Event-triggered sliding mode control of Markovian jump systems against input saturation

Abstract

In this work, we pay attention to investigating event-triggered sliding mode control (SMC) strategy with input saturation for a class of Markovian jump systems (MJSs). The state vectors of MJSs are to be sufficiently sampled by a performed event trigger mechanism in a periodic computation way. However, there are some inevitable delays, on the channel from sensor to controller, occurring on the sampling process. For the demand of updates of control input, it is necessary to keep receiving and sending delayed state signals, thus we employ a zero-order-hold (ZOH) in the proposed framework to make the event-triggered SMC strategy come true. Then, by designing an integral-type sliding surface function, combining with the prior knowledge of event trigger scheme, the sliding mode dynamics is derived and the criterion of stochastic stability with ${\mathcal{H}}_{\infty }$ attenuation performance are established. After that, an event-triggered SMC law, which aims at impelling the system trajectories to arrive on the sliding surface, is designed. Furthermore, we take input saturation into account, and specially, in this situation, we propose an adaptive control law of the integral-type sliding surface for the first time. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed results.

Introduction

The intensified research into the event-triggered scheme recently has been driven by the growing need for efficient and intelligent utilization of computation and communication resources in various control applications. The basic principle of an event-triggered scheme is that transmissions of state measurements are subject to a feedback loop and thus occur only when certain pre-designed triggering conditions are met. Nowadays, by taking the event-triggered strategy into consideration, a wide variety of practical systems have observed the expected performance by improvement of event-triggered control approach in conjunction with other control methods, such as predictive control [1], sliding mode control [2], etc. Thus, plenty of noticeable results utilizing the event-triggered strategy can be found in stochastic systems [3], chaotic Lur’e systems [4], T–S fuzzy systems [5], multi-agent systems [6], networked systems [7], [8], [9], singular systems [10], nonlinear systems [11], [12] and filter design [13], [14], [15].

Another attractive research front is sliding mode control (SMC). It has been actively utilized in a wide varieties of complex systems for its obvious advantage of strong robustness [16], especially in regard to parameter variations and exogenous disturbances, for example in [17], [18], [19] and references therein. The purpose of adopting SMC in the design of systems is to force the state trajectories onto a pre-designed subspace, on which requested properties such as stability and disturbance inhibition can be achieved. Meanwhile, a SMC law guides the trajectories to reach the designated sliding surface within a limited time and subsequently keep the trajectories on that surface. Nowadays, SMC has a large amount of significant results including switched systems [20], [21], time-delay systems [22], nonlinear systems [23], fuzzy systems [24], uncertain systems [25] and cyber–physical systems [26]. Meanwhile, recently, input saturation, as a kind of classical nonlinearity, has become one of the most popular research areas especially concerned with SMC approach fault-tolerant framework [27] and state feedback ${\mathcal{H}}_{\infty }$ performance [28]. Some significant results have been proposed, such as [29], [30], [31], [32], [33] and references there in. However, these results are not suitable for integral-type sliding surface, and thus, how to design an adaptive law for this situation is a problem to be solved in this paper.

On yet another research front, Markovian jump systems (MJSs) are of great value in widespread use in practical applications, such as economic systems, fault tolerant systems, aircraft systems etc. MJSs form a category of stochastic system with random changes of system matrices at some discrete time instances. The modes evolution is determined by a Markov process in mathematical expression. MJSs are also a kind of hybrid systems, with each operation mode corresponding to some dynamics and mode transitions governed by a Markov chain. Recently, most leading research mainly focuses on MJSs with incomplete knowledge of transition probabilities [34], stochastic stabilization [35], fault tolerant control [36], passivity analysis [37], [38], filtering [39], anti-windup design [40], and two-dimensional system [41], to name a few.

Up till now, there are some noteworthy results about event-triggered SMC approach. Su et al. proposed an SMC framework under event-trigger mechanism for solving the conundrum in load frequency control in power systems [42]. Behera et al. studied SMC strategy associated with event-triggered scheme to achieve the property of robustness stabilization of nonlinear systems [2]. Wen et al. researched event-triggered SMC method for aperiodic data sampling of fuzzy systems with induced delays [43]. Wu et al. investigated event-triggered SMC for stochastic systems via design of observer and took it into consideration over a series of limited communication networks [3]. Nevertheless, to the best knowledge of the authors, there are few results concentrating on SMC associated with event-triggered scheme and input saturation for MJSs in the literature.

