CONTROL DESIGNS FOR THE NONLINEAR BENCHMARK

INTERNATIONALJOURNALOFROBUSTANDNONLINEARCONTROLInt.J.RobustNonlinearControl8,401Ð433(1998)CONTROLDESIGNSFORTHENONLINEARBENCHMARKPROBLEMVIATHESTATE-DEPENDENTRICCATIEQUATIONMETHODCURTISP.MRACEK*ANDJAMESR.CLOUTIERU.S.AirForceResearchLaboratory,MunitionsDirectorate,EglinAFB,FL32542-6810,U.S.A.SUMMARYAnonlinearcontrolproblemhasbeenposedbyBuppetal.1toprovideabenchmarkforevaluatingvariousnonlinearcontroldesigntechniques.Inthispaper,thecapabilitiesofthestate-dependentRiccatiequation(SDRE)techniqueareillustratedinproducingtwocontroldesignsforthebenchmarkproblem.TheSDREtechniquerepresentsasystematicwayofdesigningnonlinearregulators.ThedesignprocedureconsistsofÞrstusingdirectparameterizationtobringthenonlinearsystemtoalinearstructurehavingstate-dependentcoe¦cients(SDC).Astate-dependentRiccatiequationisthensolvedateachpointxalongthetrajectorytoobtainanonlinearfeedbackcontrolleroftheformu!R~1(x)BT(x)P(x)x,whereP(x)isthesolutionoftheSDRE.AnalysisoftheÞrstdesignshowsthatintheabsenceofdisturbancesanduncertainties,theSDREnonlinearfeedbacksolutioncomparesveryfavorablytotheoptimalopen-loopsolutionoftheposednonlinearregulatorproblem,thelatterbeingobtainedvianumericaloptimization.Itisalsoshownviasimulationthattheclosed-loopsystemhasstabilityrobustnessagainstparametricvariationsandattenuatessinusoidaldisturbances.Intheseconddesignitisdemonstratedhowahardboundcanbeimposedonthecontrolmagnitudetoavoidactuatorsaturation.(1998JohnWiley&Sons,Ltd.1.INTRODUCTIONNumerousdesignmethodologiesexistforthecontroldesignofnonlinearsystems.Theseincludeanumberoflineardesigntechniques2Ð5usedinconjunctionwithgainscheduling;6Ð8nonlineardesignmethodologiessuchasdynamicinversion,9slidingmodecontrol,9andrecursivebackstep-ping,10andadaptivetechniqueswhichencompassbothlinearadaptive11andnonlinearadap-tive10control.Alesserknownnonlineardesignprocedureisthestate-dependentRiccatiequation(SDRE)technique.12ThistechniquewasusedinReference13toproduceafullstateinformation,nonlinear,minimumenergydesign,wasusedinReferences14and15toproducetwooutputfeedback,nonlinear,minimumenergywithrespecttowhitenoisedesignsandisbrießymentionedinReference16.However,Reference12istheÞrstworkthatweareawareofthatexplorestheSDREmethodindetail,providingadeeperunderstandingofthetechnique,itscapabilitiesandlimitations.AmorerecenttheoreticalinvestigationofthemethodiscontainedinReference17.AbriefoverviewoftheSDREmethodandsomeofthekeyresultsobtainedinReferences12follow.*CorrespondencetoCurtisP.Mracek,U.S.AirForceLaboratory,MunitionsDirectorate,EglinAFB,FL32542-6810,USA.CCC1049-8923/98/050401Ð33$17.50(1998JohnWiley&Sons,Ltd.1.1.TheSDREmethodforcompletestateinformationConsiderthegeneralinÞnite-horizon,input-a¦ne,autonomous,nonlinearregulatorproblemoftheform:MinimizeJ12P=tÒzTz#uTR(x)udt(1)withrespecttothestatexandcontrolusubjecttothenonlinearsystemconstraintsxRf(x)#B(x)u(2)zC(x)x(3)wherex3Rn,u3Rmandz3Rs,andwhereR(x)'0forallx.Weassumethatf(0)0andthatB(x)O0inaneighbourhoodaroundtheorigin.TheSDREapproachforobtainingasuboptimalsolutionofproblem(1)Ð(3)is(i)Usedirectparameterizationtobringthenonlineardynamicstothestate-dependentcoe¦cient(SDC)formxRA(x)x#B(x)u(4)wheref(x)A(x)x(5)(ii)Solvethestate-dependentRiccatiequationAT(x)P#PA(x)!PB(x)R~1(x)BT(x)P#CT(x)C(x)0(6)toobtainP*0,wherePisafunctionofx.(iii)Constructthenonlinearfeedbackcontrolleru!R~1(x)BT(x)P(x)x(7)AssociatedwiththeSDCform(4),wehavethefollowingdeÞnitions.DeÞnitionThepairMC(x),A(x)Nisanobservable(detectable)parameterizationofthenonlinearsysteminaregion)ifMC(x),A(x)Nispointwiseobservable(detectable)inthelinearsenseforallx3).DeÞnitionThepairMA(x),B(x)Nisacontrollable(stabilizable)parameterizationofthenonlinearsysteminaregion)ifMA(x),B(x)Nispointwisecontrollable(stabilizable)inthelinearsenseforallx3).InspectionofEquations(1)Ð(7)revealsthattheSDREnonlinearregulatorhasthesamestructureasthatofthelinearquadraticregulator(LQR)exceptthatallofthecoe¦cientmatricesarestate-dependent.Thus,inthecaseoflineardynamics,andforconstantstateandcontrol402CURTISP.MRACEKANDJAMESR.CLOUTIER(1998JohnWiley&Sons,Ltd.Int.J.RobustNonlinearControl8,401Ð433(1998)weightings,theSDREnonlinearregulatorreducestotheinÞnite-horizon,time-invariant,linearquadraticregulator.Additionally,fromLQRtheory,itiswell-knownthattherelationshipbetweentheco-statejandthestatexisgivenbyjPx,wherePisthesolutionoftheRiccatiequation.FortheSDREnonlinearregulator,wemaketheassumptionthattheco-statejisrelatedtothestatexasjP(x)x.Wewillrefertothisassumptionastheco-stateassumption.Thus,whentheSDREnonlinearregulatorreducestoLQR,theco-stateassumptionreducestojPxwhichistheoptimalvalueofjinthelinearquadraticregulator.1.2.PropertiesandcapabilitiesoftheSDREmethodThepropertiesandcapabilitiesoftheSDREmethodarediscussedbelow.Someofthepropertiesarepresentedastheorems.TheproofsofthetheoremsaregiveninAppendixAaswellasinReference12.Pertinentquantities,deÞnitions,andassumptionsusedintheproofsareasfollows.TheHamiltonianofthenonlinearregulatorproblem(1)Ð(3)isgivenbyH(x,j)12xTQ(x)x#12uTR(x)u#jT[f(x)#B(x)u],whereQ(x)CT(x)C(x)a