Relations between information theory, robustness and statistical mechanics of stochastic uncertain systems via large deviation theory

Abstract

Robustness issues of stochastic uncertain controlled systems are intimately related to concepts from information theory and statistical mechanics. These relations are unveiled by using the theory of large deviations through the solution of two fundamental optimization problems. In one of the optimization problems the aim is to minimize the relative entropy between a nominal measure induced by a reference stochastic system and the measures induced by the uncertain stochastic systems subject to energy constraints. In the other the supremum of an energy functional with respect to the measures induced by the nominal and uncertain systems, subjected to constraints on the relative entropy between the measures induced by the nominal and uncertain stochastic systems is sought. The solutions of these problems, by virtue of their formulation, satisfy the H∞ robustness criterion and through statistical mechanics arguments the associated optimal sensitivity can be characterized. © 2006 Springer-Verlag Berlin Heidelberg.