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Controllability:

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. +. 1 s. x. +. y. B. C. +. -. +. A.

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Controllability:

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  1. Controllability:

  2. PBH test for diagonal case

  3. PBH test for block Jordan diagonal case

  4. Observability

  5. PBH test for diagonal case

  6. PBH test for block Jordan diagonal case

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

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

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

  10. Example

  11. 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

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

  13. Example A = 1 2 3 4 5 6 7 8 9 >> B B = 1 0 1 >> C C = 2 1 0 Ab = 11.5 11.0 10.5 5.5 5.0 4.5 -0.5 -1.0 -1.5 >> Bb Bb = 0 1 0 >> Cb Cb = 1 2 3 Find the transformation relating the two.

  14. [P1, J1]=jordan(A) P1 = -1.2833 0.2833 1.0000 -0.1417 0.6417 -2.0000 1.0000 1.0000 1.0000 J1 = -1.1168 0 0 0 16.1168 0 0 0 0 [P2, J2]=jordan(Ab) P2 = -0.8832 -18.1168 1.0000 0.0584 -8.5584 -2.0000 1.0000 1.0000 1.0000 J2 = -1.1168 0 0 0 16.1168 0 0 0 0

  15. P0=P1*inv(P2) P0 = 0.3182 -0.8182 -0.9545 -0.3409 0.5909 -0.4773 0 0.0000 1.0000 P = 0 1 1 1 0 1 1 1 0 Abb=P0\A*P0 Abb = 11.5000 11.0000 10.5000 5.5000 5.0000 4.5000 -0.5000 -1.0000 -1.5000 Ab = 11.5 11.0 10.5 5.5 5.0 4.5 -0.5 -1.0 -1.5

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