Sliding Mode Fuzzy Control of Stochastic Nonlinear Systems Under Cyber-Attacks

In this article, the problem of integral sliding mode control (ISMC) for a class of nonlinear systems with stochastic characteristics under cyber-attack is investigated. The control system and the cyber-attack are modeled as an Itô-type stochastic differential equation. The stochastic nonlinear systems are approached by the Takagi–Sugeno fuzzy model. A dynamic ISMC scheme is applied and the states and control input are analyzed within a universal dynamic model. It is demonstrated that trajectory of the system can be confined to the integral sliding surface within finite time, and the stability of closed-loop system under cyber-attack will be guaranteed by using a set of linear matrix inequalities. Following a standard procedure of universal fuzzy ISMC, it is shown that all signals in the closed-loop system will be guaranteed bounded, and the states are asymptotic stochastic stable if some conditions are met. An inverted pendulum is applied to show the effectiveness of our control scheme.

I. Introduction

Stochastic systems [1], [2], [3], [4], [5] refer to systems containing random variables, such as internal stochastic parameters, external disturbances, or observation noise. In practice, stochastic factors inevitably exist. In this case, if the control system is designed according to deterministic control theory, it will deviate from the predetermined design requirements and generate stochastic deviations. However, a stochastic variable cannot be described by a known function of time but only by certain stochastic properties. Due to the influence of many uncertain stochastic factors in practical systems, the research on stochastic systems and fuzzy control [7], [8], [9] has received increasing attention in the literature. In [6], the stochastic differential systems are strictly characterized, and Itô’s stochastic differential equation theory is established. Subsequently, research on Itô-type stochastic differential equations has developed vigorously, and many scholars have conducted in-depth studies on them. The stochastic system model has been extensively applied in social fields, such as the economy and population systems, as well as in natural and engineering fields, including aerospace, navigation and control, and manufacturing engineering.