FRT 041 System Identification Laboratory Exercise 3

Transcription

1 FRT 041 System Identification Laboratory Exercise 3 Ulf Holmberg Revised: Kjell Gstafsson Karl Henrik Johansson Anders Wallén Johan Nilsson Rolf Johansson Johan Bengtsson Maria Henningsson Department of Atomatic Control Lnd Institte of Technology Lnd University October 1987 Revised: September 1989, March 1994, March 1995, Febrary 1996, Febrary 1997, Febrary 1999, March 2005, Febrary 2011

2 Identification and Controller Design Theprposeofidentificationisoftentogetamodelthatcanbesedfor controllerdesign.whatisaimedatisalowordermodelthatcaptresthe mainpropertiesoftheprocess,andwhichcanbesedforanalysisand synthesis. It is important that the model is accrate for a freqency band arond the crossover freqency, while higher order dynamics and weak nonlinearities often can be discarded. Also the low freqency properties are often less important, since the closed loop behavior at these freqencies is determined by having integration(high gain) in the controller. Still, the signofthestaticgainhastobeknown. The model is valid only for a limited freqency range, and probably ncertain also within this range. Therefore when doing the design of the controller a synthesis method reslting in a robst controller shold be sed, i.e., the performance of the controller shold not be affected by small changes in the model. In this laboratory exercise an identification experiment will be done. UsingMATLABamodelwillbeestimatedandsedtodesignacontroller. The controller is then tested on the real process. Throghot the laboratory exercise MATLAB is sed. This manal contains many references to MATLAB fnctions. Some of these are standard MATLAB fnctions while others were written to facilitate the exercise. Wheneveryoareindobtabotwhatacertainfnctiondoes,se help. Preparation Read throgh A Manal For System Identification [Andersson et al.(1998)] whichwassedinlab2andbringittothelaboratoryexercise. 1. The process The process tobe designed isalittle tricky to control. It consists of a rectanglarplatehanginginoneofitsedges,seefigre1.theplateis montedschthatitcanswingbackandforth.aweightismontedon onesideoftheplatetomakeitdeflectfromtheverticalplane.afanis positionedashortdistancefromtheplateandsedtoblowanairstream on the plate, ths affecting its position. The angle between the plate and theverticalplaneismeasredandactasotptfromtheprocess.itis possibletocontroltheplateanglesingthevoltagetothefanmotoras control variable. The process has the following properties Thereisatimeconstantinthefanmotorandhencetheairstream is not immediately affected when varying the motor voltage. Theprocesscontainsatimedelaysinceittakessometimefortheair stream to reach the plate. Theplateactslikeapendlmresltinginalightlydampedresonance. Trblence cases noise which affects the process. Try to control the process manally. Use a potentiometer to vary the voltage tothefanmotor.trytoestimatethetimeconstantoftheprocessandthe 1

3 Figre 1 The process. freqencyoftheresonancesothatyolatermayjdgeiftheidentified model is reasonable. 2. The identification experiment Yo are to perform an open loop identification experiment on the process. StartMATLABandtype initlab3atthecommandprompt.thissetsp MATLABandopenstheSIMULINKmodel logger,whichwillbesedto logdata,seefigre2.themodelgeneratesaprbs(psedorandombinary seqence) signal which is sed as inpt to the process. The excitation signal andtheprocessotptareloggedandsavedintheworkspaceas and y for later processing in MATLAB. measred y measred y PRBS generator Ot1 measred jitter Process To Workspace y To workspace Note: Skip initial data from the seqences de to real time problems with initialization (see the jitter plot) measred otpt jitter time Clock To Workspace2 Figre 2 Simlink model for logging data. The plate is to be controlled approximately arond its downright position, i.e. a plate angle slightly greater than zero. This corresponds to a certainvoltagefedtothefanmotorsincetheweightontheplatecases ittodeflect.adjstthemeanvaleoftheexcitationsignaltogetasmall positiveplateangle.oneeasywaytoachievethisistoseanexternal adjstable voltage added to the signal to the fan motor. When the mean vale is adjsted, set the amplitde of the excitation signal to 0.5 V. According to Åström and Wittenmark(1997) it is reasonable tochoosethesampleperiodhschthat ωh ,where ω isthe resonance freqency of the plate estimated above. Ths we may choose h=50ms.frtherwedecidetocollect1200datapoints,andtosetthe PRBSperiodto10.Starttheloggingandconsidertheloggeddatainthe scopes. 2

4 Figre 3 GUI for System Identification Toolbox. 3. The identification The System Identification Toolbox in MATLAB will be sed to estimate a model of the process. The identification may be done either by writing the commands below or by sing the graphical ser interface of the System Identification Toolbox, see Figre 3 The graphical ser interface is started by writing at the command prompt ident The rest of this section describes command-line identification. Yo can follow the same procedre in the graphical ser interface. Beginbylookingatthedatabyrnning plot([y ]) Do the signals look all right? What transients shold be neglected? Pick ot the process otpt and the excitation signal and remove the bias sing z = [y ]; z = detrend(z, constant ); whichgivesamatrixwithtwocolmns yand.wecheckinwhatfreqency interval a good model might be estimated by plotting the coherence fnction Γ y (ω)= S y (ω) S (ω)s y (ω) sing the command 3

