On the optimal control of non-expansive or passive systems

Timothy H. Hughes

Published 2017-03-23Version 1

The bounded-real and positive-real lemmas solve two important linear-quadratic optimal control problems related to the concepts of non-expansive and passive systems, respectively. The lemmas assume controllability, yet a non-expansive or passive system can be uncontrollable. In this paper, we solve these optimal control problems without assuming controllability. We obtain necessary and sufficient conditions for a system to be non-expansive (passive) which augment the bounded-real (positive-real) condition with new conditions relating to the uncontrollable modes of the system. A characterisation of the set of solutions to the linear matrix inequalities in the bounded-real and positive-real lemmas is obtained.