SISO-GPCIT
Contents
1. Ooo Consirainis and stability lt 2 s 4 au un Pee EX o X om mom Constraints combination lt s ss sas sses vom Rode m ek saai ee Dal ee en a a en a na ee we 10 16 19 CONTENTS List of Figures 2 1 Main view of the tool 4 2o MIN eos X 3o na 55986 EMRE eRe SHH OS 5 Zo Example ol modilcation NE sos cis gt eee Ehe XE Box oe xod 5 2 4 Example of set point modification 6 2 5 Example of including unmeasured disturbances 6 2 06 Example of constraints acting on the output 7 2 7 Control signal and control increment 7 2 8 Modification of N in the control increment window 8 2 9 Constraints in input amplitude and input increment 8 2 10 Location of poles and zeros in the s and z planes 9 2 11 Modification of the length of the simulation 9 LIT QUIM Pn i ee 10 2 13 Scaling of poles and zeros 10 2 14 Values in the task 11 215 Paras CON e s o er ce ew 11 215 PAU DOTES uu suu ox oo ck Xo o ox Rx E x Wo E xo e d 12 2 17 Modification of the 4 12 2 18 Sliders over control parameters 13 111 2 19 2 20 245 2 23 2 24 2 25 3 1 3 22. amp Poles Zeros Integrator Model 0 Plant Constraints in y Constraints u Constraints in Du Figure 2 1 Main view of the tool 2 1 GRAPHICS Constraints in y 0 20 40 Figure 2 2 Plant Model e Two dotted blue vertical lines represent the prediction horizons and Na To modify the value of each of them it is only required to place the mouse pointer over them and drag them to the left or to the righ to decrease or increase their values example in which the upper prediction horizon is modified is shown in figure 2 3 where it can be seen how the system response changes when Na is modified from 4 to 8 sample times Constraints in y Constraints in y 20 40 a Value of 4 b Value of 8 Figure 2 3 Example of modification of Na e Another element is the set point represented as a step using violet color small circle of the same color is placed over this curve in the place corresponding to the sample time in which the step is produced Using this circle it is possible to modify both the amplitude and the time instant in which the step is introduced in the same way explained with the prediction horizons that is using the mouse pointer and dragging it vertically amplitude modification or horizontally modification of the step time Figure 2 4 shows an example of set point modification e Over the y 0 line a set of small circles can be observed used to gener
3. e t 1 0 82 By t 0 4 0 6z u t 1 3 1 where C z is considered equal 1 without noise polynomial the prediction and control horizons 1 3 and the weighting factors 0 8 and 1 For this controller configuration the results obtained can be seen in 3 1 Constraints in y Constraints in u Constraints in Du Figure 3 1 Simple Example Initial Configuration 19 20 CHAPTER 3 EXAMPLES By modifying the values of the parameters using the interface the effect on closed loop performance can be analysed Figure 3 2 a shows the effect of reducing the control horizon in two sample instants while figure 3 2 b presents the results for the configuration 1 5 1 10 1 Constraints in y Constraints in u Constraints in Du a 1 No 3 N 1 0 8 and 1 Constraints in u Constraints in Du b 1 5 m L 10 and 1 Figure 3 2 Modification of control parameters This example is an excellent starting point to put into practice the basic achieved concepts on GPC 3 2 T polynomial As it has been commented by several authors 3 8 5 the use of the T polynomial improves the robustness of the GPC controller in the face of modelling errors and unmodelled dis turbances example where the disturbance rejection problem is analyzed can be found in 7 and it has been included in the tool under the denomi
4. a horizontal 2 3 MENUS 17 SS te Transfer Function num den 223141 Cancel Figure 2 25 Dialog box to modify the transfer function line The first part has several discretization methods Zero Order Hold Bilinear Bilinear with Prewarping Forward Rectangle Backward Rectangle y Pole Zero Matching The default method is Zero Order Hold In any time it is possible to change the method and the result will be updated in all the elements of the tool Moreover in the Discrete Time Poles graph the zone of the z plane in which the discretized poles will be placed is represented in yellow color related to stability issues An example is shown in figure 2 24 The second group of this menu shows four dialog boxes to allow modifying the plant and model transfer functions the T Polynomial and the sample time Continuos Transfer Function Model Continuos Transfer Function Plant T Polynomial and Sample Time The three dots indicate that when selecting any of these elements a dialog box will appear to allow performing the desired operation When selecting any of the two first options the appearing dialog box will allow introducing or modifying the transfer function in continuous time of the model and the plant in vectorial polynomial form following the Matlab conven tions An example of this dialog is shown in figure 2 25 other two options open dialog boxes to modify the T polinomial including its val
5. bar 11 20 40 Step amplitude 0 127 Sample instant of step 16 b Value of the disturbance in the task bar can be distinguished Process parameters control and simulation parameters Constraints and Others Control Parameters Tm 0 1 Delta 1 Lambda 1 zeros T 0 simulation Parameters Move O Add Remove Others Constraints Gots Poles Feros Integrator Clipping Ju Du DK Overshoot Monotonous WMP Modto ato Finalstate Integral Band Integra 10 Mi 4 1 TYMax 95 35 Gamma 1 1 Figure 2 15 Parameters control The first category Process parameters is shown in the left part of figure 2 2 Using this section it is possible to modify both the plant and model dynamics A set of radiobuttons permits moving adding or removing poles zeros and integrators of the model and the plant As it can be observed in the figure the radiobuttons are divided in three blocks the first of them indicating the objective over which the changes will be performed model plant or both the second indicating the operation move add or remove and the third the element over which the operation is carried out pole zero or integrator The use of this section consists on selecting the desired option in each one of the three groups and carrying it out on the Continuous time poles and zeros graph For instance to add a ple to t