Based on aforementioned retrospects, which are devoted to the inspirations of our current work, we now briefly formulate the main contributions of the paper as follows:

(i) An event-trigger scheme is performed to the control design of considered continuous-time MJSs;

(ii) With the aid of SMC strategy, a set of feasible solutions towards event driven MJSs are obtained to realize stochastic stability as well as ${\mathcal{H}}_{\infty }$ attenuation performance;

(iii) By taking input saturation into account, an adaptive SMC law which is suitable for the integral-type sliding surface designed for continuous-time MJSs is designed.

In this paper, for the considered MJSs, we concentrate on the design of an event-triggered mechanism combined with SMC. Our aim is to obtain a strict sufficient conditions, with reference to the stochastic stability with ${\mathcal{H}}_{\infty }$ disturbance attenuation property, for our considered system. The process of event-triggered SMC design includes: (i) Perform an integral-type sliding surface function under event-trigger scheme with gain matrices to be designed and thus the sliding mode dynamics of considered MJS can be obtained; (ii) Analyze the stochastic stability with disturbance attenuation performance in ${\mathcal{H}}_{\infty }$ sense to derive feasible solution for predesigned sliding surface; (iii) Synthesize an event-triggered SMC law to impel the state trajectories of our concerned system onto the switching surface in a limited time and maintain there in future. (iv) Take input saturation into consideration, then we perform an adaptive controller which is suitable for the integral-type sliding surface of considered MJSs for the first time.

The framework of the rest of this paper is that some prior knowledge about event-trigger scheme are to be performed in Section 2. After that, in Section 3, main results about SMC design under event-triggered mechanism are presented. The sliding mode dynamics is to be obtained at first, then, a strictly feasible condition concerning with desired properties of MJS will be analyzed. Meanwhile, a SMC law is to be synthesized for realizing the control action. After that, in Section 4, an adaptive SMC law for input saturation is to be proposed. Furthermore, in Section 5, two numerical examples are utilized to illustrate the effectiveness of our results in Sections 3 Event-trigger SMC design without controller saturation, 4 Event-trigger SMC design against controller saturation, respectively. Finally, Section 6 concludes the paper.

Notations. We utilize a string of normal notations in the whole paper. The notations ${\mathbb{R}}^n$ and ${\mathbb{R}}^{m\times n}$ denote the $n$-dimensional Euclidean space and the set with all $m\times n$ real matrices, respectively. For a vector $x\in$ ${\mathbb{R}}^n$, $\Vert x\Vert$ represents its Euclidean norm, while for a matrix $A,\Vert A\Vert$ represents its induced norm. Consider a given matrix $A$, the superscript ‘$T$’ and “$-1$” are used to denote matrix transposition $A^T$ and matrix inversion $A^{-1}$, respectively. Meanwhile, $A>0$ represents that $A$ is a real symmetric matrix with positive definiteness. Besides, we define the minimum eigenvalue of matrix $A$ as ${\mu }_{min}\left(A\right)$. Further, for the sake of notation simplicity, the term ${\left[A\right]}_s$ represents matrix addition $A+A^T$. In particular, $I$ and $0$ denote the identity matrix and zero matrix, respectively. $\mathcal{E}\left\{\cdot \right\}$ is the expectation operator. The complete probability space is defined in this paper as $\left(\Omega ,\mathcal{F},\mathcal{P}\right)$, where $\Omega ,\mathcal{F}$ and $\mathcal{P}$ represent the sample space, the $\sigma$-algebra subsets of space $\Omega$ and the probability measure on subsets $\mathcal{F}$, respectively. ${\ell }_2[0,+\infty )$ is used to denote the space of vector functions whose essential characteristics is square integrable. Moreover, $\mathrm{diag}\left(\dots \right)$ is introduced to denote a block-diagonal matrix. Particularly, a star notation ($\star$) stands for the terms of considered block-symmetric matrix which is induced by symmetry.