Universal Fuzzy Integral Sliding-Mode Controllers for Stochastic Nonlinear Systems

In this paper, the universal integral sliding-mode controller problem for the general stochastic nonlinear systems modeled by Itô type stochastic differential equations is investigated. One of the main contributions is that a novel dynamic integral sliding mode control (DISMC) scheme is developed for stochastic nonlinear systems based on their stochastic T-S fuzzy approximation models. The key advantage of the proposed DISMC scheme is that two very restrictive assumptions in most existing ISMC approaches to stochastic fuzzy systems have been removed. Based on the stochastic Lyapunov theory, it is shown that the closed-loop control system trajectories are kept on the integral sliding surface almost surely since the initial time, and moreover, the stochastic stability of the sliding motion can be guaranteed in terms of linear matrix inequalities. Another main contribution is that the results of universal fuzzy integral sliding-mode controllers for two classes of stochastic nonlinear systems, along with constructive procedures to obtain the universal fuzzy integral sliding-mode controllers, are provided, respectively. Simulation results from an inverted pendulum example are presented to illustrate the advantages and effectiveness of the proposed approaches.

I. Introduction

Takagi-sugeno fuzzy models [1], or the so-called fuzzy dynamic models [2], have been widely utilized in control of complex nonlinear systems during the last few decades. In the T-S fuzzy model-based methodology, local dynamics of the original nonlinear system in different state space regions are described by linear dynamic models, and the overall model of the system is then constructed by fuzzy blending of these local models through a set of fuzzy membership functions. This relatively simple structure provides great advantages in stability analysis and controller synthesis for T-S fuzzy systems in view of the powerful conventional control theory and techniques. T-S fuzzy models have been shown to be universal function approximators in the sense that they are able to approximate any smooth nonlinear functions to arbitrary degree of accuracy in any convex compact region [3]–[5]. All these results provide a solid theoretical foundation for modeling and control design of complex nonlinear systems based on T-S fuzzy models [6]–[21]. Readers can refer to several books and survey papers [22]–[24] and the references therein for the most recent advances on this topic.