6. included in the tool and checkbox correspondences The activation of the U DU Y Final state and Y Band constraints automatically produces the appearance of horizontal lines or curves for output band in the corresponding graph The first two will appear in Constraints U and Constraints DU respectively and the last three constraints in Constraints Y as was commented in the previous section Figures 2 6 and 2 9 present some examples Figure 2 6c shows the case for output band constraints in an step response exponential form first order system which amplitude is modifiable by accessing the circles on the right part of them and the time constant by using the sliders TMax and TMin for the upper and lower curves respectively These sliders represent the Je pue a parameter in a first order representation given by G s 1 An example is shown in figure 2 20 Others Others Constraints C Clipping u L SER O overshoot C Monotonous C NMP pe 1 Clipping Qu _ Final stat C Overshoot Monotonous I NMP _ Final state Integral Y Band Integral 10 Integral 10 Ni 4 TYMax 95 Fl Gamma 1 1 M 244 PTOS Termin 185 M Gamma 1 1 0 20 40 a Initial situation Tmax 9 5 Tmin 9 5 b Final situation Tmax 24 4 Tmin 18 5 Figure 2 20 Example of output band constraints In the same way for the constraints Overshoot or Integral there
7. that could lead the system to unstability In what follows a set of examples are included to show the main characteristics of these constraints 22 CHAPTER 3 EXAMPLES 3 3 1 Physical limitations and security contraints This set of constraints is constituted by those affecting the amplitudes of the input and output of the process In order to show the results achievable with these constraints an open chi 2 6 135 48 04 loop unstable plant has been selected which is described by 5 845614504 Using GPC it is possible to obtain a stable closed loop system for this process for tuning knobs 1 No 4 4 A 1 and 6 1 using a sample time of 0 1 seconds as it is shown in figure 3 4 Constraints in y Constraints in u Constraints in Du 0 20 a Unconstrained case for an unstable plant Constraints in u Constraints in Du 0 20 0 20 b Output amplitude constraints 0 8 and Ymin 0 0 Figure 3 4 Unstable plant Output amplitude constraints e Output amplitude constraints this constraints determines the maximum and minimum values within which the output signal must lie As it can be seen in figure 3 4 a the output of the process suffers from small oscillations in the transient dynamic evlu tion Using this kind of constraint it is possible to eliminate such oscillations using a maximum and minimum limiting values of 0 8 and 0 0 respectively see figure 3 4 b e Input constraints these constrai
8. the control increment is represented in brown color as it is shown in figures 2 7 a and 2 7 b respectively Constraints in u Constraints in Du 0 4 O2 4 0 2 0 4 0 20 40 0 20 40 a Control signal U b Control signal increment DU Figure 2 7 Control signal and control increment e In both figures vertical magenta line represents the control horizon lo modify its value it is only necessary to place and drag the mouse pointer over this line being possible to increase or decrease its value An example of a change from 3 to 9 sample times is shown in figure 2 8 e When the constraints in the input amplitude and increment signals are activated or the clipping option their maximum and minimum values are represented by two horizontal 8 CHAPTER 2 SECTIONS THE TOOL Constraints in Du Constraints in Du 0 4 02 4 0 2 D 20 40 0 20 40 Ny b Na 9 Figure 2 8 Modification of N in the control increment window lines in both graphics in the same way the output amplitude constraint is represented in the Constraints in Y graph In this way by modifying the position of these hori zontal lines the value of the associated constraint is changed Those constraints can be appreciated in figure 2 9 Constraints in u Constraints in Du a U constraint b DU constraint Figure 2 9 Constraints in input amplitude and input increment Two more graphics are shown in the
9. 3 3 3 4 3 9 3 6 De 3 8 3 9 3 10 3 11 3 12 SM ES 3 14 3 19 3 16 LIST OF FIGURES Modification of the number of set point 13 Example of output band constraints 14 Example of final state constraint 2 2 2 15 Example of Clipping s saas ard 2 a a omo d dia ok hd s 15 p EE uuo Som v9 X quedo dmm md pzq 423 e 16 Visualizing the discretization methods 16 Dialog box to modify the transfer 17 Simple Example Initial Configuration 0 008 19 Modification of control parameters 20 Disturbance rejection 21 Unstable plant Output amplitude constraints rn 22 Unstable plant Input 23 Nonminimum phase behavior 24 Monotonous behavior constraint cr 25 Overshoot uou nob o9 o9 99 9 3x od dex 9 9 Rd EU dcs 26 Output band constraint rios a a ER m GR Roo oos 26 Unstable output Without CRHPC 27 ad El Se we the eR ew 28 Modification or A dn CRHPCG s o uos 44684 93 eed ae 28 Constraints combination 2 4 ae ee eh 9o en ini nn 29 EXPL BRE BEA cu 4C 409 Rh deor ua 30 First stability theor
