A new result on observer-based sliding mode control design for a class of uncertain It$\widehat{\mathrm{o}}$ stochastic delay systems

Highlights

• A new technical procedure of stability analysis for the observer-based closed-loop systems is developed.

• A simplified state observer is designed without any control terms compared with the existing results on stochastic delay systems.

• A novel integral sliding surface design is shown on the basis of the new observer and the outputs.

• A new adaptive SMC law is synthesized to ensure the sliding mode phase and maintain the expected performance of the system.

Abstract

This paper develops a new observer-based sliding mode control (SMC) scheme for a general class of It$\widehat{\mathrm{o}}$ stochastic delay systems (SDS). The key merit of the presented scheme lies in its simplicity and integrity in design process of the traditional sliding mode observer (SMO) strategy, i.e., the state observer and sliding surface design as well as the associated sliding mode controller synthesis. For guaranteeing to use the scheme, a new LMIs-based criterion is established to ensure the exponential stability of the underlying sliding mode dynamics (SMDs) in mean-square sense with H_∞ performance. A bench test example is provided to numerically demonstrate the efficacy of the scheme and illustrate the application procedure for potential readers/users with interest in their ad hoc applications and methodology expansion.

Introduction

It has been widely witnessed that control of stochastic systems has increasingly received much attention as one of the most practically meaningful systems in both academic research and application fields [1]. It is worthwhile pointing out that, in recent years, the study of stochastic delay systems (SDS) has inspired a new wave of research under a very crucial factor that time-delay may frequently occur during the whole operation process, such as chemical processes, networked control systems, etc. Accordingly, a great deal of work has been devoted to the stability and stabilization of It$\widehat{\mathrm{o}}$ SDS [2], [3]. Meanwhile, the associated control design problems have been explored for the systems in parallel with the development of system control theory, e.g., H_∞ control and filtering [4], [5], [6].

Due to its various attractive features such as quick response, good transient performance, particularly, the invariance against matched uncertainties, and wide applications to various complex systems [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], sliding mode control (SMC) [18], [19] has been well known as an effective robust control strategy for uncertain and incompletely modelled systems. It is noted that a growing interest has been devoted to the extension of SMC to accommodate the SDS. Also, it is a fact that uncertain nonlinearity may occur through the system control channels (i.e., the matched uncertainty in SMC theory [18]), due to the variation of the control components and the structural parameters as well as the existence of the inevitably external disturbance, and this will also affect the systems performance directly or indirectly, and even leads to instability. Some representative results regarding the SMC of SDS include SMC of uncertain SDS [14], SMC for uncertain SDS with H_∞ performance index [15] to deal with a limitation (i.e., there exists a matrix G with appropriate dimension satisfying $Gg(t,x(t),x(t-d\left)\right)=0$ for all t ≥ 0); further, robust SMC of uncertain SDS has been considered where such restrictive condition to the most existing results is removed in [16].

It should be noted that most existing results for the SDS are obtained upon the premise that the system states are accessible, despite the efficacy of SMC. In many cases, consider that the state variables may not be totally acquired or even knotty to be estimated via output measurement, the observer-based SMC, also called sliding mode observer (SMO) strategy [20], has been developed and excellently implemented in various cases [21], [22], [23], [24], [25], [26], [27], [28], [29]. In particular, by using the SMO approach, a class of Markovian jump systems against actuator faults with quantized measurements and unknown actuator faults was concerned in [21]. In view of a new observer-based SMC design, a class of nonlinear delay systems was investigated in [25], and recently in [26] robust H_∞ control for uncertain singular time-delay systems was studied via a novel SMO synthesis. It is noted that designs of the new sliding surface and/or new-form observers were developed in those works, which may not be extended to stochastic control systems for certain technical reasons directly. To the best of the authors’ knowledge, vast majority of the existing routes for the SMO are that: state observer is designed to generate the original state with assistance of the control input and/or its compensator to restrain the uncertainties and nonlinearity of the system and make the closed-loop systems operate stably. In detail, the design principle leads to that the estimation error system does not contain the control input in general, and then the closed-loop systems can maintain the desirable characteristics on the predesigned sliding surface through the observer and its error system when the associated sliding mode controller is employed. As a result, the achievements using this SMO-idea have been widely applied for SDS [28], [29]. By following along the lines of [25], [26], a new SMO-based scheme is presented in this paper, which may be a worthy addition of the SMO approach for the SDS. The key novelty covers the following:

(a) A particular state observer is designed without any control terms compared with the existing results on SDS.

