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PARADISE User Manual

1. 3 Getting started with PARADISE This chapter explains in detail how to input plant and design specifications in PARADISE It assumes that the toolbox is installed correctly For information on installation please see the file INSTALL 3 1 Starting and quitting PARADISE session Simply start Matlab session and type paradise at the Matlab prompt The message loading libraries will appear completed by done when the initialization of the Extended symbolic toolbox finishes At the same time the PARADISE main window appears From there other graphical user interfaces can be launched which are necessary for input specifications and starting algorithms PARADISE can be shut down on different ways e PARADISE automatically quits when shutting down the Matlab session e Select File Exit from the PARADISE main window This will only shut down PARADISE not your Matlab session e Type paradise quit orparadise bye at the Matlab prompt In all cases you will be asked to save modified plants 17 18 3 GETTING STARTED WITH PARADISE Main window menus File Save current model data This saves the data of the currently active model The name of this model is displayed in the title menu of the PARADISE main window The following information will be stored Symbolic system represen tation and parameter substitutions see below in a file named system name ssr Parameter settings
2. Muhler 1997 1999 1 0 D Odenthal Preface Thank you for you interest in PARADISE new toolbox for robust parametric control Robust parametric control has long tradition with roots back to Vys86 design rules for governors for stable speed control of engines and the Boundary Crossing Theorem by FD29 in 1929 which was motivated by aircraft flutter analysis However up to now robust parametric control was rarely applied in the practically oriented field of industrial control The main reason lies simply in the fact that no software was available for this type of problems As you already can tell from the name parametric plant model is required as basis of robust parametric control algorithms Handling and manipulating such kind of models is not at all trivial task It requires symbolic computer programs like e g Maple or Mathematica to facilitate the application of robust control algorithms But even with such modern programs it still requires deeper insight into the meth ods to solve robust control problems PARADISE not only keeps you away from lengthy and annoying symbolic calculations It offers you easy to use graphical user interfaces to use algorithms Simply read in Simulink model and let PAR ADISE compute the symbolic closed loop model for you Then do your closed loop specifications graphically use an adequate design method and graphically visualize the solution Chapter gives you a short introduction to t
3. considered representatives If there exist more than two controller parame ters the ones considered fixed for the design can be modified and a new design iteration can be started with these new values In case of only two controller parameters no such modifications are possible Then it is nec cessary to reconsider the design specifications or to change the controller structure A promising approach is to start with a simple controller struc ture and relaxed design specifications such that it is more likely to find a I stabilizing solution Once an initial controller could be determined the design specifications have to be strengthened towards the final ones in sev eral design iterations Design of gain scheduled controllers Not in all cases a fixed gain control can be determined for plants with vary ing parameters though it is possible to find I stabilizing solutions for a subset of the operating domain Typically the plant dynamics are only influenced heavily by a small number of uncertain parameters while the remaining ones play a minor role Assuming that one of these parame ters with major influence can be measured the following solution is suit able Compute the set of stabilizing parameters in a plane made up by the varying parameter and a controller parameter Further uncertain parame ters are considered as nominal representatives i e the idea of simultaneous I stabilization is applied From the stable region the controller p
4. e g upper and lower bounds of un certain parameters nominal values of representatives controller and fixed parameters and description of the I regions are stored in a file named systemname dat Save all model data All models currently open will be saved Remove current model The currently active model will be removed from the PARADISE ses sion If necessary the user will be asked to save the model data Remove all models This item removes all opened models from the running session If necessary the user will be asked to save the model data Save system to workspace The system equations of the currently activated model are saved to the Matlab workspace for center of q box for vertices of q box for representatives as LTI objects systemname C systemname V 4 or systemname R 4 cor responding to the number of vertices or representatives respectively Note that this item as well as the next requires the control system toolbox Save system to file This item opens the file selection box and saves the system equations of the currently activated model for center of q box 3 1 STARTING AND QUITTING PARADISE SESSION 19 for vertices of q box for representatives as LTI objects systemname systemname V 1 or systemname 1 re spectively to the selected file Exit PARADISE Quits the PARADISE session Input Plant specification Load new model This opens a file selection box from where the desir
5. the check for I stability can be carried out by chosing arbitrary points in the controller parameter plane and per forming check on these points The region where the crosshair is located in in Figure 4 9 is the stable one 46 4 THE PARAMETER SPACE METHOD Parameter specification i x Edit Varying parameters Not permitted in MultiModel mode Controller parameters kq 0 940909 knz 0 108601 Fixed parameters empty Advanced Representatives Representative 1 Associated with Gamma region Representative 2 Associated with Gamma region Representative 3 Associated with Gamma region Representative 4 Associated with Gamma region Parameter planes Figure 4 7 Parameter settings for the F4E design example A robustness analysis can not be performed for this case since no continuous model description is available 4 3 DESIGN EXAMPLES FOR THE PARAMETER SPACE METHOD 47 Gamma E ditor Ganna region 2 7 Set construction un is el e2 e3 e4 Element 1 Circle sigma Radius 0 12 6 Inverted No Element 2 Damping Damping 5 Inverted No Element 3 Circle Yes Element 4 Real interval Lover 70 Upper 15 Inverted No Gamma region 3 Gamma region 4 Figure 4 9 Simultaneous stabilization of the four flight conditions 48 4 THE PARAMETER SPACE METHOD 5 Design in an invariance plane 5
6. 1 Introduction In case of state feedback control the number of feedback gains is determined by the plant order Thus systematic method is needed to obtain robust con troller especially for high order plants Design in an invariance plane is possi ble approach This chapter presents the theoretical background implementation in PARADISE and some examples 5 2 Theoretical background This approach AT 82 Ack85 ABK 93a ABK 93b for design of state feedback is based on the parameter space approach The main idea is to iteratively shift only the most critical eigenvalues while the other eigenvalues remain at their location For nominal operating point this can be accomplished using an extension of Ackermann s formula Hereby m dimensional cross section in the n dimensional controller parameter space is determined such that n m eigenvalues are unobservable through this feedback Hence for any controller with parameters in this m dimensional sub space only m eigenvalues are shifted while the rest of the eigenvalues remain at their location For practical applications m equals two and the stability boundaries can be vi sualized by simple 2 D plot in this two dimensional cross section in controller parameter space using the parameter space approach 49 50 5 DESIGN IN AN INVARIANCE PLANE Therefore by iterating the analysis determining the most critical eigenvalues and the synthesis step of selec
7. 4 1 SOME INTRODUCING THEORY 35 boundary of intersects the real axis e g for Hurwitz stability it is s 0 and for a circle they are s and s2 where s oo is the center of the circle and R its radius Since in these cases w always equals zero and hence 4 1 becomes linearly dependent real root boundaries fall under the category of singu lar frequencies Complex root boundaries result from all complex frequencies of the I boundary In terms of eigenvalues real root boundaries describe the cases where a real pole crosses the boundary of while complex root boundaries represent all cases where pair of complex conjugate eigenvalues crosses the boundary of The parameter space approach can be applied to different types of problems Design of fixed gain controllers The stability boundaries in a plane of two controller parameters k and ko are determined i e the set of controllers for which the characteristic polyno mial p s ki k2 is This approach yields a set of controllers which permits to incorporate further design criteria to select the final controller Examples for these additional criteria are High low gain solutions can be determined by selecting the controller from the set with maximal minimal norm Select the controller from the set with maximal distance from the sta bility boundaries This guarantees some additional robustness margins in the case
8. Con sidering the operating domain in Figure 1 2 it cannot be the aim of the design to specify precisely the location of the poles in dependency of the operating point The design goal is reached if the damping is sufficiently large This is the case if 1 3 PERFORMANCE SPECIFICATIONS 5 80 60 40 Altitude 1000ft 20 Mach number Figure 1 4 Flight envelope the roots of the closed loop system for all possible operating points have degree of damping larger then a value Do Furthermore maximal settling time might be required This corresponds to a maximal real part of the roots not larger than a certain value 09 Both conditions can be visualized graphically in the complex s plane see Figure 1 5 a b a Im b Im c p Im d X H Im Re Re fe d an Figure 1 5 Examples for l regions A line parallel to the imaginary axis guaran tees a maximal settling time a a pair of lines a minimal degree of damping b a circle limits the bandwidth c All these conditions can be satisfied simultane ously by the intersecting region d The admissible region for closed loop eigenvalues is denoted as a system is called if all its eigenvalues are located in this region and an uncertain system is called robustly if all eigenvalues for all operating conditions are contained in The definition of I stability permits arbitrary regions in the complex s plane and does not under