5 mscohere(,y,[],[],[],1/h); In a previos laboratory exercise we did freqency response analysis. An alternative is to do spectral analysis in order to get an estimated freqency response. The qality of this estimate will highly depend on the length of the Hamming window sed. This makes spectral analysis sometimes hard toseinpractice.ithasbeenshownthatforordataawindowlengthof abot100isgood.try g = spa(z,100,[],[],h); plot(g); It is often sefl to split the data into two seqences; one for identification and one for verification: nz = size(y,1); z1 = iddata(y(1:nz/2),(1:nz/2),h); z2 = iddata(y(nz/2+1:nz),(nz/2+1:nz),h); z1 = detrend(z1,0); z2 = detrend(z2,0); Use the fnction armax(z,[na,nb,nc,k]) to estimate an ARMAX model inthebackwardshiftoperator(q 1 )accordingto M : (1+a 1 q 1 + a na q na )y(t) =(b 1 q k + b nb q k nb+1 )(t) +(1+c 1 q 1 + c nc q nc )e(t) Note that nb corresponds to the nmber of b-parameters and not the degree of the B polynomial. The parameter k denotes the time delay in the system. Remembertohave k>0togetacasalmodelwithotanydirectterm. Do yo have any ideas abot initial vales for [na,nb,nc,k]? Theresltofthecommand armaxisrepresentedonaspecialform,the theta-form. The command present will list the parameter vales, their variance, the vale of the loss fnction, and the Akaike FPE(final prediction error) vale. An example is th3222 = armax(z1,[3,2,2,2]); present(th3222) UsetheFPEandthevarianceoftheparametervalestodeterminea sitabletimedelay(k)andmodelorder(na, nband nc).trytofindagood model with an order as low as possible(the order eqals max(na,nb+k-1)). Agoodwaytoverifythemodelistocompareitsotptsignalwiththe process otpt. This can be done as follows ym = idsim(z2.inptdata,th3222); t = h [1:1:length(z2.OtptData)]; plot(t,z2.otptdata,t,ym); Notethatwese z2andnotthedatathatweresedformodelestimation when we simlate the model. Yosholdalsotakealookatthepole-zeroconfigrationofthemodel. Atoolargemodelordermayshowpaspolesandzerosthatalmostcancel. Use pzmap(th3222) 4

6 Thefinalmodel,i.e.theonethatismostsatisfactory,sholdbesed todesignacontroller.thisisalsodoneinmatlab.themacrosthatdo thedesignneedaprocessmodelontheform B(q)/A(q),with B(q)and A(q) being polynomials in the forward shift operator. The two polynomials have to be extracted from the theta-form and converted to forward shift operatorform. Aand Binbackwardshiftoperatorarederivedby A = th3222.a; B = th3222.b; Rewrite the transfer fnction into forward shift representation on a piece ofpaper.matlabcommandsforwhatyohavedoneare A = [A zeros(1,k+nb-1-na)] B(1:k) = [], B = [B zeros(1,na-k-nb+1)] Make sre yo nderstand that these commands convert from backward to forward shift. Compare the zeros of A with the expected closed-loop poles from the introdction. 4. The controller design Thedesignmethodthatisgoingtobesedispoleplacement.Itmaybe hardtodecidewheretoplaceallthepoles,soforsimplicitywewillchoose apolepatternandonlyvaryitsdistancefromtheorigin.thisiseqivalent torestrictingthetimeresponseoftheclosedloopsystemtoacertainform, andthenonlyvaryits speed (why?).wewillvarythedesired speed and try to evalate the robstness of the reslting closed loop system. Finally, some promising designs will be stored for ftre tests on the real process. Hereisastepbystepdescriptionofthedesignmethod. 1. Firstlookatthefreqencyresponseofthemodeltotrytogetanidea ofhowmchitispossibletodemandfromtheclosedloopsystem. Plot the Bode diagram sing bode(th3222) AsoneexpectsthereisalargeresonanceintheBodediagram,corresponding to the plate acting as a pendlm. The freqency of this resonance tells abot the natral freqency of the open loop system andgivesahintabotwhattoexpectfromtheclosedloopsystem. The controller shold take care of the resonance and damp it. Trying to get a closed loop bandwidth differing mch from the resonance freqency will reqire a large control effort and a very accrate model. (Comparewithhowtheclosedlooppolesaremovedinarootlocs plot when a proportional controller is sed.) What do yo regard as a reasonable range for the closed loop bandwidth? PlotalsotheNyqistcrveofthemodelsing nyqist(th3222) Is proportional feedback sfficient to get a stable closed loop system with reasonable performance? 5

