控制中的數(shù)學課件

LinearMatrixInequalitiesin

ControlSystems:

Analysis,DesignandApplications

Part0:IntroductionCaiChenxiaoAutomationSchool,NUSTEmail:ccx5281@AbouttheCourseGoalContentstobecoveredMainReferencesTheassignmentGoalWhat:UnderstandthebasicconceptsandtheoryaboutLMIsWhy:UnderstandtheadvantagesofLMIanditsroleincertaincontrolsystemsanalysisanddesignproblemsHow:BeabletoformulateandsolvesomeLMI-basedproblemswiththehelpofthelearnttheoryandtechniques.ContentstobecoveredChapter1IntroductionPreliminariesChapter2PreliminariesChapter3ReviewofOptimizationTheoryControlSystemsAnalysisChapter4StabilityAnalysisChapter5PerformanceAnalysisChapter6PropertyAnalysisControlSystemsDesignsChapter7FeedbackStabilizationChapter8H2/H∞ControlChapter9StateObservationandFilteringChapter10MultipleObjectiveDesignChapter11ApplicationsMainReferencesG.R.Duan,H.H.Yu,LMIsinControlSystems:Analysis,DesignandApplications,CRCPress,2013.StephenBoyd,LaurentEiGhaoui,EricFeron,andVenkataramarianBalakrishnan,LinearMatrixInequalitiesinSystemsandControlTheory,SIAMStudiesinAppliedMathematics,1994.L.Yu.AnLMIapproachtorobustcontrol(inChinese).TsinghuaUniversityPress,2002.OtherreferencesB.AndersonandJ.B.Moore.Optimalcontrol:LinearquadraticMethods,Pretice-Hall,1990.J.B.Hiriart-UrrutyandC.Lemarechal.ConvexanalysisandMinimizationalgorithmsI,volume305ofGrundlehrendermathematischenWissenschaften.Springer-Verlag,NewYork,1993.Yu.NesterovandA.Nemirovsky.Interior-pointpolynomialmethodsinconvexprogramming,volume13ofStudiesinAppliedMathematics.SIAM,Philadelphia,PA,1994.TheassignmentStudentsarerequiredtoreadeverychapterandteachitabout45minutesbyagroup.(25%)Homework(25%)WorkoutoneortwoproblemsinvolvingLMIstosubmitthemtoProf.Caiinthefilewithyournamewhenthecourseisover.(50%)DistributionoflecturesAllthestudentsaredividedintofivegroupsandfivepersonspergroup.PartIPartIIPartIII-1PartIII-2PartIVPartIPRELIMINARIES

TechnicalLemmas

Generalizedsquareinequalities

Therestriction-freeinequality;

Inequalitieswithrestriction;

Thevariableeliminationlemma;

Schurcomplementlemma

Schurcomplements;

Matrixinversionlemma;

Schurcomplementlemma;

Eliminationofvariables

Variableeliminationinapartitionedmatrix;

Theprojectionlemma;

Thereciprocalprojectionlemma;

Someotherusefulresults

TraceofanLMI;

Themaximummodulusprinciple;

TheParsevallemma;

PartIPRELIMINARIES

ReviewofOptimizationTheory

Convexsets

Definitionsandproperties;

Hyperplanes,halfspaces,polyhedronsandpolytopes;

Convexfunctions

Definitionandproperties;

Criteria;

Mathematicaloptimization

Leastsquaresprogramming;

Linearprogramming;

Quadraticprogramming;

Convexoptimization

Theproblem;

Localandglobaloptima;

TheLMIproblem

Convexity;

Theextremeresult;

Standardproblems

PartIICONTROLSYSTEMSANALYSIS

StabilityAnalysis

HurwitzandSchurstability

Hurwitzstability;

Schurstability

D-stability

Specialcases;

GeneralLMIregions;

GeneralizedLyapunovtheorem;

Quadraticstability

Familyofsystems;

QuadraticHurwitzstability;

QuadraticSchurstability;

QuadraticD-stability

Definitionandmainresults;

Somespecialcases

Time-delaysystems

Thedelayindependentcondition

Thedelaydependentcondition

PartIICONTROLSYSTEMSANALYSISH∞/H2

Performance

H∞

andH2

indices

H∞

index;

H2

index;

Equivalentdefinitions;

LMIconditionsforH∞

index

Thebasicconditions;

Deducedconditions;

LMIconditionsforH2

index

Basicconditions;

Deducedconditions;

PropertyAnalysis

Hurwitzstabilizabilityanddetectability

Hurwitzstabilizability;

Hurwitzdetectability

Schurstabilizabilityanddetectability

Schurstabilizability;

Schurdetectability

Dissipativity

Definition;

Equivalentconditions

Passivityandpositive-realness

Definitions;

Positive-reallemma;

LMIcondition

Nonexpansivityandbounded-realness

Definitions;

Bounded-reallemma;

LMIconditionsPartIII-1CONTROLSYSTEMSDESIGNFeedbackStabilization

Statefeedbackstabilization

Caseofcontinuous-timesystems;

Caseofdiscrete-timesystems;

D-stabilization

H

(a,B)-stabilization;

D(q,r)-stabilization;

GeneralD-stabilization;

Quadraticstabilization

Familyofsystems;

QuadraticHurwitzstabilization;

QuadraticSchurstabilization;

QuadraticD-stabilization

Problemformulation;

Solution;

Specialcases

Insensitiveregiondesign

Sensitivityofmatrixeigenvalues;

Insensitivestripregiondesign;

Insensitivediskregiondesign

Robuststabilizationofsecond-ordersystems

Stabilization;

RobustStabilization

Stabilizationoftime-delaysystems

Caseofdelayindependence;

Caseofdelaydependence

PartIII-1CONTROLSYSTEMSDESIGNH∞/H2

Control

H∞

statefeedbackcontrol

Problem;

Solution;

Otherconditions;

H2

statefeedbackcontrol

Problem;

Solution;

Otherconditions;

RobustH∞/H2

statefeedbackcontrol

Problems;

SolutiontotherobustH∞

controlproblem;

SolutiontotherobustH2

controlproblem;

LQregulationviaH2

control

Problemdescription;

RelationwithH2performance;

Solution;

PartIII-2CONTROLSYSTEMSDESIGN

StateObservationandFiltering

Full-andreduced-orderstateobservers

Full-orderstateobservers;

Reduced-orderstateobserverdesign;

Full-orderH∞/H2stateobservers

Problemsformulation;

Solutionstoproblems;

Examples

H∞

filtering

Problemsformulation;

SolutiontoH∞

filtering;

H2filtering

Problemsformulation;

SolutiontoH2filtering;

MultipleObjectiveDesigns

Insensitiveregiondesignswithminimumgains

Insensitivestripregiondesignswithminimumgains;

Insensitivediskregiondesignswithminimumgains;

MixedH∞/H2designswithdesiredLMIpoleregions

Problem;

Solutionstotheproblem

MixedrobustH∞/H2designswithdesiredLMIpoleregions

Problem;

Solutionstotheproblem

PartIVAPPLICATIONS

MissileAttitudeControl

Thedynamicalmodel

Modelsfornon-rotatingmissiles

;

ModelsforBTTmissiles

;

Attitudecontrolofnon-rotatingmissiles

Problem

;

Solution;

AttitudecontrolofBTTmissiles

Problem;

Quadraticstabilization;

Numericalresultsandsimulation;SatelliteControl

Systemmodelling

Thesecond-ordersystemform;