《现代控制理论》课程教学资源——现控课件_Ch1-5 State Transformation of the LTI system

1.5.4 Obtaining a Diagonal Canonical Form byState TransformationConclusion 1:Andimension systemgovernedbythe state spacedescriptionX(t) = AX(t) + Bu(t)(1.117)y(t) = CX(t) + Du(t)If A has distinct eigenvalues ^, ,"..An , there is a nonsingulartransformationX(t) = Px(t)which transforms the general state description (1.117) intothediagonal canonical form (1.120)0M0X(t)= P-'APX(t)+ P-"Bu(t) = AX(t)+Bu(t)P-"AP =00y(t) = CPX(t) + Du(t) = CX(t) + Du(t)00元(1.120)

1.5.4 Obtaining a Diagonal Canonical Form by State Transformation ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = +  = + Conclusion 1: A n dimension system governed by the state space description: 1 2 , , If A has distinct eigenvalues   n , there is a nonsingular transformation X(t) = PX(t) which transforms the general state description (1.117) into the diagonal canonical form (1.120) 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t t − − = + = + = + = + X P APX P Bu AX Bu y CPX Du CX Du (1.117)           = − n  0 0 0 0 0 0 1 P 1 ΑP  (1.120)

Proof. A n dimension system governed by the state spacedescription:X= AX+Bu(1.117)y= CX + DuLet A,2,.., be the distinct eigenvalues of AV,bethe eigenvector of A associatedwiththeeigenvaluea,(i=l,2,.",n)Then,P can be established as a nonsingular matrixP=[vV, ... V.](1.118)Since AV, = ,V, we have:1..av]AP=[AV ... AV,]-[aVM00000MT0P-'AP =000000= P=[V .. V,]0A00000元,2

Proof. A n dimension system governed by the state space description: = + = + X AX Bu y CX Du (1.117) Let be the distinct eigenvalues of 1 ,2 , n A Vi be the eigenvector of A associated with the eigenvalue (i 1,2, ,n) i =  Then, P can be established as a nonsingular matrix   P = V1 V2 Vn (1.118) Since , we have: AVi = i Vi               =           = = = n n n n n n       0 0 0 0 0 0 0 0 0 0 0 0 [ ] 1 1 1 1 1 1      V V P AP AV AV V V           = − n  0 0 0 0 1 0 0 P 1 ΑP 

X=AX+Bu(1.117)y= CX +Du结 论: If a system described by/(1.117)and A has distincteigenvalues M,2,".. , , there is a nonsingular transformationX(t) = PX(t)which transforms the general state description (1.117) into thediagonal canonical form (1.120)X= AX+Bu(1.120)y=CX+Du[α]0Where A=P-lAP-is a diagonal matrix02n=CPB=P-'BD=D

1 2 , ,   n 结 论 : If a system described by (1.117) and A has distinct eigenvalues , there is a nonsingular transformation X(t) = PX(t) which transforms the general state description (1.117) into the diagonal canonical form (1.120) = + = + X AX Bu y CX Du (1.120) 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 (1.117)