10. 32 CHAPTER 3 EXAMPLES Constraints in y Discrete Time Poles a Initial situation for Z 0 STERNE ny Discrete Time Poles b Unstable system for A 0 Discrete Time Poles Constraints in y 2 2 c Stable system for N gt No 2 Figure 3 15 First stability theorem of Zhang 3 4 STABILITY ISSUES 33 Constraints in Discrete Time Poles Re ET NS OK 2 1e3 2 0 2 0 100 Unstable system for No 6 and N 1 Constraints in y Discrete Time Poles 2 o 1 ho 0 100 2 0 2 b Variation of A Constraints in y Discrete Time Poles 2 0 100 2 0 c Stable system with N gt 6 Figure 3 16 Second stability theorem of Zhang 34 CHAPTER 3 EXAMPLES Bibliography 1 E F Camacho and C Bord ns Model predictive control Springer 1999 2 D W Clarke C Mohtadi and Tuffs P S Generalized predictive control parts i and ii Automatica 23 1987 no 2 137 160 3 D W Clarke and C Mohtadi Properties of generalized predictive control Automatica 25 1989 no 6 859 875 4 D W Clarke and Scattolini Constrained recending horizon predictive control IEE Proc Pt D 138 1991 no 4 347 354 5 D W Clarke Designing robustness into predictive control 1991 6 J Maciejowski Constrained predictive control Academic Press 2000 Y Piguet Sysquake User manual Clerga 2000 8 B D
11. Constraints in u Constraints in Du a A 1 Nm 10 and im 39 Constraints in y Constraints in u Constraints in Du 0 2 b A 1000 Nm 10 and m 3 Figure 3 12 Modification of A in CRHPC 3 9 CONSTRAINTS 20 3 3 4 Constraints combination The tool can also be used to see how a combination of constraints affect the closed loop system performance of any linear system The same plant used in the case of Physical Security Constraints has been used in this section Figure 3 13 a shows the unconstrained output evolution for 1 4 4 1 and 1 In the first case the overshoot constraint has been included with 1 with input increments constraints max and min of 0 256 and 0 459 obtaining the result shown in figure 3 13 b When applying such constraints the kick back phenomenon is produced being possible to suppress by activating the monotonous behavior constraint as shown in figure 3 13 c Constraints in y Constraints in u Constraints in Du 0 20 0 20 a Unconstrained output Constraints in y Constraints in u Constraints in Du 0 20 b Output overshoot and constraints Constraints in u Constraints in Du 0 20 0 20 c Incorporation of monotonous behavior constraints Figure 3 13 Constraints combination 30 CHAPTER 3 EXAMPLES 3 4 Stability issues As pointed out by Maciejowski 6 predictive control using the receding horizon idea is a feedback contro
12. Robinson and D W Clarke Robustness effects of a prefilter in generalised predictive control IEE Proc Pt D 138 1991 no 1 1 8 9 J A Rossiter Model based predictive control A practical approach CRC Press 2003 10 P O M Scokaert Constrained predictive control Phd thesis University of Oxford 1994 11 J Zhang and Y Xi Some stability results Int J Control 70 1998 no 5 831 840 12 J Zhang Property analysis of gpc based on coefficient mapping IFAC 13th Triennial World Congress San Francisco C 1996 no 5 457 462 35
13. SISO GPCIT GENERALIZED PREDICTIVE CONTROL INTERACTIVE TOOL FOR SISO SYSTEMS OFTIMLCADSIE Errores wma arm b Pre en VR utt Mir Authors Jos Luis Guzm n Manuel Berenguel Sebasti n Dormido Referencia al internal report GPCIT Generalized Predictive Control Interactive Tool O Jos Luis Guzm n Manuel Berenguel Sebasti n Dormido Universidad de Almer a Dpto de Lenguajes y Computaci n rea de Ingenier a de Sistemas y Autom tica Ctra Sacramento s n 04120 Almer a Spain Tel 34 950 015683 Fax 34 950 015129 E mail joguzman beren ual es Universidad Nacional de Educaci n a Distancia Dpto de Inform tica y Autom tica Avda Senda del Rey n 7 28040 Madrid Spain Tel 34 91 Fax 34 91 E mail sdormido dia uned es submitted to IEEE Control Systems Magazine 2003 Contents Introduction Sections of the tool Bl ee Ae eee Eee ae 9 5 9 24449535 2 4 Parameters 8 eae OM 52 252 95 E WO 8 RB 4 4o 34 eX 4 Examples 3 1 Unconstrained case nos eze x 3 4 9 xe RE eue me CS dob SEA S XP d Ou WOVE oso 9 RoR RR Y Foe __ A 3 3 1 Physical limitations and security contraints 3 3 2 Performance constraints
14. and load different illustrative examples A general view of the tool can be seen in figure 2 1 In what follows the different elements of the tool are described 2 1 Graphics In this part of the tool several graphs containing the output of the system the control signal and control increments the plant and model poles and zeros locations both in s and z planes are shown The figure located at the lower left part of the screen 2 1 shows the output of the system with the name Constraints y Several elements differentiated in shape and color are shown e Both the output of the real plant in green color and that of the model in orange color are displayed This distinction allows analysing the system response in the face of modeling errors see figure 2 2 CHAPTER 2 SECTIONS OF THE TOOL fj untitled SysQuake File Edit Settings Plots Figure Layout View Help ol Continuous Time Poles and Zeros Discrete Time Poles and Zeros Control and simulation parameters Control Parameters 2 Tm 0 1 Delta 1 Lambda 0 8 Zeros T 0 Simulation Parameters Nend 25 SP 1 0 Others Constraints C Clipping Ju Jou 20 0 20 DK C Overshoot Monotonous C NMP Operations over poles zeros 2 _ Final state _ Integral Y Band Selection Model Plant Both Integral 10 IE 2 0 2 TYMax 95 MT 95 Mode Move O Add Remove Delays Gamma 141 Gp sy
15. ate unmeasured disturbances in the form of step black circles impulse red circles and noise green circles By modifying their vertical position the amplitude of the selected disturbance can be changed and modifying their horizontal position the instant from which the disturbance acts on the output of the system can be changed activation of the step and impulse disturbances is immediate To incorporate the noise disturbance a checkbox CHAPTER 2 SECTIONS OF THE TOOL Constraints in y Constraints in 0 20 40 0 20 40 Amplitude 0 8 Instant 0 b Amplitude 1 288 Instant 11 Figure 2 4 Example of set point modification has to be activated in the parameters control block as will be commented in the next section example of step disturbances in sample times 13 and 24 with amplitudes 0 293 and 0 007 respectively is shown in figure 2 5 In the instant 29 an impulse of amplitude 0 493 is also introduced and finally noise disturbance of amplitude 0 1943 is activated in the 35 sample time The composition of all these signals is shown as a black dotted line Constraints in y Constraints in y 0 20 40 Without disturbances b With disturbances Figure 2 5 Example of including unmeasured disturbances The rest of modifiable elements are the following constraints output amplitude con straints final state constraints and band constraints When those constraints are ac tivated in the cor
