《现代控制理论》课程教学资源——现控课件_Ch2-Time response of LTI system

方法3 相似变换法(similaritytransformation method)/特征值法：Case1: If a system described by (2.25) and A has distincteigenvaluesA,,"..,,thereis anonsingulartransformationX(t) = PXx(t)which transforms the general state description (2.25) into thediagonal canonical form (2.27)X=AX+BuX= AX+Bu(2.27)(2.25)y=CX+Duy=CX+DuT0Where A=P-lAP-is a diagonal matrix02n=CPD=DB=P-"B

1 2 , ,   n Case1: If a system described by (2.25) and A has distinct eigenvalues , there is a nonsingular transformation X(t) = PX(t) which transforms the general state description (2.25) into the diagonal canonical form (2.27) = + = + X AX Bu y CX Du (2.27) Where           = = − n  0 1 0 A P 1 AP  is a diagonal matrix B P B −1 = C = CP D = D = + = + X AX Bu y CX Du (2.25) 方法3 相似变换法(similarity transformation method) /特征值法：

en0[]0etr12eAt = exp(2.28)24t =020At0eteeP.Atp-I = P.eAt . P-I = P:p-1At(2.29)0

              =             = t t t n t n e e e e t       0 0 0 0 exp 2 1 2 1   A (2.28) 1 1 0 0 2 1 1   − −                = =   =  − P P P P A P A P A t t t t t t n e e e e e e     (2.29)