H_∞ control for discrete-time nonlinear stochastic systems

In this paper we develop an H_∞-type theory for a large class of discrete-nonlinear stochastic systems. In particular, we establish a bounded real lemma 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 bounded real lemma. In particular, this bounded real lemma establishes necessary and sufficient conditions, in terms of a certain Hamilton Jacobi Inequality (HJI), for a discrete-time nonlinear stochastic system to be an l ₂-gain≤ γ. The time-invariant case is also considered, where in this case the bounded real lemma guarantees necessary and sufficient conditions for the system to be l₂-gain≤ γ, by means of a solution to a certain algebraic HJI. Stability, in both the mean square sense and in probabilty, is discussed and a utilization of the Linear Matrix Inequalities (LMIs) technique is made to synthesize a controller that achieves an l₂-gain property.