Signals and Systems

9.9 CONTROLLABILITY

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

 

[s] = [B AB A²B … Aⁿ⁻¹B] must have rank n.

 

If there is no connection between a certain state and input, the state is not controllable as shown in Fig. 9.12.

 

 

Fig. ...