H_∞ control for discrete-time nonlinear stochastic systems

In this note, we develop an H_∞-type theory for a large class of discrete-time nonlinear stochastic systems. In particular, we establish a bounded real lemma (BRL) for this case. We introduce the notion of stochastic dissipative systems, analogously to the familiar notion of dissipation associated with deterministic systems, and utilize it in the derivation of the BRL. In particular, this BRL establishes a necessary and sufficient condition, in terms of a certain Hamilton Jacobi inequality (HJI), for a discrete-time nonlinear stochastic system to have l₂ - gain ≤ γ. The time-invariant case is also considered as a special case. In this case, the BRL guarantees necessary and sufficient conditions for the system to have l₂ - gain ≤ γ, by means of a solution to a certain algebraic HJI. An application of this theory to a special class of systems which is a characteristic of numerous physical systems, yields a more tractable HJI which serves as a sufficient condition for the underlying system to possess l₂ - gain ≤ γ. Stability in both the mean square sense and in probability, is also discussed. Systems that possess a special structure (norm-bounded) of uncertainties in their model are considered. Application of the BRL to this class of systems yields a linear state-feedback stabilizing controller which achieves l₂ - gain ≤ γ, by means of certain linear matrix inequalities (LMIs).