On mean‐square H ∞ control for discrete‐time nonlinear stochastic systems with (x, u, v) ‐dependent noises

Summary In this paper, the H ∞ control problem is investigated for a general class of discrete‐time nonlinear stochastic systems with state‐, control‐, and disturbance‐dependent noises (also called (x, u, v) ‐dependent noises). In the system under study, the system state, the control input, and the disturbance input are all coupled with white noises, and this gives rise to considerable difficulties in the stability and H ∞ performance analysis. By using the inequality techniques, a sufficient condition is established for the existence of the desired controller such that the closed‐loop system is mean‐square asymptotically stable and also satisfies H ∞ performance constraint for all nonzero exogenous disturbances under the zero‐initial condition. The completing square technique is used to design the H ∞ controller with hope to reduce the resulting conservatism, and a special algebraic identity is employed to deal with the cross‐terms induced by (x, u, v) ‐dependent noises. Several corollaries with simplified conditions are presented to facilitate the controller design. The effectiveness of the developed methods is demonstrated by two numerical examples with one concerning the multiplier‐accelerator macroeconomic system.