16. ays Model 1 1 Plant 11 7 Model 1 4 I Plant 1 1 p 0 20 0 20 a Initial situation model delay of 1 1 b Final situation model delay of 1 4 Figure 2 17 Modification of the delay Another category is Control and simulation parameters shown in the right upper part of figure 2 2 helping the user to modify control and simulation parameters The first group of this category is Control parameters in which GPC control parameters are changed 2 2 PARAMETERS SECTION 13 A control effort weighting factor 6 tracking error weighting factor Tm sample time and T polynomial to enhance robustness against unmodelled disturbances and dynamics These parameters are shown in figure 2 18 effect of the variation of those parameters will be studied in the section devoted to show some examples Control Parameters Tm 0 2 Delta 0 9 Lambda 3 2 zeros 0 Figure 2 18 Sliders over control parameters The other group of this category is Simulation Parameters from which it is possible to modify the final time of the simulation Nend and the number of set point step changes SP Regarding the first one an increase or decrease of the simulation time will produce the same effect that if this change should be performed in the graphics using the red triangle placed on the x axis as was commented in the previous section figure 2 11 Then when this parameter is modified in the graphs its value is automaticall
17. could give rise to collision To show the effect of this constraint the same plant used in the physical and security limits sec tion has been used Figure 3 4 a shows the oscillatory response To limit overshoot the overshoot constraint can be activated using the overshoot checkbox obtaining the result shown in figure 3 8 a for Gamma 1 1 and in figure 3 8 b for Gamma 1 01 where it can be seen how the overshoot has practically disappeared e Output band constraint this constraint is useful when it is desirable to limit the dy namical evolution of the system within band For instance in the food industry it is normal that in some operations a temperature profile must be followed within deter mined tolerances This kind of requisite can be included in the control system using the output band constraints as shown in figure 3 9 where the result of the application of this kind of constraint to the example presented in figure 3 4 a is shown The shape of the bands is that of the step response of a first order system e Integral constraint There are several situations in which more than requiring the output to exactly follow a set point it is required that the integral of the output is limited within determined time range An example can be found in the case of protected crops climate control where desired temperature integrals must be achieved 26 CHAPTER 3 EXAMPLES Constraints in y Constraints in u Constraint
18. em in the design stage of the GPC algorithm The another checkbox named dk activates the noise disturbance affecting the output of the system The green circle in the line y 0 of the graph Constraints in Y allows modifying the time instant and the amplitude of the noise disturbance but this only will affect the output if the corresponding checkbox is active 2 16 CHAPTER 2 SECTIONS OF THE TOOL Settings Plots Figure Layout View w Zero Order Hold Bilinear Bilinear with Prewarping Back Rect For Rect PolelZero Matching Continuous Time TF Model Continuous Time TF Plant T Polynomial Sampling Time wv Simple example T Polynomial Mon minimumn phase Physical Security Constraimts Oscillatory behaviour Constraints and stability CRHPC Constraints and stability GPC v Sukoscale wv Show close loop poles Tutorial in HTML and PDF Figure 2 23 Menu Settings 2 3 Menus The third and last section of teh tool is men Settings that is placed in the menu bar of the tool To access the options it is necessary to place the left mouse buttom over it obtaining the result shown in figure 2 23 Discrete Time Poles Discrete Time Poles Discrete Time Poles 2 2 2 2 a 2 0 2 0 2 Zero Order Hold method b Backward Rectangle method c Forward Rectangle method Figure 2 24 Visualizing the discretization methods As it can be seen in the figure the menu is divided into four parts separated by
19. em of 7 32 Second stability theorem of Zhang 33 List of Tables 2 1 Constraints included in the tool and checkbox correspondences vi LIST OF TABLES Chapter 1 Introduction SISO GPCIT Single Input Single Output Generalized Predictive Control Interactive Tool is an interactive tool for control education developed with Sysquake 7 Its objective is to help the students to learn and understand the basic concepts involved in Generalized Predictive Control GPC 2 3 Using this tool the student can put into practice the adquired knowledge on this control technique using simple examples of unconstrained cases effect of plant model mismatch and robustness issues disturbance rejection effect of constraints in the design and performance of the controller stability issues etc Thus using this tool the student can better understand the underlying theory interactively It is possible to analyse how the closed loop system response is affected by changes in different design parameters such as weighting factors on tracking errors and A on control effort the prediction Ni and No and control N horaizons the sample time the T polynomial etc At the same time different constraints related to physical limits or security issues desired performance or stability can be selected and activated to see their effect on the performance of the controlled system This docume
