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
1.5.1 Eigenvalue and EigenvectorConsider the LTI system such asX(t) = AX(t) + Bu(t)Where, A e Rxn is the system matrix and plays an importantrole in system properties. When u = O, the system X = AX is calleda free system.One case is X and AX have the same direction in thestate space but may differ in magnitude, and can be illustratedAX = 2XWhere is a scalar proportionality factor and called theeigenvalueofA.+
1.5.1 Eigenvalue and Eigenvector
1.5.1 Eigenvalue and EigenvectorAX = 2XIn this case (l - A)X = 0 have not the zero solutionIt means the matrix al -A must not be full rank, orrank(l - A) <n, as well as the determinant |l - Al mustbe zero. +The polynomial about aQ(2) = - A=" +α, i=0is called thecharacteristic polynomialand Q(a) = 0 iscalledthecharacteristicequationof system
1.5.1 Eigenvalue and Eigenvector
1.5.1 Eigenvalue and EigenvectorIf the polynomial Q(2) can be written in factored form as72Q(2) = det(I - A) = II(-2)-1the roots , (i=l,2,..-,n) of the characteristic equation arethe eigenvaluesof A. *Some important properties of eigenvalues are given asfollows.u
1.5.1 Eigenvalue and Eigenvector
1.5.1 Eigenvalue and Eigenvector(l) If the elements of A are real, then its eigenvalues areeither real or in complex conjugate pairs..(2)If 2, (i=l,2,..,n) arethe eigenvaluesof A, thenntr(A)=Z^i=1Thatis, thetraceof A is the sum ofall eigenvaluesof A.(3) If 2, (i=1,2,..,n) are eigenvalues of A, then they arethe eigenvalues of AT
1.5.1 Eigenvalue and Eigenvector
1.5.l Eigenvalue and Eigenvector(4) If A is nonsingular, with eigenvalues a, (i=l,2, .-,n) ,1are the eigenvalues of A-1then元(5)If the characteristic polynomial of A can be written infactored form as.n[21 -A=(-)~I(-)i=α+1of thethe parameter α is called the algebraic multiplicityeigenvalue 2 -
1.5.1 Eigenvalue and Eigenvector
1.5.l Eigenvalue and Eigenvector(6) If n-rank(2I - A)= β, the parameter β is called thegeometricalmultiplicity of the eigenvalue 2 .Any nonzero vector V, which satisfies the matrix equation(2,I-A)V, = 0 of A associated with eigenvalueis called theeigenvector元, (i=1,2,.,n). If A has distinct eigenvalues, the eigenvectors can besolved directly by the equation above.It should be pointed out that if A has multiplicityeigenvalues, not all eigenvectors can be found
1.5.1 Eigenvalue and Eigenvector
1.5.l Eigenvalue and EigenvectorAny nonzero vector V, which satisfies the matrix equation(a,I - A)V, = 0eigenvectoris called the r of A associated with eigenvalue, (i=1,2,..-,n). +Let us assume that is the m multiplicity eigenvalue ofA and the remainingdistinct eigenvaluesaren-mm+1,2Based on 入m+1, 2m+2a,, we can find n-m linearlyVVindependenteigenvectorsV171-13m+2
1.5.1 Eigenvalue and Eigenvector
l.5.l Eigenvalue and EigenvectorBut if rank(aI -A)=n-1,we can only find one linearlyVfromtheequationindependenteigenvectorslinearlyobtainm-1(2 I -A)V =0 . We may alsoYumfrom theVindependentgeneralized eigenvectorsfollowing m-l vectorequations(2 I - A)Vi2 = -V1 t(2I - A)V13 = -V12 *(I- A)Vim = -1(m-1)
1.5.1 Eigenvalue and Eigenvector
Example 1.17 Determine the eigenvalues of A and thecorrespondingeigenvectors.o65021A=324SolutionThecharacteristicequationis[1 - A| =(-2)(-1)2 = 0The eigenvalues of A are ^ =2,, =l. Thus, A has a2 multiplicity eigenvalue at l