9. Define the center of the operating domain as representative by opening the Parameter specification window and selecting Edit Add representative The nominal values of the new representa tive by default are the ones of the center of the operating domain So no further changes are necessary at this point Additionally the parameter k has to be set to 500 Now open the parameter space window from the PARADISE main window and select the k3 plane as the plane where the stability boundaries shall be com puted Selecting Run Execute representatives starts computation of the stability boundaries The result is shown in Figure 4 2 The controller param eter plane is separated by the stability boundaries into a finite number of regions To find out about the stable regions check an arbitrary point of each region This can be done by selecting Stability checks Check stability for grid The cursor changes to a crosshair while being in this mode Select ing points in the parameter plane by clicking with the mouse cursor in the plane performs a stability check The result of the check I stable or not is displayed at the Matlab prompt Once a I stable point is found the entire region to which this point belongs is I stable For the example the region in Figure 4 2 where the cursor is located in turned out to be the P stable region The controller k 500 2192 8358 0 is selected as a solution from this set whic
10. Multiplexer Mux Demultiplexer Demux Summation Sum Selector all Source blocks e g Step all Sink Blocks e g Display Scope PID Controller Matrix Gain Model Info You can use Matlab matrix creation functions e g eye 2 or ones 2 3 Please do not use any submodels 22 CHAPTER 3 GETTING STARTED WITH PARADISE 3 2 2 Simulink is not available Of course you can use PARADISE without Simulink In this case the input pro cedure is not as comfortable but offers more flexibility if you are familiar with Maple The steps are as follows 1 Open your favorite text editor and create a file which has the same format as the one illustrated on page 20 You only have to specify either the system matrices or the corresponding transfer function e g the two files _ 0 1 0 0 b2 k1 b2 k2 a23 b2 k3 b2 k4 O Og 1 b4 kl b4 k2 a43 b4 k3 b4 k4 b 0 b2 0 4 1 0 Oy OG 194 dy 0 0 Oy O Ty OJ 197 07 0 1 1 0 0 0 0111 and num 1 den a43 b2 k1 a23 b4 kl a43 b2 k2 s a23 b4 k2 s a43 872 b2 kl s 2 b4 k3 s 2 b2 k2 s 3 b4 k4 8 3 _s 4 represent the same system where in the latter case numerator and denom inator of the transfer function have been given The characteristic polyno mial does not have to be specified it will be calculated automatically Also the parameter subst
11. controller from the Tools menu a cross hair is displayed and waits for a mouse button to be pressed The cross hair can be positioned with the mouse Pressing a mouse button selects the controller at the current location and marks the controller parameters by a big red asterisk At the same time the values of the controller parameters k kn in the Parameter specification GUI are updated Figure 5 5 shows that the controller k 572 3388 33699 2587 was selected from the set which stabilizes the plant for both representatives 5 3 EXAMPLE CRANE Parameter space Pez Figure 5 4 Parameter space window Parameter specification k2 3388 57 k3 33699 kl 572 813 Fixed parameters Advanced Figure 5 5 Parameter specification window 55 56 5 DESIGN IN AN INVARIANCE PLANE The next step is to evaluate the robustness of the controller with respect to the op erating domain This is done by applying the parameter space approach For this purpose the l m plane is selected via the Parameter selection popup menu and computation of the stability boundaries is started by selecting Run Execute grid in the parameter space window Figure 5 6 shows the result for the example The plot shows that for the selected controller the real root boundary still intersects the operating domain edges Thus we have to repeat the described procedure to find a controller which I stabili
12. matrices e g for the flight control example the nominal values of the four flight conditions are given in Table 1 1 Specify the nominal matrices as cell elements 1 0 9896 17 4100 96 1500 0 2648 0 8512 11 3900 0 0 14 0000 bil 97 7800 0 14 0000 4 0 0 0 and save them in a file named systemname mat using the Matlab command save f4e b dif the sytem should be called 4e The state space box has then to be filled with the substitutes ANom BNom CNom and DNom see Figure 3 1 Now execute Input Plant specification Load WIS Eile Edit View Simulation Format Tools E esse r State Space State space model d dt Bu y Cx Du Parameters BNom c CNom D DNom Initial conditions 0 _ Cancel Boal Ready 10 ode45 Figure 3 1 Simulink model of the F4E aircraft new model from the PARADISE main window and chose the corresponding Simulink file The closed loop system equations will now be calculated for each representative The number of resulting nominal representatives is of course iden tical to the number of plants specified in the mat file If you do not have Simulink available you need to generate the ssr file as de scribed above You have to specify all the representatives in closed loop form For the aircraft example assuming state feedba
13. of PARADISE see Figure 2 3 Block Parameters Crane r State Space State space model dx dt Bu y Cx Du Parameters 0 b2 0 b4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 Initial conditions 0 Cancel Help Figure 2 3 Parametric representation of plants This block contains the description of the crane dynamics which is of fourth or der for the linearized crane model as it is used here The state space matrices A B C D were given in a very general form for example ri 0 0 0 23 0 a 00 0 1 lo 0 agg 0 The entries of this matrix depend on crane parameters like crab mass load mass and rope length This dependency could of course be declared in the Simulink model However if the system order is large the detailed specification of the matrices in the Simulink block would be quite awkward and could easily lead to typing errors To facilitate this procedure it is possible to substitute Simulink parameters after the symbolic system equations are determined the parameters contained in the Simulink model are determined from the symbolic equations see Figure 2 4 The user now has the possibility to substitute these parameters by their actual dependency In the example was replaced by g Of course not all parameters have to be replaced like for example the controller parameters k to k4 12 2 TOUR THROUGH PARADISE Parameter specifi
14. robust controller designed in section 4 3 1 Namely the controller is given by k 500 2740 18788 0 and the operating domain was given by mr 1000 2000 kg 8 16 m Now consider the real world operating domain mr 50 2000 kg l 8 16 m 52 5 DESIGN IN AN INVARIANCE PLANE Parameter space DPI Bun Stability checks Tools Options Output anng zoom off 2000 ml 1500 1000 103000 Figure 5 1 Robustness analysis which includes the case of an empty hook First we evaluate robustness of the given controller with respect to the new op erating domain Figure 5 1 shows that the system is not I stable because the operating domain is intersected by I stability boundaries As a first step to design a controller by use of invariance planes an operating point in the parameter space must be determined for which the invariance plane is computed This is done by specifying a representative in the Advanced section of the Parameter specification interface Figure 5 2 shows the interface with an appropriate representative for the crane example Hint Usually an operating point which lies closes to or on a I stability bound ary is a good choice since this ensures that the origin of the resulting Ka plane is in the vicinity of the I stable controller parameter This holds at least for the boundaries of the same operating point 5 3 EXAMPL
15. state x4 Although we had to use non zero k4 values during intermediate design steps in the invariance plane we could achieve k4 0 for the final con troller Bibliography ABK 93a ABK 93b Ack85 AT82 Cel91 FD29 S69 Vys86 J Ackermann A Bartlett D Kaesbauer W Sienel and R Stein hauser Robust control Systems with uncertain physical parameters Springer London 1993 v 49 50 J Ackermann A Bartlett D Kaesbauer W Sienel and R Stein hauser Robuste Regelung Analyse und Entwurf von lin earen Regelungssystemen mit unsicheren physikalischen Parame tern Springer Berlin 1993 49 J Ackermann Sampled data control systems analysis and synthe sis robust system design Springer Berlin 1985 49 J Ackermann and S T rk A common controller for a family of plant models In Proc 21st IEEE Conf Decision and Control pages 240 244 Orlando 1982 49 F Cellier Continuous system modelling Springer New York 1991 3 R A Frazer and W J Duncan On the criteria for the stability of small motions In Proc Royal Society A volume 124 pages 642 654 1929 v D D Siljak Nonlinear systems the parameter analysis and design Wiley New York 1969 I A Vyshnegradsky Sur la theorie generale des regulateurs Comptes Rendus 83 318 321 1886 61
16. tation As an example consider the crane example introduced in Chapter 1 The simulink model is shown in Figure 2 2 The state space block contains the plant description of the crane with its parameters in algebraic form The Simulink model has to be created in advance and stored in a file To read it in within PARADISE select Input Plant specification This opens a file selection box from where you can select the desired model When the model is loaded for the first time the symbolic computation of the closed loop equations will be started automatically You are informed about this action by the message Connecting system When finished the re quired computation time will be displayed For larger systems this step may be quite time consuming Therefore the symbolic closed loop equations can be saved to a file when selecting File Save current model data The re sulting file will be named systemname ssr It is an ASCII file for the crane its contents is 0 1 0 0 b2 k1 b2 k2 a23 b2 k3 b2 k4 0 0 0 1 b4 k1 b4 k2 a43 b4 k3 b4 k4 b 0 b2 0 b4 Je LL Og 0 Ol O Ty Oy OG 195 Oy FT Ole Or 0 LIII 0 0 0 0111 num Cranenum den Craneden _ a43 b2 k1 a23 b4 kl a43 b2 k2 s a23 b4 k2 s a43 s 2 b2 kl s 2 b4 k3 s 2 b2 k2 8 3 b4 k4 s 3 5 4 3 2 PLANT SPECIFICATION
