Problem -

(a) The linear continuous-time system

x^· = Ax + Bu, x ∈ Rⁿ, u ∈ Rⁿ and y = Cx                    (1)

is controllable if and only if the controllability matrix C has full rank, i.e., rank (C) = n. Given the linear continuous-time system

Determine the controllability matrix C for this third-order system and find the rank condition.

(b) Consider the vector input from of (1) and the similarity transformation x^{^} = Px such that

x^{^} = A^{^}x^{^} + B^{^}u,                                                    (2)

where A^{^} = PAP^-1 and B^{^} = PB. Then prove that the following theorems hold:

Theorem -1: The pair (A, B) is controllable if and only if the pair (A^{^}, B^{^}) is controllable. Defining C^{^} = CP^-1, a similar theorem is valid for observability.

Theorem-2: The pair (A, C) is observable if and only if the pair (A^{^}, C^{^}) is observable.

Attachment:- Problem.rar