Question 10.21: Controllability and Observability Determine the controllabil...
Controllability and Observability
Determine the controllability and observability for the second-order systems given in Example 10.20.
For the system in Part (a), the controllability matrix \mathbf{P} is
\mathbf{P}=\left[\begin{array}{ll} \mathbf{B} & \mathbf{A B} \end{array}\right]=\left[\begin{array}{cc} 2 & -8 \\ 0 & 0 \end{array}\right],
which is singular. Thus, the system is uncontrollable. The observability matrix \mathbf{Q} is
\mathbf{Q}=\left[\begin{array}{c} \mathbf{C} \\ \mathbf{C A} \end{array}\right]=\left[\begin{array}{cc} 1 & 3 \\ -4 & -3 \end{array}\right]
which is nonsingular. Thus, the system is observable.
For the system in Part (b), the controllability matrix \mathbf{P} is
\mathbf{P}=\left[\begin{array}{ll} \mathbf{B} & \mathbf{A B} \end{array}\right]=\left[\begin{array}{ll} 2 & -8 \\ 1 & -1 \end{array}\right],
which is nonsingular. Thus, the system is controllable. The observability matrix \mathbf{Q} is
\mathbf{Q}=\left[\begin{array}{c} \mathbf{C} \\ \mathbf{C A} \end{array}\right]=\left[\begin{array}{cc} 1 & 0 \\ -4 & 0 \end{array}\right],
which is singular. Thus, the system is unobservable.
Note that the two systems in Example 10.20 have the same transfer function representation but different controllability and observability properties. This implies that controllability and observability are functions of the state of the system and cannot be determined from a transfer function.