17. 21 sub a23 a43 b2 b4 kl k2 k3 k4 g mL mC g mC mLE 3 L mC mo 1 1 L mC KL k2 k3 k4 g mC mL 1 g mC mL 1 To avoid naming conflicts the internal variables start with _ When launching a new PARADISE session with this model the computation of the closed loop equations will be skipped if the file systemname ssr is found Instead the information contained in this file will be read in by the Extended Symbolic Toolbox and is available in the session When the Simulink model is modified the closed loop equations are inconsistent with the current model and have to be regenerated This can be performed selecting Input Plant specification Reload current model Note The model cannot be used for numeric simulation in Simulink as long as no specific values for the plant parameters are given However you can be treat this model with PARADISE which computes the parametric closed loop model via the Extended Symbolic Toolbox If you want to simulate the model you have to specify the plant parameters at the Matlab prompt for example mL 500 mC 1000 Also note that this does not affect the parametric model i e though a numeric value is defined for example for mz in the Matlab workspace mz will still be treated as a symbol within PARADISE The following list of blocks is currently supported by PARADISE Transfer Function State Space Zero Pole Integrator Gain
18. 3 2 2 Simulink is not available 3 2 3 Multi model specification PGS PN uses ket REAR E EKER KOR EQ The region editor ke RR REIR RERO 10 10 12 12 13 4 The parameter space method 4 1 Some introducing theory 4 2 The parameter plane menu 4 3 Design examples for the parameter space method 4 3 1 4 3 2 Design example 2 F4E fighter aircraft 5 Design in an invariance plane 5 1 Introduction 5 2 Theoretical background Sad 5 3 Example Crane Bibliography Design example 1 Crane Design in an Invariance Plane CONTENTS 1 Introduction to robust control Almost all technical systems depend on varying or uncertain parameters Just consider the velocity or mass of vehicles the oil temperature of hydraulic systems or the rope length and load mass of the crane illustrated in Figure 1 1 These parameters may vary more or less significantly within certain bounds and they influence the system dynamics Traditional control design approaches however consider a fixed operating point in the hope that the resulting controller is robust enough to stabilize the plant for different operating conditions These approaches definitely yield good results if the parameter variations are small or the system dynamics is not too sensitive with respect to these parameters For significant parameter variations these control design methods reach their performance limits New de
19. 7 Parameter specifications for second iteration The next step is to choose the two most critical eigenvalues in the Gamma Editor window invoked from the Parameter space window as described for the first iteration Figure 5 8 shows selection of the most critical eigenvalues In this case an eigenvalue lying on the real axis is the most critical Since two eigenvalues have to be selected the second eigenvalue on the real axis must be chosen Next determine the stability boundaries in the invariance plane to find a suit able controller by selecting the command Execute Representatives from the Run menu in the Parameter space window Figure 5 9 shows the stability boundaries in the invariance plane Using the Check stability 58 CHAPTER 5 DESIGN IN AN INVARIANCE PLANE Gamma E ditor ol x Edit Options Set value Zoom on Figure 5 8 Selection of most critical eigenvalues 2nd iteration for gridcommand from the Stability checks menu the region of con troller parameters which I stabilizes the plant for both representatives can be identified The big red asterisk in figure 5 9 shows the selected controller k 550 3300 28451 2000 Figure 5 10 shows the stability boundaries in the mz l parameter plane Now we have found a I stabilizing controller for the entire operating domain because there is no stability boundary crossing the oper ating domain Actually there is a wide margin between the stabil
20. E CRANE 53 Parameter specification Edit Varying parameters Paraneter 1 Paraneter mL Controller parameters 0 2740 18788 4 0 WN gt ow ow n mC 1000 dvanced Representatives Associated with Gamma 1 16 mL 800 Representative 2 Associated with Gamma region 1 1 16 region mL 2000 Representative 3 Associated with Gamma region 1 mL 2000 Representative 4 Associated with Gamma region 1 mL 400 Paraneter planes Set value Figure 5 2 Nominal point for invariance plane After we have specified an operating point for the invariance plane in this example see Fig 5 2 then next task is to select the most critical eigenvalues This step is initiated from the Parameter space window by choosing the parameter invplane in the parameter selection popup menu After choosing invplane in either of the two menus the Select Eigenvalues pushbutton in the lower left corner is displayed which allows selection of the eigenvalues in the Gamma Editor window Figure 5 3 shows the corresponding Parameter space and Gamma Editor windows All eigenvalues are displayed in the Gamma Editor window and marked by a blue cross Two eigenvalues have to be selected for the design in an invariance plane A critical eigenvalue can be selected by moving the mouse pointer above the eigenvalue and pressing the left mouse button An eigenvalue is displayed by a red cross after select
21. Ellipsis Mathematical description EEE 2 o A 2 1 o 09 a oo a The parameters a b and sigma have to be specified Pair of ellipses Mathematical description OV el acosa jbsin a jwo a 0 2 The parameters 09 Sigma wo omega a b and y Rotation have to be specified Delete element The currently selected basic element will be removed from the region description Redraw region Refreshes the displayed I region Add new region A new I region will be added Delete region The currently selected I region will be removed Close Closes the I region editor window Options Display eigenvalues Single Plant Use this item if there is no Q box e g for multi model represen tat ions Center of Q box The eigenvalues of the center of the Q box will be calculated and displayed in the currently displayed I region 3 4 THE EDITOR 31 Vertices The eigenvalues of the vertices of the Q box will be calculated and displayed in the currently displayed I region Representatives The eigenvalues of all representatives will be calculated and dis played in the currently displayed region Hold eigenvalues When selecting one of the above items the already displayed eigen values will be cleared This feature prevents this action and allows to simultaneously display for example eigenvalues of the cen
22. P DLR Institute of Robotics and Mechatronics Robust Control PARADISE 2 0 PARAMETRIC ROBUSTNESS ANALYIS AND DESIGN INTERACTIVE SOFTWARE ENVIRONMENT USER S MANUAL PARADISE 2 0 USER S MANUAL How to Contact PARADISE 8153 28 1847 Fax Robuste Regelung Mail DLR Postfach 1116 82230 Wessling http www op dlr de FF DR RR paradise Web paradise dlr de Technical support Product enhancement suggestions Bug reports Paradise User Manual 1999 2000 by DLR Oberpfaffenhofen The user should be aware that this product is protected by copyright law and in ternational treaties The authors and DLR shall under no circumstances be liable for any indirect incidental or consequential damages caused by the provided soft ware Furthermore the authors and DLR assure that the software tool and its documen tation was developped with great care However the user should recognize that software and documentation still might contain errors and omissions for which the authors shall not be responsible The software is not recommended to be used for applications in which errors or omissions could threaten life injury or significant loss All rights reserved The software may be used or copied only under the terms of the license agreement Please note that most of the referred soft and hardware names in this document are registered trademarks and are subject to legal requirements Printing History Date Version Author 2000 2 0
23. again makes the textfields invisible Display Q box The Q box will be displayed in the selected parameter plane The Q box can be removed by re selecting this feature This feature only has effect if at least one of the parameters is of varying type Hold current plot Starting a new computation of stability boundaries from the Run menu erases by default earlier computed stability boundaries Select ing this feature preserves these stability boundaries Display individual stability This option controls the used marker symbol for the stabilty checks If this option is selected a point in the current plane for which some but not all grid points representatives are l stable is marked by a blue 40 4 THE PARAMETER SPACE METHOD triangle Otherwise these points are marked by red symbol since they are not robustly I stable 4 3 Design examples for the parameter space method 4 3 1 Design example 1 Crane The initial design in the parameter space will be carried out for the crane As shown in Chapter 2 the controller parameters k and k4 had been fixed by some a priori considerations to k 500 and ky 0 The goal of the design is to de termine the remaining parameter ky and using the parameter space approach In the first step only the center of the operating domain will be considered To solve the problem with PARADISE the system equations of the crane have to be specified first as explained in Chapter 3
24. ane For this purpose the plane has to be fixed before any calcula tions can be executed The user can select the desired plane using the two popup 14 2 TOUR THROUGH PARADISE ditor Of x Edit Options Gamma region 1 Element 1 Hyperbola sigma 0 Damping 0 447214 0 25 Inverted No zi Set value 55 Figure 2 6 region editor menues displayed in Figure 2 7 After the plane was specified the user starts the computations of the stability boundaries from the Run menu An example is illustrated in Figure 2 7 Dashed lines identify real root boundaries solid lines complex root see section 4 1 boundaries The stability boundaries divide the parameter plane into a finite num ber of regions By checking I stability of each region in the parameter plane the set of simultaneously I stabilizing parameters can be identified This is ac complished by selecting the appropriate function from the Opt ions menu and selecting different points in the parameter plane by mouse click The checked point is colored green if this pont is stable A red colored star indicates instability Several others functionalities are implemented for example scaling and identification of curves 2 3 ALGORITHMS THE PARAMETER SPACE APPROACH Parameter space Figure 2 7 Parameter space 16 2 TOUR THROUGH PARADISE
