nbar

CTM:StateSpaceTutorial

StateSpaceTutorial

State-spaceequations

Controldesignusingpoleplacement

Introducingthereferenceinput

Obrverdesign

KeyMatlabcommandsudinthistutorial:acker,lsim,place,plot,rscale

Matlabcommandsfromthecontrolsystemtoolboxarehighlightedinred.

Non-standardMatlabcommandsudinthistutorialarehighlightedingreen.

State-spaceequations

Therearevte-spacereprentationisgivenbytheequations:

wherexisannby1vectorreprentingthestate(commonlypositionandvelocityvariablesinmechanical

systems),uisascalarreprentingtheinput(commonlyaforceortorqueinmechanicalsystems),ricesA(nbyn),B(nby1),andC(1byn)determinethe

attherearenfirst-orderdifferential

pacereprentationcanalsobeudforsystemswithmultipleinputsandoutputs(MIMO),butwewillonlyusingle-input,single-output(SISO)systemsinthetutorials.

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CTM:StateSpaceTutorial

Tointroducethestatespacedesignmethod,wewilluthe

rentthrough

thecoilsinducesamagneticforcewhichcanbalancetheforceof

gravityandcautheball(whichismadeofamagneticmaterial)

elingofthissystemhasbeenestablishedinmanycontroltextbooks(includingAutomatic

,theventhedition).Theequationsforthesystemaregivenby:

wherehistheverticalpositionoftheball,iisthecurrentthroughtheelectromagnet,Vistheappliedvoltage,

Misthemassoftheball,gisgravity,Listheinductance,Ristheresistance,andKisacoefficientthat

plicity,wewillchoovaluesM=0.05Kg,K=

0.0001,L=0.01H,R=1Ohm,g=9.81m/c^tem感叹词英语 isatequilibrium(theballissuspendedin

midair)wheneverh=Ki^2/Mg(atwhichpointdh/dt=0).Welinearizetheequationsaboutthepointh=0.01m(wherethenominalcurrentisabout7amp)andgetthestatespaceequations:

where:isthetofstatevariablesforthesystem(a3x1vector),uistheinputvoltage(deltaV),and

y(theoutput),hesystemmatricesintoam-file.

A=[010

9800-2.8

00-100];

B=[0

0100];

C=[100];

Oneofthefirstt糯米粉的做法大全点心 hingsyouwanttodowiththestateequationsisfindthepolesofthesystem;thearethevaluesofswheredet(sI-A)=0,ortheeigenvaluesoftheAmatrix:

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CTM:StateSpaceTutorial

poles=eig(A)

Youshouldgetthefollowingthreepoles:

poles=

31.3050

-31.3050-100.0000

Oneofthepolesisintheright-halfplane,whichmeansthatthesystemisunstableinopen-loop.

Tocheckoutwhathappenstothisunstablesystemwhenthereisanonzeroinitialcondition,addthefollowinglinestoyourm-file,

t=0:0.01:2;u=0*t;

x0=[0.00500];

[y,x]=lsim(A,B,C,0,u,t,x0);

h=x(:,2);%Delta-histheoutputofinterest

plot(t,h)andrunthefileagain.

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CTM:StateSpaceTutorial

Itlookslikethedistancebetweentheballandtheelectromagnetwillgotoinfinity,butprobablytheballhitsthetableorthefloorfirst(andalsoprobablygoesoutoftherangewhereourlinearizationisvalid).

Controldesignusingpoleplacement

Let'ematicofafull-statefeedbacksystemisthefollowing:

Recallthatthecharacteristicpolynomialforthisclod-loopsystemisthedeterminantof(sI-(A-BK)).Since

thematricesAandB*Kareboth3by3matrices,gfull-state

dutheMatlabfunctionplacetofindthe

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CTM:StateSpaceTutorial

controlmatrix,K,whichwillgivethedesiredpoles.

Beforeattemptingthismethod,ethe

criteriaforthecontrollerwerettlingtime<0.5candovershoot<5%,thenwemighttrytoplacethetwo

dominantpolesat-10+/-10i(atzeta=0.7or45degreeswithsigma=10>4.6*2).Thethirdpolewemight

placeat-50tostart,the

lsimcommandfromyourm-fileandeverythingafterit,thenaddthefollowinglinestoyourm-file,

p1=-10+10i;

p2=-10-10i;p3=-50;

K=place(A,B,[p1p2p3]);

lsim(A-B*K,B,C,0,u,t,x0);

Theovershootistoolarge(therearealsozerosinthetransferfunctionwhichcanincreatheovershoot;有趣的文字 you

donotethezerosinthestate-spaceformulation).Tryplacingthepolesfurthertothelefttoeifthetransientresponimproves(thisshouldalsomaketheresponfaster).

p1=-20+20i;

p2=-20-20i;p3=-100;

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CTM:StateSpaceTutorial

K=place(A,B,[p1p2p3]);

lsim(A-B*K,B,C,0,u,t,x0);

tyou空桐 rtextbookforfurthersuggestionsonchoosingthedesiredclod-looppoles.

