Asynchronous non-fragile control for persistent dwell-time switched singularly perturbed systems with strict dissipativity

Abstract

This paper focuses on the problem of asynchronous non-fragile dissipativity control for a class of switched singularly perturbed systems (SPSs) governed by the persistent dwell-time (PDT) switching mechanism in the discrete-time context. Unlike some previous results, the modes of system and controller in this paper are assumed to be asynchronized, which conforms better with the practical scenarios. Besides, considering the case that the controllers may be affected by uncertain factors and can not be realized accurately during system operation, the non-fragile mechanism is introduced in the process of controller design to enhance the reliability and security of the SPSs. Based on Lyapunov stability theory and stochastic analysis theory, some sufficient conditions are obtained, which can ensure the exponentially mean-square stable (EMSS) and strict dissipative performance of the closed-loop system. Furthermore, the asynchronous non-fragile slow state variables feedback (SSVF) controller gains are obtained by solving a set of linear matrix inequalities (LMIs). Finally, a numerical example and an inverted pendulum model are applied to demonstrate the superiority and the practicability of the developed control mechanism.

Introduction

It is acknowledged that multiple time-scales phenomena often occur in the dynamical industrial systems and electronic power systems [1], [2], [3]. In fact, these intractable phenomena are mainly due to some unavoidable small parasitic parameters, for example, the-small time constant of the actuator in the process of industrial systems and some small inductance in the circuit models. The existence of these small parameters leads to the coexistence of “slow” and “fast” time scales, which may lead to higher-order dynamic equations and even ill-conditioned mathematical problems. It has been widely proven that the singularly perturbed systems [3] are an effective tool to analyze the multiple time-scales phenomena. Compared with the traditional research methods which ignore these small parasitic parameters, SPSs as an effective method of modeling such systems, regard these parameters as singularly perturbed parameters (SPP) and handle them skillfully, which can naturally overcome the “stiffness” problem to a certain extent. Therefore, a large number of research efforts have been made toward the analysis and synthesis of SPSs including stability analysis [4], [5], sliding control [6], [7], filtering analysis [8], and references therein. The theoretical and practical importance of such systems is one of the motivations for the current study.

The theoretical researches of SPSs have achieved rich results, but because of the complexity of the analysis, most of these results are limited to the study of non-switched SPSs. However, due to device damage, environmental mutation, component connection failure, and other similar factors, the parameters and structure changes of the physical systems are often encountered in practical applications. If the switching phenomenon is ignored, the established model will not conform to the reality. Therefore, it is of great significance to explore a suitable switching mechanism to describe the switching characteristics. Nevertheless, in the existing literature, the switching among subsystems of SPSs is generally considered to be controlled by Markov chain [9], [10] or Semi-Markov chain [11], [12], which is very restrictive since it is too complicated or costly to obtain enough samples of the transitions in practice. For much practical application of SPSs, such as the electric power systems, chemical reactions and reactors systems, a typical switching mechanism is that the subsystem switches intermittently between slow switching and fast switching. Therefore, there is an interesting question: Is there a suitable switching mechanism that can be introduced to describe the switching phenomenon in switched SPSs? Fortunately, the persistent dwell-time switching mechanism proposed by Hespanha in [13] can solve this problem well. Therefore, the study of PDT switched discrete-time SPSs is one of the main motivations of this paper.

Dissipativity is another popular research frontier based on the input-output energy consideration, introduced in [14], [15]. It involves various fundamental knowledge including the bounded real lemma, the circle criterion, and the passivity theorem. The dissipativity problem can be turned into the $H_{\infty }$ performance [16], [17] or passive performance [18], [19] by setting some parameters. Therefore, it is widely used in control systems such as chemical processes, power systems, and electrical networks. Numerous fundamental and classic works on this subject have been published [20], [21], [22], [23], [24] and references therein. For example, a dissipative asynchronous fuzzy sliding mode control issue has been studied for the T-S fuzzy hidden Markov jump systems model in [22]. As for switched stochastic systems, the event-triggering dissipative control has been developed in [23]. By the dissipative knowledge, [24] has concerned with investigating stability conditions for discrete-time Markov jump systems with mixed time delays. According to our knowledge, the dissipative design problem for discrete-time SPSs with PDT switching has not been analyzed, which is another motivation of this paper.