20. exist a series of sliders that allow modifying the value of the associate values In the Overshoot case a slider for the Gamma parameters is included to determine the percentage of allowed overshoot For the Integral constraint two sliders are at hand to modify the valued desired for the integral of the output slider Integral and the horizon over which this constraint has to be fulfilled When activating the constraint of Final State two vertical cyan lines appear in the graph Constraints in y representing the horizons Nm Nm m for which the error has to be zero 2 3 MENUS 15 Constraints in sy 0 20 Figure 2 21 Example of final state constraint Constraints in u Others x Clipping DK 0 20 40 Figure 2 22 Example of clipping figure 2 21 Finally the Monotonous and NMP constraints avoid when activated allow avoiding oscillatory and nonminimum phase behavior Notice that each time a constraint is activated these are introduced within the optimizer that tries to minimize the GPC cost function J subject to such constraints The last category others is composed by two checkboxes one of them associate to the clipping parameter that permits activating the saturation of the control signal using the same horizontal lines that for the U and DU constraints An example is shown in figure 2 22 in following sections it will be observed the difference between clipping the control signal or integrating th
21. ge in N from 1 to 3 Figure 3 14 GPC ystrategy Other interesting theorems as those developed by Zhang 121111 can be practically demonstrated to the students using the tool The first of this theorems says 1 Assume that the system is open loop stable if N 1 and 0 then there exists No 3 4 STABILITY ISSUES 31 so that GPC system is stable for all Na gt No For testing this theorem the Simple example available in the tool has been used Figure 3 15 a shows the response of the system with 1 1 1 4 0 8 and 6 1 where 0 The response for the same configuration but with 0 is shown in figure 3 15 b where the closed loop system is unstable Using the theorem the value for No must be found and this value has been set to 2 When Na gt the system is stable as can be seen in Figure 3 15 c The second theorem says that 1 To stable plant if 1 then there exists No so that the closed loop system is stable when N2 N1 gt NO To test this theorem the Non minimum phase example available in the menu Settings of the tool has been used With 1 6 1 0 3 and 1 the system is unstable as it is shown in Figure 3 16 a It can be proved that modifying the value of A the system stability is not reached as can be seen in Figure 3 16 b Figure 3 16 c shows that a value for No 6 has been found and with Na 7 and 1 the system is stable
22. he plant it should 12 CHAPTER 2 SECTIONS OF THE TOOL be only necessary to select in the first group the option Plant Add in the second and Poles in the third After this selection the mouse pointer can be placed on the location in which the insertion of the pole has to be performed in the s plane and then a single click of the left mouse button will perform the action This example is shown in figure 2 16 Transfer function 20 20 0 220 20 20 0 20 0 Selection 4 Model Plant gt Both Selection 2 Model Plant gt Both Made Move Q Add gt Remove Mode gt Move Add gt Remove Gate Poles O Zeros gt Integrator ips Poles Zeros C Integrator a Initial situation 1 pole b Final situation 2 poles Figure 2 16 Adding poles Below the radiobuttons two sliders can be found allowing the change of the delay of the plant and the model independently The use of sliders consists of clicking with the left mouse button on the left or on the righ of the black thin vertical bar appearing in each of them to increase or decrease its value Another way is to place the mouse over the vertical bar and maintaining the left mouse buttom pressed perform a displacement to the left or to the right An example of the modification of the delay is presented in figure 2 17 where a change in the delay of the model from 1 1 to 1 4 sampling instants is performed Constraints in y Constraints in y Delays Del
23. ing the NMP checkbox in the parameters group Constraints in y Constraints in u Constraints in Du 0 100 0 100 Nonminimum phase behavior Constraints in y Constraints in u Constraints in Du G 0 100 0 100 b Avoiding nonminimum phase behavior Figure 3 6 Nonminimum phase behavior constraint e Monotonous behavior constraint many systems present an oscillatory behavior during the rising phase of the transient that is before reaching the steady state regime kick back To appreciate this phenomenon an example extracted from 1 has been used with a plant given by G s E For a sample time of 0 1 seconds and 1 11 11 50 and 6 1 the closed loop output presented in figure 3 7 a is obtained By activating the monotonous checkbox in the constraints section such 3 9 CONSTRAINTS 25 behavior is suppressed as it is shown in figure 3 7 b This example is included in the tool under the denomination Oscillatory behavior Constraints in y Constraints in u Constraints in Du 100 0 100 a kick back in the output Constraints in y Constraints in u Constraints in Du 0 100 b kick back suppression Figure 3 7 Monotonous behavior constraint e Overshoot constraint under several circumstances it is necessary or desirable to sup press such behavior or reduce to a desired value example is mobile robotics where for a robot moving around obstacles an overshoot
