by the following parameters Target point The target point is a 3D vector which defines the position where the camera is oriented to Projection kind Two kinds of projections are supported ortho graphic and perspective View point The view point is a 3D vector which defines the posi tion of the camera For orthographic projection it defines a direc tion independent from the target position for perspective projec tion it defines a position and the view orientation is defined by the vector from view point to target point Up vector The up vector is a 3D vector which fixes the orientation of the camera around the view direction The projection is such that the up vector is in a plane which is vertical in the 2D projection Changing it makes t
289. nt See also 6 17 Array Functions cat Array concatenation Syntax cat dim Al A2 Description cat dim A1 A2 concatenates arrays Al A2 etc along dimen sion dim Other dimensions must match cat is a generalization of the comma and the semicolon inside brackets Examples cat 2 1 2 3 4 5 6 7 8 1 2 5 6 3 4 7 8 cat 3 1 2 3 4 5 6 7 8 2x2x2 array w See also operator 1 LME Reference arrays 249 cell Cell array of empty arrays Syntax C cell n C cell nl1 n2 C cell nl1 n2 Description cell builds a cell array whose elements are empty arrays The size of the cell array is specified by one integer for a square array or several integers either as separate arguments or in a vector for a cell array of any size Example cell 2 3 2x3 cell array See also operator cellfun Function evaluation for each cell of a cell array Syntax A cellfun fun C A cell fun C Description cellfun fun C evaluates function fun for each cell of cell array C Each evaluation must give a scalar result of numeric logical or char acter type results are returned as a non cell array the same size as C First argument is a function reference an inline function or the name of a function as a string With more than two input arguments cellfun calls function fun as feval fun C i other where C i is each cell of C in turn and other stands
290. ntax c hypot a b Description hypot a b gives the square root of the square of a and b or of their absolute value if they are complex The result is always real hypot avoids overflow when the result itself does not overflow Examples hypot 3 4 5 hypot 1 2 3 4j inf 5 5 099 5 3852 5 831 inf See also 180 Sysquake for TEX ij Imaginary unit Syntax i j 1 23e4i 1 23e4j Description i or j are the imaginary unit i e the pure imaginary number whose square is 1 i and j are equivalent Used as a suffix appended without space to a number i or j mark an imaginary number They must follow the fractional part and the exponent if any for single precision numbers they must precede the single suffix To obtain a complex number i you can write either i or 1i or j or 1j The second way is safer because variables i and j are often used as indices and would hide the meaning of the built in functions The expression 11 is always interpreted as an imaginary constant number Imaginary numbers are displayed with i or j depending on the op tion set with the format command Examples i 1j format i 2i 2i 2single 5jsingle 2 5i single See also icdf Inverse cumulative distribution function LME Reference mathematical functions 181 Syntax X X X icdf distribution p icdf distribution p al icdf distribution p al a2 Description icdf distribution p calculates th
291. ntents of a MATLAB MAT file Syntax mat filedecode fd mat filedecode data matfiledecode ignoreErr var var var Description matfiledecode fd reads data from file descriptor fd until the end of the file The data must be the contents of a MATLAB compatible MAT file They are made of 8 bit bytes no text conversion must take place The result is a structure whose fields have the name and the contents of the variables saved in the MAT file Instead of a file descriptor the data can be provided directly as the argument In that case the argument data must be an array which can be read from the actual file with freadjor obtained from a network connection Only arrays are supported scalar matrices arrays of more than two dimensions real or complex numerical logical or char A second input argument can be used to specify how to handle data of unsup ported types with false default value unsupported types cause an error with true they are ignored Example fd fopen data mat s matfiledecode fd fclose fd S S x real 1x1024 y real 1x1024 See also matfileencode LME Reference shell 381 matfileencode Encode the contents of a MATLAB MAT file Syntax matfileencode fd s matfileencode s Description matfileencode fd s writes the contents of structure s to file de scriptor fd as a MATLAB compatible MAT file Each field of s corre sponds to a separate variable in the MA
292. ntext in which they are created Their main use is as argu ment to other functions They are obtained with operator 48 Sysquake for ATEX Inline and anonymous functions Inline and anonymous functions encapsulate executable code They differ only in the way they are created inline functions are made with function inline while anonymous functions have special syntax and semantics where the values of variables in the current context can be captured implicitly without being listed as argument Their main use is aS argument to other functions Sets Sets are represented with numerical arrays of any type integer real or complex double or single character or logical or lists or cell ar rays of strings Members correspond to an element of the array or list All set related functions accept sets with multiple values which are always reduced to unique values with function unique They imple ment membership test union intersection difference and exclusive or Numerical sets can be mixed the result has the same type as when mixing numerical types in array concatenation Numerical sets and list or cell arrays os strings cannot be mixed Objects Objects are the basis of Object Oriented Programming OOP an ap proach of programming which puts the emphasis on encapsulated data with a known programmatic interface the objects Two OOP languages in common use today are C and Java The exact definition of OOP varies from person to p
293. nts on the same row and semicolons to separate the rows themselves gt 1 2 5 3 ans 12 5 3 Column vectors are matrices with one column and row vectors are matrices with one row You can also use the colon operator to build a row vector by specifying the start and end values and optionally the step value Note that the end value is included only if the range is a multiple of the step Negative steps are allowed There are functions to create special matrices The zeros ones rand and randn functions create matrices full of zeros ones random num bers uniformly distributed between 0 and 1 and random numbers nor mally distributed with a mean of 0 and a standard deviation of 1 re spectively The eye function creates an identity matrix i e a matrix with ones on the main diagonal and zeros elsewhere All of these func tions can take one scalar argument n to create a square n by n matrix or two arguments m and n to create an m by n matrix Tutorial 23 ans 0 1386 0 9274 0 3912 0 8219 gt randn 2 ans 0 2931 1 2931 2 3011 0 9841 gt eye 3 ans 10 01 0 0 gt eye ans 0 1 0 0 1 2 3 Ors 0 0 You can use most scalar functions with matrices functions are applied to each element gt sin 1 2 ans 0 8415 0 9093 There are also functions which are specific to matrices For example det calculates the determinant of a square matrix gt det 1 2 5 3 ans 7 Arithmetic
294. nvex solid whose six faces are squares By default edges are not drawn An optional string input argument specifies the edge style With four output arguments cube produces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also dodecahedron plotpoly Libraries polyhedra 569 dodecahedron Create a regular dodecahedron Syntax dodecahedron dodecahedron style X Y Z ind dodecahedron Description Without output argument dodecahedron displays a regular convex dodecahedron i e a convex solid whose twelve faces are regular pen tagons By default edges are not drawn An optional string input argument specifies the edge style With four output arguments dodecahedron produces the X Y Zand ind arrays expected by plotpoly and it does not display anything See also cube octahedron greatstellateddodecahedron greatdodecahedron smallstel lateddodecahedron greatdodecahedron Create a great dodecahedron Syntax greatdodecahedron greatdodecahedron style X Y Z ind greatdodecahedron Description Without output argument greatdodecahedron displays a great do decahedron i e a regular nonconvex solid whose twelve faces are regular pentagons By default edges are not drawn An optional string input argument specifies the edge style With four output arguments greatdodecahedron produces the X Y Z and ind arrays expected by plotpoly and it
295. o while Loop controlled by a boolean expression Syntax while expr sl end Description The statements between the while statement and the corresponding end are executed repeatedly as long as the expression of the while statement yields true nonempty and all elements different from 0O and false If a break statement is executed in the scope of the while loop i e not in an enclosed loop the loop is terminated 94 Sysquake for TEX If a continue statement is executed in the scope of the while loop statements following continue are ignored and a new loop is per formed if the while statement yields true Example e l i 2 while true forever eNew 1 1 i i if abs e eNew lt 0 001 break end eNew 2 i 2 717 See also 6 12 Miscellaneous Functions This section describes functions related to programming function ar guments error processing evaluation memory assert Check that an assertion is true Syntax assert assert assert assert expr expr str expr format argl arg2 expr identifier format argl arg2 nannan Description assert expr checks that expr is true and throws an error otherwise Expression expr is considered to be true if it is a non empty array whose elements are all non zero LME Reference miscellaneous functions 95 With more input arguments assert checks that expr is true and throws the error specified by remaining argum
296. object is an object of class class directly or by inheritance Example isa pi double true See also Class isobject methods isobject Test for an object Syntax b isobject a Description object a returns true if a is an object created with class See also methods List of methods for a class 342 Sysquake for TEX Syntax methods classname list methods classname Description methods classname displays the list of methods defined for class classname Inherited methods and private methods are ignored With an output argument methods gives produces a list of strings See also 6 26 Logical Functions all Check whether all the elements are true Syntax all A all A dim all v v v b Description all A performs a logical AND on the elements of the columns of array A or the elements of a vector If a second argument dim is provided the operation is performed along that dimension all can be omitted if its result is used by if or while because these statements consider an array to be true if all its elements are nonzero Examples all 1 2 3 2 false all 1 2 3 gt 0 true LME Reference logical functions 343 See also any Check whether any element is true Syntax v any A any A dim any v v b Description any A performs a logical OR on the elements of the columns of array A or the elements of a vector If a second argu
297. ode45 filterfun 0 5 0 opt plot t y Function filterfun is integrated in the normal way until t 1 inclu sive with u 0 This is why the conditional expression in filterfun is less than or equal to and not less than Then the event occurs and integration continues from t 1 with u 0 Non integrated state For the last example we will consider a system made of an integrator and a relay with hysteresis in a loop Let y 1 be the output of the integrator and y 2 the output of the relay Only y 1 is integrated 304 Sysquake for ATEX Relay with hysteresis 1 An event occurs when the integrator is larger or smaller than the hys teresis value y 1 y 2 isTerminal false direction sign y 2 When the event occurs a new value is computed for the 2nd state yn y 2 To integrate this the following functions are defined function yp relayfun t y yp y 2 function v te d relayevents t y v y 1 y 2 te false d sign y 2 function yn relayonevent t y i yn y 2 The initial state is 0 1 O for the integrator and 1 for the output of the relay State is integrated during 5 s see Fig 6 4 with t y ode45 relayfun 0 5 0 1 odeset Events relayevents plot t y OnEvent relayonevent LME Reference non linear numerical functions 305 See also optimset Options for minimization and zero finding Syn
298. odecahedron smallstel Lateddodecahedron plotpoly icosahedron Create a regular icosahedron Syntax icosahedron icosahedron style X Y Z ind icosahedron Description Without output argument icosahedron displays a regular convex icosahedron i e a convex solid whose twenty faces are equilateral triangles By default edges are not drawn An optional string input argument specifies the edge style With four output arguments icosahedron produces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also dodecahedron plotpoly octahedron Create a regular octahedron Syntax octahedron octahedron style X Y Z ind octahedron 572 Sysquake for ATEX Description Without output argument octahedron displays a regular octahedron i e a convex solid whose eight faces are equilateral triangles By de fault edges are not drawn An optional string input argument specifies the edge style With four output arguments octahedron produces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also tetrahedron dodecahedron icosahedron plotpoly smallstellateddodecahedron Create a small stellated dodecahedron Syntax smallstellateddodecahedron smallstellateddodecahedron style X Y Z ind smallstellateddodecahedron Description Without output argument smallstellateddodecahedron displays a small stellated dodecahed
299. odephase Ad Bd Cd Dd Ts dbodephase Ad Bd Cd Dd Ts w dbodephase Ad Bd Cd Dd Ts opt dbodephase Ad Bd Cd Dd Ts w opt dbodephase style dbodephase style id phase w dbodephase Description dbodemag numd dend Ts plots the phase of the frequency response of the discrete time transfer function numd dend with sampling period Ts The range of frequencies is selected automatically or can be spec ified in an optional argument w a vector of frequencies Further options can be provided in a structure opt created with responseset field Range is utilized The optional arguments style and id have their usual meaning dbodephase Ad Bd Cd Dd Ts plots the phase of the frequency response Y jw UGw of the discrete time state space model Ad Bd Cd Dd defined as 2zX Z Y z AaX z BgU z CaX z DgU z where z els With output arguments dbodephase gives the phase and the fre quency as column vectors No display is produced Example dbodephase 1 poly 0 9 0 7 0 6j 0 7 0 6j 1 See also bodephase dbodemag dimpulse Impulse response plot of a discrete time linear system LME Reference graphics for dynamical systems 439 Syntax dimpulse numd dend Ts dimpulse numd dend Ts opt dimpulse Ad Bd Cd Dd Ts dimpulse Ad Bd Cd Dd Ts opt dimpulse style dimpulse style id y t dimpulse Description dimpulse numd dend Ts plots the impulse
300. of points in triangulation simplices Syntax ix tsearchn x t xi Description tsearchn x t xi searches in which simplex each point given by the rows of array xi is located Rows of array x represent the vertices of the simplices and rows of array t represent their indices in x array t is usually the result of deLaunayn Dimensions must match in a space of n dimensions x and xi have n columns and t has n 1 columns The result is a column vector with one element for each row of xi which is the row index in t of the first simplex which contains the point or NaN if the point is outside all simplices i e outside the convex hull of points x if t is a proper triangulation of x such as the one computed with delaunayn Example Search for simplices containing points 0 0 and 0 1 corresponding to Delauny triangulation of 20 random points x randn 20 2 t delaunayn x xi 0 0 0 1 ix tsearchn x t xi 286 Sysquake for ATEX See also delaunayn voronoin voronoi 2 d Voronoi tessalation Syntax v p voronoi x y Description voronoi x y calculates the Voronoi tessalation of the set of 2 d points given by arrays x and y Both arrays must have the same number of values m The first output argument v is an array of two columns which contains the coordinates of the vertices of the Voronoi cells one row per vertex The first row contains infinity and is used as a marker for unbounded Voro
301. oint number Syntax x double r Description double r converts ratio r to a floating point number of class double Example r ratio 2 3 double r 0 6666 See also ratio ratio 7 5 bitfield Library bitfield implements the constructor and methods of class bitfield for bit fields binary numbers Basic arithmetic operators and functions are overloaded to support expressions with the same syntax as for numbers and matrices Contrary to integer numbers bitfield objects have a length between 1 and 32 and are displayed in binary The following statement makes available functions defined in bitfield use bitfield bitfield beginning First bit position in a bitfield Syntax a beginning 510 Sysquake for TEX Description In the index expression of a bitfield beginning is the position of the least significant bit i e 0 See also bitfield bitfield bitfield end bitfield bitfield Bitfield object constructor Syntax bitfield bitfield n bitfield n wordlength a a a Description bitfield n wordlength creates a bitfield object initialized with the wordlength least significant bits of the nonnegative integer number n The default value of wordlength is 32 if n is a double an int32 or a uint32 number 16 is n is an int16 or uint16 number or 8 if nis an int8 or uint8 number Without argument bitfield gives a bit field of 32 bits 0 Like any integer number in LME n may
302. olynomial are the eigenvalues of M With a vector argument poly r returns the polynomial whose roots are the elements of the vector r The first coefficient of the polynomial is 1 If the complex roots form conjugate pairs the result is real Examples poly 1 2 3 4 1 5 2 roots poly 1 2 3 4 5 3723 0 3723 eig 1 2 3 4 0 3723 5 3723 poly 1 3 1 6 11 6 See also polyder Derivative of a polynomial or a polynomial product or ratio Syntax Al polyder A Cl polyder A B N1 D1 polyder N D Description polyder A returns the polynomial which is the derivative of the poly nomial A Both polynomials are given as vectors of their coefficients highest power first The result is a row vector LME Reference linear algebra 237 With a single output argument polyder A B returns the derivative of the product of polynomials A and B It is equivalent to polyder conv A B With two output arguments N1 D1 polyder N D returns the derivative of the polynomial ratio N D as N1 D1 Input and output ar guments are polynomial coefficients Examples Derivative of x 2x2 5x 2 polyder 1 2 5 2 345 Derivative of x 2x2 5x 2 2x 3 N D polyder 1 2 5 2 2 31 N 4 13 12 11 D 4 12 9 See also polyint Integral of a polynomial Syntax pol2 polyint pol1 pol2 polyint poll c Description polyint pol1 returns the polynomial which is t
303. on would probably fail mfunc X X 1 lt 1 X 1 exp X 1 X 2 2 X 1 X 2 O 1 X 1 2 inf fminsearch mfunc 1 2 1 0 7071 See also fzero Zero of a function Syntax x fzero fun x0 x fzero fun xlow xhigh x fzero options x fzero options Description fzero fun finds numerically a zero of function fun fun is either specified by its name or given as an anonymous or inline function or a function reference It has at least one input argument x and it returns one output argument also a real number fzero finds the value x such that fun x 0 up to some tolerance Second argument tells where to search it can be either a starting point or a pair of values xLow and xhigh which must bracket the zero such that fun xlow and fun xhigh have opposite sign The optional third argument may contain options It is either the empty array for the default options or the result of optimset Additional input arguments of fzero are given as additional input arguments to the function specified by fun They permit to parame terize the function LME Reference non linear numerical functions 295 Examples Zero of a sine near 3 displayed with 15 digits fprintf 15g n fzero sin 3 3 141592653589793 To find the solution of e c x between 0 and 100 with c 10 a function f whose zero gives the desired solution is written function y f x c y exp
