Any linear time invariant system can be represented by the dynamic equation
and y(t) = Cx(t) + Du(t)
The state x(t) will be controllable at t = t0 when there exists a piecewise continuous input u(t) which will drive the state to any final state x(t) for finite time interval (t − t0) ≥ 0. The system is called controllable if every state of the system is controllable in finite time interval.
For controllability of the system the matrix
If there is no connection between a certain state and input, the state is not controllable as shown in Fig. 9.12.