Lecture 9: Controllability and Observability - Illinois Institute of ...

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Controllability We would like to prove that To do this, we will prove that • Im(W t ) ⊂ R t • R t ⊂ Im(C(A, B)) • Im(C(A, B)) ⊂ Im(W t ) So far, we have show that R t = Im(W t ) = Im(C(A, B)) Next, we will show that R t ⊂ C AB Im(W t ) ⊂ R t M. Peet Lecture 9: Controllability 2 / 15
This proof M. Peet is especially useful, as it gives theLecture control 9: Controllability input which will get us to x3 / 15 Controllability: Im(W t ) ⊂ R t Theorem 1. Im(W t ) ⊂ R t Proof. First, suppose that x ∈ Im(W t ) for some t > 0. Then x = W t z for some z. • Now let u(s) = B T e AT (t−s) z. Then Γ t u = = ∫ t 0 ∫ t = W t z = x • Thus x ∈ Im(Γ t ) = R t . We conclude that Im(W t ) ⊂ R t 0 e A(t−s) Bu(s)ds e A(t−s) BB T e AT (t−s) zds
Page 1: Modern Control Systems Matthew M. P
Page 5 and 6: Proof by Contradiction: Im(C(A, B))
Page 7 and 8: Summary: R t = Im(W t ) = Im(C(A, B
Page 9 and 10: Representation and Controllability
Page 11 and 12: Invariant Subspaces Proposition 2.
Page 13 and 14: Controllability Form When a system
Page 15: Controllability Form Example Contin

Lecture 9: Controllability and Observability - Illinois Institute of ...

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