304. ond to red green and blue components as real numbers between 0 to 1 maximum inten sity The default value of n is 256 See also black2orangecm black2red2whitecm blue2greencm blue2yellow2redcm cyan2magentacm graycm green2yellowcm interprgbc tecm red2yellowcm Colormap with shades from black to orange Syntax cm red2yellowcm cm red2yellowcm n Description red2yellowcm n creates a color map with n entries corresponding to color shades from red to yellow The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 Libraries colormaps 567 See also colormap black2orangecm black2red2whitecm joblue2greencm blue2yellow2redcm cyan2magentacm green2yellowcm magenta2yellowcm sepiacm Colormap with shades of sepia Syntax cm sepiacm cm sepiacm n Description sepiacm n creates a color map with n entries corresponding to shades of sepia The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also black2orangecm black2red2whitecm blue2greencm blue2yellow2redcm igraycm green2yellowcm magenta2yellowcm whitecm Colormap with shades from black to orange Syntax cm whitecm cm whitecm n
305. one of them is a scalar then it is replicated as needed Lightangle sets multiple light sources See also material LME Reference 3D graphics 427 lines Plot straight lines in 3D space Syntax Line3 A b line3 A b style line3 A b style id Description line3 displays one or several straight line s in the 3D space Each line is defined by two equations of the form a x a2y a3z b The first argument of line3 is a matrix which contains the coefficients a in the first column az in the second column and a3 in the third column two rows define a different line The second argument is a column vector which contains the coefficients b If one of these arguments has two rows and the other has several pairs the same rows are reused multiple times The optional third and fourth arguments are the as for all graphical commands Example Vertical line at x 5 y 6 line3 1 0 0 1 5 6 See also material Surface reflexion properties Syntax material p 428 Sysquake for ATEX Description material p sets the reflexion properties of the Blinn Phong model of following surfaces drawn with surf and plotpoly Argument p is a scalar or a vector of two real values between 0 and 1 The first or only element ka is the weight of ambiant light the second element kd is the weight of diffuse light reflected from all light sources See also lightangle mesh Plot a mesh in 3D space Syntax
306. ons match For instance three arguments of size 3x1x2 1x5 and 1x1 are replicated into arrays of size 3x5x2 Output arguments are assigned to Y1 Y2 etc Function fun is specified by either by its name as a string a function reference or an inline function Example fevalx plus 1 5 10 10 30 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 See also fun2str Name of a function given by reference or source code of an inline function Syntax fun2str funref fun2str inlinefun str str Description fun2str funref gives the name of the function whose reference is funref fun2str inlinefun gives the source code of the inline function inlinefun Examples fun2str sin sin fun2str inline x 2 y x y function y f x y y x 2 y LME Reference miscellaneous functions 103 See also info Information about LME Syntax info info builtin info functions info methods info variables info global info persistent info libraries info usedlibraries info threads str info list info kind Description info displays the language version With an output argument the language version is given as a string info builtin displays the list of built in functions with their mod ule name modules are subsets of built in functions A letter u is displayed after each untrusted function functions which cannot be ex ecuted in the sandbox With an output argument info built
307. operations can also be applied to matrices with their usual mathematical behavior Additions and subtractions are performed on each element The multiplication symbol x is used for the product of two matrices or a scalar and a matrix gt 1 2 3 4 2 7 ans 16 34 24 Sysquake for ATEX The division symbol denotes the multiplication by the inverse of the right argument which must be a square matrix To multiply by the inverse of the left argument use the symbol This is handy to solve a set of linear equations For example to find the values of x and y such that x 2y 2 and 3x 4y 7 type gt 1 2 3 4 277 ans 3 0 5 Hence x 3 and y 0 5 Another way to solve this problem is to use the inv function which return the inverse of its argument It is sometimes useful to multiply or divide matrices element wise The and operators do exactly that Note that the and operators do not need special dot versions because they perform element wise anyway gt 1 2 3 4 x 2 1 5 3 ans 12 7 26 15 gt 1 2 3 4 2 1 5 3 ans 2 2 15 12 Some functions change the order of elements The transpose operator tick reverses the columns and the rows gt 1 2 3 4 5 6 ans 135 246 When applied to complex matrices the complex conjugate transpose is obtained Use dot tick if you just want to reverse the rows and columns The flipud function flips a matrix upside down and fliptr
308. or 416 Sysquake for TEX instance you can plot data in radians the standard angle unit in LME and display ticks and labels in degrees by using a factor of 180 pi With an output argument scalefactor gives the current factors as a 2 elements vector Example Display the sine with a scale in degrees phi 0 0 01 2 pi plot phi sin phi scalefactor 180 pi 1 See also text Display text in a figure Syntax text x y string text x y string justification text font Description With three arguments text x y string displays a string centered at the specified position An optional fourth argument specifies how the string should be aligned with respect to the position x y Itisa string of one or two characters from the following set Char Alignment c Center may be omitted Left r Right t Top b Bottom For instance l means that the string is displayed to the right of the given position and is centered vertically and rt that the string is to the bottom left of the given position An optional trailing argument specifies the font size type face and color to use It is a structure which is typically created with fontset LME Reference 3D graphics 417 Examples A line is drawn between 1 1 and 1 1 with labels at both ends plot 1 1 1 1 text 1 1 pl tr text 1 1 p2 bl Text with font specification font fontset Fo
309. or each line Automatic scaling is performed the same way as for cartesian plots after polar coordinates have been converted The figure axis ticks and grids are specific to polar plots Polar plots can be mixed with other graphical commands based on cartesian coordinates such as plot line and circle Example theta 0 0 01 20 pi rho exp 0 1 x theta sin 5 x theta polar theta rho See also plot quiver Quiver plot Syntax quiver x y u v quiver u v quiver scale quiver style 412 Sysquake for ATEX Description quiver x y u V displays vectors u v starting at x y If the four arguments are matrices of the same size an arrow is drawn for each corresponding element If x and y are vectors they are repeated x is transposed to a row vector if necessary and repeated to match the number of rows of u and v and y is transposed to a column vector if necessary and repeated to match their number of columns The absolute size of arrows is scaled with the average step of the grid given by x and y so that they do not overlap if the grid is uniform If x and y are missing their default values are 1 2 m and 1 2 n respectively where m and n are the number of rows and columns of u and v With a 5th or 3rd argument quiver scale multiplies the arrow lengths by the scalar number scale If scale is zero arrows are not scaled at all u and v give directly the absolute value of the
310. or cell arrays of character strings Multiple elements of input arguments are considered as single mem bers the result is always sorted and has unique elements The optional second and third output arguments are vectors of indices such that if c ia ib setxor a b then c is the union of a ia and b ib LME Reference arrays 273 Example a a bc bbb de b z bc aa bbb c ia ib setxor a b C a aa de z ia 14 ib 31 union a ia b ib a aa de z Set exclusive or can also be computed as the union of a and b minus the intersection of a and b setdiff union a b intersect a b a aa de z See also size Size of an array Syntax v size A m n size A m size A i Description Size A returns the number of rows and the number of elements along each dimension of array A either in a row vector or as scalars if there are two output arguments or more size A i gives the number of elements in array A along dimen sion i size A 1 gives the number of rows and size A 2 the num ber of columns Examples M ones 3 5 size M 274 Sysquake for TEX w atu wi own Il size M 3 3 size M 1 3 size M 2 5 See also sort Array sort Syntax A_sorted ix sort A A_sorted ix sort A dim A_sorted ix sort A dir A_sorte
311. or dimensions in dims dimensions of cells are the same as dimensions of A for dimen sions in dims and 1 for dimensions not in dims Argument A can be a numerical array of any dimension and class a logical array or a char array Examples num2cell 1 2 3 4 1 2 3 4 num2cell 1 2 3 4 1 1 3 2 4 num2cell 1 2 3 4 2 1 2 3 4 See also numel Number of elements of an array Syntax n numel A Description numel A gives the number of elements of array A It is equivalent to prod size A Examples numel 1 5 5 numel ones 2 3 6 numel 1 1 6 abc 4 266 Sysquake for TEX numel 2 vwxyz 2 See also ones Array of ones Syntax A ones n A ones nl1 n2 A ones nl n2 A ones type Description ones builds an array whose elements are 1 The size of the array is specified by one integer for a square matrix or several integers either as separate arguments or in a vector for an array of any size An additional input argument can be used to specify the type of the result It must be the string double single int8 int16 int32 int64 uint8 uint16 uint32 or uint64 64 bit arrays are not supported on all platforms Example ones 2 3 111 111 ones 2 int32 2x2 int32 array 11 11 See also permute Permutation of the dimensions of an arra
312. or3 Make graphical element sensitivive to 3D interactive displacement Syntax sensor3 type param id sensor3 type param typeAlt paramAlt id Description sensor3 type param id specifies how a 3D element can be dragged interactively Contrary to 2D graphics where the mapping between the mouse cursor and the graphical coordinates depends on two separate scaling factors manipulation in 3D space must use a surface as an additional constraint sensor3 specifies this surface for a graphical object whose ID is the same as argument id The constraint surface is specified with string type and numeric array param It always contains the selected point For instance if the user clicks the second point of plot3 1 2 5 3 2 4 1 and sensor3 defines a horizontal plane the move lies in horizontal plane z 4 In addition to position _p1 parameters specific to the constraint surface are provided in special variable _g a vector of two elements type plane The constraint surface is the plane defined by the selected point _p0 and two vectors vxl vyl vz1 and vx2 vy2 vz2 given in argument param vxl vyl vzl vx2 vy2 vz2 During the drag q contains the coefficients of these two vectors such that p1 _p0 _q param 432 Sysquake for ATEX type sphere The constraint surface is a sphere whose center is defined by a point param px py pz Its R is such that the surface contains the selected point _p0 Duri
313. orm suitable for other applications such as source code or HTML files Functions specific to files are described in this sec tion Input output and control are done with the following generic functions Function Description fclose close the file feof check end of file status fflush flush I O buffers fgetl read a line fgets read a line fprintf write formatted data fread read data fscanf read formatted data fseek change the current I O position ftell get the current I O position fwrite write data redirect redirect output fopen Open a file Syntax fd fd fopen path fopen path mode Description fopen opens a file for reading and or writing The first argument is a path whose format depends on the platform If it is a plain file name the file is located in the current directory what Current means also 378 Sysquake for TEX depends on the operating system The output argument a real num ber is a file descriptor which can be used by many input output func tions such as fread fprintf or dumpvar The optional second input argument a string of one or two charac ters specifies the mode It can take one of the following values Mode Meaning none sameas r r read only binary mode seek to beginning w read write binary mode create new file a read write binary mode seek to end rt read only text mode seek to beginning wt read write text mode create new file at
314. outputs with the syntax var expr or var expr is much more efficient than specifying both subscripts or a single index The syntax for field access var field is defined for the following fields for state space models A B C and D matrices of the state space model for transfer functions num and den cell arrays of co efficients for both var string and Ts scalar or empty array for continuous time systems The syntax with braces var i is not supported See also subsasgn operator subsasgn Iti tfdata Get transfer functions Syntax num den tfdata a num den Ts ssdata a Description tfdata a where a is any kind of LTI object gives the numerator and denominator of the transfer function model and optionally the sampling period or the empty array for continuous time systems The numerators and denominators are given as a cell array of power descending coefficient vectors the rows of the cell arrays correcpond to the outputs and their columns to the inputs See also zpk LTI transfer function constructor using zeros and poles 544 Sysquake for ATEX Syntax a zpk z p k a zpk z zeroslist poleslist gainlist a zpk Ts a zpk TS var a zpk b a zpk b Description zpk creates transfer function LTI systems like tf Instead of using transfer function coefficients as input it accepts a vector of zeros a vector of poles and a gain for a simple
315. over shoots some value In the general case ni states are integrated and n ni states are kept constant between events The total number of states n is given by the length of the initial state vector y0 and the number of integrated states ni is given by the size of the output of the integrated function Function OnEvent can produce a vector of size n to replace all the states of size n ni to replace the non integrated states or empty to replace no state this can be used to display re sults or to store them in a file for instance Event times are computed after an integration step has been ac cepted If an event occurs before the end of the integration step the step is shortened event information is stored in the output arguments of ode te ie and ye and the OnEvent function is called The output arguments t and y of ode contain two rows with the same time and the state right before the event and right after it The time step used for integration is not modified by events Additional arguments PreArg is a list of additional input arguments for all functions called during integration they are placed before normal arguments For ex ample if its value is 1 abc the integrated function is called with fun 1 abc t y the output function as outfun 1 abc tn yn and so on Examples Default options odeset AbsTol 1le 6 LME Reference non linear numerical functions Figure 6 2 Van der Pol equation with
316. ow err end end See also lasterror try error str2fun Function reference Syntax funref str2fun str Description 113 str2fun funref gives a function reference to the function whose name is given in string str It has the same effect as operator which is preferred when the function name is fixed Examples str2fun sin sin sin sin a cos str2fun a cos See also str2obj Convert to an object its string representation Syntax obj str2obj str 114 Sysquake for ATEX Description str2obj str evaluates string str and gives its result It has the in verse effect as dumpvar with one argument It differs from eval by restricting the syntax it accepts to literal values and to the basic con structs for creating complex numbers arrays lists structures objects and other built in types Examples str2obj 1 2j 1 2j str dumpvar 1 abc 1 100 str 1 abc 1 100 str2obj str 1 abc real 1x100 eval str 1 abc real 1x100 str2obj sin 2 Bad argument str2obj eval sin 2 0 9093 See also varargin Remaining input arguments Syntax function fun varargin l varargin Description varargin is a special variable which may be used to collect input ar guments In the function declaration it must be used as the last or unique input argument When the function is ca
317. owing operators and functions may be used with polynom arguments with results analog to the corresponding functions of LME plus rem minus mpower mtimes uminus mrdivide uplus mldivide roots Examples p polynom 3 0 1 4 2 3x 4 x 2 4x 2 q 3 p 2 8 27X 8 18x 6 72x 5 39x 4 24x 3 60x 2 48x 20 See also polynom disp polynom diff jratfun ratfun polynom double polynom int polynom subst polynom feval polynom inline polynom disp Display a polynom object Syntax disp a 500 Sysquake for TEX Description disp a displays polynomial a It is also executed implicitly when LME displays the polynom result of an expression which does not end with a semicolon Example p polynom 3 0 1 4 2 p 3x 4 x 2 4x 2 See also polynom polynom polynom double Convert a polynom object to a vector of coefficients Syntax coef double a Description double a converts polynomial a to a row vector of descending power coefficients Example p polynom 3 0 1 4 2 double p 301 42 See also polynom polynom polynom subst Substitute the variable of a polynom object with another polynomial Libraries classes 501 Syntax subst a b Description subst a b substitutes the variable of polynom a with polynom b Example p polynom 1 2 3 x 24 3Xx 9 q polynom 2 0 2x r subst p q r 4x
318. p 1 1 2 3 4 label t psi Legend psi t int_0 infty g tau s t tau rm d tau end sysquake caption Step response end center end figure Here is an example of a plot in a document with class beamer Note the use of the option fragile required by the frame environment documentclass beamer usepackage lLatexingraphics sysquake usepackage graphicx usepackage epstopdf begin document begin frame fragile frametitle Graphics begin sysquake 280 200 fplot x x 0 3 2 7 2 exp 3 x 2 2 3 r end sysquake end frame end document Chapter 6 LME Reference This chapter describes LME Lightweight Matrix Engine the inter preter for numerical computing used by Sysquake 6 1 Program format Statements An LME program or a code fragment typed at a command line is com posed of statements A statement can be either a simple expression a variable assignment or a programming construct Statements are separated by commas semicolons or end of lines The end of line has the same meaning as a comma unless the line ends with a semicolon When simple expressions and assignments are followed by a comma or an end of line the result is displayed to the standard output when they are followed by a semicolon no output is produced What follows programming constructs does not matter When typed at the command line the result of simple expre
319. p p sets the up vector to p p is a 3D vector camup auto sets the up vector to 0 0 1 camup manual does nothing With an output argument camup gives the current up vector See also cando tty camorbit amVa CamZoom O camva View angle Syntax camva va va camva Description camva va sets the view angle to va which is expressed in degrees The projection mode is set to perspective The scale is adjusted so that the graphics have about the same size With an output argument camva gives the view angle in degrees which is O for an orthographic projection See also camdolly camorbit campan campos camproj camro camtarget Camup Camzoom camzoom Zoom in or out Syntax camzoom f LME Reference 3D graphics 425 Description camzoom f scales the projection by a factor f The image grows if f is larger than one and shrinks if it is smaller See also contour3 Level curves in 3D space Syntax contour3 z contour3 z xmin xmax ymin ymax contour3 z xmin xmax ymin ymax levels contour3 z xmin xmax ymin ymax levels style Description contour3 z plots in 3D space seven contour lines corresponding to the surface whose samples at equidistant points 1 size z 2 in the x direction and 1 size z 1 on the y direction are given by z Contour lines are at equidistant levels With a second non empty argument xmin xmax ymin ymax the samples are at eq
