State feedback \(H_{\infty}\) control for a class of nonlinear stochastic systems. (English) Zbl 1157.93019

Zhang, Weihai; Chen, Bor-Sen

SIAM J. Control Optim. 44, No. 6, 1973-1991 (2006).

This paper investigates an \(H_{\infty}\) control problem for a class of nonlinear stochastic systems with state- and disturbance-dependent noise. This problem is discussed in both the finite and infinite horizon cases. In this regard, an approach involving Hamilton-Jacobi equations is used in order to develop infinite and finite horizon nonlinear stochastic \(H_{\infty}\) control designs.
In the main, the authors generalize some results on nonlinear \(H_{\infty}\) control for deterministic systems to a stochastic setting. In order to treat the infinite horizon nonlinear stochastic \(H_{\infty}\) control problem, they introduce definitions for the concepts of “zero-state observability” and “zero-state detectability.” Another tool to solve the aforementioned problem is the stochastic LaSalle invariance principle.
Reviewer’s remark: The paper is well-written and of interest to experts in both \(H_{\infty}\) control as well as nonlinear stochastic systems.

Reviewer: Mark A. Petersen (Potchefstroom)

MSC:

93C10	Nonlinear systems in control theory
93D09	Robust stability
93E15	Stochastic stability in control theory

Keywords:

\(H_{\infty}\) control; nonlinear stochastic systems; Hamilton-Jacobi equations; zero-state observability; zero-state detectability