What’s more, there are two shortcomings in some existing methods about SPSs. On the one hand, they ignore the fragility problem according to an implicit assumption that the controller may be exactly implemented. Such an assumption, sometimes, is unavailable owing to the fact that uncertainties or inaccuracies may occur when a designed controller is implemented [25], [26]. On the other hand, the aforementioned SPSs are required that the controller’s switching is synchronized with the subsystem’s switching. However, due to the time delay of mode identification and activation, such restriction is not satisfied in many practical SPSs, one can refer to [27], [28], [29] for more details. Regrettably, the inclusion of non-fragile control and asynchronous switching in the study of SPSs has not been taken into account so far. As a consequence, there are still some interesting questions requiring further investigation for the asynchronous non-fragile control of PDT switched discrete-time SPSs as follows: (1) How to design an asynchronous non-fragile mode-dependent controller when only applying slow state variables for the switched SPSs? (2) How to ensure that the obtained results are $\epsilon$-independent and how to develop a technique to estimate the upper bound of singularly perturbed parameters? (3) Compared with the matrix decoupling method to determine the controller gains in [4], [10], [11], is it possible to propose an improved matrix decoupling method to establish less conservative conditions?

Motivated by the above discussions, we shall study the asynchronous non-fragile control issue for PDT switched SPSs under dissipative performance. In view of the switching characteristics, more and more literature have considered the PDT switching mechanism [30], [31], [32], [33], which is more general than dwell-time (DT) [34] or average dwell-time (ADT) switching [35]. A mode-dependent asynchronous non-fragile SSVF controller is constructed via the slow state variables and asynchronous model. We apply the Lyapunov functions to achieve some $\epsilon$-independent sufficient conditions, which ensure that EMSS and strict dissipative performance of the closed-loop system. The availability of the obtained results is finally illustrated by applying a numerical example and a modified inverted pendulum model. The contributions of this paper are as follows: (1) To accommodate the model characteristics and reduce design conservatism, a mode-dependent and $\epsilon$-dependent non-monotonic Lyapunov function is adopted to obtain controller design algorithms. (2) Unlike some existing results [4], [10], [11], an improved matrix decoupling method is proposed in this paper, which can be served for systems to acquire less conservatism. Moreover, the matrix decoupling methods presented in [4], [10], [11] can be viewed as a special case of our method. (3) The effect of the singularly perturbed parameters upon the system performance is fully addressed, and a technique based on the convex optimization algorithm is developed to estimate the upper bound of singularly perturbed parameters. (4) Compared with the literature [16], [17], [18], [19], the proposed design technique in this paper is more general, which can be transformed into some special cases, including mode-independent, synchronous, $H_{\infty }$, and passive controllers.

The remainder of this work is organized as follows. In Section 2, the system model and some preliminaries are introduced. In Section 3, two main results and two corollaries are given: the exponentially mean-square stable and strict dissipative performance conditions. In Section 4, two examples are provided to demonstrate the effectiveness of the method. Finally, the conclusion of our work is drawn in Section 5.

Notation: Throughout this paper, the mathematic notations are standard. $``{*}^{″}$ represents the term that is induced by symmetry and $l_2[0,\infty )$ is the space of square-summable vector function over $[0,\infty )$. ${\mathbb{R}}^n$ stands for $n$-dimensional real space. $\parallel ·\parallel$ means the usual Euclidean norm. The notation diag $\{·\}$ represents a diagonal matrix. If $A$ is a square matrix, ${\lambda }_{\max }\left\{A\right\}$ $\left({\lambda }_{\min }\left\{A\right\}\right)$ stands for the maximum (minimum) eigenvalue of $A$. $I_n$ (or $0_n$) is the $n\times n$ identity (or zero) matrix. $0_{a\times b}$ denotes the $a\times b$ zero matrix. If $b>a$, $I_{a\times b}$ denotes the matrix $\left[\begin{array}{cc}{I}_a\hfill & 0_{a\times (b-a)}\hfill \end{array}\right]$. If $b=a$, $I_{a\times b}$ denotes the matrix $I_a$. If $b<a$, $I_{a\times b}$ denotes the matrix $\left[\begin{array}{c}{I}_b\\ 0_{(a-b)\times b}\end{array}\right]$. $\mathbb{E}\left\{x\right\}$ denotes the expectation of the stochastic variable $x$. The superscribe $T$ is matrix transposition. The square matrix $X>0$ indicates that matrix $X$ is positive definite.