滚动时域控制原理-Receding-Horizon-Control-principle

RecedingHorizonControlMar´ıaM.SeronSeptember2004CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleFixedhorizonoptimisationleadstoacontrolsequencefui;:::;ui+N1g,whichbeginsatthecurrenttimeiandendsatsomefuturetimei+N1.Thisfixedhorizonsolutionsuffersfromtwopotentialdrawbacks:(i)Somethingunexpectedmayhappentothesystematsometimeoverthefutureinterval[i;i+N1]thatwasnotpredictedby(orincludedin)themodel.Thiswouldrenderthefixedcontrolchoicesfui;:::;ui+N1gobsolete.(ii)Asoneapproachesthefinaltimei+N1,thecontrollawtypically“givesuptrying”sincethereistoolittletimetogotoachieveanythingusefulintermsofobjectivefunctionreduction.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleTheabovetwoproblemsareaddressedbytheideaofrecedinghorizonoptimisation.Thisideacanbesummarisedasfollows:(i)Attimeiandforthecurrentstatexi,solveanoptimalcontrolproblemoverafixedfutureinterval,say[i;i+N1],takingintoaccountthecurrentandfutureconstraints.(ii)Applyonlythefirststepintheresultingoptimalcontrolsequence.(iii)Measurethestatereachedattimei+1.(iv)Repeatthefixedhorizonoptimisationattimei+1overthefutureinterval[i+1;i+N],startingfromthe(now)currentstatexi+1.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleIntheabsenceofdisturbances,thestatemeasuredatstep(iii)willbethesameasthatpredictedbythemodel.Nonetheless,itseemsprudenttousethemeasuredstateratherthanthepredictedstatejusttobesure.Theabovedescriptionassumesthatthestateismeasuredattimei+1.Inpractice,onewouldusesomeformofobservertoestimatexi+1basedontheavailabledata.Morewillbesaidabouttheuseofobserversinthenextlecture.Forthemoment,wewillassumethatthefullstatevectorismeasuredandwewillignoretheimpactofdisturbances.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleIfthemodelandobjectivefunctionaretimeinvariant,thenthesameinputuiwillresultwheneverthestatetakesthesamevalue.Thatis,therecedinghorizonoptimisationstrategyisreallyaparticulartime-invariantstatefeedbackcontrollaw:PSfragreplacementsukxkxk+1=f(xk;uk)RHCInparticular,wecanseti=0intheformulationoftheopenloopcontrolproblem.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleMoreprecisely,atthecurrenttime,andforthecurrentstatex,wesolve:PN(x):VoptN(x),minVN(fxkg;fukg);(1)subjectto:xk+1=f(xk;uk)fork=0;:::;N1;(2)x0=x;(3)uk2Ufork=0;:::;N1;(4)xk2Xfork=0;:::;N;(5)xN2XfX;(6)whereVN(fxkg;fukg),F(xN)+N1Xk=0L(xk;uk):(7)CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleThesetsURm,XRn,andXfRnaretheinput,stateandterminalconstraintset,respectively.Allsequencesfukg=fu0;:::;uN1gandfxkg=fx0;:::;xNgsatisfyingtheconstraints(2)–(6)arecalledfeasiblesequences.Apairoffeasiblesequencesfu0;:::;uN1gandfx0;:::;xNgconstituteafeasiblesolution.ThefunctionsFandLintheobjectivefunction(7)aretheterminalstateweightingandtheper-stageweighting,respectively.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleInthesequelwemakethefollowingassumptions:f,FandLarecontinuousfunctionsoftheirarguments;URmisacompactset,XRnandXfRnareclosedsets;thereexistsafeasiblesolutiontoproblem(1)–(7).BecauseNisfinite,theseassumptionsaresufficienttoensuretheexistenceofaminimumbyWeierstrass’theorem.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleTypicalchoicesfortheweightingfunctionsFandLarequadraticfunctionsoftheformF(x)=xtPxandL(x;u)=xtQx+utRu;whereP=Pt0,Q=Qt0andR=Rt0.Moregenerally,onecouldusefunctionsoftheformF(x)=kPxkpandL(x;u)=kQxkp+kRukp;wherekykpwithp=1;2;:::;1,isthep-normofthevectory.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleDenotetheminimisingcontrolsequence,whichisafunctionofthecurrentstatexi,byUoptxi,fuopt0;uopt1;:::;uoptN1g;(8)thenthecontrolappliedtotheplantattimeiisthefirstelementofthissequence,thatis,ui=uopt0:(9)Timeisthensteppedforwardoneinstant,andtheaboveprocedureisrepeatedforanotherN-step-aheadoptimisationhorizon.ThefirstelementofthenewN-stepinputsequenceisthenapplied,andsoon.Theaboveprocedureiscalledrecedinghorizoncontrol(RHC).CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrinciplePSfragreplacements000111222333444556iiiUoptx0Uoptx1Uoptx2ThefigureillustratestheRHCprincipleforhorizonN=5.EachplotshowstheminimisingcontrolsequenceUoptxigivenin(8),computedattimei=0;1;2.Notethatonlytheshadedinputsareactuallyappliedtothesystem.Wecanseethatwearecontinuallylookingaheadtojudgetheimpactofcurrentandfuturedecisionsonthefutureresponse.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrincipleTheaboverecedinghorizonprocedureimplicitlydefinesatime-invariantcontrolpolicyKN:X!UoftheformKN(x)=uopt0:(10)Therecedinghorizoncontrollerisimplementedinclosedloopasfollows.CentreforComplexDynamicSystemsandControlTheRecedingHorizonControlPrinciplePSfragreplacementsukukxkxkxk+1=f(xk;uk)KN(
)uopt0266666666664uopt0:::uoptN1377777777775Finitehorizonoptimisationsolver[I0:::0]Figure:RecedinghorizoncontrolCentreforComplexDynamicSystemsandControlTheRecedingHorizonContro