25. arame ter can be determined as a function of the varying parameter such that the controller parameter always lies within the l stable region Figure 4 1 illus trates this procedure Here it is assumed that a plant parameter varies within the interval q q q The result of Figure 4 1a shows that there exists a b c Figure 4 1 Design of gain scheduled controllers a No gain scheduling necessary b gain scheduling control no I stabilizing solution 4 2 THE PARAMETER PLANE MENU 37 a solution kg which lies in the I stable region independently of the varying parameter q i e no gain scheduling is necessary at this point However in Figure 4 1b no such fixed gain control is possible The dashed line indi cates a solution for which the controller gain in dependency of the varying plant parameter is always located in the I stable region yielding a I stable solution in dependency of the varying plant parameter Figure 4 1c shows a solution for which no robustly I stabilizing solution is possible Robustness analysis The stability boundaries in a plane of two uncertain parameters q and q2 are computed The system is robustly I stable if the entire operating domain is contained in the I stable parameter set The parameter plane menu The parameter plane window is opened from the PARADISE main window by selecting Algorithms Parameter space The desired parameter plane can be set via the two popup me
26. c e iret p Parameters B BNom To Workspace E JENom D DNom Initial conditions accel gyro 0 feedback Read 1002 ode45 E Cancel Apply Figure 4 6 F4E simulink model For the design the plant description has to be specified as explained in Chapter 3 Since the flight dynamics changes with varying flight condition it is obvious that different flight conditions may have to fulfill different eigenvalue specifica tions This leads to I region specification depending on the flight condition This requirement can be satisfied within PARADISE by defining a I region for each flight condition In a second step the I region has to be associated with the corre sponding representative This step is performed in the Parameter specification window as shown in Figure 4 7 and Figure 4 8 After the parameter settings are completed the controller design can be carried out in the controller parameter plane Open the parameter space window Since the two controller parameters are the only parameters involved in this example the displayed plane is the desired one By selecting the item Run Execute representative the computation of the stability boundaries is started where now for each representative the I region associated with this operating point is used for generating the mapping equations and computing the stability bound aries The result is shown in Figure 4 9 Again
27. cation ol x Edit Direct substitution a43 g mC mL 1 mC a23 Indirect substitution Set value t1 FmC Figure 2 4 Substitution of Simulink parameters 2 2 2 The operating domain After a substitution was performed the resulting parameters have to be classified Three classes of parameters exist e Varying parameters These are plant parameters which are uncertain for example the crab mass or vary but can be measured e Fixed parameters These are plant parameters which will not change their value for example the wheel base of a car e Controller parameters to be determined by the design process The uncertain parameters are assumed to vary within given intervals The interface in Figure 2 5 allows the specification of these values Also the values for fixed and controller parameters have to be set 2 2 3 I region For technical applications Hurwitz stability is mostly not sufficient Further spec ifications like settling time damping and bandwidth have to be met Several specifications can be translated into locations of eigenvalues which leads to a re stricted set of eigenvalues in the left half plane This set of admissible eigenvalue 2 3 ALGORITHMS THE PARAMETER SPACE APPROACH 13 Parameter specification 151 Edit Varying parameters Parameter Lower bound 8 Upper bound 16 grid points 2 Parameter mL Lower bound 1000 Upper bound 2000 grid points 2 Controller param
28. ck control the file looks as follows 3 3 PARAMETER SPECIFICATION 25 _ 1237 1250 4889 knz 50 1741 100 4889 kq 50 1923 20 331 1250 532 625 1139 100 14 knz 14 kq 14 851 500 1361 knz 5 1268 25 1361 kq 5 527 2 2201 10000 709 500 3199 100 14 knz 14 kq 14 667 1000 8509 knz 100 1811 100 8509 kq 100 4217 50 8201 100000 6587 10000 1081 100 14 knz 14 kq 14 2581 5000 878 knz 5 674 25 878 kq 5 1789 10 431 625 49 40 1519 50 14 knz 14 kq 14 _b 4889 50 0 14 1361 5 0 141 8509 100 0 14 878 5 0 14 64 gt PII 0 0 0 1 0 0 0 1 Lii 0 0 O 1 0 0 O 1 1 0 0 De 1 0 O O 1 1 0 0 0 1 0 0 0 EDD The matrices are organized in the form X1 X to be specified in Maple format 3 3 Parameter specification Once the plant was read in you can start with specifying the parameter settings PARADISE distinguishes three different parameter types e Varying parameters These parameters are specified by their upper and lower bounds Additionally the number of grid points has to be specified which is used for algorithms applying a grid on varying parameters By default parameters starting with q are categorized as varying parameters The initial lower bound is zero the upper bound i
29. e cases for which eigenvalues are located exactly on the boundary of the admissible set of eigenvalues In case of two parameters the boundaries can be visualized graphically in this parameter plane PARADISE has implemented the algorithms for this case But even for higher number of parameters the approach is still applicable to the problem of robustness analysis Two parameters are selected for a graphical rep 33 34 4 THE PARAMETER SPACE METHOD resentation while the remaining ones are being gridded For each grid point the stability boundaries are computed and projected into the selected parameter plane Assuming that the uncertain parameters are independent of each other the uncer tainty domain is hypercube Then the image of the projection of the operating domain in the selected plane is always the same rectangle independent of the grid point In case of controller design it was already demonstrated in Chapter I how to keep low the number of controller parameters to be determined in the design process For the special case of full state feedback where the number of controller parame ters equals the system order and is fixed the invariance plane approach introduced in Chapter 5 is suited well for design For the following the case of two uncertain parameters q and q2 will be consid ered to illustrate the procedure Furthermore Hurwitz stability will be assumed for this introduction without loss of generali
30. e desired model the information contained in the Simulink model is passed to the symbolic computation part where parametric representation of the closed loop is calculated If the Simulink model is changed the system equa tions have to be re calculated In order to save computation time especially for larger systems the symbolic system equations can be saved When re reading the Simulink model in later session the saved data will be passed to the symbolic computation part without re calculating the system equations from the Simulink model If Simulink is not available the closed loop system equations state space or transfer function representation have to be typed in manually Figure 2 2 shows an example of a Simulink model It illustrates the block diagram of a crane positioning control The example represents a continuous plant fam ily PARADISE also allows to handle multi model representations where a finite number of linear plants are given as representatives of a nonlinear plant Typical applications are flight control problems where only linearized models for various flight conditions are given cranebild OF x File Edit View Simulation Format Tools Dienas e PARADISE application Crane state feedback Ready 100 ode45 7 Figure 2 2 Simulink model of crane 2 2 INPUT SPECIFICATIONS 11 The description of the block Crane in the Simulink model points to another fea ture
31. e roots of the polynomial p s The desired characteristic closed loop polynomial p s is now written as a product p s h s t s where h s represents the eigenvalues which remain fixed and t s denotes the eigenvalues to be shifted Equation 5 1 becomes k e M A HA ef HA 5 3 5 3 EXAMPLE CRANE 51 where e e h A Further assuming that only two eigenvalues should be shifted at a time i e t s to t s 5 equation 5 3 yields 2 k el toI t A4 A to ti 1 ef A 5 4 ef For the open loop i e k 07 the eigenvalues represented by t s are denoted by d s do dis 3 ie 07 el d A el 41 di A A 5 5 Forming the difference of 5 3 and 5 5 yields k 5 6 T ka ko E with amp to do and t di By arbitrary Ka a feedback vector k is determined in the two dimensional cross section defined by the vectors e7 and e A such that only the two eigenvalues of d s are shifted while h s remains fixed Applying the parameter space approach allows to determine the set of parameters in the Ka amp 5 plane which stabilize the system 5 3 Example Crane In this section we show how to use PARADISE to design a simultaneously stabilizing controller by using the invariance plane approach We use the same model for the crane already introduced in chapter and 2 The starting point for this design example is the
32. ed model can be selected Reload model The symbolic closed loop equations will be regenerated for ex ample if the associated Simulink model has been changed by the user Gamma editor Starts the region editor Parameter specification Opens a GUI where all parameter settings for the active plant can be specified Algorithms Parameter space This command opens a window for computation of stability bound aries in parameter space Help Opens the help window to be done About Paradise System name This is a dynamic menu item listing the names of currently opened models If only one model is opened its name will be listed but cannot be activated If 20 3 GETTING STARTED WITH PARADISE more than one system is loaded the menu item changes to a pulldown menu from where the current model can be selected Note that due to Matlab bug the wrong system name is displayed To find about the currently active system open the pulldown menu The active system is marked Algorithms usually apply to the currently active model Parameter settings will be up dated in the corresponding windows if the active model is being changed 3 2 Plant specification 3 2 Using Simulink The most comfortable way to specify a plant is to use Simulink PARADISE reads in the specified plant and computes the closed loop equations of the model via the Extended Symbolic Toolbox Simply specify the plant in its parametric represen
