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