State Controllability

Clearly, in order to control a system, it must be controllable. Similar to state observability, complete state controllability can be checked by means of Kalman’s controllability matrix or by means of the controllability Gramian matrix. As to be expected, state observability as well as state controllability are invariant under a non-singular transformation of the state-space model. In the case of a system with repeated eigenvalues, the state-space representation can be transformed into Jordan canonical form. The state-space model of a controllable SISO can be transformed into the controller canonical form, from which a transfer function can be obtained directly by inspection. Non-controllable system modes can be found by means of the Hautus criterion.