Introduction:

Linear time invariant system may be represented in state space form by the following equations:
State equation: $$ \dot{x}(t)=A x(t)+B u(t) \tag{1a} $$ Output equation: $$ y(t)= C x(t) + D u(t) \tag{1b} $$
where, x(t) is state vector, y(t) is output vector, u is input or control vector, A is system matrix, B is input matrix, C is output matrix and D is feed forward matrix.

The concepts of controllability and observability were introduced by Kalman. The conditions of controllabilty and observability may govern the existence of a complete solution to the control system design problem. The solution to this problem may not exist if the system considered is not controllable. Although most physical systems are controllable and observable, corresponding mathematical models may not possess the property of controllability and observability. Then it is necessary to know the conditions under which a system is controllable and observable.

A system is said to be controllable at time t₀ if it is possible by means of an unconstrained control vector to transfer the system from any initial state x(t₀) to any other state in a finite time interval of time.

A system is said to be observable at time t₀ if, with the system is state x(t₀), it is possible to determine this state from the observation of the output over a finite time interval.