33. esets the controller parameters Identify curve Once several I stability boundaries have been computed for some rep resentatives or several grid points it is not possible to tell for which 4 2 THE PARAMETER PLANE MENU 39 representative or for which grid point specific boundary has been generated Clicking on curve with the left mouse button displays information about the type of the selected boundary at the Matlab prompt Pressing the right mouse button terminates this mode Scale curve The stability boundaries are computed by solving equation 4 2 for a sweep of the complex frequency s along the boundary of This fea ture parametrizes the computed I stability boundaries with the com plex frequency which resulted in this specific point contributing to the l stability bounary in parameter space Remove scaling Removes scaling from all stability boundaries Clear plot Erases all displayed stability boundaries Mark stable region Selecting this item starts an algorithm which automatically detects the I stable region for the computed I stability boundaries The bound ary of this region will be marked with solid lines Attention This algorithm may be time consuming Options Alter axis limits You can alter or predifine before starting the calculation the axis limits of both parameters If you select this item two textfields wil appear were you can manually change the axis limits Pushing this buttom
34. eters g 10 mC 1000 Advanced Set value r Figure 2 5 Specification of plant parameters locations of the closed loop is referred to as A graphical editor for the con struction of such regions is part of PARADISE It offers a set of basic elements e g Real part limitation pair of lines of constant damping hyperbolas circles ellipses etc from which the region can be composed The example in Fig ure 2 6 illustrates the functionality of the I editor The region consists simply of a hyperbola This guarantees a certain degree of damping and a maximal settling time of the system The basic elements can be combined arbitrarily using the operations intersection and union Additionally each element can be used in its inverted form The basic elements can be modified by clicking on the specific element and dragging it with the mouse to the new location Multiple I regions can be loaded and edited in a PARADISE session 2 3 Algorithms The parameter space approach The parameter space approach is implemented as an algorithm in the toolbox Once the input specifications have been accomplished the parameter space ap proach can be selected from the Algorithms menu in the main control window This will fire up a new window titled Parameter Space see Figure 2 7 The pa rameter space approach is used to determine the set of stabilizing parameters in a parameter pl
35. g conditions like vertex points of the operating domain It is the goal of the second design step to I stabilize the four vertices of the op erating domain Again select the k2 k3 plane in the parameter plane window To compute the stability boundaries for the four vertices select Run Execute grid This will grid the remaining uncertain parameters in the example mz and with the number of grid points specified in the Parameter specification win dow In order to get only the vertices the number of grid points has to be set to two which is the default The result is shown in Figure 4 4 The I stable region can again be determined by checking arbitrary points of each set for I stability The region for 42 4 THE PARAMETER SPACE METHOD Parameter space Ol x Bun Stability checks Tools Options Output ann zoom off 2500 2000 1500 mL Figure 4 3 Robustness analysis The system is not robustly I stable since the operating domain is intersected by I stability boundaries this example is the one where the cursor is located in From this set the controller k 500 2740 18788 0 was selected Of course it is of interest now to evaluate robustness of this new controller The procedure is the same as for the first design The I stability boundaries in the plane of uncertain parameters are displayed in Figure 4 5 For the second design the controller turns out to be robustl
36. h I stabilizes the center of the operating domain The robustness of the controller will be evaluated now in robustness analysis The analysis can also be carried out by applying the parameter space approach 4 3 DESIGN EXAMPLES FOR THE PARAMETER SPACE METHOD 41 Parameter space olx zoom off 0 Identity curve GEFEN Ga EE Scale curve 1 scaling Clear plot Mark stable region TC CIC cb D D 1 1 D D A 4 stecsoogereecemgaiedieecep ieecusenpeseosedececcbma ep 1 1 D 1 1 1 D D 1 i i D D 500 1000 1500 2000 2500 3000 3500 4000 4500 ex Figure 4 2 Set of I stabilizing controllers for the center of the operating domain For this purpose select the Z m plane and start the computation of the stability boundaries by selecting Run Execute grid The result is displayed in Figure 4 3 The system is not robustly I stable since by design the center of the operating domain is I stable and the operating domain is intersected by I stability boundaries This result emphasized that a robust design can only be successful if the entire operating domain is considered for the design A design which not only consid ers one nominal point will more likely be robust enough to stabilize the entire operating domain A suitable selection of representatives are extreme operatin
37. he basis of robust control Users with some knowledge of robust control may want to skip this chapter Chapter 2 takes you on a short trip through PARADISE to give you an impression about the possi bilities of this toolbox Chapter 3 explains how to input your plant and controller structure and your problem specifications The remaining chapters explain the ro bust control algorithms in more detail and demonstrate how you can solve these problems using PARADISE Of course a complete introduction to robust paramet ric control goes beyond the scope of this manual For a deeper insight the reader is refered to the literature for example ABK 93a 69 Contents Ld 12 1 3 1 4 Al Pa 23 ad 2 9 3 4 Introduction to robust control Parametric models and uncertainty Multi model representation Periormance specifications lt oc evo go xc oe eee P Robust controller design and analysis A tour through PARADISE Starting a PARADISE session Input specifications 22 22 9445 baade det Zel MN s ir a be s p gaia PER EY 2 2 2 The operating domain u Te PN ccoo eh ees va ee bee be Algorithms The parameter space approach Getting started with PARADISE Starting and quitting a PARADISE session Plant specification 24 o sa 2404454 Ree ae 3 21 Using Simulink oc ses ea oe we
38. ing domain A more practically oriented way of robust control design is simultaneous I stabilization which is as follows Step 1 Select a finite number of nominal operating points which adequately represent the operating domain In case of a symbolic model description chose a number of operating points and calculate the nominal system equations A good choice are the vertices of the operating domain see Figure 1 2 In case of a multi model descrip tion these nominal system descriptions are already given Step 2 Determine a controller which simultaneously stabilizes the rep resentatives of Step 1 Despite of this simplification the design step still remains a complicated task The parameter space approach which you will get to know in Chapter 4 yields as its result the entire set of controller pa rameters for which a nominal operating point is I stable Computing this set for each representative and forming the intersection of all sets results in the set of all controller parameters which simultaneously stabilize the representatives To complete this step select an adequate solution from the set This procedure is illustrated in Figure 1 6 For a given region the set of l stabilizing controller parameters K and are calculated for the two representatives b and b2 The intersection Kr K 0 NK of both sets simultaneously stabilizes the two representatives Step 3 Since the controller was designed only to sim
39. ion The other n 2 eigenvalues are constant for any controller parameter in the invariance plane and the operating point After the most critical eigenvalues have been selected the stability boundaries are computed for the representatives This is done by selecting the command Execute Representatives from the Run menu in the Parameter Space window 54 CHAPTER 5 DESIGN IN AN INVARIANCE PLANE Gamma E ditor Edit Options Parameter space Bun Stability checks Tools 15 16 Select eigenvalues Inv x Figure 5 3 Selection of most critical eigenvalues To ensure robustness not only for representative which was selected as the operating point for the invariance plane it is useful to add extra representatives e g points for which the previous controller is I stable or vertices of the op erating domain In our example we add the representatives R2 and Ry as a representative see Fig 5 2 to ensure that we increase the portion of the operating domain which is I stable in the next design step Figure 5 2 shows the corre sponding Parameter specification window and the stability boundaries for the representatives in the Parameter space window Using the Check stability for grid command from the Stability checks menu we can check the stability of different coherent regions The little green cross in figure 5 4 indicates that this region is I stable After invok ing the command Select
