New Approach to General Nonlinear Discrete-Time Stochastic $H_\infty$ Control

In this paper, a new approach based on convex analysis is introduced to solve the H
_∞
problem for discrete-time nonlinear stochastic systems. First, by using the disintegration property of the conditional expectation, sufficient/necessary conditions are given for the internal stability of the concerned systems. Second, in order to separate the unknown exogenous disturbance from the state variables, the properties of convex functions are applied to prove the stochastic version of bounded real lemma (BRL). Third, the state feedback H
_∞
control problem is studied, and an H
_∞
control is designed based on the BRL. Finally, two numerical examples and one real-world regulation control example of synthetic genetic circuit are presented to show the effectiveness of our developed theory.