320. pattern 390 daspect datestr datevec 557 dbodemag 436 dbodephase deal 97 deblank 309 deconv 212 define 80 delaunay 281 n det 2 diag 2 diff 214 diln 166 dimpulse 438 dinitial 439 dir 383 disp 362 dlsim ae dlyap 2 dna ST dnichols 441 dnyquist 443 dodecahedron 569 dos 384 dot 215 double 167 Hum dpcmdeco 549 dumpvar 98 eig ellip ellipam 582 ellipap ellipe 168 elseif end endfunction 82 eq 122 erf erfc 172 erfinv 1 3 erlocus 446 error eval 100 exist 100 factor 4 7 factorial 175 false 349 fclose 363 feof 363 feval 101 fevalx 101 ff Lush 364 fft 217 fft2 218 fftn 218 fftshift 478 fgetl fgets fieldnames 335 filled shape 391 filter filter2 478 find 252 findstr 309 fix 175 flipdinm fliplr Sysquake for ATEX flipud floor fminbnd 291 fminsearch 292 fontset 400 aa eed for format 365 fplot fprintf 366 fread fscanf B68 fseek B69 ftell 370 fun2str 102 function inline 48 reference function 82 funm 220 fwrite 370 fzero 294 gamma gammainc 177 gammaln gcd ge geomean greatdodecahedron 569 greaticosahedron 5 0 greatstellateddodecahedron 570 green2yellowcm 564 grid 393 griddata griddatan gt 122 hankel 478 Index harmmea
321. perator for objects The colon character is also used to separate the alternatives of a conditional expression b x y Example 2 D u 2 WW w D ul w TNMNASENWWNUNY Ro o ul m D N w N 0 1j 10j 1j 2j 3j 4j 5j 6j 7j 8j 9j 10j 1 1 1j 5 4j 1 2 1j 3 2j 4 3j 5 4j 0 1 1j 5 O 1 1j 2 2j 3 3j 4 4j 5 5j See also Operator Function reference or anonymous function Syntax fun arguments expression Description fun gives a reference to function fun which can be used wherever an inline function can Its main use is as the argument of functions like feval or quad but it may also be stored in lists cell arrays or LME Reference operators 151 structures A reference cannot be cast to a double unlike characters or logical values nor can it be stored in an array Anonymous functions are an alternative more compact syntax for inline functions In args expr args is a list of input arguments and expr is an expression which contains two kinds of variables input arguments provided when the anonymous expression is executed captured variables all variables which do not appear in the list of input arguments which have the value of variables of the same name existing when and where the anonymous function is created These values are fixed Anonymous functions are a convenient way to provide the glue be tween functions like fzero and ode45 and the function they accept as argu
322. ples sqrt 4 2 sqrt 1 4 9 3 4j 1 2 3j 2 1j LME Reference mathematical functions 201 See also swapbytes Conversion between big endian and little endian representation Syntax Y swapbytes X Description swapbytes X swaps the bytes representing number X If X is an array each number is swapped separately The imaginary part if any is discarded X can be of any numerical type swapbytes is its own inverse for real numbers Example swapbytes luint32 16777216uint32 See also typecast cast tan Tangent Syntax y tan x Description tan x gives the tangent of x which is complex if x is complex Example tan 2 2 185 202 Sysquake for TEX See also atan tanh tanh Hyperbolic tangent Syntax y tanh x Description tanh x gives the hyperbolic tangent of x which is complex if x is complex Example tanh 2 0 964 See also atanh tan typecast Type conversion with same binary representation Syntax Y typecast X type Description typecast X type changes the numeric array X to the type given by string type which can be double single int8 or any other signed or unsigned integer type char or logical The binary representation in memory is preserved The imaginary part if any is discarded Depending on the conversion the number of elements is changed so that the array size in bytes in preserved The
323. ponse of the continuous time state space model Ac Bc Cc Dc defined as t Acx t B u t y t C x t D u t 452 Sysquake for TEX where u is a Dirac impulse The state space model must have a scalar input and may have a scalar or vector output With output arguments impulse gives the output and the time as column vectors No display is produced Example impulse 1 1 4 m See also initial Time response plot for a continuous time state space model with initial conditions Syntax initial Ac Bc Cc Dc x0 initial Ac Bc Cc Dc x0 opt initial style initial style id y t initial Description initial Ac Bc Cc Dc x plots the output s of the continuous time state space model Ac Bc Cc Dc with null input and initial state x0 The model is defined as dx a Acx t B u t y t C x t Dc u t where u t is null The state space model may have a scalar or vector output The simulation time range can be provided in a structure opt cre ated with responseset It is a vector of two elements the start time and the end time Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time The optional arguments style and id have their usual meaning With output arguments initial gives the output and the time as column vectors No display is produced LME Reference graphics for dynamical systems 453 initi
324. pose a a b plus a b Operator which do not have a corresponding function are amp amp and because unlike functions they do not always evaluate all of their operands Operator Parenthesis Syntax expr v v index v index1 index2 v index v index v select v selectl select2 v Description A pair of parenthesis can be used to change the order of evaluation The subexpression it encloses is evaluated as a whole and used as if it was a single object Parenthesis serve also to indicate a list of input or output parameters see the description of the function keyword The last use of parenthesis is for specifying some elements of an array or list variable Arrays In LME any numerical object is considered as an array of two dimensions or more Therefore at least two indices are required LME Reference operators 123 to specify a single element the first index specifies the row the sec ond the column and so on In some circumstances however it is sometimes convenient to consider an array as a vector be it a column vector a row vector or even a matrix whose elements are indexed row wise For this way of handling arrays a single index is specified The first valid value of an index is always 1 The array whose ele ments are extracted is usually a variable but can be any expression an expression like 1 2 3 4 1 2 is valid and gives the 2nd element of the first row i e 3
325. ppercase E same as g with as few characters as possible same as k with an uppercase E character string NDORRXTOQMOomM AO lt x HA L The flag forces octal numbers to begin with 0 nonzero hexadec imal numbers to begin with 0x and floating point numbers to always have a decimal point even if they do not have a fractional part Instead of decimal digits the width n and or the precision d can be replaced with character then one or two additional arguments or elements of an array are consumed and used as the width or preci sion Examples sprintf d 2f 2e 2E 2g pixones 1 5 3 3 14 3 14e0 3 14E0 3 14 sprintf 1k 0 001 0 11 111 1000 le 3 0 11 111 1e3 sprintf 8 3f 8 65 8 6s pi abcdefgh abcdefgh 3 142 abcdefxabcdef sprintf c_ a z a_b_c_d_e_f_g_h_i_j_k_l_m_n_o_p_q_r_s_t_u_v_w_x_y_z sprintf f 15 7 pi 3 1415927 Caveat Exotic formats unsupported See also fprintf sscanf 374 Sysquake for TEX sread Raw input from a string or an array of bytes Syntax a count sread str size precision a count sread str precision a count sread bytes Description sread str reads data from string str or array of class uint8 or int8 the same way as fread reads data from a file Examples data count sread abc data 97 98 99 count 3 data count sread abcdef 2 2 data
326. pt for commands 50 Sysquake for TEX You can use these file descriptors without calling any opening func tion first and you cannot close them For instance to display the value of m you can use fprintf fprintf 1 pi 6f n pi pi 3 141593 Some functions use implicitly one of these file descriptors For in stance disp displays a value to file descriptor 1 and warning displays a warning message to file descriptor 2 File descriptors for files and devices are obtained with specific func tions For instance fopen is used for reading from or writing to a file These functions have as input arguments values which specify what to open and how file name host name on a network input or output mode etc and as output argument a file descriptor Such file de scriptors are valid until a call to fclose which closes the file or the connection 6 6 Error Messages When an error occurs the execution is interrupted and an error mes sage explaining what happened is displayed unless the code is en closed in a try catch block The whole error message can look like gt factor 2 Wrong type stdlib primes 164 ones gt stdlib factor 174 The first line contains an error message the location in the source code where the error occurred and the name of the function or oper ator involved Here stdlib is the library name primes is the function name and 164 is the line number in the file which contains the library
327. quake Several options can be specified in a comma separated list in square brackets for example usepackage lLatexingraphics noseparatelogfile sysquake Here is a list of all supported options latexingraphics By default text in graphics is embedded in EPS files without further processing In addition to axis scale labels graphical commands which display explicit text are Label Legend text and title With option Latexingraphics all the text which appears in graph ics in handled by LaTeX This has two benefits first text font and size mix much better with the document text for a higher overall quality Second LaTeX constructs can appear anywhere in graph ics with greek letters math symbols and equations A small draw back is that some characters such as and have a special meaning for LaTeX which make the code incompatible with other versions of Sysquake 34 Sysquake for TEX noexec Each time the document is typeset Sysquake code frag ments which have been extracted during the previous iteration are evaluated by a platform dependant program sysquakelatextool This program produces output data fragments which are inserted in the document and EPS files which contain graphics With option noexec sysquakelatextool is not executed The typesetting result will still be correct if all its output files lmeout and eps files already exist and code fragments and the size of all text strings displayed in graphics if
328. r Stats false statistics display Time steps and output Several options control how the time step is tuned during the numeri cal integration Error is calculated separately on each element of y if NormControl is false or on norm y if it is true time steps are chosen so that it remains under AbsTol or RelTol times the state whichever is larger If this cannot be achieved for instance if the system is stiff and requires an integration step smaller than MinStep integration is aborted Refine specifies how many points are added to the result for each integration step When it is larger than 1 additional points are interpolated which is much faster than reducing MaxStep The output function OutputFcn if defined is called after each step It is a function name in a string a function reference or an anonymous or inline function which can be defined as function stop outfun tn yn LME Reference non linear numerical functions 299 where tn is the time of the new samples yn their values and stop a logical value which is false to continue integrating or true to stop The number of new samples is given by the value of Refine when multiple values are provided tn is a row vector and yn is a matrix whose columns are the corresponding states The output function can be used for incremental plots for animations or for managing large amounts of output data without storing them in variables Events Events are additional time
329. r TEX Description mod x y gives the modulo of x divided by y i e a number m between O and y such that x q y m with integer q Imaginary parts if they exist are ignored Examples mod 10 7 3 mod 10 7 4 mod 10 7 4 mod 10 7 3 See also rem nan Not a Number Syntax nan NaN nan n nan n1 n2 nan n1 n2 nan type x X XXX X tod i id ouweou Description NaN Not a Number is the result of the primitive floating point func tions or operators called with invalid arguments For example 0 0 inf inf and O inf all result in NaN When used in an expression NaN propagates to the result All comparisons involving NaN result in false except for comparing NaN with any number for inequality which re sults in true Contrary to built in functions usually found in the underlying oper ating system many functions which would result in NaNs give complex numbers when called with arguments in a certain range LME Reference mathematical functions 191 With integer non negative arguments nan creates arrays whose elements are NaN Arguments are interpreted the same way as zeros and ones The last argument of nan can be a string to specify the type of the result double for double precision default or single for single precision Examples nan nan Oxnan nan nan nan false nan nan true log 1 0 3 1416j See also nchoosek Binomial coefficient S
330. r arrays are also accepted by most functions Unless a conversion function is used explicitly numbers are repre sented by double or complex values Most mathematical functions ac cept as input any type of numerical value and convert them to double they return a real or complex value according to their mathematical definition Basic element wise arithmetic and comparison operators accept di rectly integer types element wise means the operators and the functions mod and rem as well as operators with a scalar multiplicand or divisor If their arguments do not have the same type 44 Sysquake for ATEX they are converted to the size of the largest argument size in the fol lowing order double gt uint64 gt int64 gt uint32 gt int32 gt uint16 gt int16 gt uint8 gt int8 Functions which manipulate arrays such as reshape which changes their size or repmat which replicates them preserve their type To convert arrays to numerical char or logical arrays use func tions unary operator char or Logical respectively To convert the numerical types use functions double single or uint8 and similar functions Numbers Double and complex numbers are stored as floating point numbers whose finite accuracy depends on the number magnitude During computations round off errors can accumulate and lead to visible ar tifacts for example 2 sqrt 2 sqrt 2 which is mathematically 0 yields 4 4409e 16
331. rands are matrices with a size different from 1 by 1 their size must be equal the division is performed element wise If one operand is a scalar it is repeated to match the size of the other operand ldivide x y is equivalent to x y It can be used to redefine this operator for objects Example 1 2 3 10 11 12 10 5 5 4 See also Operator Matrix power Syntax x y M k x 7M mpower x y Description x y calculates x to the y power provided that either both operands are scalar the first operand is a square matrix and the second operand is an integer or the first operand is a scalar and the second operand is a square matrix Other cases yield an error mpower x y is equivalent to x y It can be used to redefine this operator for objects 136 Examples 2 3 8 1 2 3 4 2 7 10 15 22 2 1 2 3 4 10 4827 14 1519 21 2278 31 7106 See also Operator Scalar power Syntax M1 M2 x M M x power x y Description Sysquake for ATEX M1 M2 calculates M1 to the M2 power element wise Both arguments must have the same size unless one of them is a scalar power x y is equivalent to x y It can be used to redefine this operator for objects Examples 1 2 3 4 2 1 4 9 16 1 2 3 5 4 3 1 16 27 See also LME Reference operators 137 Operator Complex conjugate transpose Syntax M ctranspose M Description M
332. rdinates given by the corresponding elements of arrays xi and yi in a 2 dimension plane Original data are defined by corresponding elements of ar rays x y and v which must have the same size such that the value at coordinate x i y i is v i The result is an array with the same size as xi and yi where vi j is the value interpolated at xi j yi j All coordinates are real imaginary components are ignored Val ues v and vi can be real or complex The result for coordinates outside the convex hull defined by x and y is NaN griddata is based on Delaunay triangulation The interpolation method used in each triangle is linear by default or can be specified with an additional input argument a string Argument Meaning 0 or nearest nearest value 1 or linear linear See also griddatan Data interpolation in N d space Syntax vi griddatan x v xi vi griddatan x v xi method 284 Sysquake for ATEX Description griddatan x v xi interpolates values at coordinates given by the p rows of p by n array xi in an n dimension space Original data are defined by m by n array x and m by 1 column vector v such that the value at coordinate x i is v i The result is a p by 1 column vector vi where vi j is the value interpolated at xi j Coordinates x and xi are real imaginary components are ignored Values v and vi can be real or complex The result for coordinates outside the convex hull defi
333. re opt created with responseset fields Delay NegFreq and Range are utilized The op tional arguments style and id have their usual meaning With output arguments nichols gives the phase and magnitude of the frequency response and the frequency as column vectors No display is produced In Sysquake when the mouse is over a Nichols diagram in addition to the magnitude and phase which can be retrieved with _y0 and _x0 the frequency is obtained in _q Example Nichols diagram of a third order system see Fig 6 20 458 Sysquake for ATEX scale lindb ngrid nichols 20 poly 1 2 1j 2 1j See also nyquist Nyquist diagram of a continuous time system Syntax nyquist numc denc nyquist numc denc w nyquist numc denc opt nyquist numc denc w opt nyquist style nyquist style id w nyquist re im nyquist re im w nyquist Description The Nyquist diagram of the continuous time transfer function given by polynomials numc and denc is displayed in the complex plane In con tinuous time the Nyquist diagram is the locus of the complex values of the transfer function evaluated at jw where w is real positive other definitions include the real negative values which gives a symmetric diagram with respect to the real axis The range of frequencies is selected automatically or can be speci fied in an optional argument w a vector of frequencies Further options can
334. rectangular in expression UxS V some columns of U or V are multiplied by rows or columns of zeros in S respectively U S V svd M false produces U S and V where these columns or rows are discarded relationship M U S V still holds Size of A Size of U Size ofS Size of V mbyn m lt n mbym m by m n by m mbyn m gt n m by n n by n n by n svd M true produces the same result as svd M With one output argument s svd M returns the vector of sin gular values s diag S The singular values of M can also be computed with s sqrt eig M M but svd is faster and more robust Examples U S V svd 1 2 3 4 U _ 0 4046 0 9145 0 9145 0 4046 S 5 465 0 0 0 366 V 0 576 0 8174 0 8174 0 576 U S V 12 3 4 LME Reference linear algebra 247 svd 1 2 1 2 3 1623 3 4697e 19 See also trace Trace of a matrix Syntax tr trace M Description trace M returns the trace of the matrix M i e the sum of its diagonal elements Example trace 1 2 3 4 5 See also var Variance of a set of values Syntax s2 var A s2 var A p s2 var A p dim 248 Sysquake for ATEX Description var A gives the variance of the columns of array A or of the row vector A The variance is normalized with the number of observations minus 1 or by the number of observations if a second argument is true The dimension along which var proceeds may be specified with a third argume
