1. Similarity transformation
same
system
as(#)

2. Controllability:

3. Example:

4. Controller
Canonical
Form:
Completely
Controllable

5. Controllability:
Only need to check this for eigenvalues

6. Controllability:

7. PBH test for diagonal case

8. PBH test for block Jordan diagonal case

9. Are the following (A, B) pairs C.C.?

10. Are the following (A, B) pairs C.C.?

11. Observability

12. Example:

13. Observability

15. PBH test for diagonal case

16. PBH test for block Jordan diagonal case

17. Are the following (C, A) pairs C.O.?

18. Are the following (C, A) pairs C.O.?

19. Controllability and Observability

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

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

24. Pole placement
Solve this to get k’s.

25. Example

26. 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=[λ1; λ2; λ3;…; λn]
④Compute:
K=place(A,B,ev)
A+Bk will have eigenvalues at these values

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