State Feedback $H_\infty$ Control for a Class of Nonlinear Stochastic Systems

Abstract

This paper discusses the H∞ control problem for a class of nonlinear stochastic systems with both state-and disturbance-dependent noise. By means of Hamilton-Jacobi equations, both infinite and finite horizon nonlinear stochastic H∞ control designs are developed.Some results on nonlinear H∞ control of deterministic systems are generalized to a stochastic setting. We introduce some useful concepts such as "zero-state observability" and "zero-state detectability" which, together with the stochastic LaSalle invariance principle, yield some valuable consequences in infinite horizon nonlinear stochastic H∞ control. Introduction.In practice, when the exogenous disturbance enters the system, an H ∞ control design is often first considered, when a control law is sought to efficiently eliminate the effect of the disturbance; see [12], [13] and the references therein. Theoretically, study of H ∞ control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard H ∞ control leads to a breakthrough; details can be found in the prize-winning paper [17]. From the viewpoint of the state space, the linear H ∞ control problem can be converted into the study of a game-theoretic Riccati equation, and the "completion of square methodology," similar to the linear quadratic (LQ) and linear quadratic Gaussian (LQG) theories, can be applied; see [15], [16], [20], [21], [25], and [30]. Soon after the appearance of [17], the nonlinear H ∞ control problem (on deterministic systems) was investigated by many authors; see [5], [14], [18], and [19]. From the time-domain perspective, an H ∞ norm of the transfer function is nothing else but the L 2 -induced norm of the input-output operator with initial state zero. This important feature makes it possible to develop nonlinear or stochastic H ∞ theory. Van der Schaft [5] made a contribution to the state feedback H ∞ control for nonlinear deterministic systems with infinite time horizon, where a relatively deep tool, i.e., the strict relation between Hamilton-Jacobi equations (HJEs) and invariant manifolds of Hamiltonian vector fields, was applied. Using this tool, he showed that a local solution to the primal nonlinear H ∞ control exists if its linearized H ∞ control problem is solvable. The authors of [18] and [19] dealt with the output feedback H ∞ control of nonlinear deterministic systems with incomplete state information, and a separation principle was obtained. In [5], [18], and [19], the differential geometric approaches

On the Information Transmission Ability of Nonlinear Stochastic Dynamic Networks

“…That is, the stochastic part of fluctuation is absorbed by with , where is a standard Wiener process [16][17][18] (Brownian motion) and the change of linear dynamic network by the random fluctuation source is denoted by .…”

“…For convenience of analysis, the origin of the nonlinear stochastic dynamic network is shifted to the equilibrium point e x to simplify the measuring procedure of information transmission ability of the nonlinear stochastic dynamic network. Let us denote ( ) ( ) e x t x t x    , then the following shifted nonlinear stochastic dynamic network can be obtained as follows [17,18]: …”

“…Then, let us consider the information transmission ability of the nonlinear stochastic dynamic network in (15). According to the stochastic Lyapunov theory [17], we get the following result.…”

“…In accordance with system gain viewpoint, the ratio of output to input signal energy of a stochastic dynamic network is determined for all possible input signals. Then the maximum ratio is denoted as the information transmission ability, i.e., from the system (network) gain perspective [16][17][18][19][20]. This measure of information transmission ability is more dependent on the system characteristics of the studied dynamic network rather than the input signals.…”

Robust sliding mode design for uncertain stochastic systems based on H∞ control method

“…For example, the state feedback controller in [10] was designed for uncertain stochastic system such that the closed-loop systems is robustly asymptotically stable and satisfies a prescribed H ∞ performance. Zhang, et al [11] further presented the stochastic H 2 /H ∞ control design for nonlinear stochastic systems with state-dependent noise.…”