335. rection eventsfun t y Input arguments are the same as for EventTime Output arguments are column vectors where each element i corresponds to an event stream An event occurs when value i crosses zero in either direction if direction i 0 from negative to nonnegative if direction i gt 0 or from positive to nonpositive if direction i lt 0 The event terminates integration if isterminal i is true The Events function is evaluated for each state obtained by integration 300 Sysquake for TEX intermediate time steps obtained by interpolation when Refine is larger than 1 are not considered When an event occurs the integration time step is reset to the initial value and new events are disabled during the next integration step to avoid shattering MaxStep should be used if events are missed when the result of Events is not monotonous between events When an event occurs function OnEvent is called if it exists It can be defined as function yn onevent t y i where i identifies the event stream positive for events produced by Events or negative for events produced by EventTime and the out put yn is the new value of the state immediately after the event The primary goal of odex functions is to integrate states However there are systems where some states are constant between events and are changed only when an event occurs For instance in a relay with hysteresis the output is constant except when the input
336. rete time systems z transform or delta transform Example A B C D tf2ss 2 5 2 3 8 A 1 5 4 10 B 1 0 C 12 5 D 0 See also 6 28 Input Output Functions bwrite Store data in an array of bytes Syntax bwrite data bwrite data precision S S Description bwrite data stores the contents of the matrix data into an array of class uint8 The second parameter is the precision whose meaning is the same as for fread Its default value is uint8 362 Sysquake for ATEX Examples bwrite 12345 uint32 1l 1x4 uint8 array 57 48 0 0 bwrite 12345 uint32 b 1x4 uint8 array 0 0 48 57 See also clc Clear the text window or panel Syntax clc clc fd Description clc clear console clears the contents of the command line window or panel clc fd clears the contents of the window or panel associated with file descriptor fd disp Simple display on the standard output Syntax disp obj Description disp obj displays the object obj Command format may be used to control how numbers are formatted Example disp hello hello LME Reference input output 363 See also format fprintt fclose Close a file Syntax fclose fd fclose all Description fclose fd closes the file descriptor fd which was obtained with functions such as fopen Then fd should not be used anymore fclose all closes all the open file descriptors f
337. rgument dim is provided the operation is performed along that dimension A can be a double or an integer array For double arrays bitany uses the 32 least significant bits Examples bitany 5 3 7 bitany Quint8 6uint8 3uint8 6uint8 2 2x1 uint8 array 6 7 See also bitcmp Bit complement bitwise NOT Syntax b bitcmp i b bitcmp a n 346 Description Sysquake for ATEX bitcmp i gives the 1 complement bitwise NOT of the integer i bitcmp a n where a is an integer or a double gives the 1 complement of the n least significant bits The result has the same type as a The inputs can be scalar arrays of the same size or a scalar and an array If a is of type double so is the result and the operation is performed on at most 32 bits Examples bitcmp 1 4 14 bitcmp 0 1 8 13 7 15 31 63 127 255 bitcmp Ouint8 luint8 255uint8 1x3 uint8 array 255 254 0 See also bitget Bit extraction Syntax b bitget a n Description bitget a n gives the n th bit of integer a a can be an integer or a double The result has the same type as a n 1 corresponds to the least significant bit The inputs can be scalar arrays of the same size or a scalar and an array If a is of type double so is the result and n is limited to 32 Examples bitget 123 5 1 bitget 7 1 8 11100000 LME Reference logical functions 347 bitget 5uint8 2 Ouint8 See also bitset bitan
338. ron i e a regular nonconvex solid whose twelve faces are regular star pentagons and where each vertex is com mon to five faces By default edges are not drawn An optional string input argument specifies the edge style With four output arguments smallstellateddodecahedron pro duces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also dodecahedron greatdodecahedron greatstel Lateddodecahedron plotpoly tetrahedron Create a regular tetrahedron Libraries solids 573 Syntax tetrahedron tetrahedron style X Y Z ind tetrahedron Description Without output argument tetrahedron displays a regular tetrahe dron i e a solid whose four faces are equilateral triangles By default edges are not drawn An optional string input argument specifies the edge style With four output arguments tetrahedron produces the X Y Z and ind arrays expected by plotpoly and it does not display anything See also dodecahedron plotpoly 7 14 solids Library solids implements functions which create solid shapes in 3D Solids are generated with parametric equations and displayed with surf When called without output argument with an optional trail ing string argument for the edge style the solid is displayed with the current scaling and color map With output arguments arrays X Y Z expected by surf or mesh are produced They can be modified to move scale or stretch the solids
339. rotated In rot90 A1 k the second argument is the number of times the array is rotated 90 degrees counter clockwise With k 2 the array is rotated by 180 degrees with k 3 ork 1 the array is rotated by 90 degrees clockwise Examples rot90 1 2 3 4 5 6 3 6 25 14 rot90 1 2 3 4 5 6 1 rot90 1 2 3 4 5 6 1 654 Ww setdiff Set difference Syntax c setdiff a b c ia setdiff a b Description setdiff a b gives the difference between sets a and b i e the set of members of set a which do not belong to b Sets are any type of nu merical character or logical arrays or lists or cell arrays of character 272 Sysquake for TEX strings Multiple elements of input arguments are considered as single members the result is always sorted and has unique elements The optional second output argument is a vector of indices such that if c ia setdiff a b then cis a ia Example a a bc bbb de b z bc aa bbb c ia setdiff a b C a de ia 14 a ia a de See also setxor Set exclusive or Syntax c setxor a b c ia ib setxor a b Description setxor a b performs an exclusive or operation between sets a and b i e it gives the set of members of sets a and b which are not mem bers of the intersection of a and b Sets are any type of numerical character or logical arrays or lists
340. ructor Syntax r ratio r ratio n r ratio num den r ratio r Description ratio num den creates a rational fraction object whose value is num den Arguments num and den may be double integer numbers or longint Common factors are canceled out With one numeric input argument ratio n creates a rational fraction whose denominator is 1 Without input argument ratio creates a rational number whose value is 0 With one input argument which is already a ratio object ratio returns it without change The following operators and functions may be used with ratio ob jects with results analog to the corresponding functions of LME plus ne minus lt lt x mtimes lt le mrdivide gt gt mldivide gt ge s mpower inv uminus min a b uplus max a b eq 508 Sysquake for ATEX Examples r ratio 2 3 r 2 3 q 5 gt gt r 1 7 3 See also ratio disp ratio double ratio char Display a ratio object Syntax char r Description char r converts ratio r to a character string See also ratio ratio ratio d1sp jchar ratio disp Display a ratio object Syntax disp r Description disp r displays ratio r with the same format as char It is also ex ecuted implicitly when LME displays the ratio result of an expression which does not end with a semicolon See also disp Libraries bitfield 509 ratio double Convert a ratio object to a floating p
341. s for lists when s type is and for structures when s typeis Examples A 1 2 3 4 1 subsref A struct type subs 1 12 See also Operator 6 11 Programming Constructs Programming constructs are the backbone of any LME program Ex cept for the variable assignment all of them use reserved keywords which may not be used to name variables or functions In addition to the constructs described below the following keyword is reserved for future use goto break Terminate loop immediately Syntax break Description When a break statement is executed in the scope of a loop construct while repeat or for the loop is terminated Execution continues at the statement which follows end Only the innermost loop where break is located is terminated The loop must be in the same function as break It is an error to execute break outside any loop 80 Sysquake for TEX See also case Conditional execution of statements depending on a number or a string See also switch catch Error recovery See also try continue Continue loop from beginning Syntax continue Description When a continue statement is executed in the scope of a loop con struct while repeat or for statements following continue are ig nored and a new loop is performed if the loop termination criterion is not fulfilled The loop must be in the same function as
342. s Syntax theta subspace A B Description subspace A B gives the angle between the two subspaces spanned by the columns of A and B Examples Angle between two vectors in R 2 a 3 2 b 1 5 subspace a b 0 7854 488 Sysquake for TEX Angle between the vector 1 1 1 and the plane spanned by 2 5 3 and 7 1 0 in R 3 subspace 1 1 1 2 7 5 1 3 0 0 2226 toeplitz Toeplitz matrix Syntax X X toeplitz c r toeplitz c Description toeplitz c r creates a Toeplitz matrix whose first column contains the elements of vector c and whose first row contains the elements of vector r A Toeplitz matrix is a matrix whose diagonals have the same value In case of conflict the first element of r is ignored With one argument toeplitz gives a symmetric square matrix Example See also trapz Numerical integration with trapezoidal approximation Syntax S S S trapz Y trapz X Y trapz X Y dim Libraries stat 489 Description trapz Y calculates an approximation of the integral of a function given by the samples in Y with unit intervals The trapezoidal approxi mation is used If Y is an array integration is performed along the first non singleton dimension trapz X Y specifies the location of the samples A third argument may be used to specify along which dimension the integration is per formed Example trapz 2 3 5 6 5 trapz 1 2 5 2 3 5 14 5
343. s an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also colormap pluezye graycm green2yelTowcm h red2yellowcm Libraries colormaps 563 blue2yellow2redcm Colormap with shades from blue to yellow and red Syntax cm blue2yellow2redcm cm blue2yellow2redcm n Description blue2yellow2redcm n creates a color map with n entries corresponding to color shades from blue to yellow and red The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0O to 1 maximum intensity The default value of n is 256 See also colormap black2orangecm black2red2whitecm jblue2greencm cyan2magentacm graycm g green2yellowcm huecm interprgbcm magenta2yellowcm red2yel Lowcm cyan2magentacm Colormap with shades from cyan to magenta cyan2magentacm cyan2magentacm n Description cyan2magentacm n creates a color map with n entries corresponding to color shades from cyan to magenta The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum inten sity The default value of n is 256 See also colormap blue2yellow2redcm graycm green2yellowcm huecm interprgbcm 564 Sysquake for TEX graycm Colormap
344. s in SI units date date implements functions for date and time manipulation and conversion to and from strings filter filter implements functions for the design of analog and digital filters lti lti implements constructors and methods for Linear Time Invariant models whcih may represent dynamical systems as continuous time or discrete time state space models or transfer functions With them you can use standard operator 472 Sysquake for ATEX notations such as or array building operators such as A B C D connection functions such as parallel or feedback and much more polyhedra polyhedra implements functions which create solid shapes with polygonal faces in 3D ratio ratio implements constructors and methods for rational numbers based on long integers Standard arithmetic and boolean operators can be used Sigenc sigenc implements functions related to signal encoding to and decoding from a digital representation solids solids implements functions which create solid shapes in 3D Solids are generated with parametric equations and displayed with surf stat stat provides more advanced statistical functions stdlib stdlib is the standard library of general purpose functions for LME Functions span from array creation and manipulation to coordinates transform and basic statistics wav wav implements functions for reading and writing WAV files or encoding and decoding data encoded as WAV in memory wsserver wsser
345. s phi X sin phi Y Yi sin phi X cos phi Y D E E intersect Set intersection Syntax c intersect a b c ia ib intersect a b Description intersect a b gives the intersection of sets a and b i e it gives the set of members of both sets a and b Sets are any type of nu merical character or logical arrays or lists or cell arrays of character strings Multiple elements of input arguments are considered as single members the result is always sorted and has unique elements The optional second and third output arguments are vectors of in dices such that if c ia ib intersect a b then cis a ia as well as b ib 258 Sysquake for ATEX Example a a bc bbb de b z bc aa bbb c ia ib intersect a b C bbb bc ia 3 2 ib 42 a ia bbb bc b ib bbb bc Set exclusive or can also be computed as the union of a and b minus the intersection of a and b setdiff union a b intersect a b a aa de z See also ipermute Inverse permutation of the dimensions of an array Syntax B ipermute A perm Description ipermute A perm returns an array with the same elements as A but where dimensions are permuted according to the vector of dimensions perm It performs the inverse permutation of permute perm must con tain integers from 1 to n dimension perm
346. s string str as an error mes sage and the computation is aborted With more arguments error use the first argument as a format string and displays remaining argu ments accordingly like fprintf In a try block error str throws a user error without displaying anything An error identifier may be added in front of other arguments It is a string made of at least two segments separated by semicolons Each segment has the same syntax as variable or function name i e it begins with a letter or an underscore and it continues with letters digits and underscores The identifier can be retrieved with Lasterr or Lasterror in the catch part of a try catch construct and helps to identify the error For errors thrown by LME built in functions the first segment is always LME Examples error Invalid argument Invalid argument o ground error robot hit The robot is going to hit s o The robot is going to hit ground lasterror message The robot is going to hit ground identifier robot hit See also fprint 100 Sysquake for TEX eval Evaluate the contents of a string as an expression or statements Syntax x eval str_expression eval str_statement Description If eval has output argument s the input argument is evaluated as an expression whose result s is returned Without output arguments the input argument is evaluated as statement s eval can evaluate and assign to e
347. s2tf A B C D converts the model from state space to transfer function If the state space model has multiple outputs num is a matrix whose lines correspond to each output the denominator is the same for all outputs If the state space model has multiple inputs a fifth input argument is required and spec ifies which one to consider For a sampled time model exactly the same function can be used The derivative is replaced by a forward shift and the variable s of the Laplace transform is replaced by the variable z of the z transform But as long as coefficients are concerned the conversion is the same The degree of the denominator is equal to the number of states i e the size of A The degree of the numerator is equal to the number of states if D is not null and one less if D is null Example num den ss2tf 1 1 1 0 num 1 de e Il n 1 See also tf2ss Conversion from transfer function to state space Syntax A B C D tf2ss num den LME Reference input output 361 Description tf2ss num den returns the state space representation of the trans fer function num den which is given as two polynomials The transfer function must be causal i e num must not have more columns than den Systems with several outputs are specified by a num having one row per output the denominator den must be the same for all the outputs tf2ss applies to continuous time systems Laplace transform as well as to disc
348. set Remaining input arguments of fminbnd if any are given as addi tional input arguments to function fun They permit to parameterize the function For example fminbnd fun x0 2 5 calls fun as fun x 2 5 and minimizes its value with respect to x The first output argument of fminbnd is the value of x at optimum The second output argument if it exists is the value of fun x at op timum The third output argument if it exists is set to true if fminbnd has converged to an optimum or to false if it has not in that case other output arguments are set to the best value obtained With one or two output arguments fminbnd throws an error if it does not con verge Examples Minimum of a sine near 2 displayed with 15 digits fprintf 15g n fminbnd sin 2 4 712389014989218 To find the minimum of ce sinx between 1 and 10 with c 0 1 the expression is written as an inline function stored in variable fun 292 Sysquake for ATEX fun inline c exp x sin x x c Then fminbnd is used with the value of c passed as an additional argument xX fminbnd fun 1 10 0 1 x 1 2239 With an anonymous function this becomes wg x cxexp x sin x fminbnd fun 1 10 1 ll I c fun x x Il 2239 Attempt to find the minimum of an unbounded function x y didConverge fminbnd exp 10 x inf y 0 didConverge false See also fminsearch
349. signal processing Here is the list of these commands 2D low level drawing commands LME Reference graphics activeregion colormap area contour bar fplot barh image circle line 389 pcolor plot polar quiver text 2D high level drawing commands bodemag dsigma bodephase dstep dbodemag erlocus dbodephase hgrid dimpulse hstep dinitial impulse dlsim initial dnichols lsim dnyquist ngrid Scaling and labels label plotoption Legend scale 3D contour3 plot3 line3 plotpoly mesh sensor3 3D scaling and lighting camdolly camroll camorbit camtarget campan camup campos camva camproj camzoom nyquist nyquist plotroots rlocus sgrid sigma step zgrid scalefactor title surf daspect Llightangle material 390 Sysquake for ATEX 6 34 Remarks on graphics Most functions which produce the display of graphical data accept two optional arguments one to specify the style of lines and symbols and one to identify the graphical element for interactive manipulation Style The style defines the color the line dash pattern for continuous traces or the shape for discrete points of the data The possible values are given below Note that the color is ignored on some output devices such as black and white printers and the dash pattern is used only on high resolution devices such as printers or EPS output The color code is lowercase for thin lines and uppercase for thicker lines on devices which suppor
