Fall 2001 16.3117-2 Can fix this problem by introducing a new variable a a and then converting the closed-loop system dynamics using the similarity transformation T △ I 0 I-1| Note that T=T-1 Now rewrite the system dynamics in terms of the state cl TAct=A Note that similarity transformations preserve the eigenvalues, so we are guaranteed that 入(A4)≡A(Aa . Work through the math BK Acl I 0 A Ⅰ-1LCA-BK-LCI-I A BK I 0 A-LC-A+LCII-I A-BK BK 0 A-LC Because Ac is block upper triangular, we know that the closed DOD poles of the system are given by det(sI -Ac)= det(sI -(A- BK). det(sI-(A-LC))=0Fall 2001 16.31 17—2 • Can fix this problem by introducing a new variable ˜x = x − xˆ and then converting the closed-loop system dynamics using the similarity transformation T x˜cl , ∙ x x˜ ¸ = ∙ I 0 I −I ¸ ∙ x xˆ ¸ = T xcl — Note that T = T −1 • Now rewrite the system dynamics in terms of the state ˜xcl Acl ⇒ T AclT −1 , A¯cl — Note that similarity transformations preserve the eigenvalues, so we are guaranteed that λi(Acl) ≡ λi(A¯cl) • Work through the math: A¯cl = ∙ I 0 I −I ¸ ∙ A −BK LC A − BK − LC ¸ ∙ I 0 I −I ¸ = ∙ A −BK A − LC −A + LC ¸ ∙ I 0 I −I ¸ = ∙ A − BK BK 0 A − LC ¸ • Because A¯cl is block upper triangular, we know that the closed-loop poles of the system are given by det(sI − A¯cl) , det(sI − (A − BK)) · det(sI − (A − LC)) = 0