The optimality property of an optimal regulator incorporating an observer

Abstract

The optimal control law for a linear system and a quadratic performance index is, as is well known, given by state feedback. In the case where there exist some inaccessible state variables, and where an observer is incorporated in a feedback loop to estimate them, the resultant control law is no longer optimal almost always for the original performance index. In the present paper we discuss whether or not it is optimal for some quadratic performance index. More precisely, we study the optimality of an optimal feedback control law incorporating the observer, in the sense that it minimizes some quadratic performance index for the system augmented by the observer. The main purpose of this paper is to obtain necessary and sufficient conditions for the optimality to hold irrespectively of the choice of observer.