40. ion and un for union arbitrary set operations can be carried out In the example of Figure 3 3 the set operation is un is el e2 e3 e4 The index i of the argument ei refers to the i th basic element used to describe the l region To restore the default simply select the corresponding entry and replace it by the string default Chapter 4 The parameter space method 4 1 Some introducing theory The typical question of robust control is Given an uncertain system represented by its characteristic polynomial p s q Is the system robustly I stable for the given uncertainty range Q The answer can be given by finding the solution to the reformulated problem Determine the entire set of uncertain parameters for which the characteristic poly nomial p s is robustly I stable Only if the operating domain is entirely con tained in the resulting set of I stable parameters then the system is robustly stable The field of application of this method is not only robustness analysis but also controller design In this case the set of stabilizing controller parameters is deter mined All controllers from this set stabilize the plant thus allowing to incorpo rate further design criteria to select the final controller The set of I stable parameters can be determined by mapping the region via the characteristic equation p s q 0 4 1 into the desired parameter space Equation 4 1 is fulfilled for all critical cases i
41. ituted and edit the term in the edit box at the bottom of the window After pressing the return key the substituted parameter will appear in the right column of the listbox New parameters introduced by this substitution again can be substituted They appear in the item Indirect substitution and can also be edited Parameter specification window menus II Edit Edit operating domain Switches back to the parameter settings Close Closes the window 3 4 The editor The performance specifications for a specific control problem have to be speci fied in PARADISE via the closed loop eigenvalue location The specifications are fulfilled if the closed loop eigenvalues are contained in a plant specific region re ferred to as A graphical editor for editing this region is part of PARADISE It offers several basic elements which can be combined to obtain the desired region An example of the I region editor was illustrated in Figure 2 6 Red lines indicate the part of the boundary contributing to the region dashed lines indicate the side of the basic element not belonging to A basic element can be modified on two ways e Select the value to be changed in the listbox the parameter a in Figure 2 6 Edit the value in the edit box at the bottom of the window e Click on the basic element and move it with the mouse cursor to the desired position After releasing the cursor button the numeric values displayed in
42. itutions can be performed later within the corresponding user interface however you are free to specify it already in the text file as shown in the example on page 20 From the samples it should be easy to tell the required syntax Experts The file is interpreted by the Extended Symbolic Toolbox i e it is of pure Maple syntax You do not need to specify the state space representa tion explicitly It can be the result of arbitrary symbolic calculations where at the end the system matrices _a _b _c dor num and respectively result 2 Save the file to disk using the extension ssr 3 2 PLANT SPECIFICATION 23 3 Select the command Input Plant specification Load new model This opens a file selection box from where you chose the file just specified Note for Simulink users Chosing the file systemname md1 automatically starts Simulink If you want to omit this and the symbolic closed loop equations are already saved i e systemname Ssx exists you can of course use the ssr file This feature is especially useful when only a limited number of Simulink licenses is available in a computer network and you only want to read in a model In some cases the symbolic representation of the plant may be quite lengthy which can really make the plant specification in a Simulink state space box pretty an noying For this case the symbolic description of the state space model can be specified in a file This is convenient es
43. ity boundaries and the operating domain 5 3 EXAMPLE CRANE 59 Parameter space Figure 5 9 Selection of a I stabilizing controller 2nd iteration Parameter space Figure 5 10 Robustness analysis 2nd iteration 60 5 DESIGN IN AN INVARIANCE PLANE Final step In the first design example in section 4 3 1 one design specification was k4 0 This enables the implementation of controller without sensor which measures state x4 As a final step we analyze if we can modify the controller such that kj 0 without losing the I stabilizing property for the operating domain Therefore we determine the stability boundaries in the k2 k4 parameter plane Figure 5 11 shows a plot of the stability boundaries It can be easily seen from figure 5 11 that changing the parameter k by a small amount enables us to choose k4 0 Using the controller k 550 0 3200 00 28451 5 0 I stabilizes the whole operating domain Parameter space 151 Bun Stability checks Tools Options Output i i TA i i i i i i i 6000 4000 2000 2000 4000 6000 8000 10000 12000 v Figure 5 11 Parameter space approach final step Thus using the invariance plane approach we could find controller which stabilizes the whole operating domain Using a rather easy subsequent analysis and design step we could actually find controller which does not require feed back of
44. lie any restrictions It also includes the spe cial cases of the left halfplane for Hurwitz stability and the unit circle for Schur stability 6 1 INTRODUCTION TO ROBUST CONTROL FC 2 Mach 0 5 0 85 Altitude ft 5000 5000 35000 35000 0 667 18 11 84 34 0 08201 0 6587 10 81 85 09 81 3 07 4 90 187 Table 1 1 Model data for an F4 E aircraft for four typical flight conditions The eigenvalues s and s result from s a11 8 0 The region is specified by its boundary Besides lines adequate elements to describe the boundary of a I region are circles hyperbolas and ellipses Lines parallel to the imaginary axis restrict the settling time circles the bandwidth Cir cles can be used to limit the bandwidth or in its inverted version to guarantee a minimal bandwidth In order to fulfil several specifications simultaneously the intersection of basic elements can be formed see Figure 1 5d The I region is not necessarily simply connected Just imagine a plant with a pole pair close to the origin whose effect on the system dynamics is almost can celled by a close pair of zeros It would not be wise to move this pole pair to the left such that it satisfies a certain degree of damping A better approach is to stabilize only the remaining poles while the pole pair remains more at its open loop location A pair of circles with small radius around the
45. n and controller parameters This engineering art approach is justified by the resulting controller which exploits the highest possi ble potential due to the parametric basis of the design and the mapping of sharp boundaries i e whenever a boundary in parameter space is crossed then also an eigenvalue crosses the boundary OI of the chosen I stability region 2 tour through PARADISE 2 1 Starting a PARADISE session To start up the toolbox simply type paradise at the Matlab prompt This will start up the main control window and automatically load the symbolic libraries which are used to perform symbolic computations From the menus of the main control window all further inputs can be accomplished and in general the user does not have to return to the Matlab window PARADISE 2 0 File Input Algorithms Help Figure 2 1 PARADISE main window The main window is illustrated in Figure 2 1 The logo of the toolbox represents the color coded value set of a track guided vehicle for fixed frequency 10 2 TOUR THROUGH PARADISE 2 2 Input specifications 2 2 1 The plant Different kinds of inputs are necessary The most obvious one is the plant itself For this purpose graphical user interface GUI was developed PARADISE of fers two different ways to specify the plant and controller structure The more comfortable manner is the plant specification via Simulink Once the user has selected th
46. nt model These plant representatives can be used for the controller design A typical application is flight control Especially modern high performance fighter aircrafts are aerodynamically unstable and need to be controlled Figure 1 3 shows a sketch of such a plane equipped with additional canards to increase maneuver ability The task here is to control the short period mode which is described by the 4 1 INTRODUCTION TO ROBUST CONTROL Canard Elevator Figure 1 3 Sketch of a fighter aircraft states normal acceleration n and pitch rate q The linearized state equations are Q12 413 Nz by T Q22 a23 q 0 U 1 4 0 0 14 14 where the elevator actuator state with time constant of 1 14 is considered The operating domain of the aeroplane the so called flight envelope illustrated in Figure 1 4 describes the admissible range of altitude in dependency of the Mach number Nominal values for the system description 1 4 are given in Table 1 1 for the four representative flight conditions indicated in Figure 1 4 1 3 Performance specifications Back to the crane example To stabilize this plant it would suffice to shift the poles only little bit into the left complex halfplane In view of practical realization this is certainly not sufficient since the damping still could be arbitrarily small Here comes in the notion of I stability to assure sufficient stability margins
47. ntative has to be checked in a later stage the associated I region will be used for testing stability This menu item is not available in the case of multi model represen tation since then the number of representatives is fixed It cannot be modified since the number of nominal plants is then part of the plant specification Only the I region associated with each of the represen tatives can be modified see Figure 3 2 Remove current representative Removes the representative selected in the listbox Not available in the case of multi model representation Remove all representatives Removes all representatives Not available in the case of multi model representation Reset operating domain Resets all parameters to their initial values i e zero for controller pa rameters and lower bounds for varying parameters all other parame ters are set to one Close Closes the window PARADISE offers comfortable feature to substitute Simulink parameters This is especially useful if term appears more than once in the system equations Simply specify a substituting term in the Simulink model After the model was read in into PARADISE select Edit Edit substitution list from the Parameter specification window This replaces the listbox by another one where all Simulink parameters are contained in the item Direct substitution 28 CHAPTER 3 GETTING STARTED WITH PARADISE Simply select the parameter to be subst
