Robust Stability and Stabilization of a Class of Nonlinear Discrete-time Stochastic Systems with State and Controller Dependent Noise

A problem of robust state feedback stability and stabilization of nonlinear discrete-time stochastic processes is considered. The linear rate vector of a discrete-time system is perturbed by a nonlinear function that satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of nonlinear discrete-time systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable.The state dependent diffusion is modeled by a normal sequence of identically independently distributed random variables. The new formulation provides a suitable setting for robust stabilization of nonlinear discrete-time systems where the underlying deterministic system satisfy the generalized matching conditions. Our method which is based on linear matrix inequalities (LMI’s) is distinctive from the existing robust control and absolute stability techniques. Examples are given to demonstrate the results.