350. so accepted by 2D graphical commands are used unchanged Colors specified by a single component value RGB colors or implicit are pro cessed differently whether Lightangle and or material have been executed or not In the first case colors depend directly on the col ors specified or the default value in the second case the Blinn Phong reflection model is used with flat shading In both cases single color values are mapped to colors using the current color map set with colormap Commands which accept a color argument are mesh surf and plotpoly Direct colors If neither Lightangle nor material has been executed colors depend only on the color argument provided with x y and z coordinates If the this argument is missing color is obtained by mapping linearly the z coordinates to the full range of the current color map Blinn Phong reflection model In the Blinn Phong reflexion model the color of a surface depends on the intrinsic object color the surface reflexion properties and the relative positions of the surface the viewer and light sources 6 39 Functions camdolly Move view position and target Syntax camdolly d 420 Sysquake for ATEX Description camdolly d translates the camera by 3x1 or 1x3 vector d moving the target and the view point by the same amount See also ampan camorbit Camva Camzoom pE axisbounds Bounds of display area along each axis Syntax axisbounds bnds bnds
351. solve cross references and update the table of contents and the index So there is nothing new 3 4 Formatted results Our first expression could have been computed in plain LaTeX which gives access to the computation capabilities of TeX So our next exam ple will use an advanced LME function magic which produces a magic square in a matrix Matrices are displayed in mathematical notation Since matrices are too large to fit nicely in a paragraph we put it in an equation Magic square M_3 of order 3 is begin equation M_3 sqexpr magic 3 end equation 3 5 Programs sqexpr is very convenient for simple expressions However it is not suited to more complicated programs for two reasons first only a single expression can be evaluated which gives a single result Sec ond LME syntax may interfer with LaTeX syntax the backslash for in stance is a normal character used for premultiplying with the inverse of a matrix in LME while it is an escape character in LaTeX We can always escape special characters in LaTeX but Sysquake for LaTeX s aim is to make things simpler not overly complicated The answer to sqexpr s limitations is the sysquake environment which can contain any kind of code Let us write another magic square with it Magic square M_4 of order 4 is begin equation M_4 begin sysquake magic 4 end sysquake end equation 16 Sysquake for ATEX The sysquake environment can contain an
352. space model Ad Bd Cd Dd defined by dx ae A x t B u t y t C x t Dcu t and x k 1 Agx k Bgu k y k Cax k Dau k Method matched is not supported for state space models Examples numd dend c2dm 1 1 1 0 1 numd 0 0952 dend 1 0 9048 numd dend c2dm 1 1 1 0 1 foh numd 0 0484 dend 1 0 9048 numd dend c2dm 1 1 1 0 1 tustin numd 0 0476 0 0476 dend 1 0 9048 See also 356 Sysquake for ATEX d2cm Discrete to continuous time conversion Syntax numc denc d2cm numd dend Ts denc d2cm numd dend Ts numc denc d2cm numd dend Ts method denc d2cm numd dend Ts method Description numc denc d2cm numd dend Ts method converts the discrete time transfer function numd dend with sampling period Ts to a continuous time transfer function numc denc The continuous time transfer function is given by two polynomials in s and the discrete time transfer function is given by two polynomials in z all as vectors of coefficients with highest powers first Method is tustinor t Tustin bilinear transformation default The input and output arguments numc denc numd and dend can also be matrices in that case the conversion is applied separately on each row with the same sampling period Ts d2cm Ad Bd Cd Dd Ts method performs the conversion from discrete time state space model Ad Bd Cd Dd to continuous time state space
353. sponse filter given by polynomial predictor and it quantizes the residual error with codebook and partition like quantiz The output i is an array of codes with the same size and dimension as input The prediction y k for sample k s degpredictor y k gt predictor yq k 0 i 1 where yg k is the quantized reconstructed signal The predictor must be strictly causal predictor must be zero To encode the difference between in k and yq k 1 predictor 0 1 Note that there is no drift between the reconstructed signal and the input E contrary to the case where the input is differentiated quantized and integrated lActually there may be a drift if the arithmetic units used for encoding and decoding do not produce exactly the same results Libraries sigenc 551 Example t 0 0 1 10 x Sin t codebook 1 01 1 partition 0 01 09 predictor 0 1 i dpcmenco x codebook partition predictor y dpcmdeco i codebook predictor See also dpcmopt Differential pulse code modulation decoding Syntax predictor codebook partition dpcmopt in order n predictor codebook partition dpcmopt in order codebook0 predictor codebook partition dpcmopt in predictor predictor codebook partition dpcmopt tol predictor dpcmopt in order Description dpcmopt in order n gives the optimal predictor of order order codebook of size n and partition to en
354. ssions is assigned to the variable ans this makes easy reusing intermediate results in successive expressions Continuation characters A statement can span over several lines provided all the lines but the last one end with three dots For example 1 2 is equivalent to 1 2 After the three dots the remaining of the line as well as empty lines and lines which contain only spaces are ig nored 40 Sysquake for ATEX Comments Unless when it is part of a string enclosed between single ticks a single percent character or two slash characters mark the beginning of a comment which continues until the end of the line and is ignored by LME Comments must follow continuation characters if any a 2 comment at the end of a line X25 another comment comment spanning the whole line b comment after the continuation characters a a 3 no need to put spaces before the percent sign S percent characters in a string Comments may also be enclosed between and in that case they can span several lines Pragmas Pragmas are directives for LME compiler They can be placed at the same location as LME statements i e in separate lines or between semicolons or commas They have the following syntax pragma name arguments where name is the pragma name and arguments are additional data whose meaning depends on the pragma Currently only one pragma is defined Pragmas with unknown names are ignore
355. step Ad Bd Cd Dd Ts plots the step response of the discrete time state space model Ad Bd Cd Dd defined as x k 1 y k where u k is a unit step The state space model must have a scalar input and may have a scalar or vector output With output arguments dstep gives the output and the time as column vectors No display is produced Aadx k Bgu t Cgx k Dgu k Example Step response of a discrete time third order system see Fig 6 13 dstep 1 poly 0 9 0 7 0 6j 0 7 0 6j 1 g See also erlocus Root locus of a polynomial with coefficients bounded by an ellipsoid Syntax erlocus CQ P erlocus C0 P sizes colors LME Reference graphics for dynamical systems 447 dstep 1 poly 0 9 0 7 0 6 0 7 0 6j 0 2 s Figure 6 13 dstep 1 poly 9 7 6j 7 6j 0 2 s Description erlocus displays the set of the roots of all the polynomial whose coef ficients are bounded by an ellipsoid defined by CO and P The polyno mials are defined as CO 0 dC where dCxinv P dC lt 1 If sizes and colors are provided sizes must be a vector of n val ues and colors an n by 3 matrix whose columns correspond respec tively to the red green and blue components The locus correspond ing to dCxinv P dC lt sizes i 2 is displayed with colors i The vector sizes must be sorted from the smallest to the largest ellip soid The default values are sizes 0 1 0 5 1 2 and colors 0 0
356. steps at controlled time to change instan taneously the states and to base the termination condition on the states Time instants where events occur are either given explicitly by EventTime or implicitly by Events There can be multiple streams of events which are checked independently and are identified by a positive integer for Events or a negative integer for EventTime For instance for a ball which bounces between several walls the inter section between each wall and the ball trajectory would be a different event stream For events which occur at regular times EventTime is an n by two matrix for each row the first column gives the time step ts and the second column gives the offset to Non repeating events are specified with an infinite time step ts Events occur at time t to k ts where k is an integer When event time is varying EventTime is a function which can be defined as function eventTime eventtimefun t y where t is the current time y the current state and the ellipsis stand for additional arguments passed to ode The function returns a col umn vector whose elements are the times where the next event oc curs In both cases each row corresponds to a different event stream For events which are based on the state the value of a function which depends on the time and the states is checked the event occurs when its sign changes Events is a function which can be defined as function value isterminal di
357. t the rate given by the system and t is ignored or can be omitted See also Iti nichols Nichols plot Syntax nichols a nichols a Description nichols a plots the Nichols diagram of system a Libraries lti 547 See also bodemag bodephase Iti nyquist Nyquist plot Syntax nyquist a nyquist a Description nyquist a plots the Nyquist diagram of system a See also N1icnoOls DOGeMag jOOdephnase Iti step Step response Syntax step a step a Description step a plots the step response of system a See also Iti uminus Negative 548 Sysquake for TEX Syntax b a b uminus a Description a multiplies all the outputs or all the inputs of system a by 1 If a is a transfer functions or a matrix of transfer functions this is equivalent to the unary minus See also Iti uplus Negative Syntax b b a uplus a Description a gives a See also 7 8 sigenc sigenc is a library which adds to LME functions for encoding and de coding scalar signals It implements quantization DPCM differential pulse code modulation and companders used in telephony The following statement makes available functions defined in sigenc use sigenc alawcompress A law compressor Libraries sigenc 549 Syntax output alawcompress input output alawcompress input a Description alawcompress inpu
358. t a compresses signal input with A law method using parameter a The signal is assumed to be in 1 1 values out side this range are clipped input can be a real array of any size and dimension The default value of a is 87 6 The compressor and its inverse the expander are static nonlin ear filters used to improve the signal noise ratio of quantized signals The compressor should be used before quantization or on a signal represented with a higher precision See also alawexpand uLawcompress alawexpand A law expander Syntax output alawexpand input output alawexpand input a Description alawexpand input a expands signal input with A law method using parameter a input can be a real array of any size and dimension The default value of a is 87 6 See also alawcompress ulLawexpand dpcmdeco Differential pulse code modulation decoding 550 Sysquake for ATEX Syntax output dpcmdeco i codebook predictor Description dpcmdeco i codebook predictor reconstructs a signal encoded with differential pulse code modulation It performs the opposite of dpcmenco See also dpcmenco dpcmenco Differential pulse code modulation encoding Syntax i dpcmenco input codebook partition predictor Description dpcmenco input codebook partition predictor quantizes the signal in vector input with differential pulse code modulation It predicts the future response with the finite impulse re
359. t before the end of the list you have to interpret the meaning of the variables yourself LME always sets the nargin first arguments There are two other ways to let a function accept a variable num ber of input arguments to define default values directly in the func tion header or to call varargin to collect some or all of the input arguments in a list With one argument nargin fun returns the maximum number of input arguments a function accepts fun may be the name of a built in or user function a function reference or an inline function Func tions with a variable number of input arguments such as fprintf give 1 Examples A function with a default value pi for its second argument LME Reference miscellaneous functions 111 function x multiplyByScalar a k if nargin lt 2 multiplyByScalar x k pi same as multiplyByScalar x pi end x kxa A function with a default value standard output for its first argument Note how you have to interpret the arguments function fprintstars fd n if nargin fprintstars n to standard output fprintf repmat 1 fd n is actually stored in fd else fprintf fd repmat 1 n end Number of input arguments of function plus usually written nargin plus 2 See also nargout varargin function nargout Number of output arguments Syntax n nargout n nargout fun Description A function may be called with between
360. t can be very slow if the data length has large prime factors or is a prime number The coefficients of the DFT are given from the zero frequency to the largest frequency one point less than the inverse of the sampling 218 Sysquake for TEX period If the input f is real its DFT has symmetries and the first half contain all the relevant information Examples fft 1 4 10 2 2j 2 2 2j fft 1 4 3 6 1 5 0 866j 1 5 0 866j See also fft2 2 d Fast Fourier Transform Syntax F fft2 f F fft2 f size F fft2 f nr nc F fft2 f n Description fft2 f returns the 2 d Discrete Fourier Transform DFT along dimen sions 1 and 2 of array f With two or three input arguments fft2 resizes the two first dimen sions by cropping or by padding with zeros fft2 f nr nc resizes first dimension to nr rows and second dimension to nc columns In fft2 f size the new size is given as a two element vector nr nc fft2 F n is equivalent to fft2 F n n If the first argument is an array with more than two dimensions fft2 performs the 2 d DFT along dimensions 1 and 2 separately for each plane along remaining dimensions fftn performs an DFT along each dimension See also ifft2 fft FF tn fftn n dimension Fast Fourier Transform LME Reference linear algebra 219 Syntax F fftn f F fftn f size Description fftn f returns the n dimension Discrete Fourier Transform of array f DFT along each dimension of
361. t error message LME Reference miscellaneous functions 109 Syntax msg lasterr msg identifier lasterr Description lasterr returns a string which describes the last error With two out put arguments it also gives the error identifier It can be used in the catch part of the try construct Example xXx 2 x 3 Index out of range msg identifier lasterr msg Index out of range identifier LME indexOutOfRange See also tasterror try error lasterror Last error structure Syntax s lasterror Description Lasterror returns a structure which describes the last error It con tains the following fields identifier string short tag which identifies the error message string error message The structure can be used as argument to rethrow in the catch part of a try catch construct to propagate the error further 110 Sysquake for ATEX Example X 2 x 3 Index out of range lasterror message Index out of range identifier LME indexOutOfRange See also nargin Number of input arguments Syntax n n nargin nargin fun Description Calling a function with less arguments than what the function expects is permitted In this case the trailing variables are not defined The function may use the nargin function to know how many arguments were passed by the caller to avoid accessing the undefined variables Note that if you want to have an optional argumen
362. t it Color String black blue green cyan red magenta yellow white RGB RGB rrggbb rgb IIx Bsan0Qao R Dash Pattern String solid _ underscore dashed dotted i dash dot Shape String none invisible space point circle cross plus star triangle up triangle down square diamond gt X O A7 lt LME Reference remarks on graphics 391 Miscellaneous String stairs S fill f arrow at end a arrows at beginning and end A Color h rrggbb specifies a color by its red green and blue components each of them is given by two hexadecimal digits from 00 minimum brightness to ff maximum brightness Color h rgb specifies each component with a single hexadecimal digit For exam ple h 339933 and h 393 both specify the same greenish gray Like for other colors an uppercase H means that the line is thick Style s stairs is supported only by the plot dimpulse dstep dlsim and dinitial functions It is equivalent to a zero order hold i e two points are linked with a horizontal segment followed by a vertical segment Style f fill fills the shape instead of drawing its contour Exactly how the shape is filled depends on the underlying graphics architecture if the contour intersects itself there may be holes Style a adds an arrow at the end of lines drawn by plot and style A adds arrows to the beginning and the end The arrow size
363. t to represent arbitrary rotations in the 3d space They are more compact than matrices and are easier to normalize This makes them suitable to simulation and control of me chanical systems and vehicles such as flight simulators and robotics Functions below are specific to quaternions LME Reference quaternions 321 Function Purpose isquaternion test for quaternion type q2mat conversion to rotation matrix q2rpy conversion to attitude angles q2str conversion to string qimag imaginary parts qinv element wise inverse qnorm scalar norm qslerp spherical linear interpolation quaternion quaternion creation rpy2q conversion from attitude angles Operators below accept quaternions as arguments Function Operator Purpose ctranspose conjugate transpose eq element wise equality horzcat horizontal array concatenation ldivide left division ne element wise inequality minus difference mldivide matrix left division mrdivide matrix right division mtimes matrix multiplication plus addition rdivide division times x multiplication transpose i transpose uminus unary minus uplus unary plus vertcat 3 vertical array concatenation Most of these operators work as expected like with complex scalars and matrices Multiplication and left right division are not commuta tive Matrix operations are not supported operators and are defined as a convenience they are equivalent to
364. tandardized euclidean distance is the euclidean distance after each column of M has been divided by its standard deviation The Minkowski metric is based on the p norm of vector differences Examples pdist 1 3 121 squareform pdist 1 3 012 496 Sysquake for TEX 101 210 squareform pdist 1 2 6 3 1 7 6 1 2 0 2 4495 6 4807 2 4495 0 5 831 6 4807 5 831 0 See also squareform prctile Percentile Syntax m m prctile A prc prctile A prc dim Description prctile A prc gives the smallest values larger than prc percent of the elements of each column of array A or of the row vector A The dimension along which prctile proceeds may be specified with a third argument Example prctile rand 1 1000 90 0 8966 See also range Mean absolute deviation Syntax m range A m range A dim Libraries stat 497 Description range A gives the differences between the maximum and minimum values of the columns of array A or of the row vector A The dimension along which range proceeds may be specified with a second argu ment Example range rand 1 100 0 9602 See also squareform Resize the output of pdist to a square matrix Syntax D squareform d Description squareform d resize d which should be the output of pdist into a symmetric square matrix D so that the distance between observations iand j is D i j See also trimmean Trimmed mean of a set o
365. tax num str weekday year month day num str weekday datetime num str weekday jd Description weekday finds the week day of the date given as input The date can be given with three input arguments for the year the month and the day or with one input argument for the date or date and time vector or julian date Libraries constants 559 The first output argument is the number of the day from 1 for Sunday to 7 for Saturday and the second output argument is its name as a String of 3 characters such as Mon for Monday Example Day of week of today num str weekday clock num str Mon See also cal2julian 7 11 constants constants is a library which defines physical constants in SI units me ter kilogram second ampere The following statement makes available constants defined in constants use constants The following constants are defined 560 Sysquake for ATEX Name Value Unit avogadro_number 6 0221367e23 1 mole boltzmann_constant 1 380658e 23 J K earth_mass 5 97370e24 kg earth_radius 6 378140e6 m electron_charge 1 60217733e 19 C electron_mass 9 1093897e 31 kg faraday_constant 9 6485309e4 C mole gravitational_constant 6 672659e 11 N m 2 kg 2 gravity_acceleration 9 80655 m s 2 hubble_constant 3 2e 18 1 s ice_point 273 15 K induction_constant 1 256e 6 Vs Am molar_gaz_constant 8 314510 J K mole molar_volume 22 41410e 3 m 3 mole muon_mass 1 8835327e 28 kg neutron