48. nus integrated in the window see Figure 4 2 AII varying and controller parameters are available for selection For the additional items 1 Invariance plane and User defined the reader is refered to the next chapter Run Execute grid This item starts computation of the stability boundaries in the selected plane The varying parameters not considered in the parameter plane are being gridded where the number of grid points has to be speci fiedinthe Parameter specification window for each param eter The stability boundaries are computed for each grid point and displayed graphically For the generation of mapping equations the I region with index one will be used also if more than one I region has been defined by the user Execute representatives Starts computation of the stability boundaries in the selected plane The values for varying parameters not considered in the parameter plane are taken from the representatives defined in the Parameter specification window The stability boundaries are computed 38 CHAPTER 4 THE PARAMETER SPACE METHOD for all representatives and displayed graphically For each representa tive the I region specified by its index in the Parameter specification window will be used for the generation of the mapping equations Quit Closes the parameter plane window Stability checks From this submenu you can perform different I stability checks of arbitrary points in the displayed pa
49. on 3 Representative 4 ssociated with Gamma region 4 Parameter planes Set value Figure 3 2 Parameter specification window for multi model representation To change a parameter value select the adequate line The associated value will appear in the edit box at the bottom of the window where it can be edited The new value will be set by hitting the return key or by moving the cursor out of the editor field To move a parameter to another category select the desired parameter and chose Edit Classify parameter Controller parameter ifthe parameter should be moved to the category of controller parameters Parameter specification window menus I Edit Edit substitution list Switches to the list of Simulink parameters for substitution Classify parameters 3 3 PARAMETER SPECIFICATION 27 Varying parameter Moves the selected parameter to the class of varying parameters Controller parameter Moves the selected parameter to the class of controller parame ters Fixed parameter Moves the selected parameter to the class of fixed parameters Add representative Defines nominal operating point which can be used later for anal ysis or controller design new item is inserted in the listbox in Advanced Representatives The initial values for the pa rameters are the center of the operating domain Additionally region can be associated with this representative i e if I stability of this represe
50. ontrol system requires in the first step a model of the plant A 1 2 MULTI MODEL REPRESENTATION 3 linear state space representation of the plant dynamics is 0 1 0 0 0 0 0 zi 0 0 0 0 1 x 0 1 1 2 mol mot where the states are crab position crab velocity 22 rope angle x3 and rope angle rate x4 A stability analysis of the crane can be performed using its characteristic polyno mial It is p s mr mc Det sI A s s 1 9 0 1 2 The plant parameters enter into the characteristic polynomial and hence make the plant dynamics depending on these parameters For this simple example the roots of the polynomial can be calculated explicitly They are 12 0 1 3 53 4 1 4 mz mc g It is a fourth order system with a double pole at the origin and a complex con jugate pole pair varying with the operating parameters along the imaginary axis from J 1 mz mc g C to 1 mj mc g e For details about parametric modeling the reader is referred to the literature for example C619 1 1 2 Multi model representation Not in all cases symbolic system equations can be retrieved for instance if the plant is highly nonlinear and complex In those cases it is often possible to obtain a finite set of local linear models for a number of distinct operating points for example by means of system identification of the open loop plant or by lineariza tion of a nonlinear pla
51. pecially in those cases where the state space model is the result of a Maple session and already available in the desired format The state space model has to be stored in advance to a file named system name Sym as shown in the following sample ASymb 0 1 0 0 0 0 a23 0 POLO 40 7 TG 0 0 a43 011 BSymb 0 b2 0 b4 CSymb convert array l 4 1 4 identity listlist DSymb 0 0 0 01 In this example the Maple commands array and 1istlist were used instead of 1 0 0 0 0 0 0 1 for the output matrix The Simulink state space box has to be filled then with the substitutes ASymb BSymb CSymb and DSymb Note that the representation may be mixed e g only a substitute for the input matrix is used while the other matrices are specified con ventionally in the Simulink state space box 3 2 3 Multi model specification The multi model approach pointed out in Chapter I requires special format for the plant input Consider the case that a plant is not represented by contin uous plant description but by a set of nominal state space models Aj n B Bn sius D This is for example the case if no sym bolic plant model is available and the nominal matrices were obtained by some 24 3 GETTING STARTED WITH PARADISE identification procedures from technical plant Then construct cells in Matlab built up from the system
52. poles forming a union with the remaining I region guarantees that the poles do not move too far away from the zeros 1 4 Robust controller design and analysis Back to the crane design example The task is to determine a controller which robustly stabilizes the crane for the entire operating domain In this example state feedback u will be investigated 1 4 ROBUST CONTROLLER DESIGN AND ANALYSIS 7 In case of a step input the initial force is proportional to the controller gain k It can be adjusted according to the efficiency of the used actuator for the design example it will be set to 500 The gain k feeds back the rope angle rate which is difficult to measure Since the rope angle rate is observable from the rope angle its measurement can be neglected which leads to k4 0 Nothing can be said at this moment about k and With these simple considerations it was possible to reduce the number of unde termined controller parameters and hence simplify the design The closed loop characteristic polynomial is now with g 10 m s and mc 1000 kg p s mz L ka 5000 10kas 10000 500 10m ka s 4 kols 10005 1000 The design step has to assure that for the resulting nominal values of kz and the roots of the characteristic polynomial lie in the region for all possible operating conditions In general this cannot be accomplished in one design step for the entire operat
53. rameter plane The points are selected with mouse clicks using the left mouse button Pressing the right mouse button quits this mode If the selected point is I stable the point will be marked with a green star in the parameter space window If the Check individual stability Option from the Opt ions Menu is selected points which are I stable for some but not all points are marked by a blue triangle If the Check individual stability Option is not selected these points will be marked by a red symbol since they are not robustly Points which are not I stable will be marked by a red symbol in all cases Check stability for Center of Q box Using this check I stability is only checked with respect to the Center of the specified Q Box Check stability for grid This entry checks I stability with respect to all grid points of the given Q Box Check stability for representatives This entry checks stability with respect to all defined representa tives Tools Select controller If at least one of the parameters of the selected parameter plane is a controller parameter this feature allows to select controller parame ters graphically with the mouse cursor The selected point is marked by a cross and the corresponding controller parameter values will be automatically set to the selected values The cross can be moved by clicking on it with the mouse cursor and dragging it to the new posi tion This action automatically r
54. s one e Controller parameters By default parameters starting with k are catego rized as controller parameters Their initial value is zero e Fixed parameters These are parameters assumed constant By default all parameters initially not categorized as controller or varying parameters are fixed parameters Their initial value is one To change parameter settings select Input Parameter specification from the main window This opens the window illustrated in Figure 2 5 It con tains a listbox which can be unfolded to display the desired information For 26 3 GETTING STARTED WITH PARADISE example to inspect varying parameters perform double click on Varying parameters This will unfold the desired information and display the refering parameter names In order to inspect or change the settings of a specific parameter perform double click on the specific line In the case of multi model representation as described in Section 3 2 3 varying parameters cannot exist since the plant is described by a set of nominal system equations see Figure 3 2 Parameter specification OP Edit Varying parameters lt Not permitted in MultiModel mode gt Controller parameters kq 0 940909 knz 0 108601 Fixed parameters lt enpty gt Advanced Representative 1 Associated with Gamma region 1 Representative 2 Associated with Gamma region 2 Representative 3 Associated with Gamma regi
55. sign approaches which already incorporate the plant uncertainty in the design step are then required 1 1 Parametric models and uncertainty Robust parametric control offers several design approaches for this type of prob lems The basis of these approaches is parametric model As an example con sider the crane in Figure 1 1 The task of the operator of such a crane is to pick up a load and transport it to another location while preventing the load from sway ing Especially when placing a load precisely for example on top of a truck or on a container ship the load should have stopped completely from swaying which requires training and full attention of the operator An anti sway control system certainly will support the operator and help him to operate the loads faster The design of such a control system has to handle the uncertainty in the rope length and the load mass mz and guarantee stability 1 2 1 INTRODUCTION TO ROBUST CONTROL mr Figure 1 1 crane and its schematic representation for all possible operating conditions The load mass may vary from the hook mass up to the maximal load mass m and also the rope length varies within known limits other crane parameters like the crab mass mc are assumed tot be known The resulting rectangular operating domain designated as is illustrated in Figure 1 2 ML gr Figure 1 2 Rectangular operating domain of the crane The design of a c