24. l policy There is therefore a risk that the resulting closed loop might be unstable mostly in the presence of constraints Some examples can be shown in 9 The tool has the possibility of showing the roots of characteristic polynomial and therefore to study the stability of GPC One of the technique that can be tested in the tool is GPCoo 10 which is a compromise between the LQ law obtained with infinite horizons and finite horizon strategies where by an infinite upper costing horizon is still used but the control horizon is reduced to a finite value mean level control is for example a special case of GPCoo where only one future control increment is postulated An example extracted from 10 of the application of this technique has been included in the tool menu Settings under the denomination Constraints and Stability GPC with the transfer function G z 1 DER mean level control is shown in figure 3 14 a for oo when using the tool a sufficiently high value has been selected and 1 To show the effect of modifying control parameters a change from 1 to 3 in the control horizon is shown in figure 3 14 b Notice that in the simulation the value of infinite for the final prediction horizon cannot be included using a value of 100 as a good approximation Constraints in y Constraints in u Constraints in Du a Mean level control Constraints in y Constraints in u Constraints in Du 0 50 b Chan
25. le and in b how the closed loop system roots lie out of the unitary circle Constraints in y Constraints in u Constraints in Du Blob CT A A A A ee Se Se A y 0 50 0 50 a Unconstrained unstable output Discrete Time Poles 0 50 2 0 2 b Poles out of the unitary circle Figure 3 10 Unstable output Without CRHPC By activating the final state constraint with Nm 4 and m 3 the system can be stabilized as shown in figure 3 11 a To activate this constraint in the tool it is only required to activate the Final state checkbox As it was commented in the previous chapter Nm and m are modifiable from the Constraints in Y graph accessing the dotted vertical cyan lines In 4 it can be observed that for small values of A the effect of modifying N Na Ny Nm does not affect the system output as can be seen in figure 3 11 5 Nevertheless small modifications in produce an immediate effect as shown in figure 3 12 When A is increased the oscillations and velocity of the system response decrease This example has been included in the tool under the denomination Constraints and Stability CRHPC 28 CHAPTER 3 EXAMPLES Constraints in y Constraints in u Constraints in Du 50 a Stable output with CRHPC 10 9 Nm 4andm 3 Constraints in y Constraints in u Constraints in Du 0 5 0 50 b Effect of N N Na Nm 10 Figure 3 11 CRHPC activation Constraints in y
26. nation of T polynomial The plant is described by G z 1 012 and for a configuration defined by 1 Na 10 N 1 0 and 6 1 the response shown figure 3 3 a is obtained By using the tool two disturbances are included one of step shape with amplitude 0 25 in the instant 34 and the other of random noise after instant 77 where for a value of T 1 the result shown in figure 3 3 b is obtained In this figure it can be seen how the error in the face of load changes is corrected very fast but the control signal posses a high variance inducing continuous oscillations in the output To improve this behavior 1 0 827 can be used 3 9 CONSTRAINTS 21 deteriorating the load disturbance rejection but reducing the variance of the control signal figure 3 3 c Constraints in y Constraints in u Constraints in Du 100 0 100 a Output without disturbances Constraints in u Constraints in Du b Output with disturbances 1 Constraints in u Constraints in Du c Output with disturbances 1 0 827 Figure 3 3 Disturbance rejection T Polynomial 3 3 Constraints One of the principal features of predictive control is the possibility of including constraints in the design process of the controller These can be associate to physical limitations or security issues and even they can be used for performance objective The nature of allows anticipating the constraints violation
27. nt is tutorial of the tool The second chapter presents a description of its main parts and how to use it Once the user is familiarized with it the third chapter introduces a set of examples with increasing complexity which objective is to facilitate the student see some interesting features of these examples are available in the tool CHAPTER 1 INTRODUCTION Chapter 2 Sections of the tool Three types of elements can be distinguished in the tool graphics section of parameters and menus The first one is composed by the different graphs that show the output and input to the system disturbances set points and poles and zeros locations for both the model and the plant and in the s and z planes On these graphs it is possible to interactively modify a lot of parameters control and prediction horizons graphs scaling poles and zeros location constraints etc second set of elements allows modifying the control and simulation parameters those corresponding to the plant and model transfer functions as far as those constraints imposed to the system and other options The third set of elements is linked to the menu Settings from which it is possible to modify the discretization method for the plant and model transfer functions the value of the sample time the T polynomial the values of the transfer function coefficients of the plant and the model select automatic or manual scaling of figures accessing this tutorial
28. nts Y Constraint U and Constraint DU graphics which role has not been commented yet two red and one green triangles These are useful for allowing manual scaling of the figures modification of the length of the simulation and y axis scaling The red triangle close to the x axis allows modifying the final time of the simulation By clicking with the left mouse buttom on the rigth hand of the triangle the final time will be increased and decreased if clicking on the left hand of the triangle When this action is performed in any one of the graphs the value is automatically updated in the other ones An example can be observed in figure 2 11 Constraints in u b Simulation time of 80 sample instants Figure 2 11 Modification of the length of the simulation The green and red triangles that are placed on the right hand of each graph allow modi fying the y axis scale By clicking on the upper part of this triangle the scale is augmented from the actual value The scale is diminished when clicking on the lower part The green 10 CHAPTER 2 SECTIONS OF THE TOOL triangle operates inversely Another way of using these triangles is placing the mouse over them and dragging displacing to the desired place to the left or to the right to modify the length of the simulation and upstair downstair to modify the scale Figure 2 12 presents an example of their use Constraints in y Constraints in y a Unscaled output b Scaled ou