366. tax options optimset options optimset namel valuel options optimset optionsO namel valuel Description optimset namel valuel creates the option argument used by fminbnd fminsearch and fzero Options are specified with name value pairs where the name is a string which must match exactly the names in the table below Case is significant Options which are not specified have a default value The result is a structure whose fields correspond to each option Without any input argument optimset creates a structure with all the default options Note that fminbnd fminsearch and fzero also interpret the lack of an option argument or the empty array as a request to use the default values When its first input argument is a structure optimset adds or changes fields which correspond to the name value pairs which fol low Here is the list of permissible options empty arrays mean auto matic Name Default Meaning Display false detailed display MaxFunEvals 1000 maximum number of evaluations MaxIter 500 maximum number of iterations TolXx maximum relative error The default value of TolX is eps for fzero and sqrt eps for fminbnd and fminsearch Examples Default options optimset Display false 306 Sysquake for ATEX MaxFunEvals 1000 MaxIter 500 TolxX Display of the steps performed to find the zero of cosx between 1 and 2 fzero cos 1 2 optimset Display true Checking lower
367. tch while for or catch block case or otherwise without switch The case or otherwise statement is not inside a switch block catch without try The catch statement does not match a try block break or continue not in a loop The break or continue statement is not inside a while or for block Variable name reused Same variable used twice as input or as output argument Too many user functions Not enough memory for that many user functions Attempt to redefine a function A function with the same name already exists Can t find function definition Cannot find a function definition during compilation Unexpected end of expression Missing right parenthesis or square bracket Unexpected statement Expression expected but a statement is found e g if Null name Name without any character when given as a string in functions like feval and struct Name too long More than 32 characters in a variable or function name Unexpected function header A function header keyword func tion has been found in an invalid place for example in the argu ment of eval Function header expected A function header was expected but not found Bad variable in the left part of an assignment The left part of an assignment does not contain a variable a structure field a list element or the part of an array which can be assigned to Bad variable in a for loop The left part of the assignment of a for loop is not a variable
368. te that its argument is pro cessed by LaTeX contrary to the contents of sysquake environment Sysquake for ATEX Tutorial 17 3 6 Graphics Creating graphics for LaTeX documents is among the most cumber some tasks an author is facing Sysquake for LaTeX makes it as easy as in Sysquake itself Graphics are created by adding the size of the EPS image to be created after begin sysquake The result is inserted automatically ina picture environment of the same size Package graphicx is used as well as package epstopdf in PDFLaTex they should be imported explicitly with usepackage Here is an example of a function plot begin sysquake 300 200 a 7 2 fplot x x 0 3 2 a xexp 3 x 2 2 3 r end sysquake In LaTeX figures are uSually floating This is done with the figure environment begin figure begin center begin sysquake 300 200 a 7 2 fplot x x 0 3 2 a xexp 3 x 2 2 3 r end sysquake caption Function plot end center end figure 3 7 Functions and libraries Code fragments may rely on the large set of built in operators and functions as well as on additional functions stored in libraries Libraries are text files with a m suffix which contain the source code of sets of related functions You can use those which come with Sysquake for LaTeX or write develop new ones which you can share with Sysquake To make functions defined in a library available to Sysq
369. teger int8 8 bit signed integer uint64 64 bit unsigned integer int64 64 bit signed integer LME Reference 43 64 bit integer numbers are not supported by all applications on all platforms These basic types can be used to represent many mathematic ob jects Scalar One by one matrix Vector n by one or one by n matrix Functions which return vec tors usually give a column vector i e n by one Empty object 0 by 0 matrix 0 by n or n by 0 matrices are always converted to 0 by 0 matrices Polynomial of degree d 1 by d 1 vector containing the coeffi cients of the polynomial of degree d highest power first List of n polynomials of same degree d n by d 1 matrix con taining the coefficients of the polynomials highest power at left List of n roots n by 1 matrix List of n roots for m polynomials of same degree n n by m matrix Single index One by one matrix List of indices Any kind of matrix the real part of each element taken row by row is used Sets Numerical array or list or cell array of strings see below Boolean value One by one logical array 0 means false and any other value including nan means true comparison and logical operators and functions return logical values In programs and expressions constant boolean values are entered as false and true Scalar boolean values are displayed as false or true in arrays respectively as F or T String Usually 1 by n char array but any shape of cha
370. the mouse position onto the line _nb is the index of the line in A and b and _ix is empty Examples Vertical line at x 5 line 1 0 5 Draggable horizontal lines at y 2 and y 3 line 0 1 2 3 b 1 See also LME Reference base graphics 407 pcolor Pseudocolor plot Syntax pcolor C pcolor X Y C pcolor style pcolor style id Description Command pcolor C displays a pseudocolor plot i e a rectangular array where the color of each cell corresponds to the value of elements of 2 D array C These values are real numbers between 0 and 1 The color used by pcolor depends on the current color map the default is a grayscale from black 0 to white 1 pcolor X Y C displays the plot on a grid whose vertex coordi nates are given by arrays X and Y Arrays X X and C must all have the same size With an additional string input argument pcolor style specifies the style of the lines drawn between the cells The following argument if it exists is the ID used for interactivity During interactive manipulation the index obtained with _ix corre sponds to the corner of the patch under the mouse with the smallest index Example use colormaps n 11 x y meshgrid 1 n phi pi 8 X x cos phi y sin phi Y x sin phi y cos phi C magic n n 2 pcolor X Y C k colormap blue2yellow2redcm plotoption noframe See also 408 Sysquake for TEX plot Gen
371. the string shape you can force a row vector with char A char sl1 s2 concatenates vertically the arrays given as ar guments to produce a string matrix If the strings do not have the same number of columns blanks are added to the right LME Reference strings 309 Examples char 65 70 ABCDEF char 65 66 67 68 ABCD char ab cde ab cde char abc de fg abc de fg See also deblank Remove trailing blank characters from a string Syntax s2 deblank s1 Description deblank s1 removes the trailing blank characters from string s1 Blank characters are spaces code 32 tabulators code 9 carriage returns code 13 line feeds code 10 and null characters code 0 Example double tAB CD r n 0 32 9 65 66 32 32 67 68 13 10 0 double deblank tAB CD n r 0 32 9 65 66 32 32 67 68 See also findstr Find a substring in a string 310 Sysquake for TEX Syntax pos findstr str sub Description findstr str sub finds occurrences of string sub in string str and returns a vector of the positions of all occurrences or the empty vector if there is none Occurrences may overlap Examples findstr ababcdbaaab ab 13 10 findstr ababcdbaaab ac findstr aaaaaa aaa 123 See also ischar Test for a string object Syntax b ischar obj Description ischar obj is true if t
372. the code consists of a single expression args is the list of arguments which must be supplied when the inline function is called paramstruct is a structure whose fields define fixed param eters Anonymous functions are an alternative often easier way of creat ing inline functions The result is the same Since inline is a normal function it must be used in contexts where fixed parameters cannot be created as separate variables Examples A simple expression evaluated at x 1 and x 2 fun inline cos x exp x y feval fun 2 y 5 6319e 2 y feval fun 5 1 9113e 3 A function of x and y fun inline exp x 2 y 2 x y A function with two output arguments the string is broken in three lines to have a nice program layout a mean v b prod v 1 length v 7J am gm feval fun 1 10 am 5 5 gm 4 5287 fun inline function a b f v Simple expression with fixed parameter a fun inline cos a x x struct a 2 feval fun 3 0 9602 LME Reference miscellaneous functions 107 An equivalent function where the source code of a complete function is provided fun inline function y f a x y cos a x 2 feval fun 3 0 9602 A function with two fixed parameters a and b whose values are pro vided in a list inline function y f p x a b deal p y a x b 2 3 See also inmem List of fun
373. the control sequences which all begin with the character They have the form fn dt where f is zero one or more of the following flags Flag Meaning left alignment default is right alignment display of a sign for positive numbers 0 zero padding instead of spaces alternate format see below space sign replaced with space for positive numbers n is the optional width of the field as one or more decimal digits default is the minimum width to display the data d is the number of digits after the decimal separator for a number or the number of characters for a string one or more decimal digits by default it is 4 for a number or the length of the string for a string and t is a single character denoting the type of conversion In most cases each control sequence corresponds to an additional argument All elements of arrays are used sequentially as if they were provided separately strings are used as a whole The table below gives the valid values of t LME Reference input output 373 Oo gt g Conversion single decimal number as an integer same as d hexadecimal number for integers between 0 and 2732 1 same as x with uppercase digits octal number for integers between 0 and 2 32 1 fixed number of decimals exp notation if abs x gt 1e18 same as f with an uppercase E scientific notation such as 1e5 scientific notation such as 1E5 decimal or scientific notation same as g with an u
374. ti object a by connecting some of the outputs to some of the inputs and by keeping some of the inputs and some of the outputs Connections are specified by the rows of matrix Link In each row the first element is the index of the system input where the connection ends other elements are indices to system outputs which are summed The sign of the indices to outputs gives the sign of the unit weight in the sum Zeros are ignored Arguments in and out specify which input and output to keep See also Iti d2c Conversion from discrete time to continuous time Syntax b d2c a b d2c a method Libraries lti 531 Description d2c a converts the discrete time system a to a continuous time sys tem d2c a method uses the specified conversion method method is one of the methods supported by d2cm See also Iti end Last index Syntax var end Description In an expression used as an index between parenthesis end gives the last valid value for that index It is size var 1 or size var 2 Example Time response when the last input is a step P ss 1 2 3 4 1 0 0 1 3 5 P1 P end continuous time LTI state space system A 1 3 4 B step P1 See also beginning 532 Sysquake for ATEX Iti evalfr Frequency value Syntax y evalfr a x Description evalfr a x evaluates system a at complex value or values x If x is a vector of values results are stacked along
375. tic amplitude phi Parameter m must be in 0 1 The Jacobi elliptic amplitude is the inverse of the Jacobi integral of the first kind such that u F g m ellipam u m tol uses tolerance tol the default tolerance is eps 168 Sysquake for ATEX Example phi ellipam 2 7 0 6 u llipf phi 0 6 N Il See also ettip ettiri ellipe Jacobi elliptic integral of the second kind Syntax u ellipe phi m Description ellipe phi m gives the Jacobi elliptic integral of the second kind defined as o E g m V1 msin tdt 0 Complete elliptic integrals of first and second kinds with phi pi 2 can be obtained with ellipke See also ellipf ellipke ellipf Jacobi elliptic integral of the first kind Syntax u ellipf phi m LME Reference mathematical functions 169 Description ellipf phi m gives the Jacobi elliptic integral of the first kind de fined as Fom m sin Complete elliptic integrals of first and second kinds with phi pi 2 can be obtained with ellipke See also ellipj Jacobi elliptic functions Syntax sn cn dn sn cn dn ellipj u m ellipj u m tol Description ellipj u m gives the Jacobi elliptic function sn cn and dn Parame ter m must be in 0 1 These functions are based on the Jacobi elliptic amplitude the inverse of the Jacobi elliptic integral of the first kind which can be obtained with ellipam u F m s
376. ting points in such a way that no point is included in the circum scribed circle of any other triangle Triangles are as equilateral as possible 282 Sysquake for TEX Example Delaunay triangulation of 20 random points Xx rand 20 1 rand 20 1 t e delaunay x y With Sysquake graphical functions points belonging to the convex hull are displayed as crosses and interior points as circles clf scale equal plot x e y e plot x e y e 0 Array of vertex indices is modified to have closed triangles Triangles are displayed plot x t y t See also delaunayn N d Delaunay triangulation Syntax t delaunayn x t e delaunayn x Description delaunayn x calculates the Delaunay triangulation of points given by the rows of array x in a space of dimension size x 2 The result is an array with one more column Each row corresponds to a simplex values are row indices in x and give the vertices of each polyhedron The second output argument if requested is a logical vector with as many elements as rows in x elements are true if the corresponding point in x belongs to the convex hull of the set of points LME Reference triangulation 283 See also delaunay tsearchn voronoin griddata Data interpolation in 2 d plane Syntax vi vi griddata x y v xi yi griddata x y v xi yi method Description griddata x y v xi yi interpolates values at coo
377. tion aware of Sysquake for LaTeX You should add a new root directory to the list of TDS directories 10 Sysquake for ATEX In the Start menu click MikKTeX gt Settings and select the Roots tab Click the Add button select directory TexFiles of Sysquake for La TeX and confirm clicking the main OK button of the MikKTeX Options window For other TeX distributions please read their documentation Sysquake for LaTeX installer is totally independent from any other software you should just configure TeX so that it finds the file sysquake sty 2 2 Mac OS X On Mac OS X Sysquake for LaTeX is distributed as a package To install it just double click its icon and accept everything You will need an administrator password An alias of the documentation entry file is put on the Desktop The Mac OS X package stores the distribution in opt SysquakeLx and creates the following symbolic links also creating directories if they do not already exist ln s opt SysquakeLx bin i386 darwin sysquakelatextool usr bin ln s opt SysquakeLx TexFiles tex latex sysquake sty Library texmf tex latex ln s opt SysquakeLx TexFiles tex latex sq example tex Library texmf tex latex ln s opt SysquakeLx TexFiles doc latex sysquake index html Desktop SQ LaTeX html 2 3 TeX Live on Linux and other Posix sys tems Install the latest release of TeX Live e g TeX Live 2007 Then follow the steps below Unarchive the distribution
378. tion fun is specified either by its name as a String or by an inline function The number of expected output arguments can be specified with an optional third input argument nargout Examples apply min 5 7 5 apply size magic 3 2 3 3 apply inline 2 x 3 y x y 5 10 40 See also join List concatenation Syntax list join ll 12 Description join 1l1 12 joins elements of lists l1 12 etc to make a larger list Examples join 1 a 2 5 4 2 xxx 1 a 2 3 4 5 14 2 Xxx 332 Sysquake for ATEX See also islist Test for a list object Syntax b islist obj Description islist obj is true if the object obj is a list false otherwise Examples islist 1 2 x true islist true islist false ischar false See also isstruct isnumeric ischar is logical isempty list2num Conversion from list to numeric array Syntax A lList2num list Description List2num list takes the elements of List which must be numbers or arrays and concatenates them on a row along second dimension as if they were placed inside brackets and separated with commas Element sizes must be compatible LME Reference lists 333 Example List2num 1 2 3j 4 6 1 2 3j 4 5 6 See also operator T map Function evaluation for each element of a list Syntax listoutl m
379. tion order follows the usual rules which state that the multiplication denoted with a star and division slash have a higher priority than the addition and subtraction You can change this order with parenthesis gt 24 3 4 ans 20 20 Sysquake for TEX The result of expressions is automatically assigned to variable ans more about variables later which you can reuse in the next expres sion gt 3xans ans 60 Power is represented by the symbol gt 25 ans 32 LME has many mathematical functions Trigonometric functions as sume that angles are expressed in radians and sqrt denotes the square root gt sin pi 4 sqrt 2 ans 1 4 2 Complex Numbers In many computer languages the square root is defined only for non negative arguments However it is extremely useful to extend the set of numbers to remove this limitation One defines i such that i 1 and applies all the usual algebraic rules For instance 1 V i and 4 V4 1 2i Complex numbers of the form a bi are the sum of a real part a and an imaginary part b It should be mentioned that i the symbol used by mathematicians is called j by engineers LME accepts both symbols as input but it always writes it j You can use it like any function or stick an i or j after a number gt 2 3 j ans 2 3j gt 3j 2 ans 2 3j Many functions accept complex numbers as argument and return a complex result when the input requ
380. tions class Object creation Syntax object class strct classname object class strct classname parentl str class object Description class strct classname makes an object of the specified class with the data of structure strct Object fields can be accessed only from methods of that class i e functions whose name is classname methodname Objects must be created by the class constructor classname classname class strct classname parent1 makes an object of the specified class which inherits fields and methods from one or several other object s parent1 Parent objects are inserted as additional fields in the object with the same name as the class Fields of parent objects cannot be directly accessed by the new object s methods only by the parent s methods class object gives the class of object as a string The table below gives the name of native types Class Native type double real complex or logical scalar or array char character or character array list list or structure inline inline function funref function reference Examples ol class struct fld1 1 fld2 rand 4 cl1 o2 class struct fld3 abc c2 ol class o2 c2 LME Reference objects 341 See also isa Test for an object of a given class Syntax b isa object classname Description isa object classname returns true of
