Controllability PBH test for diagonal case PBH test

Controllability:

PBH test for diagonal case

PBH test for block Jordan diagonal case

Observability

PBH test for diagonal case

PBH test for block Jordan diagonal case

C. C. , C. O. and TF poles/zeros

State Feedback D r + u - + B 1 + s A x C + + K feedback from state x to control u y

Pole placement Solve this to get k’s.

Example

Pole placement In Matlab: Given A, B, C, D ①Compute QC=ctrb(A, B) ②Check rank(QC) If it is n, then ③Select any n eigenvalues(must be in complex conjugate pairs) ev=[λ 1; λ 2; λ 3; …; λn] A+Bk will ④Compute: have K=place(A, B, ev) eigenvalues at these values

Invariance under state feedback Thm: Controllability is unchanged after state feedback. But observability may change!

Example A= 1 4 7 2 5 8 3 6 9 >> B B= >> Bb 1 0 1 Bb = >> C C= Ab = 11. 5 11. 0 10. 5 5. 0 4. 5 -0. 5 -1. 0 -1. 5 0 1 0 >> Cb 2 1 0 Cb = Find the transformation relating the two. 1 2 3

[P 1, J 1]=jordan(A) [P 2, J 2]=jordan(Ab) P 1 = -1. 2833 -0. 1417 1. 0000 P 2 = -0. 8832 -18. 1168 1. 0000 0. 0584 -8. 5584 -2. 0000 1. 0000 0. 2833 1. 0000 0. 6417 -2. 0000 1. 0000 J 1 = -1. 1168 0 0 16. 1168 0 0 J 2 = 0 0 0 -1. 1168 0 0 16. 1168 0 0 0

P 0=P 1*inv(P 2) P 0 = 0. 3182 -0. 8182 -0. 9545 -0. 3409 0. 5909 -0. 4773 0 0. 0000 1. 0000 P= 0 1 1 1 0 Abb=P 0A*P 0 Abb = 11. 5000 11. 0000 10. 5000 5. 0000 4. 5000 -0. 5000 -1. 0000 -1. 5000 Ab = 11. 5 11. 0 10. 5 5. 0 4. 5 -0. 5 -1. 0 -1. 5