Comparethecontroleffortrequired(K)ral,thefartheryoumovethepoles,themorecontroleffortittakes.

Note:Ifyouwanttoplacetwoormorepolesatthesameposition,ua

functioncalledackerwhichworkssimilarlytoplace:

K=acker(A,B,[p1p2p3])

Introducingthereferenceinput

Now,wewilltakethecontrolsystemasdefinedaboveandapplyastepinput(wechooasmallvalueforthe

step,soweremainintheregionwhereourlinearizationisvalid).Replacet,uandlsiminyourm-filewiththefollowing,

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CTM:StateSpaceTutorial

t=0:0.01:2;

u=0.001*ones(size(t));

lsim(A-B*K,B,C,0,u,t)

Thesystemdoesnottrackthestepwellatall;notonlyisthemagnitudenotone,butitisnegativeinsteadofpositive!

Recallth怎样让孩子爱上学习 eschematicabove,wedon'tcomparetheoutputtothereference;insteadwemeasureallthestates,

multiplybythegainvectorK,snoreason血压高可以喝酒吗 toexpect

thatK*inatethisproblem,wecansca华南农业大学排名 lethereferenceinputto

makeitequaltoK*x_alefactorisoftencalledNbar;itisintroducedasshowninthefollowingschematic:

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CTM:StateSpaceTutorial

WecangetNbarfromMatlabbyusingthefunctionrscale(placethefollowinglineofcodeafterK=...).

Nbar=rscale(A,B,C,0,K)

ere

,ifwewanttofindtheresponofthesystemunder

statefeedbackwiththisintroductionofthereference,wesimplynotethefactthattheinputismultipliedbythisnewfactor,Nbar:

lsim(A-B*K,B*Nbar,C,0,u,t)

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CTM:StateSpaceTutorial

andnowastepcanbetrackedreasonablywell.

Obrverdesign

Whenwecan'tmeasureallthestatesx(asiscommonlytheca),wecanbuildanobrvertoestimatethem,

whilemeasuringonlytheoutputy=magneticballexample,wewilladdthreenew,ematicisasfollows:

Theobrverisbasicallyacopyoftheplant;ithasthesameinputandalmostthesamedifferentialequation.

Anextratermcomparestheactualmeasuredoutputytotheestimatedoutput;thiswillcauthe

ordynamicsoftheobrveraregivenbythepolesof(A-L*C).

ewantthedynamicsoftheobrvertobemuchfaster

thanthesystemitlf,wenee销售英文怎么说 dtoplacethepolesatleastfivetimesfarthertotheleftthanthedominantpoles

nttouplace,weneedtoputthethreeobrverpolesatdifferentlocations.

op1=-100;

op2=-101;op3=-102;

Becauofthedualitybetweencontrollabilityandobrvability,wecanuthesametechniqueudtofind

thecontrolmatrix,butreplacingthematrixBbythematrixCandtakingthetransposofeachmatrix(consultyourtextbookforthederivation):

L=place(A',C',[op1op2op3])';

Theequationsintheblockdiagramabovearegivenfor

.Itisconventionaltowritethecombined

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CTM:StateSpaceTutorial

equationsforthesystemplusobrverusingtheoriginalstatexplustheerrorstate:e=x-.Weuas

statefeedbacku=-littlebitofalgebra(consultyourtextbookformoredetails),wearriveatthecombinedstateanderrorequationswiththefull-statefeedbackandanobrver:

At=[A-B*KB*K

zeros(size(A))A-L*C];

Bt=[B*Nbarzeros(size(B))];

Ct=[Czeros(size(C))];

Toehowtheresponlookstoanonzeroinitialconditionwithnoreferenceinput,addthefollowinglines

callyassumethattheobrverbeginswithzeroinitialcondition,=vesusthattheinitialconditionfortheerroriqualtotheinitialconditionofthestate.

lsim(At,Bt,Ct,0,zeros(size(t)),t,[x0x0])

thatlsimgivesusxande;togetx-e.

weneedtocompute

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CTM:StateSpaceTutorial

Zoomintoesomedetail:

Thebluesolidlineistheresponoftheballposition

ThegreensolidlineistheresponoftheballspeedTheredsolidlineistheresponofthecurrent

,thebluedottedlineistheestimatedstate,thegreendottedlineistheestimatedstate

.

;;

,thereddottedlineistheestimatedstate

Wecanethattheobrverestimatesthestatesquicklyandtracksthestatesreasonablywellinthesteady-state.

Theplotabovecanbeobtainedbyusingtheplotcommand.

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CTM:StateSpaceTutorial

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