(a) Determine the state observability of the system described by $$ \dot{x}=\left[\begin{array}{ccc} -3 & 1 & 1 \\ -1 & 0 & 1 \\ 0 & 0 & 1 \end{array}\right] x+\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \\ 2 & 1 \end{array}\right] u \quad \dot{y}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right] x $$
Explanation
To determine the observability of the system, we need to calculate the observability matrix O. The observability matrix is given by O = [C; CA; CAB]. We need to calculate CA and CAB. CA = A * C = [-3 1 1; -1 0 1; 0 0 1] * [0 0 1; 1 0 0] = [-1 1 1; 0 1 1; 0 0 1]. CAB = A * B = [-3 1 1; -1 0 1; 0 0 1] * [0 1; 0 0; 2 1] = [-1 1 3; 0 1 2; 0 0 2]. The observability matrix is O = [0 0 1; 1 0 0; -1 1 -1; 0 1 1]. The rank of O is 3, which is equal to the number of states. Therefore, the system is observable.
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