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Similarity transformation

Similarity transformation. same system as(#). Controllability:. Example:. Controller Canonical Form:. Completely Controllable. Controllability:. Only need to check this for eigenvalues. Controllability:. PBH test for diagonal case. PBH test for block Jordan diagonal case.

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Similarity transformation

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  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

  14. PBH test for diagonal case

  15. PBH test for block Jordan diagonal case

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

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

  18. Controllability and Observability

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

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

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

  22. Example

  23. 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

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

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