《现代控制理论》课程教学资源课程教学资源——教学课件_Chap1_1.6 Obtaining a Jordan Canonical Form by State Transformation

CHAPTER1 STATE SPACE MODELCONTENT> 1.1 Definition of State Space> 1.2 Obtaining State Space Model from I/O Model> 1.3ObtainingTransferFunctionMatrixfromStateSpace Model>1.4 ModelofCompositeSystems> 1.5 State Transformation of the LTI system>1.6Obtaininga Jordan CanonicalFormby StateTransformation

CHAPTER1 STATE SPACE MODEL • CONTENT  1.1 Definition of State Space  1.2 Obtaining State Space Model from I/O Model  1.3 Obtaining Transfer Function Matrix from State Space Model  1.4 Model of Composite Systems  1.5 State Transformation of the LTI system  1.6 Obtaining a Jordan Canonical Form by State Transformation

1.6 Obtaining a Jordan Canonical Form byState TransformationConsider the LTI system such asy(t) = CX(t) + Du(t)X(t) = AX(t) + Bu(t)In this section, we will find the nonsingular transformationX(t) = Px(t), by which the state space description can betransform to Jordan canonical form from some general formCase 1 The eigenvaluesof A are all distinct

1.6 Obtaining a Jordan Canonical Form by State Transformation

1.6 Obtaining a Jordan Canonical Form byState TransformationCase 1 The eigenvalues of A are all distinct.Let ,2,..-2, be the distinct eigenvalues of A, and letV, be the eigenvector of A associated with the eigenvalue2, (i=1,2,...,n) ..Then, the matrix P =, V, ... V, I is anonsingular matrixSince AV, = a,VAP =[AV[ava.y.1...AV]=00[00M0000=P=[VV.2002n00

1.6 Obtaining a Jordan Canonical Form by State Transformation

1.6 Obtaining a Jordan Canonical Form byState TransformationCase 1 The eigenvalues of A are all distinct.AP =[AVa.VAV,]=[[avD00<M[000000=P=[VV.F00^n00MT0000P-1AP =Hence.002

1.6 Obtaining a Jordan Canonical Form by State Transformation

1.6 Obtaining a Jordan Canonical Form byState TransformationCase1 The eigenvalues of A are all distinct.y(t) = CX(t) + Du(t)X(t) = AX(t) + Bu(t)P-[iv, ... V,]LetorandX(t) = PX(0)X(t) = P-1 X(t)It can transform the general state space model into thediagonalcanonicalformX(t) = P-1APX()+ P-1Bu(t) = AX(t)+ Bu(t)y(t) = CPX(t) + Du(t) = CX(t) + Du(t)

1.6 Obtaining a Jordan Canonical Form by State Transformation