(b) A novel integral sliding surface design is established on the basis of the new observer and the outputs in such a way that specific sliding mode dynamics (SMDs) of the closed-loop systems is reconstructed.

(c) A sufficient criterion for expected performance of the underlying SMDs is proposed with an easy-to-test LMI framework.

(d) A novel associated reaching motion controller is then synthesized to adaptively ensure the sliding mode phase so as to accommodate the desirable effects of the control strategy.

As such, the proposed scheme is feasible for analysing the stability of the unmeasured system state through the original system itself and its error system. In the other words, an improved procedure is created from the fact that if the stability of the original system and its error system can be ensured, then the observer state can also tend to be stable as it is. All these features distinguish the present scheme from the existing literatures on SDS.

The rest of the paper is organized as follows. Section 2 describes the research problems and preliminaries. Section 3 presents the main results of the new scheme. Section 4 selects a bench test example to demonstrate the results with computational experiments. Section 5 draws brief summary of the study and potential research expansion.

Notations: Throughout the paper and unless specified, let $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathcal{P})$ be a completed probability space with a natural filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},$ where Ω is a sample space, $\mathcal{F}$ is the σ-algebra of subset of the sample space, and $\mathcal{P}$ is the probability measure on $\mathcal{F}$. $\mathcal{E}\{·\}$ is the expectation operator with respect to the probability measure $\mathcal{P}$. If A is a vector or matrix, its transpose is denoted by A^T, and the symmetric elements of the matrix is denoted by “ * ”. X > Y means that the matrix $X-Y$ is positive definite. sym{X} is denoted as sym $\left\{X\right\}=X+X^{\mathrm{T}}$. If M is a matrix, its operator norm is denoted by $\parallel M\parallel =\text{sup}\{\parallel Mx\parallel :\parallel x\parallel =1\},$ λ_max(M) and λ_min(M) represent its maximum and minimum eigenvalues, respectively. Tr{ · } denotes the trace of a matrix. diag{ · } represents a block-diagonal matrix. ${\mathbb{L}}_2[0,\infty )$ stands for the space of square integral vector functions over [0, ∞). Let d > 0 and $\mathcal{C}([-d,0];{\mathbb{R}}^n)$ denote the family of all continuous ${\mathbb{R}}^n$-valued functions on $[-d,0]$. Let ${\mathcal{C}}_{{\mathcal{F}}_0}^b([-d,0];{\mathbb{R}}^n)$ be the family of all ${\mathcal{F}}_0$-measurable bounded $\mathcal{C}([-d,0];{\mathbb{R}}^n)$-valued random variables, and ${\mathbb{L}}^2([a,b];{\mathbb{R}}^n)$ be the family of all ${\mathbb{R}}^n$-valued ${\mathcal{F}}_t$-adapted process ${\left\{{\mathcal{F}}_t\right\}}_{a\le t\le b}$ such that ${\int }_a^b{\parallel f\left(t\right)\parallel }^2\mathrm{d}t<\infty$ a.s. Let ${\mathcal{M}}^2([a,b];{\mathbb{R}}^n)$ be the family of processes ${\left\{{\mathcal{F}}_t\right\}}_{a\le t\le b}$ in ${\mathbb{L}}^2([a,b];{\mathbb{R}}^n)$ such that $\mathcal{E}\{{\int }_a^b\parallel f\left(t\right){\parallel }^2\mathrm{d}t\}<\infty$.