Nonlinear Control(FRTN05)

Transcription

1 Nonlinear Control(FRTN05) troduction to Simulink Last updated: Spring of 204 Contents Te exercise is intended as an introduction into Simulink and te Control System Toolbox. It can be performed in Matlab and Simulink, eiter at te department or at any oter computer tat as Matlab wit te Control SystemToolbox,andSimulink.Tofindoutifyouavetecontrolsystem toolbox write elp control.(we will use te commands ss, tf, bode, nyuist, evalfr. Use elp to find out ow tey work.) Te introduction contains small examples of ow Simulink works.. troduction to Simulink Simulink is a simulation program based upon Matlab. Tere are severalwaystodefineamodel.onecanworkgrapicallyandconnectblockdiagrams wit predefined blocks. Alternatively one can give te matematical description in forms of differential euations in an m-file(te format for programs written in te Matlab programming language). Matlab/Simulink supports bot tese representations as well as combinations. Furtermore one can use descriptions tat include a ierarcy of connected subsystems. To understand ow models are described and simulated using block diagrams,itisbesttorunsmallexamplesonacomputer.. How to Start Simulink Start Matlab. Ten give te command simulink in Matlab. Tis gives a windowwitblocksasinfigure.eacblockrepresentsalibrarytat contains several building blocks. Out Sources Sinks Continuous Discrete Mat Functions & Tables Nonlinear Signals & Systems Simulink Block Library 3.0 Copyrigt (c) by Te MatWorks, c. Figure Available Simulink block diagram libraries

2 .2 A Simple System Click on File in te Simulink-window and coose New->Model. Click on te library Continuous and move a Transfer Fcn to te new window called Untitled. Do te same wit Sources->Step Fcn and Sinks->Scope. Draw arrows(left mouse button) and connect te ports on te block. You sould nowaveablockdiagramasinfigure2. Step s+ Transfer Fcn Scope Figure 2 A simple Simulink system Coose Simulation->Parameters in te window called Untitled. Set Stop timeto5.openawindowforte Scopebydoubleclickingonit.Start a simulation by Simulation->Start(or by pressing Ctrl-t in te window called "Untitled"). HowtoCangeaSystem Tocangetesystemto s s+2 you double-click on te block Transfer Fcn and cange Denominator to [ 0.5 2]. Simulate te new system(simulation->start or Ctrl-t). Cange parameters in te Simulation menu and te scales in te block Scope until you are satisfied. HowtoCange anputsignal Tocangeteinputsignal,start witremovingteblock Step Fcnbyclickingonitanddeleteitbypressing Delete(or using Edit->Cut or pressing Ctrl-x). Replace it by a Sources->Signal Generator block. Double-click on Signal Generator and select a wave form, amplitude and freuency. Also cange Simulation->Parameters->Stop Time to and press Simulation->Start. Tis gives an"infinite" simulation tat can be stopped by pressing Simulation->Stop(or Ctrl-t). Can te amplitude of te input signal be canged during simulation? Also try to cange te block Transfer Fcn during simulation. How to Use Matlab Variables in Blocks Note tat variables defined in te Matlab environment can be used in Simulink. Define numerator and denominator by writing te following in te Matlab window. num=[ ] den=[ 2 3 4] Cange Transfer Fcn->Numerator to num and Transfer Fcn->Denominator to den. 2

3 HowtoSaveResultstoMatlabVariables Tosaveinputandoutput, move two copies of te block Sinks->To Workspace. Connect tese wit te input and output to te block Transfer Fcn. Get a Sources->Clock and connect it to a Sinks->To Workspace. Double click on te Workspace blocks tobe able to cange te variable names to u,y,and trespectively. Also cange Save format to te value Array. Te window sould look someting like Fig. 3. Figure3 HowtosaveresultstoMatlabvariablesinSimulink How to Use Simulation Results in Matlab Calculations Let te input signal be a sinusoidal wit freuency 0. rad/s and amplitude. Doasimulationtatislongenougforteoutputtobecomestationary. Compute n=lengt(y) max(y(n/2:n)) and compare tis wit te teoretical value G(0.i). >>g=tf(num,den) Transfer function: s s^3 + 2 s^2 + 3 s + 4 >> abs(evalfr(g,0.*i)) ans = How to Save Systems Use File-Save As or File->Save..3 AFlowSystem Considerasimpletankasintebasiccontrolcourse ḣ = A (u ) = a 2. 3

4 TiscanbeimplementedinSimulinkasinFigure4.Tefunction f(u) /A Gain s tegrator f(u) Fcn 2 Figure4 Atanksystem as te value a*srt(2*g*u[]). Te summation block as been given two inputs wit different signs by assigning te string -+ to Sum->List of Signs.Tesummationandte Gainblocksarefoundinte Matlibrary andte Fcnblockisfoundinte Functions & Tableslibrary.Tesmall ellipses, tat are contained in te Signals & Systems library, tell Simulink wat sould be considered inputs and outputs to tis(sub)system. Te blocktitlescanbecangedbyclickingontem.markteentiresystem by olding te left mouse bottom pressed and drawing a rectangle around it. Ten coose Edit->Create Subsystem. Te result is tat te system is represented by one block. Use Edit->Copy to create te following doubletanksystem.usetecommands trimand linmodtofindalinearizedmodel Subsystem Out Subsystem Figure 5 Two tanks and some connections oftedoubletankaround 0 =0 2 =0..UseteparametersA =A 2 = ,a =a 2 = , =9.8.PlotteNyuistcurveusingte command nyuist. >> A=2.7e-3;a=7e-6;g=9.8; >> [x0,u0,y0]=trim( flow,[0. 0.],[],0.) Warning: Output port 2 of block twotank/subsystem is not connected. Warning: Output port of block twotank/subsystem is not connected. 4

5 x0 = u0 = e-06 y0 = >> [aa,bb,cc,dd]=linmod( flow,x0,u0); >> sys=ss(aa,bb,cc,dd); >> bode(sys) Alternative: Linearization in Simulink; Byrigt-clickingonasignalconnectorinaSimulinkmodelyoucanadd Linearization points (inputs and/or outputs). Use tis for te two water tanks. Start a Control and Estimation Tool Manager by Tools -> Control Design ->Linear analysis..., see Fig. 6. Here you can set te desired operating points, export linearized model to Workspace (Model-> Export to Workspace) and muc more. Repeat te linearization of te system at te same euilibrium point as above. Figure 6 View of"control and Estimation Tool Manager" in Matlab/Simulink. 5