Controllability and observability

Controllability and observability are important structural properties of a dynamic system. Controllability can be defined as follows:

The system of Eqn. (12.2) is controllable if there exists a control signal that will take the state of the system from any initial state to any desired final state in a finite time interval.

This condition is equivalent to the following condition:

The system of Eqn. (12.2) is controllable if the controllability matrix

has full rank .

The concept of observability is parallel to that of controllability and all can be transformed to statements about observability by invoking the property of duality, as discussed later in section 13.4.2. The observability definition analogous to those for controllability are as follows:

The system of Eqs. (12.2) and (12.3) is observable if, for any , there is a finite time such that can be determined from and for

This condition is equivalent to the following:

The system of Eqs. (12.2) and (12.3) is observable if the observability matrix

has full rank .