381. to a digital discrete time transfer function given by its zeros 518 Sysquake for TEX poles and gain in the domain of the forward shift operator q The sampling frequency is fs Conversion is performed with the bilinear transormation Za 1 Zc 2fs 1 Zc 2fs If the analog transfer function has less zeros than poles additional digital zeros are added at 1 to avoid a delay With three input arguments bilinear numc denc fs uses the coefficients of the numerators and denominators instead of their ze ros poles and gain buttap Butterworth analog filter prototype Syntax z p k buttap n Description buttap n calculates the zeros the poles and the gain of a Butter worth analog filter of degree n with a cutoff angular frequency of 1 rad s See also butter Butterworth filter P K butter n w0 um den butter n w0 butter n wl wh butter n w0 high butter n wl wh stop erie butter eS Libraries filter 519 Description butter calculates a Butterworth filter The result is given as zeros poles and gain if there are three output arguments or as numera tor and denominator coefficient vectors if there are two output argu ments butter n w where w0 is a scalar gives a nth order digital low pass filter with a cutoff frequency of w0 relatively to half the sampling frequency butter n wl wh where the second input argument is a vector of two n
382. tories altname alternate name Windows only Field isdir may be missing on some patforms On Windows altname contains the DOS compatible name a k a 8 3 if it exists or an empty string otherwise 384 Sysquake for ATEX See also dos Execute a command under Windows Syntax status dos str Description dos str executes a command with the system Windows function No input can be provided and output is discarded dos returns the status code of the command which is normally O for successful execution Example dos del C tmp data txt See also unix getenv Get the value of an environment variable Syntax value getenv name Description getenv name gives the value of the environment variable of the spec ified name If no such environment variable exists getenv returns an empty string Example user getenv USER LME Reference shell 385 See also setenv unsetenv ispc Check whether platform is a PC Syntax b ispc Description ispc returns true on Windows where dos is defined false on other platforms See also dos isunix Check whether platform runs under unix Syntax b isunix Description isunix returns true on unix or unix like platforms where unix is defined false on other platforms See also pwd Get the current directory Syntax str pwd 386 Sysquake for ATEX Description pwd print working directory gives t
383. trb digest md5 fd Description md5 strb calculates the MD5 digest of strb which represents binary data strb can be a string only the least significant byte of each character is considered or an array of bytes of class uint8 or ints The result is a string of 32 hexadecimal digits It is believed to be hard to create the input to get a given digest or to create two inputs with the same digest md5 fd calculates the MD5 digest of the bytes read from file de scriptor fd until the end of the file The file is left open MD5 digest is an Internet standard described in RFC 1321 Examples MD5 of the three characters a b and c md5 abc 900150983cd24fb0d6963f7d28e17Ff72 This can be compared to the result of the command tool md5 found on many unix systems echo n abc md5 900150983cd24fb0d6963f7d28e17F72 The following statements calculate the digest of the file somefile fd fopen somefile digest md5 fd fclose fd 314 Sysquake for TEX See also setstr Conversion of an array to a string Syntax str setstr A Description setstr A converts the elements of array A to characters resulting in a string of the same size Characters are stored in unsigned 16 bit words Example setstr 65 75 ABCDEFGHIJK See also shal Calculate SHA1 digest Syntax digest shal strb digest shal fd Description shal strb calculates the SHA1 digest of strb which repr
384. ts also accept cell arrays functions which expect cell arrays treat lists of n elements as 1 by n cell arrays To create a cell array with 2 dimensions cells are written between braces where rows are separated with semicolons and row elements with commas 1 abc 27 true Since the use of braces without semicolon produces a list there is no direct way to create a cell array with a single row or an empty cell array Most of the time this is not a problem since lists are accepted where cell arrays are expected To force the creation of a cell array the reshape function can be used reshape ab cde 1 2 Structures Like lists and cell arrays structures are sets of data of any type While list elements are ordered but unnamed structure elements called fields have a name which is used to access them There are two ways to make structures with the struct function or by setting each field in an assignment s f refers to the value of the field named f in the structure s Usually s is the name of a variable but unless it is in the left part of an assignment it can be any expression a struct name Sysquake os Windows Mac OS X Linux b x 200 b y 280 b radius 90 c s b With the assignments above a os 3 is Linux and c s radius is 90 Function references Function references are equivalent to the name of a function together with the co
385. two output arguments ellip n w0 where w0 is a scalar gives a nth order digital low pass filter with a cutoff frequency of w0 relatively to half the sampling frequency ellip n wl wh where the second input argument is a vector of two numbers gives a 2nth order digital bandpass filter with passband between wl and wh relatively to half the sampling frequency ellip n w0 high gives a nth order digital highpass filter with a cutoff frequency of w0 relatively to half the sampling frequency ellip n wl wh stop where the second input argument is a vector of two numbers gives a 2nth order digital bandstop filter with stopband between wl and wh relatively to half the sampling frequency With an additional input argument which is the string s ellip gives an analog elliptic filter Frequencies are given in rad s See also ellipap Elliptic analog filter prototype Libraries filter 523 Syntax z p k ellipap n rp rs Description ellipap n rp rs calculates the zeros the poles and the gain of an elliptic analog filter of degree n with a cutoff angular frequency of 1 rad s Ripples have a peak to peak magnitude of rp dB in the passband and of rs dB in the stopband See also Ip2bp Lowpass prototype to bandpass filter conversion Syntax z p k lp2bp z0 pO kO wc ww num den lLp2bp num0 denO wc ww Description lp2bp convert a lowpass analog filter prototype with un
386. uake a use statement should be placed before any call in a sysquake environ ment or a Sqeval command begin verbatim begin figure begin center begin sysquake 300 200 use polyhedra smallstellateddodecahedron 18 Sysquake for ATEX colormap blue2yellow2redcm plotoption nomargin plotoption noframe plotoption fill3d end sysquake caption Great dodecahedron end center end figure end verbatim Chapter 4 LME Tutorial The remainder of this chapter introduces LME TM Lightweight Matrix Engine the interpreter for numerical computing used by Sysquake and shows you how to perform basic computations It supposes you can type commands to a command line interface You are invited to type the examples as you read this tutorial and to experiment on your own For a more systematic description of LME please consult the LME Reference chapter In the examples below we assume that LME displays a prompt gt This is not the case for all applications You should never type it your self Enter what follows the prompt on the same line hit the Return key or tap the Eval or Execute button and observe the result 4 1 Simple operations LME interprets what you type at the command prompt and displays the result unless you end the command with a semicolon Simple ex pressions follow the syntactic rules of many programming languages gt 2 3 4 ans 14 gt 2 3 4 ans 2 75 As you can see the evalua
387. uaternion imaginary parts Syntax b qimag q 326 Sysquake for TEX Description qimag q gives the imaginary parts of quaternion q as a quaternion i e the same quaternion where the real part is set to zero real q gives the real part of quaternion q as a double number Example q quaternion 1 2 3 4 q 1 2i 3j 4k real q 1 qimag q 2i 3j 4k See also quaternion qinv Quaternion element wise inverse Syntax b qinv q Description qinv q gives the inverse of quaternion q If its input argument is a quaternion array the result is an quaternion array of the same size whose elements are the inverse of the corresponding elements of the input The inverse of a normalized quaternion is its conjugate Example q quaternion 0 4 0 1 0 2 0 2 0 4 0 1i 0 2j 0 2k p qinv q p 1 6 0 4i 0 8j 0 8k abs q 0 5 LME Reference quaternions 327 abs p 2 See also quaternion qnorm conj qnorm Quaternion scalar norm Syntax n qnorm q Description qnorm q gives the norm of quaternion q i e the sum of squares of its components or the square of its absolute value If q is an array of quaternions qnorm gives a double array of the same size where each element is the norm of the corresponding element of q See also quaternion jabs qslerp Quaternion spherical linear interpolation Syntax q qslerp ql q2 t Description qslerp q1 q2 t performs spherical lin
388. uations have been converted from text to real mathematic notation and a table of contents and an index have been added The same method has been used to create the material for the help command Thanks to the make utility the whole process is completely automatic This system has proved to be very flexible to maintain three useful formats in parallel two for on line help and one for high quality printing World Wide Web http www calerga com E mail sysquake calerga com Mail alerga Sar Av de la Chabli re 35 1004 Lausanne Switzerland Typesetting 2007 11 29 Contents 7 9 2 1 Windows 0 00 eee ees 9 22 Ma OS X aoe hoe eo ee a e ee Ge ke ee ela amp eke Ads 10 2 3 TeX Live on Linux and other Posix systems 10 3 Sysquake for TeX Tutorial 13 eee ee ee ee 13 at iea aea He ok aoe a ak h 14 ee ee eee 14 3 4 Formatted results 00 0 15 3 9 PFOOrAMS o cs s gioii ew Gob Pee ke de ee i a 15 3 6 Graphics ow eo eG bw aoe Ra ee aren BE a o 17 3 __ Functions and libraries 17 4 LME Tutorial 19 4 1 Simple operations n saaa a 19 4 2 Complex Numbers saasaa aaa a 20 4 3 Vectors and Matrices naaa a 22 4 4 Polynomials 000000 a 25 45 SINGS e i ed e a aa E e e eae iE a ead 26 4 60 Narlables sas eave fee ad a a hae E e ead ed ees 26 4 7 Loops and Conditional Execution 27 4 8 Functions 0 0 00 ee eee 27 4
389. uidistant points between xmin and xmax in the x direction and between ymin and ymax in the y direction The optional third argument levels if non empty gives the num ber of contour lines if it is a scalar or the levels themselves if it is a vector The optional fourth argument is the style of each line from the minimum to the maximum level styles are recycled if necessary The default style is kbrmgcy See also surf daspect Scale ratios along x y and z axis 426 Sysquake for TEX Syntax daspect rx ry rz daspect R daspect Description daspect R specifies the scale ratios along x y and z axis Argument Ris a vector of 3 elements rx ry and rz Coordinates in the 3D space are divided by rx along the x axis and so on before the projection is performed For example a box whose size is 2 5 3 would be displayed as a cube with daspect 2 5 3 daspect sets the scale ratios so that the bounding box of 3D elements is displayed as a cube With an output argument R daspect gives the current scale ratios as a vector of 3 elements See also lightangle Set light sources in 3D world Syntax lightangle lightangle az el Description Lightangle az el set lighting source s at infinity with asimuth az and elevation el both in radians With missing input argument the default azimuth is 4 and the default elevation is 1 If az and el are vectors they must have the same size except if
390. umbers gives a 2nth order digital bandpass filter with pass band between wl and wh relatively to half the sampling frequency butter n w0 high gives a nth order digital highpass filter with a cutoff frequency of w0 relatively to half the sampling frequency butter n wl wh stop where the second input argument is a vector of two numbers gives a 2nth order digital bandstop filter with stopband between wl and wh relatively to half the sampling frequency With an additional input argument which is the string s butter gives an analog Butterworth filter Frequencies are given in rad s See also cheblap Chebyshev type 1 analog filter prototype Syntax z p k cheblap n rp Description cheblap n rp calculates the zeros the poles and the gain of a Chebyshev type 1 analog filter of degree n with a cutoff angular fre quency of 1 rad s Ripples in the passband have a peak to peak mag nitude of rp dB See also 520 Sysquake for ATEX cheb2ap Chebyshev type 2 analog filter prototype Syntax z p k cheb2ap n rs Description cheb2ap n rs calculates the zeros the poles and the gain of a Chebyshev type 2 analog filter of degree n with a cutoff angular fre quency of 1 rad s Ripples in the stopband have a peak to peak mag nitude of rs dB See also chebyl1 Chebyshev type 1 filter Syntax z p k chebyl n w0 num den cheby1 n w0 chebyl n wl wh
391. un object constructor Syntax a ratfun a ratfun coefnum a ratfun coefnum coefden Description ratfun coefnum coefden creates a ratfun object initialized with the coefficients in vectors coefnum and coefden given in descending pow ers of the variable Without argument ratfun returns a ratfun object initialized to 0 If omitted coefden defaults to 1 504 Sysquake for ATEX The following operators and functions may be used with ratfun ar guments with results analog to the corresponding functions of LME inv mpower minus plus mtimes uminus mrdivide uplus mldivide Example r ratfun 3 0 1 4 2 2 5 0 1 r 3x 4 x 2 4x 2 2x 34 5x 2 1 See also ratfun disp ratfun inline ratfun feval polynom polynom ratfun disp Display a ratfun object Syntax disp a Description disp a displays rational function a It is also executed implicitly when LME displays the ratfun result of an expression which does not end with a semicolon See also ratfun ratfun disp ratfun num Get the numerator of a ratfun as a vector of coefficients Syntax coef num a Libraries classes 505 Description num a gets the numerator of a as a row vector of descending power coefficients See also ratfun den ratfun ratfun ratfun den Get the denominator of a ratfun as a vector of coefficients Syntax coef den a Description den a gets the
392. upported supported supported supported Windows supported undefined supported supported supported true supported not supported supported not supported not supported for the status code of the command Syntax cd str str cd Description cd str changes the current directory Without input argument cd gives the current directory like pwd The current directory is the root directory where files specified by relative paths are searched by functions like fopen and dir LME li braries are specified by name not by path the places where they are searched is specified by a list of search paths typically specified with a path command or a dialog box in a graphical user interface Example cd usr include Sysquake for TEX LME Reference shell 383 See also cputime Amount of processing time since the beginning of the process Syntax t cputime Description cputime gives the amount of processing time spent since the applica tion has been launched See also dir List of files and directories Syntax dir dir path r dir r dir path Description dir displays the list of files and directories in the current path A string input argument can specify the path With an output argument dir gives a list of files and directories Each element is a structure with the following fields Name Value name file name or directory name isdir false for files true for direc
393. ure false otherwise Structures are lists whose at least one field has a name Examples isstruct struct a 123 true isstruct 1 2 x false a f 3 isstruct a true See also orderfields Reorders the fields of a structure Syntax strctout orderfields strctin strctout orderfields strctin structref strctout orderfields strctin names strctout orderfields strctin perm strctout perm orderfields Description With a single input argument orderfields strctin reorders struc ture fields by sorting them by field names With two input arguments orderfields reorders the fields of the first argument after the second argument Second argument can be a permutation vector containing integers from 1 to Llength strctin another structure with the same field names or a list of names In the last cases all the fields of the structure must be present in the second argument The first output argument is a structure with the same fields and the same value as the first input argument the only difference is the field order An optional second output argument is set to the permu tation vector 338 Sysquake for ATEX Examples s struct a 123 c 1 3 b 123 7 a 123 c real 1x3 b abcde t p orderfields s t a 123 b abcde c real 1x3 p 1 3 2 t orderfields s c b a t c real 1x3 b abcde a
394. vector of two elements the start time and the end time Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time tOnly When opt tOnly is true lsim produces output only at the time instants defined in t The logical value false gives the default interpolated values The optional arguments style and id have their usual meaning With output arguments Lsim gives the output and the time as col umn vectors or an array for the output of a multiple output state space model where each row represents a sample No display is produced Example Response of continuous time system given by its transfer function with an input defined by linear segments see Fig 6 18 t 0 10 20 30 50 u 1 1 0 1 1 lsim 1 1 2 3 4 u t b LME Reference graphics for dynamical systems 455 lsim 0 2 Figure 6 18 lsim 1 1 2 3 4 u t See also ngrid Nichols chart grid Syntax ngrid ngrid style Description ngrid plots a Nichols chart in the complex plane of the Nichols dia gram see Fig 6 19 The Nichols chart is a set of lines which corre spond to the same magnitude of the closed loop frequency response The optional argument specifies the style The whole grid is displayed only if the user selects it in the Grid menu or after the command plotoption fullgrid By default only the lines corresponding to unit magnitude and to a ph
395. vector v Length A gives the num ber of elements along the largest dimension of array A Length list gives the number of elements in a list Examples length 1 5 kama T E E T 1 6 abc Length 09 See also magic Magic square Syntax M magic n Description A magic square is a square array of size n by n which contains each integer between 1 and n2 and whose sum of each column and of each line is equal magic n returns magic square of size n by n There is no 2 by 2 magic square If the size is 2 the matrix 1 3 4 2 is returned instead 262 Example magic 3 8 16 357 492 See also meshgrid Arrays of X Y coordinates Syntax X Y meshgrid x y X Y meshgrid x Description Sysquake for ATEX meshgrid x y produces two arrays of x and y coordinates suitable for the evaluation of a function of two variables The input argument x is copied to the rows of the first output argument and the input argument y is copied to the columns of the second output argument so that both arrays have the same size meshgrid x is equivalent to meshgrid x x Example X X lt l meshgrid 1 5 2 4 N Ul NN Ww w w D AUN ul 0 4636 0 7854 0 9828 0 3218 0 5880 0 7854 0 2450 0 4636 0 6435 1 1903 1 0304 0 8961 LME Reference arrays 263 See also ndgrid Arrays of N dimension coordinates Syntax X1 Xn ndgrid xl xn X1 Xn