56. ter of the Q box and of the representatives Clear eigenvalues The currently displayed eigenvalues will be removed from the T region window In Figure 3 3 the regions are displayed for the flight control example Each of ditor iO x Edit Options iGamma region 1 a Ganna region 2 Set construction un is el e2 e3 e4 Element 1 Circle 0 Radius 12 6 Inverted No Element 2 Damping Damping 0 35 Inverted No Element 3 Circle signa Radius 3 5 Inverted Yes Element 4 Real interval Lower 70 Upper 15 Inverted No Gamma region 3 Ganna region 4 Set value Figure 3 3 Set of regions for the flight control example the I regions is associated with a representative flight condition This figure also reveals two other features of the graphical editor 32 3 GETTING STARTED WITH PARADISE 1 Each basic element can be used in its default version or in its inverted form For instance the I stable part of a circle is by default its interior If the entry Inverted is changed to Yes the exterior of the cirlce describes the region If more then one basic element is used to construct a I region the resulting set will be by default the intersection of the two basic elements If however also the union of basic elements is required it can be specified using the set construction feature Using the two functions is for intersect
57. that the model does not cover the entire plant uncertainty If the plant is highly nonlinear the computed set will represent only those controllers for which the linearized plant is stabilized Simula tions of the nonlinear plant for various controllers from the computed set can then provide further criteria for selecting the final controller If the controller has to be designed for an uncertain plant the first step is to determine the set of stabilizing controllers for a nominal point Qo For fixed gain control it is obvious that the resulting controller also has to stabilize the plant for other nominal operating points q Q A straight forward approach is to compute the set of stabilizing controllers for other operating points for example all vertices of the operating domain Then the intersection of all these sets guarantees stability for the considered operating points called representatives of the operating domain This approach of simultaneous I stabilization does not guarantee robustness of the entire operating domain this has to be verified by a robustness analysis However due to the fact that an arbitrary number of representatives can be considered in the design phase it is a powerful tool for robust controller design 36 4 THE PARAMETER SPACE METHOD Of course it might happen that the set of stabilizing controllers is empty i e there does not exist a controller which simultaneously I stabilizes the
58. the listbox will be updated automatically I region editor window menus Edit 3 4 THE EDITOR 29 Add basic element Real part limitation Mathematical description OV o jw o m w 0 co The parameter 09 sigma has to be specified Imag part limitation Mathematical description Or a jw e 0 v wo The parameter omega has to be specified Real interval Mathematical description OF fo jw o 07 0 w 0 The parameters c Lower and ot Upper have to be speci fied Damping Mathematical description Or fo jw o 0 sign oo v J1 10 The parameter Damping has to be specified Also negative values are allowed Then the boundary is contained in the right half plane Hyperbola Mathematical description OV o jw 2 2 1 w 0 The parameters a and b have to be specified Alternatively the pa rameters intersection of the hyperbola with the real axis sigma and the damping Damping can be specified Circle Mathematical description OV e jw o 0 R o cog co RI The parameters 09 Sigma and Radius have to be speci fied 30 3 GETTING STARTED WITH PARADISE Pair of circles Mathematical description OV Re juo a 0 27 The parameters 09 Sigma wo omega and Radius have to be specified
59. ting a suitable controller which shifts these eigenvalues a controller can be determined for which all eigenvalues are located in the desired I region For uncertain plants suitable cross section in controller parameter space is de termined for nominal operating point e g the center of the Q box and the set of simultaneously stabilizing controllers is determined in the given invariance plane for a number of representatives in the operating domain By iteratively changing or adding representatives a robust controller for a growing portion of the operating domain can be found A detailed example will illustrate this process in detail For a complete reference and case studies see 93a 5 2 1 Design in an Invariance Plane Assume an n th order state space model 2 bu with proportional state feedback control u x For this system it is possible to determine an m dimensional subspace in the n dimensional controller parameter space such that only m specific eigenvalues of the given plant are shifted by arbitrary selection of controller gains from this subspace while the remaining n 7n eigenvalues remain at their original locations This approach is based on Ackermann s Formula ABK 93a Theorem Ackermann For a controllable single input system the feedback vector k e p A 5 1 with T 2 n 1 l 0 0 1 Ab Ab 5 2 assigns the eigenvalues of A bk to th
60. ty i e w Then the characteristic equation 4 1 can be separated into real and imaginary part Re p jw q1 avla q2 an qi 0 Im p jw 9 4 w a q q2 aa q1 42 0 an 1 q1 0 4 2 Here n was assumed even the case of odd n is straight forward For fixed frequency w wo the set of equations can be solved for q and q2 The result de scribes exactly those points in parameter space for which the characteristic poly nomial has roots at s jw on the boundary of which is the imaginary axis in this case sweep of yields the complete set of critical points which forms the stability boundaries in the space of uncertain parameters For fixed frequency wo the set of equations may have different types of solutions e No solutions This means that there do not exist points which yield eigen values at s e Finite number of solutions For each of these points the characteristic poly nomial has roots at s e Indefinite number of solutions Both equations of 4 1 become linearly dependent Then either real or imaginary part represent the set of solutions for this specific frequency Frequencies of this type are also refered to as singular frequencies Besides this categorization the solutions are typically distinguished between real and complex root boundaries The first ones result from frequencies where the
61. ultaneously I stabilize the representatives it has to undergo a precise analysis which has to verify 8 CHAPTER 1 INTRODUCTION TO ROBUST CONTROL jo Ai gt b ay Ap be ky S plane k space Figure 1 6 Simultaneous I stabilization of representatives that the controller is robustly I stable for the entire operating domain If this test passes the design is finished Otherwise go back to Step 1 and add more representatives especially from the regions where the analysis indicated I instability The controller structure assumed for the design step is crucial for its success High degree controllers may give more freedom in the design directions how ever within an interactive design procedure the design engineer easily may loose survey of the influence of each controller parameter on the system dynamics For this reason it is recommended to start with the most simple controller structure If a design fails relax the design specifications and try again If this design suc ceeds investigate how to augment the controller structure to meet more restrictive specifications As you can already tell from this recipe robust control design is mostly not a straight forward design procedure It is much more a learning pro cess of the design engineer supported by robust control methods In this learning process the engineer will get much more insight into the plant dynamics and its dependency on the uncertai
62. y I stable By design the four vertex points are stable and the operating domain is not intersected by stability boundaries 4 3 DESIGN EXAMPLES FOR THE PARAMETER SPACE METHOD 43 Parameter space Bun Stability checks Tools Options Output 4 x 10 3500 4000 4500 k2 m Figure 4 4 Set of simultaneously stabilizing controller parameters for the ver tices of the operating domain red 8m mz 1000 kg green 16m mr 1000 kg blue l 8m 2000 kg cyan 16m 2000 kg 44 4 THE PARAMETER SPACE METHOD Parameter space Bun Stability checks Tools Options Output anng zoom off 2500 1500 ml Y Figure 4 5 Robustness analyis The controller is robustly I stable since the ver tices are stable and the operating domain is not intersected by stability bound aries 4 3 DESIGN EXAMPLES FOR THE PARAMETER SPACE METHOD 45 4 3 2 Design example 2 F4E fighter aircraft A different type of example is the F4E aircraft design Here a finite number of represenatives is given While for the crane design example the representatives considered for the design could be modified arbitrarily the user is fixed to the specified nominal representatives The Simulink model for this example is shown in Figure 4 6 Block Parameters Aircraft model m State Space Fie Edit View Simulation Format Tools State space modet WII ioj x jB sua o5e 2
63. zes the whole operating domain Parameter space Op Bun Stability checks Tools Options Output 1 i i i i i i i 6 8 10 12 14 16 18 20 0 7 20 4 rz Figure 5 6 Analysis Parameter space window 5 3 EXAMPLE CRANE 57 2 Iteration Using the same steps which we used for the first iteration we specify a new oper ating point for the invariance plane in order to robustly stabilize a bigger portion of the operating domain We choose m 300 kg 16 m as the operating point for the invariance plane Figure 5 6 showes a stability boundary very close to the vertex mz 150 kg I 8m Therefore we include this point and two vertex points as representatives for the following design step to avoid that this vertex becomes unstable since the invariance plane holds only for the operating point Figure 5 7 shows the Parameter specification window with the four selected representatives and the invariance plane Parameter specification Joj x Edit Varying parameters Controller parameters 4 2587 04 N ow nw M Fixed parameters dvanced Representatives Associated with Gamma region 1 1 16 mL 300 Representative 2 Associated with Gamma region 1 mL 2000 Representative 3 Associated with Gamma region 1 mL 2000 Representative 4 Associated with Gamma region 1 1 mL 150 Parameter planes Set value Figure 5

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