On the existence of Luenberger reduced order observer

In this paper the explicit necessary and sufficient conditions for the existence of Luenberger reduced order observer are established. In particular, it is proven that for the given linear time-invariant system of order n, having p linearly independent outputs and m inputs, a Luenberger observer of order (n − p) can be constructed if and only if the given system is detectable. Furthermore, a procedure is given for the construction of the observer. Our approach is based on the properties of real and polynomial matrices.