Robust $H_{\infty }$ sliding mode control for uncertain polynomial fuzzy stochastic singular systems

Abstract

This paper considers the robust $H_{\infty }$ sliding mode control (SMC) problem for uncertain polynomial fuzzy stochastic singular systems. The interest is focused on designing a polynomial fuzzy sliding mode controller, and two assumptions in most integral SMC methods for fuzzy stochastic systems are removed. By constructing an integral sliding surface, the equivalent control law is obtained, and the sliding mode dynamics is derived. With the help of the sum-of-squares method, for sliding mode dynamics, the mean-square admissibility and the $H_{\infty }$ performance are guaranteed. Furthermore, the sliding mode controller is established. The effectiveness and advantages of the proposed design method are finally revealed via a practical example of the modified bio-economic model and a numerical example.

Introduction

Sliding mode control (SMC) is a special robust control method. Due to its numerous advantages such as insensitivity to parameter changes, simple physical implementation and fast response, SMC has attracted tremendous attention in the academic community over the past few decades. It is also more and more widely used in underwater robots, aircraft, diesel engines, motors, and other fields. Generally speaking, the SMC strategy is divided into two stages: the reaching phase and the sliding phase. At the very beginning, the proposed sliding surface cannot guarantee insensitivity in the reaching phase. To tackle this, the concept of integral sliding surface (ISS) is introduced in [1], [2]. Its characteristic is that sliding mode dynamics (SMD) can ensure the same order as the original system, and the robustness can be guaranteed during the whole system response. The ISS-based SMC method in [1] and [2] is extended to study the SMC problem of mismatched uncertain systems in [3]. To date, it is noted that the integral SMC problem is still of great concern [4], [5].

Stochastic phenomena, important factors affecting system performance, are widely exist in biology, finance, communication, and other fields. In view of this, the SMC issue of stochastic systems is developed. The observer-based SMC for stochastic time-delay systems is proposed in [6]. The robust SMC for uncertain stochastic systems with time-varying delay is considered in [7]. The robust $H_{\infty }$ SMC for nonlinear stochastic systems with the disturbance input and the matched uncertainties is investigated in [8]. It is a remarkable fact that [7] and [8] need to satisfy an assumption that the product of the parameter matrix of the ISS and the diffusion term of the stochastic system is equal to zero. It is an attempt to make the selected ISS deterministic. However, the assumption is very restrictive for many practical stochastic systems. In this connection, a novel ISS-based SMC approach for uncertain time-delay stochastic systems is proposed, and the above assumption is removed in [9]. It is worth pointing out that the aforementioned research only focuses on nonlinear or stochastic systems that are easy to model.

However, some practical systems have the characteristics of high coupling, time delay, nonlinearity and environmental noises, which makes it difficult for traditional mathematical methods to establish accurate mathematical models. The T-S fuzzy (TSF) model is regarded as a remarkable tool for approaching nonlinear systems [10]. Subsequently, the TSF system theory has been applied to more complex stochastic systems. The SMC approach based on ISS for TSF stochastic systems is discussed in [11], which needs to satisfy two assumptions, each subsystem has the same input matrix and the product of the parameter matrix of the ISS and the diffusion term of the stochastic system is equal to zero. By introducing a new ISS, these two assumptions are eliminated in [12] and [13].

In addition, due to the characteristics of singular systems, TSF singular systems have attracted much attention because they can describe some complex nonlinear singular systems more accurately and conveniently. For almost a decade, a growing number of works have been given to the ISS-based SMC issue for TSF singular systems [14], [15], [16], [17], [18], [19]. In order to avoid the assumption that the subsystem input matrix is the same, a novel vector ISS is proposed in [20]. With the continuous improvement of the theory of stochastic singular systems [21], [22], [23], [24], some scholars focus on the SMC problem of stochastic singular systems. For stochastic singular Markovian jump systems, by means of strict linear matrix inequality, the observer-based SMC problem is carried out in [25], and the results in [26] are modified by Feng and Shi [25]. The robust SMC is developed for stochastic singular Markovian jump systems satisfying a one-sided Lipschitz condition in [27]. For TSF stochastic singular Markovian jump systems, the robust SMC is considered based on two assumptions in [28]. Under the assumption that all local input matrices are full column rank, the sliding mode control problem for nonlinear stochastic singular semi-Markov jump systems is studied in [29]. Although some constructive approaches are presented to relax these two restrictive assumptions in [30] and [31], it is problematic to directly generalize the results to TSF stochastic singular systems under external disturbance. On the other hand, as an extension of the TSF model, the polynomial fuzzy (PF) model can more generally and effectively describe nonlinear systems [32]. With the sum-of-squares (SOS) technique, the SMC and the observer-based SMC problems for PF singular systems based on the restrictive assumption are studied via the ISS in [33] and [34], respectively.

It is necessary to highlight the model differences between the current work and the existing related work. The stochastic systems in [6], [7], [8], [9] and T-S fuzzy systems in [10], [11], [12], [13] are nonsingular, while the system considered in this study is singular. The SMC method for stochastic singular systems in [25], [26], [27] is not applicable to more complex nonlinear singular systems, and the influence of time delay on systems is not considered. The stochastic disturbance is not involved in [33], [34]. Compared with SMC for PF singular systems without stochastic disturbance, it is difficult to treat SMC for PF stochastic singular systems. Based on the aforementioned discussions, the purpose of this study is to reduce the restrictions in the existing stochastic fuzzy SMC problem. An appropriate control method is established to obtain better control performance of PF stochastic singular systems.

The robust $H_{\infty }$ SMC problem is addressed for uncertain PF stochastic singular systems with time-varying delay and external disturbance in this study. The main contributions can be summarized as:

(1) In many works of existing fuzzy stochastic systems, two assumptions need to be satisfied: (i) each subsystem has the same input matrix; (ii) the product of the parameter matrix of the ISS and the diffusion term of the stochastic system must be zero. These two assumptions are removed in this study.

(2) Compared with [29], [30], [35], [36], [37], it is no longer required that the mean value of the local matrix is full column rank, or all local input matrices are full column rank.

(3) An equivalent set is used to transform the nonconvex SOS condition with equality constraints into the convex SOS condition for mean-square (MS) admissibility of SMD, which makes the condition easy to check by using SOSTOOLS.

The remainder of this paper is arranged as follows. Problem formulation is presented in Section 2. Section 3 develops the MS admissibility analysis and an SMC scheme. The efficiency of the proposed control scheme is revealed by simulation examples in Sections 4. Section 5 gives the conclusions.

Notations: $\text{sym}\left\{A\right\}$ is defined as $A+A^T$; $\text{diag}\{·\}$ represents a block-diagonal matrix; $\mathbb{E}\{·\}$ denotes the expectation operator; $\parallel ·\parallel$ denotes the Euclidean norm of a vector or its induced matrix norm; $\mathfrak{S}$ is the set of all SOS polynomials; $*$ means the symmetric terms in symmetric matrix.