29. nts allow establishing the maximum and minimum values of both the input amplitude and input increment In figures 3 5 a and b two examples showing the use of these constraints are included typical industrial practice consists of saturating clipping the control signal when this violates the physical ranges of the control equipment sometimes leading to unstability as it is shown in figure 3 5 c where the saturation limits have been the same used for and Umin in the previous example In order to let the students analyse the effect of saturation in the 3 9 CONSTRAINTS 28 tool this can be activated by using the clipping checkbox This example was suggested by Prof Camacho from the Seville University This example has been included in the menu Settings of the tool under the name of Physical Security Constraints Constraints in y Constraints in u Constraints in Du 0 20 a Au Constraints Aumax 0 383 and Aumin 0 383 Constraints in y Constraints in u Constraints in Du 0 20 b U contraints Unas 1 0 and Umin 0 4 Constraints in u Constraints in Du 0 20 c Clipping Umax 1 0 and Umin 0 4 Figure 3 5 Unstable plant Input constraints 3 3 2 Performance constraints The optimal operating points in industry usually lie near the constraints because of econom ical reasons There are a set of constraints in GPC that permit not only establish limits within which selected variables mus
30. responding checkboxes see section 2 2 their values are represented using horizontal vertical and exponential lines respectively being possible to modify their values positions by dragging them vertically output amplitude and band or hor izontally final state The modification of the band constraint is carried out using the circles that appear at the end of those curves while the rest can be moved by selecting and dragging any point of the lines as was previously commented in the case of predic tion horizons The output amplitude constraints are represented by two red horizontal lines figure 2 6 a those of the final state with two cyan vertical lines figure 2 6 b and those of band type with two curves of the same colour with an exponential shape 2 1 GRAPHICS first order system step response figure 2 6 c Constraints in y Constraints in y Constraints in y a Output amplitude constraint b Final state constraint c Output band constraint Figure 2 6 Example of constraints acting on the output The other graphics that are included in the tool allow the visualization of the control signal and its increment resulting from the minimization of the GPC cost function Such graphs are those appearing in figure 2 1 named Constraints in U and Constraints in DU respectively The different elements shown in these graphs are e The input of the system control signal is represented in ligth green color while
31. s in Du 0 20 a Overshoot reduction Gamma 1 1 Constraints in u Constraints in Du 0 20 b Overshoot reduction Gamma 1 01 Figure 3 8 Overshoot constraint Constraints in y Constraints in u Constraints in Du 20 Figure 3 9 Output band constraint for different periods This constraint can be activated using the corresponding ntegral checkbox in the parameters section of the tool The value desired for the value of the integral and the horizon in which the value must be satisfied can be modified using the sliders Integral and N respectively 3 3 3 Constraints and stability One of the techniques relating stability with constraints is CRHPC Constrained Receding Horizon Predictive Control 1 that explicitly uses final state constraints to guarantee closed loop stability The predicted output of the system are imposed to track the reference during 3 9 CONSTRAINTS 27 a number of sample times m after a determined horizon Nm Some degrees of freedom of the future control signals are used to fit such constraint while the others are used to minimize the cost function To illustrate the effect of this constraint a set of examples extracted from 4 are shown The first of them is a system described by G z gt where the closed loop system for 1 No 11 4 10 and 1 is unstable as can be seen in figure 3 10 In a it can be observed how the output of the system is unstab
32. t lie but also avoid undesired behaviors to improve the performance There are a variety of phenomena in the dynamics of the systems that are not desirable and their suppression usually lead an overall system improvement Some of these 24 CHAPTER 3 EXAMPLES phenomena are nonminimum phase behavior kick back responses oscillations etc hese types of behavior can be avoided by including constraints sometimes leading to feasibility problems In the same way it is also possible to impose desired characteristics to the dy namical response of the system for instance that the output evolution is always within a band that the output integral matches a determined value after prescribed time etc set of examples is included in what follows to analyze the effect of these constraints e Nonminimum phase constraints this type of behavior is observed when applying a step input to the system and the transient evolution of the system is in opposite direction that the steady state value An example has been used from 1 and it has been included in the menu Settings of the tool with the denomination Non minimum Phase The process is given by G s d The output of the system for N 1 Na 30 Nu 10 0 1 6 1 and a sample time of 0 3 seconds can be observed in figure 3 6 a By activating this constraint to avoid such kind of behavior the result shown in figure 3 6 b is obtained The activation of this constraint is performed us
