Reference no: EM131762327
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
![1560_figure.png](https://secure.expertsmind.com/CMSImages/1560_figure.png)
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