Generalized Fixed-Structure Optimality Conditions for H2 Optimal Control

Abstract

Over the last several years, researchers have shown that when it is assumed a priori that a fixed-order optimal compensator is minimal, the necessary conditions can be characterized in terms of coupled Riccati and Lyapunov equations, usually termed "optimal projection equations." When the optimal projection equations for H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> optimal control are specialised to full-order control, the standard LQG Riccati equations are recovered. This paper relaxes the minimality assumption on the compensator and derives necessary conditions for fixed-structure H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> optimal control that reduce to the standard optimal projection equations when the optimal compensators are assumed to be minimal. The results are then specialized to full-order control. The results show that the standard LQG Riccati equations can be derived using fixed-structure theory even without the minimality assumption. They also show for the first time that a reduced-order optimal projection controller is a projection of one of the extremals to the full-order H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> optimal control problem. This latter result is used to discuss suboptimal projection methods that are able to produce minimal-order realizations of nonminimal LQG compensators. For this special case, the similarity transformation relating the projection matrix used by these suboptimal methods to the optimal projection matrix from the standard optimal projection theory is explicitly defined.