7 Figre 4 The continos time pole pattern sed in the design. 2. Poleplacementwillbesedasdesignmethod,seeÅströmandWittenmark(1997).Thedesiredcharacteristicpolynomial A m andthe observerpolynomiala o havetobechosen.togetacasalcontroller itmstholdthat dega o 2degA dega m degb + 1 whereb + isthecanceled(stable)partofb.forsimplicityletsdecide nottocancelanyzeros,i.e.b + =1, B =B,andchoosedegA m = dega n.then dega o n 1 ChoosingdegA o =n 1givesacontrollerwithdirectterm.Toget anintegratorinthecontroller,thedegreeoftheobserverhastobe increased(why?). Ths dega o n Itmaybehardtorelatediscretetimepolestothepropertiesofthe closedloopsystem(atleasttheathorbelievesthisishard).wewill thereforerelatethechoiceofa m anda o tocontinostimepoles.for simplicitychoosethepoleseqallyspacedonasegmentofacirclein thelefthalfplaneofthes-plane.theradisofthecircleis ω m and halftheopeningangleis45 (akindofbtterworthconfigration, seefigre4).theobserverpolesarechoseninthesamewaybt attwicethedistancefromtheorigin,i.e. ω o =2ω m.thecontinos timepoles are translated todiscrete time throgh z= e sh.for a non-integrating controller the commands are Amc = polybtt(n,wm,45); Aoc = polybtt(n-1,2*wm,45); Am = real(poly(exp(roots(amc)*h))); Ao = real(poly(exp(roots(aoc)*h))); Thereisonlyoneparameter, ω m,tovaryinthedesign.itcorresponds totheclosedloopbandwidth,andwewillchooseitinrelationtoor observations in step 1. The choice of pole pattern is natrally jst a sggestion. It is probably possible to make a better design by choosing a different pattern. If 6

8 yo have time, try other configrations, e.g. mltiple poles on the real axis. 3. The controller is calclated by solving the DAB-eqation(Diophantine- Aryabhatta-Bezot) AR 1 +B S=A m A o with B=B + B and B m = B mb.thecontrollerpolynomialsare given by R=R 1 B + S=S T=t 0 A o B m wheret 0 issedtoadjstthestaticgainoftheclosedloopsystemto one. In MATLAB the calclations are done sing(do help rstd): [R,S,T] = rstd(1,b,a,1,am,ao,ar) ThepolynomialA r isforcedintorwhensolvingthedab-eqation. This makes it possible to inclde integral action in the controller, i.e. A r =q 1orinMATLABnotation Ar = [1-1].Ifnointegratoris wantedjstpt Ar = When the controller has been calclated it needs to be evalated. This canbedonebyplottingthenyqistcrveofthelooptransferfnctiong o (q)=b(q)s(q)/a(q)r(q).ifthenyqistcrvepassesclose to 1 or design is probably not very good. Natrally, the controller stabilizes the model, bt since the model does not exactly describe the treprocessitmayverywellbethattherealsystemwillbenstable.thereforetrytofindan ω m thatgivesareasonableclosedloop bandwidth, bt withot having a Nyqist crve passing too close to 1.Ifthiscanbesatisfieditislikelythatthecontrollerwillperform well even if the tre process shold differ slightly from or model. When interpreting closeness to 1, think in terms of gain and phase margin.trytogetaloopwithgainmarginapproximatelyeqalto2 andphasemarginapproximatelyeqalto60. PlottheNyqistcrveofG o (q) nyqist(series(tf(s,r,h),tf(b,a,h))); Evalate the controller first by simlation. To save a controller for later se, do save regname R S T This command saves yor controller in a file regname.mat. Of corse, the controller shold be casal. Check the polynomial degrees so that this reqirement is flfilled. Iterate the design steps ntil a cople of good controllers are fond. At least, try to find two with integration and two withot integration. 7

9 5. Testing the controller Wearenowreadytotestthecontrollersdesignedabove.Ifnotalready closed, close the logger-model and then type lab3_controller to bring p thesimulinkmodelwhichwillbesedforcontrol, seefigre5.the model consists of a reference generator, a controller on RST-form, the I/O connection to the process, and some scopes for displaying the signals of the system. Mx yref and y Signal Generator 0.2 Sm y_ref y Discrete RST controller measred y measred jitter Process offset jitter Figre 5 Simlink model for real-time control. Startbychoosingasqarewavewithmean0.1V,amplitde0.05V,and period 20 s as reference signal. Load a controller in workspace by execting load regname Finally, test the performance of the controller by varying the process parameters: change the distance between the fan and the plate, or change the weight on the plate. Compare yor different controllers. Do they behave as yo expected from the design phase above? Which controller wold yo prefer? 6. References Andersson, L., U. Jönsson, and K. H. Johansson(1998): A manal for system identification. In Laboratory Exercises in System Identification. KF Sigma i Lnd AB. Department of Atomatic Control, Lnd Institte of Technology, Box 118, S Lnd, Sweden. Åström, K. J. and B. Wittenmark(1997): Compter-Controlled Systems. Prentice Hall. 8