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The positive-real and bounded-real lemmas solve two importantlinear-quadratic optimal control problems for passive and non-expansivesystems, respectively. The lemmas assume controllability, yet a passive ornon-expansive system can be uncontrollable. In this paper, we solve theseoptimal control problems without making any assumptions. In particular, we showhow to extract the greatest possible amount of energy from a passive but notnecessarily controllable system (e.g., a passive electric circuit) using statefeedback. A complete characterisation of the set of solutions to the linearmatrix inequalities in the positive-real and bounded-real lemmas is alsoobtained. In addition, we obtain necessary and sufficient conditions for asystem to be non-expansive that augment the bounded-real condition with newconditions relevant to uncontrollable systems.

On the optimal control of non-expansive or passive systems