By substituting(1.108)and(1.109)intoLTl system(1.107),wehavethenew stateequationintermsof thenewstateX(1.110)X(t) = P-' APX(t) + P- Bu(t) = AX(t) + Bu(t)A = P-' AP → similarity transformation of A(1.111)where(1.112)B = P-'BBy substituting (1.108) into (1.107), we have the output equation interms of the new state Xy(t) = CPX(t) + Du(t) = CX(t) + Du(t)(1.113)C =CP(1.114)where(1.115)D=D进而得到系统关于状态 X 的状态空间描述 (A,B,C,D)应用：对于给定的状态空间描述A,B,C,D}，我们可以通过一个线性状态变换 X(t)=PX(t)来求出变换后的新的状态空间描述(A,B,C,D)特别的，可以是约当规范形或对角规范形
By substituting (1.108) and (1.109) into LTI system (1.107), we have the new state equation in terms of the new state X ( ) ( ) ( ) ( ) ( ) 1 1 X t = P APX t + P Bu t = AX t + Bu t − − (1.110) where A P AP −1 = similarity transformation of A (1.111) B P B −1 = (1.112) By substituting (1.108) into (1.107), we have the output equation in terms of the new state X y(t) = CPX(t) + Du(t) = CX(t) + Du(t) (1.113) where C = CP (1.114) D = D (1.115) 进而得到系统关于状态 X 的状态空间描述 { , , , } A B C D 应用：对于给定的状态空间描述{A,B,C,D}，我们可以通过一个线性状 态变换 来求出变换后的新的状态空间描述 ，特 别的，可以是约当规范形或对角规范形。 X(t) = PX(t) { , , , } A B C D
1.5.3softheStateInvariancePropertiesTransformation（状态变换的不变性）The characteristic equation, eigenvalues, eigenvectors andtransfer function are all preserved by the nonsingularstate transformation.特征方程、特征值、特征向量和传递函数在状态变换下是保持不变的
1.5.3 Invariance Properties of the State Transformation （状态变换的不变性） •The characteristic equation, eigenvalues, eigenvectors and transfer function are all preserved by the nonsingular state transformation. 特征方程、特征值、特征向量和传递函数在状态变换下 是保持不变的
X(t) = PX(t)X(t) = AX(t) + Bu(t)X(t) = P-'APX(t) + P-'Bu(t) = AX(t) + Bu(t)y(t) = CX(t) + Du(t)y(t)= CPX(t) + Du(t) = CX(t)+ Du(t)1、 The characteristic equation of the system (A,B,C,D)sI-A=sI-P-"AP=sP-"P-P-"AP=P-'(sI-A)P= 0sI - A = |P-I|sI - A|P| = sI - A| = 0注：特征方程在状态变换下是保持不变的，因此特征值、特征向量在状态变换下也是保持不变的
1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t t − − = + = + = + = + X P APX P Bu AX Bu ( ) ( ) ( ) y CPX Du CX Du ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + X(t) = PX(t) 1、The characteristic equation of the system { , , , } A B C D ( ) 0 1 1 1 1 − = − = − = − = − − − − sI A sI P A P sP P P A P P sI A P 0 1 − = − = − = − sI A P sI A P sI A 注：特征方程在状态变换下是保持不变的，因此特征值、特征向量 在状态变换下也是保持不变的
2、 The transfer function of the system described by(A, B,C, D)G(s)=C(sI - A)-'B +D =CP(sI - P-"AP)-'P-B+ D= C[P(sI - P-'AP)P-'}-"B + D =C[P(sP-'P- P-'AP)P-I}-'B +D=C[PP-'(sI - A)PP-'I-'B + D = C(sI - A)-'B + D= G(s)注：传递函数在状态变换下是保持不变的。事实上，能控性（controllability）和能观性（observability）在状态变换下是保持不变的
2、The transfer function of the system described by { , , , } A B C D ( ) [ ( ) ] ( ) [ ( ) ] [ ( ) ] ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s sI s s s s s s G C P P A P P B D C I A B D C P I P A P P B D C P P P P A P P B D G C I A B D C P I P A P P B D = = − + = − + = − + = − + = − + = − + − − − − − − − − − − − − − − − 注：传递函数在状态变换下是保持不变的。 事实上，能控性（controllability）和能观性（observability）在状态 变换下是保持不变的
X(t) = AX(t) + Bu(t)y(t) = CX(t) + Du(t)X(t) = PX(t)Nonsingular lineartransformationX(t) = AX(t) + Bu(t) = P-'AP X(t)+P-'B u(t)y(t)=CX(t)+ Du(t) =CP X(t)+D u(t)Invariance Properties: The characteristic equation, eigenvalues,eigenvectors andtransferfunction
( ) ( ) ( ) ( ) ( ) ( ) t t t t t t = + = + X AX Bu y CX Du ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + X(t) = PX(t) Nonsingular linear transformation 1 1 ( ) ( ) t t − − = + P AP X P B u = + CP X D u ( ) ( ) t t Invariance Properties: The characteristic equation, eigenvalues, eigenvectors and transfer function