396. ver implements a server for web services using the XML RPC or SOAP protocol It requires the TCP IP and WebSer vices functions 7 1 stdlib stdlib is a library which extends the native LME functions in the fol lowing areas creation of matrices blkdiag compan hankel linspace logspace toeplitz geometry cart2sph cart2pol pol2cart sph2cart subspace functions on integers factor isprime primes statistics corrcoef median perms data processing circshift cumtrapz fftshift filter2 hist ifftshift polyfit polyvalm trapz other isreal sortrows The following statement makes available functions defined in stdlib use stdlib Libraries stdlib 473 cart2pol Cartesian to polar coordinates transform Syntax phi r cart2pol x y phi r z cart2pol x y z Description phi r cart2pol x y transforms Cartesian coordinates x and y to polar coordinates phi and r such that x rcos and x rsin phi r z cart2pol x y z transform Cartesian coordinates to cylindrical coordinates leaving z unchanged Example phi r cart2pol 1 2 phi 1 1071 r 2 2361 See also cartZsph pol2cart sphzcart cart2sph Cartesian to spherical coordinates transform Syntax phi theta r cart2sph x y zZ Description phi theta r cart2sph x y z transforms Cartesian coordinates x y and z to polar coordinates phi theta and r such that x rcos g cos 9 y rsin
397. ward by default with a third input argument dir find gives the n first nonzero values if dir is first or f and the n last nonzero values if dir is last or UL LME Reference arrays 253 Examples ix find 1 2 0 0 3 6 1x 1 4 s1 s2 find 1 2 0 0 3 6 sl 1 2 s2 1 2 s1 s2 x find 1 2 0 0 3 6 sl 1 2 s2 2 6 rand 3 riworeil nw 0 5599 0 3074 0 5275 0 3309 0 8077 0 3666 0 7981 0 6424 0 6023 find A gt 0 7 2 last 7 5 See also flipdim Flip an array along any dimension Syntax B flipdim A dim Description flipdim A dim gives an array which has the same size as A but where indices of dimension dim are reversed 254 Sysquake for ATEX Examples flipdim cat 3 1 2 3 4 5 6 7 8 3 2x2x2 array 1 5 6 7 8 2 2 4 w See also fliplr Flip an array or a list around its vertical axis Syntax A2 fliplr A1 list2 fliplr list1 Description fliplr A1 gives an array A2 which has the same size as Al but where all columns are placed in reverse order fliplr list1 gives a list List2 which has the same length as list1 but where all top level elements are placed in reverse order Elements themselves are left unchanged Examples fliplr 1 2 3 4 2 1 43 fliptr 1 x 1 2 3 1 2 3 x 1 See also flipud flipdim flipud Flip an array upside down LME Reference arrays 255
398. which contains the coordinates of the vertices of the Voronoi cells one row per vertex The first row contains infinity and is used as a marker for unbounded Voronoi cells The second output argument p is a list of vectors of row indices in v each element describes the Voronoi cell corresponding to a point in x In each cell vertices are sorted by index See also 6 19 Integer Functions uint8 uint16 uint32 uint64 int8 int16 int32 int64 Conversion to integer types 288 Sysquake for TEX Syntax uint8 A uint16 A v vuv uvu wW tou ow id te ud 1 Description The functions convert a number or an array to unsigned or signed integers The name contains the size of the integer in bits To avoid a conversion from double to integer constant literal num bers should be written with a type suffix such as 12int32 This is the only way to specify large 64 bit numbers because double precision floating point numbers have a mantissa of 56 bits uint64 and int64 are not supported on platforms with tight mem ory constraints Examples uint8 3 3uint8 3uint8 3uint8 uint8 50 50 400 1x8 uint8 array 50 100 150 200 250 44 94 144 int8 50 50 400 1x8 int8 array 50 100 106 56 6 44 94 112 See also intmax Largest integer Syntax intmax intmax type i i LME Reference integers 289 Description Without input argument intmax gives the largest signed 32 bit inte ger intmax type gives th
399. which is especially useful for object meth ods see below use the equivalent function such as plus for operator The complete list is given in the Section about operators To define a method which is executed when one of the input argu ments is an object of class class or a child in the classes hierarchy add class before the method function name To call it use only the method name not the class name Examples Function with optional input and output arguments function Sum Prod calcSumAndProd x y if nargout return nothing to be computed end if nargin make something to be computed xX 0 end if nargin lt 1 sum of elements of x Sum sum x else sum of x and y Sum xX y end if nargout also compute the product if nargin product of elements of x Prod prod x else product of x and y Prod Xx y end end Two equivalent definitions function S area a b a ellipse false S ellipse pixta xb 4 ax b LME Reference programming constructs 85 function S area a b ellipse if nargin lt 2 b a end if nargin lt 3 ellipse false end S ellipse pixaxb 4 ax b See also if elseif else end Conditional execution depending on the value of one or more boolean expressions Syntax if expr s1 end if expr s1 else s2 end if exprl sl elseif expr2 s2 else s3 end Description If the expression follo
400. wing if is true nonempty and all elements dif ferent from 0 and false the statements which follow are executed 86 Sysquake for TEX Otherwise the expressions following elseif are evaluated until one of them is true If all expressions are false the statements following else are executed Both elseif and else are optional Example if x gt 2 disp large elseif x gt 1 disp medium else disp small end See also include Include libraries Syntax include lib Description include lib inserts the contents of the library file Lib Its effect is similar to the use statement except that the functions and constants in lib are defined in the same context as the library where include is located Its main purpose is to permit to define large libraries in multiple files in a transparent way for the user include statements must not follow other statements on the same line and can reference only one library which is searched at the same locations as use They can be used only in libraries Since LME replaces include with the contents of lib one should be cautious about the public or private context which is preserved between the libraries It is possible to include a fragment of function without a function header See also LME Reference programming constructs 87 otherwise Conditional execution of statements depending on a number or a string See also private Mark the beginning of a s
401. with shades of gray Syntax cm graycm cm graycm n Description graycm n creates a color map with n entries corresponding to gray shades from black to white The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also pluezyet tow2redcnl cyan2magentacm green2yellowcm huecm magenta2yellowcm TOAT green2yellowcm Colormap with shades from green to yellow Syntax green2yellowcm green2yellowcm n Description green2yellowcm n creates a color map with n entries corresponding to color shades from green to yellow The color map is an n by 3 array with one color per row columns correspond to red green and blue components as real numbers between 0 to 1 maximum intensity The default value of n is 256 See also colormap blue2yellow2redcm e OREAL OE interprgbcm magenta2yellowcm Libraries colormaps 565 huecm Colormap with hue from red to red throw green and blue Syntax cm huecm cm huecm n Description huecm n creates a color map with n entries corresponding to color shades with hue varying linearly from red back to red throw green and blue In HSV hue saturation value space saturation and value are 1 maximum The color map is an n by 3 array with one color per row columns correspond to red green and blue components as re
402. wo arguments or in a vector of two elements for a rectangular matrix An additional input argument can be used to specify the type of the result It must be the string double single int8 int16 int32 int64 uint8 uint16 uint32 or uint64 64 bit arrays are not supported on all platforms 252 Sysquake for ATEX Examples eye 3 100 010 001 eye 2 3 100 010 eye 2 int8 2x2 int8 array 10 01 See also find Find the indices of the non null elements of an array Syntax ix find v s1 s2 find M s1 s2 x find M find n find n dir Description With one output argument find v returns a vector containing the indices of the nonzero elements of v v can be an array of any dimen sion the indices correspond to the internal storage ordering and can be used to access the elements with a single subscript With two output arguments find M returns two vectors contain ing the subscripts row in the first output argument column in the second output argument of the nonzero elements of 2 dim array M To obtain subscripts for an array of higher dimension you can convert the single output argument of find to subscripts with ind2sub With three output arguments find M returns in addition the nonzero values themselves in the third output argument With a second input argument n find limits the maximum number of elements found It searches for
403. x c sqrt x Then fsolve is used with the value of c passed as an additional argu ment x fzero f 0 100 10 x 2 4479 f x 10 1 9984e 15 An anonymous function can be used to avoid passing 10 as an ad ditional argument which can be error prone since a dummy empty option arguments has to be inserted zero x f x 10 0 100 See also ode23 ode45 Ordinary differential equation integration Syntax ode23 fun t0 tend yO ode23 fun t0 tend y0 options ode23 fun t0 tend y0 options ode45 fun t0 tend yO ode45 fun t0 tend y0 options ode45 fun t0 tend y0 options oo ct et et et et lt lt lt M lt tou on uo ue a 296 Sysquake for TEX Description ode23 fun tQ tend y and ode45 fun t0 tend yO integrate numerically an ordinary differential equation ODE Both functions are based on a Runge Kutta algorithm with adaptive time step ode23 is low order and ode45 high order In most cases for non stiff equations ode45 is the best method The function to be integrated is either spec ified by its name or given as an anonymous or inline function or a function reference It should have at least two input arguments and exactly one output argument function yp f t y The function calculates the derivative yp of the state vector y at time t Integration is performed over the time range specified by the sec ond argument t0 tend starting from the
404. x y and uminus x to x They can be used to redefine these operators for objects Example 2 3 1 1 2 3 5 LME Reference operators 131 2 3 3 4 2 12 2 2 3j 2 2 3j See also operator conj Operator Matrix multiplication Syntax xX Yy M1 x M2 M x mtimes x y Description xx y multiplies the operands together Operands can be scalars plain arithmetic product matrices matrix product or mixed scalar and matrix mtimes x y is equivalent to x y It can be used to redefine this operator for objects Example 2 3 6 1 2 3 4 3 5 See also prod Operator Scalar multiplication 132 Sysquake for TEX Syntax xX y M1 M2 M x times x y Description x y is the element wise multiplication If both operands are matrices with a size different from 1 by 1 their size must be equal the mul tiplication is performed element wise If one operand is a scalar it multiplies each element of the other operand times x y is equivalent to x y It can be used to redefine this operator for objects Example 1 2 3 5 3 10 3 4 2 6 8 See also Operator Matrix right division Syntax a b A B A b mrdivide a b Description a b divides the first operand by the second operand If the second operand is a scalar it divides each element of the first operand If the second operand is Otherwise it must be a square matrix
405. xample P H hess 1 2 3 4 5 6 7 8 9 P 1 0 0 0 0 4961 0 8682 0 0 8682 0 4961 H 1 3 597 0 2481 8 0623 14 0462 2 8308 0 0 8308 4 6154e 2 PxHxP ans 1 2 3 4 5 6 7 8 9 See also 224 Sysquake for TEX inv Inverse of a square matrix Syntax M2 inv M1 Description inv M1 returns the inverse M2 of the square matrix M1 i e a matrix of the same size such that M2 M1 M1 M2 eye size M1 M1 must not be singular otherwise its elements are infinite To solve a set of linear of equations the operator is more efficient Example inv 1 2 3 4 2 1 1 5 0 5 See also kron Kronecker product Syntax M kron A B Description kron A B returns the Kronecker product of matrices A size m1 by n1 and B size m2 by n2 i e an m1 m2 by n1 n2 matrix made of m1 by n1 submatrices which are the products of each element of A with B LME Reference linear algebra Example kron 1 1 2 3 4 ones 2 1 2 1 2 1 2 3 4 3 4 W W e A ANNS See also kurtosis Kurtosis of a set of values Syntax k kurtosis A k kurtosis A dim Description 225 kurtosis A gives the kurtosis of the columns of array A or of the row vector A The dimension along which kurtosis proceeds may be specified with a second argument The kurtosis measures how much values are far away from the mean It is 3 for a normal distribution and positive for a distribution which has more
406. xisting variables but cannot create new ones Examples eval 1 2 3 a eval 1 2 a 3 eval a 2 3 a 5 Il See also feval exist Existence of a function or variable Syntax b exist name b exist name type Description exist returns true if its argument is the name of an existing function or variable or false otherwise A second argument can restrict the lookup to builtin functions builtin user functions function or variable variable LME Reference miscellaneous functions Examples exist sin true exist cos function false See also feval Function evaluation Syntax argoutl feval fun arginl Description 101 yl y2 feval fun x1 x2 evaluates function fun with in put arguments x1 x2 etc Output arguments are assigned to yl y2 etc Function fun is specified by either its name as a string a function reference or an inline function Examples y feval sin 3 5 0 1411 0 7568 0 9589 y feval inline sin 2 x 3 5 0 2794 0 9894 0 544 See also fevalx Function evaluation with array expansion Syntax Y1 fevalx fun X1 102 Sysquake for ATEX Description Y1 Y2 fevalx fun X1 X2 evaluates function fun with input arguments X1 X2 etc Arguments must be arrays which are ex panded if necessary along singleton dimensions so that all dimensi
407. y LME Reference arrays 267 Syntax B permute A perm Description permute A perm returns an array with the same elements as A but where dimensions are permuted according to the vector of dimensions perm It is a generalization of the matrix transpose operator perm must contain integers from 1 to n dimension i in A becomes dimen sion perm i in the result Examples size permute rand 3 4 5 2 3 1 453 See also rand Uniformly distributed random number Syntax x rand M rand n M rand nl n2 M rand nl n2 rand seed s Description rand builds a scalar pseudo random number uniformly distributed be tween 0 and 1 The lower bound O may be reached but the upper bound 1 is never The current implementation is based on a scalar 64 bit seed which theoretically allows 2 64 different numbers This seed can be set with the arguments rand seed s where s is a scalar or a vector of two components rand seed s returns the empty array as output argument To discard it the statement should be followed by a semicolon rand n rand n1 n2 and rand nl1 n2 return an n by n square matrix or an array of arbitrary size whose elements are pseudo random numbers uniformly distributed between 0 and 1 268 Sysquake for ATEX Examples rand 0 2361 rand 1 3 0 6679 0 8195 0 2786 rand seed 0 rand 0 2361 See also randn Normally distri
408. y number of lines with any code unescaped like verbatim The only invalid character sequence end sysquake which marks the end of Sysquake code If for an unlikely reason you have to write Sysquake code which contains this sequence of characters add a space somewhere or if it is in a string write a character with an escape sequence e g end sysquak 145 Code in sysquake environment does not have the restrictions of expressions written with sqexpr You can define variables have con ditional executions and loops use libraries and more Contrary to sqexpr where a single result is formatted for LaTex the sysquake environment only inserts any text produced by the code it contains in the LaTeX source This text is still processed by LaTex To format it you can either do it yourself inserting LaTeX commands or use the Sysquake function sqlxvaLue In the code fragment above magic 4 without semicolon writes the result as an assignment to vari able ans as raw text ans 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 In a LaTeX equation this does not look like a matrix With sqlxvalue magic 4 we obtain ensuremath left begin array cccc 16 amp 2 amp 3 amp 13 5 amp 11 amp 10 amp 8 9 amp 7 amp 6 amp 12 4 amp 14 amp 15 amp l end array right which is rendered more nicely like with sqexpr Short code fragments such as variable assignments can be in cluded with the sqeval command No
409. yntax b nchoosek n k Description nchoosek n k gives the number of combinations of n objects taken k ata time Both n and k must be nonnegative integers with k lt n Examples nchoosek 10 4 210 nchoosek 10 6 210 192 Sysquake for ATEX See also factorial pdf Probability density function Syntax pdf distribution x pdf distribution x al pdf distribution x al a2 y y y Description pdf distribution x gives the probability of a density function The distribution is specified with the first argument a string case is ig nored t and T are equivalent Additional arguments must be pro vided for some distributions See cdf for the list of distributions See also cdf pi Constant Tr Syntax x pi Description pi is the number r up to the precision of its floating point represen tation Example exp 1j pi 1 LME Reference mathematical functions 193 See also real Real part of a complex number Syntax re real z Description real z is the real part of the complex number z or z if z is real Examples real 1 2j 1 real 1 1 See also reallog Real natural base e logarithm Syntax y reallog x Description reallog x gives the real natural logarithm of x It is the inverse of exp for real numbers The imaginary part of x is ignored The result is NaN if x is negative Example reallog 1 0 1 10 1 2j nan
410. ype 2 filter The result is given as zeros poles and gain if there are three output arguments or as nu merator and denominator coefficient vectors if there are two output arguments cheby2 n w0 where w0 is a scalar gives a nth order digital low pass filter with a cutoff frequency of w0 relatively to half the sampling frequency cheby2 n wl wh where the second input argument is a vector of two numbers gives a 2nth order digital bandpass filter with pass band between wl and wh relatively to half the sampling frequency cheby2 n w0 high gives a nth order digital highpass filter with a cutoff frequency of w0 relatively to half the sampling frequency cheby2 n wl wh stop where the second input argument is a vector of two numbers gives a 2nth order digital bandstop filter with stopband between wl and wh relatively to half the sampling frequency With an additional input argument which is the string s cheby2 gives an analog Chebyshev type 2 filter Frequencies are given in rad s 522 Sysquake for TEX See also ellip Elliptic filter P k ellip n w0 um den ellip n w0 ellip n wl wh ellip n w0 high ellip n wl wh stop ellip s Description ellip calculates a elliptic filter or Cauer filter The result is given as zeros poles and gain if there are three output arguments or as numerator and denominator coefficient vectors if there are
411. ystem difference Syntax c a b c minus a b Description a b computes the system whose inputs are fed to both a and b and whose outputs are the difference between outputs of a and b Ifa and b are transfer functions or matrices of transfer functions this is equivalent to a difference of matrices See also parallel plus minus Libraries lti Iti mldivide System left division Syntax c c a b mldivide a b Description a b is equivalent to inv a b See also tines Iti mrdivide System right division Syntax C C a b mrdivide a b Description a b is equivalent to a inv b See also times Iti mtimes System product Syntax a x b mtimes a b c c 537 538 Sysquake for TEX Description axb connects the outputs of lti object b to the inputs of lti object a If a and b are transfer functions or matrices of transfer functions this is equivalent to a product of matrices See also series Syntax h2 norm a Description norm a gives the H2 norm of the system a See also ss obsv Observability matrix Syntax 0 obsv a Description obsv a gives the observability matrix of system a which is full rank if and only if a is observable See also Libraries lti 539 Iti parallel Parallel connection Syntax c c parallel a b parallel a b ina inb outa outb Description parallel a b connects lti obje

Sysquake Remote User Manual

Contents

Download Pdf Manuals

Related Search

Related Contents