Control system theory problem - applied mathematics

I need help with the following problem:

(A) Show that a system (A, B) is controllable if and only if [sI-A B] has rank N = dim X for all s ∈ C.

(B) Show that (a) is equivalent to saying that every eigenvector on the left of A is not

Orthogonal to B, that is, if h ∈ Cn Such that h´A = λh´ , For λ ∈ C then h´B 6 not equal to 0

(Hautus' controllability criterion).

(C) Show that a system (C, A) is observable if and only if the vector



Has post N = dim X for all s ∈ C.

(D) Show that (c) is equivalent to saying that every eigenvector of A is not in the nucleus Of C, that is, if h ∈ Cn 'and such that Ah = λh´ , For λ ∈ C then Ch not equal to 0 (criterion of Observability of Hautus).

Квалификация: Техника, Математика

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