Linear-Matrix-Inequalities-in-Control-[控制中的线性矩阵不等式

1/23LinearMatrixInequalitiesinControlCarstenSchererDelftCenterforSystemsandControl(DCSC)DelftUniversityofTechnologyTheNetherlandsSiepWeilandDepartmentofElectricalEngineeringEindhovenUniversityofTechnologyTheNetherlandsIntroduction2/23CarstenSchererSiepWeiland•Overview:WhatcanbeexpectedfromLMItechniques?•WhatareLMI’sandwhataretheygoodfor?•Example:Trusstopologydesign•Software•SomeaspectsoflinearalgebraMergingControlandOptimization3/23CarstenSchererSiepWeilandIncontrasttoclassicalcontrol,H∞-synthesisallowstodesigncontrollersinanoptimalfashion.HowevertheH∞-paradigmisrestricted:•Performancespecintermsofcompleteclosed-looptransfer-matrix.Sometimesonlyparticularchannelsarerelevant.•Onemeasureofperformancewithclearinterpretationinfrequencydomain.Oftenparticulartime-domainspecshavetobeimposed.•Noincorporationofstructuredtime-varying/nonlinearuncertainties.•CanonlydesignLTIcontrollers.Viewcontrollerasdecisionvariableofoptimizationproblem.Desiredspecificationsareconstraintsoncontrolledclosed-loopsystem.MajorGoalsforOptimizationandControl4/23CarstenSchererSiepWeiland•Distinguisheasyfromdifficultproblems:Convexityiskey.•Whataretheconsequencesofconvexityinoptimization?•Whatisrobustoptimization?•Howcanwecheckrobuststabilitybyconvexoptimization?•Whichperformancemeasurescanbeincorporated?•Howcancontrollersynthesisbeconvexified?•Whatarethelimitsforthesynthesisofrobustcontrollers?•Howcanweperformsystematicgain-scheduling?LinearMatrixInequalities(LMIs)5/23CarstenSchererSiepWeilandAnLMIisaninequalityoftheformF0+x1F1+···+xnFn≺0whereF0,F1,...,Fnarerealsymmetricmatricesandx1,...,xnarerealscalarunknowns.LMIfeasibilityproblem:Testwhetherthereexistx1,...,xnthatrendertheLMIsatisfied.LMIoptimizationproblem:Minimizec1x1+···+cnxnoverallx1,...,xnthatsatisfytheLMI.Onlysimplecasescanbetreatedanalytically→Numericaltechniques.Recap6/23CarstenSchererSiepWeilandForarealorcomplexmatrixAtheinequalityA≺0meansthatAisHermitianandnegativedefinite.•AisdefinedtobeHermitianifA=A∗=¯AT.IfAisrealthenthisamountstoA=ATandAiscalledsymmetric.Setofn×nHermitianandsymmetricmatrices:HnandSn.AlleigenvaluesofHermitianmatricesarereal.•SupposeAisHermitian.BydefinitionAisnegativedefiniteifx∗Ax0forallcomplexvectorsx6=0.Aisnegativedefiniteiffallitseigenvaluesarenegative.•A≺B,A4B,AB,ABdefined/characterizedanalogously.Observation:SystemofLMI’sisLMI7/23CarstenSchererSiepWeilandThesystemofmindividualLMI’sF10+x1F11+···+xnF1n≺0...Fm0+x1Fm1+···+xnFmn≺0isequivalenttothesingleLMIF100...0Fm0+nXk=1xkF1k0...0Fmk≺0.AssumingsingleLMIconstraintcausesnolossofgenerality.Goodsolversexploitdiagonalstructureforcomputationalefficiency!Whataretheygoodfor?8/23CarstenSchererSiepWeiland•ManyengineeringoptimizationproblemcanbeeasilytranslatedintoLMIproblems.•Variouscomputationallydifficultoptimizationproblemscanbeeffec-tivelyapproximatedbyLMIproblems.•Inpracticedescriptionofdataisaffectedbyuncertainty.RobustoptimizationproblemscanbeeithertranslatedorapproximatedbystandardLMIproblems.EssentialtopicofthiscourseHowtotranslate/approximateagiven(uncertain)optimizationprobleminto/byanLMIproblem?TrussTopologyDesign9/23CarstenSchererSiepWeiland050100150200250300350400050100150200250Example:TrussTopologyDesign10/23CarstenSchererSiepWeiland•ConnectnodeswithNbarsoflengthl=col(l1,...,lN)(fixed)andcross-sectionss=col(s1,...,sN)(to-be-designed).•Imposeboundsak≤sk≤bkoncross-sectionandlTs≤vontotalvolume(weight).Abbreviatea=col(a1,...,aN),b=col(b1,...,bN).•Ifapplyingexternalforcesf=col(f1,...,fM)(fixed)onnodestheconstructionreactswiththenodedisplacementd=col(d1,...,dM).Mechanicalmodel:A(s)d=fwhereA(s)isthestiffnessmatrixwhichdependslinearlyonsandhastobepositivedefinite.•GoalistomaximizestiffnesswhatamountstominimizingtheelasticstoredenergyfTd.Example:TrussTopologyDesign11/23CarstenSchererSiepWeilandFinds∈RNwhichmaximizesfTdsubjecttotheconstraintsA(s)0,A(s)d=f,lTs≤v,a≤s≤b.Features•Data:Scalarv,vectorsf,a,b,l,andsymmetricmatricesA1,...,ANwhichdefinethelinearmappingA(s)=A1s1+···+ANsN.•Decisionvariables:Vectorssandd.•Objectivefunction:d→fTdwhichhappenstobelinear.•Constraints:Semi-definiteconstraintA(s)0,nonlinearequalityconstraintA(s)d=f,andlinearinequalityconstraintslTs≤v,a≤s≤b.Latterinterpretedelementwise!FromTrussTopologyDesigntoLMI’s12/23CarstenSchererSiepWeilandRenderLMIinequalitystrict.EqualityconstraintA(s)d=fallowstoeliminatedwhichresultsinminimizefTA(s)−1fsubjecttoA(s)0,lTs≤v,a≤s≤b.Pushobjectivetoconstraintswithauxiliaryvariable:minimizeγsubjecttoγfTA(s)−1f,A(s)0,lTs≤v,a≤s≤b.LinearizewithSchurlemmatoequivalentLMIproblemminimizeγsubjecttoγfTfA(s)!0,lTs≤v,a≤s≤b.CongruenceTransformations13/23CarstenSchererSiepWeilandGivenaHermitianmatrixAandasquarenon-singularmatrixT,A→T∗ATiscalledacongruencetransformationofA.Congruencetransformationspreservenegative/positivedefinitenessofamatrix.Thefollowinggeneralstatementiseasytoremember.IfAisHermitianandTisnonsingular,thematricesAandT∗AThavethesamenumberofnegative,zero,positiveeigenvalues.WhatistrueifTisnotsquare?...ifThasfullcolumnrank?Schur-Lemma14/23CarstenSchererSiepWeilandT