33. tool those corresponding to the location of poles and zeros both in the s and z planes figure 2 10 as far as the T polynomial roots and the closed loop system poles These graphics have the name continuous time poles zeros and discrete time poles and zeros lhe poles are represented using the x symbol and the zeros using o poles and zeros of the plant are of green blue color and those of the model in orange color using the same color convention that for the dynamical evolution of the plant model in figure 2 2 The closed loop system poles are shown in the z plane in blue color and the roots of the T polynomial are represented using small black squares It is possible to modify the location of poles and zeros in the s plane To do so the user has to place the mouse pointer over the desired pole or zero and drag it to the desired location At the same time the location of the pole or zero is changed in the graph corresponding to the s plane the change is simultaneously produced in the z plane graph and in the figures in which the time response of the system is represented Regarding the relation between 2 1 GRAPHICS 9 the continuous time and discrete time representations different discretization methods are at hand in the menu Settings as it is shown in section 2 3 Transfer function Discrete Time Poles Figure 2 10 Location of poles and zeros in the s and z planes There are three elements the Constrai
34. tput Figure 2 12 Output scaling Transfer function Discrete Time Poles 1 24 0 1 Initial scaling b Final scaling Figure 2 13 Scaling of poles and zeros In the graphs showing the location of poles and zeros in the s and z planes also appears a small red triangle in the x axis When clicking on the righ the scale is augmented and diminished when clicking on the left An example can be found in figure 2 13 To end this section it is convenient to mention that for all the manipulable elements placed in the graphics when the mouse pointer is placed over them their value is shown in the lower task bar of the tool For certain elements only their value is shown as happens with the prediction and control horizons value of poles and zeros constraints etc while for other both their value and the time instant in which their values are reached are shown as is the case for disturbances An example is shown in figure 2 14 It is worthwhile to comment that the effect that any change performed on these parameters is automatically updated in all the graphs and elements in the tool 2 2 Parameters section As it was previously commented this section of the tool allows modifying a lot of parameters as far as to activate or desactivate a number of options shown in figure 2 2 Four categories 2 2 PARAMETERS SECTION Constraints in y Constraints in y a Value of Na in the task bar Figure 2 14 Values in the task
35. ue in vectorial polynomial form and in the case of the sample time the actual value is shown being modifiable from the dialog box set of examples are available below this menu see next section to allow studying different control situations At the bottom last group of options if the first option Autoscale is on the scale of all graphics is automatically updated on the other hand off the scale is manual When this occurs several triangles appear in the graphs and the scale is changed dragging over them The second option Show close loop poles allows showing or hiding the closed loop roots in the Discrete Poles Zeros plot The last option Tutorial opens a web page using the default browser and shows this tutorial in html and pdf formats 18 CHAPTER 2 SECTIONS OF THE TOOL Chapter 3 Examples The objective of this chapter is to show a battery of examples that facilitate the student to study practice and understand the acquired knowledge on GPC Such examples are presented in incremental level of difficulty All are available in the menu Settings of the tool as was previously commented 3 1 Unconstrained case example The first simple example is extracted from the text of Camacho and Bordons 1 in which the objective is to see how the selection and variation of the tuning knobs 0 affect the system performance without taking constraints into account process is de scribed by
36. y updated in the sliders and vice versa The another slider allows modifying the number of set point changes in such way that each time this values is increased or decreased the changes in set point are produces in the graph Constraints y Figure 2 19 shows the results of changing from one to three changes in set point As it can be seen a new circle is generated for each step change induced in the set point signal in the sample instant corresponding to the change in the signal By displacing these circles the amplitudes and instants of change can be modified figure 2 4 Simulation Parameters Simulation Parameters Nend 40 SP 1 Nend 80 SP 3 Constraints in y Constraints in y 0 20 40 a One set point change b Three set point changes Figure 2 19 Modification of the number of set point changes The following category is Constraints corresponding with a set of checkboxes placed under the title Constraints in figure 2 2 and the sliders below these This set of checkboxes allows activating the different constraints used in the GPC framework The correspondence among the nomenclature and the complete denominations of the constraints are shown in table 2 1 The study of constrained will be done in the section devoted to explain some examples Here only a brief description of how the user can activate and modify the different constraints is included 14 CHAPTER 2 SECTIONS OF THE TOOL Table 2 1 Constraints
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