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1、1ELEMENTSOFLINEARSYSTEMTHEORY1.1INTRODUCTIONThisbookdealswiththeanalysisanddesignoflinearcontrolsystems.Aprerequisiteforstudyinglinearcontrolsystemsisaknowledgeoflinearsystemtheory.Wethereforedevotethisfirstchaptertoareviewofthemostimportantingredient
2、soflinearsystemtheory.TheintroductionofcontrolproblemsispostponeduntilChapter2.Themainpurposeofthischapteristoestablishaconceptualframework,introducenotationalconvenlions,andgiveasurveyofthebasicfactsoflinearsystemtheory.Thestartingpointisthestatespac
3、edescriptionoflinearsystems.Wethenproceedtodiscussionsofthesolutionoflinearstatedifferentialequations,thestabilityoflinearsystems,andthetransformanalysisofsuchsystems.Thetopicsnextdealtwithareofamoreadvancednature;theyconcerncontrollability.reconstruc
4、tibility,duality,andphase-variablecanonicalformsoflinearsystems.Thechapterconcludeswithadiscussionofvectorstochasticprocessesandtheresponseoflinearsystemstowhitenoise.Thesetopicsplayanimporlantroleinthedevelopmentofthetheory.Sincethereaderofthischapte
5、risassumedtohavehadaninlroducliontolinearsystemtheory,theproofsofseveralwell-knowntheoremsareomitted.Referencestorelevanttexlbooksareprovided,however.Sometopicsaretreatedinsectionsmarkedwithanaslerisk,nolablycontrollability,recon-struclibility,duality
6、andphase-variablecanonicalforms.Theasteriskindicatesthatthesenotionsareofamoreadvancednature,andneededonlyinthesectionssimilarlymarkedintheremainderofthebook.1.2STATEDESCRIPTIONOFLINEARSYSTEMS1.2.1StateDescriptionofNonlinearandLinearDifferentialSystem
7、sManysystemscanbedescribedbyasetofsimultaneousdifferentialequalionsoftheformw=f[~(f),u(t),fl.1-112El~mcntsofLincnrSystemTheoryHeretisthetimevariable,x(t)isarealn-dimensionaltime-varyingcolumnvectorwhichdenotesthestateofthesystem,andtr(t)isareallc-dime
8、nsionalcolumnvectorwhichindicatestheinptrtvariableorcorttrolvariable.Thefunctionfisrealandvector-valued.Formanysystemsthechoiceofthestatefollowsnaturallyfromthephysicalstructure,and1-1,whichwillbecalledthestatedijererttialequation,usuallyfollo
