Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7920394, 14 pages

http://dx.doi.org/10.1155/2016/7920394

## Observer-Based Event-Triggered Control for Switched Systems with Time-Varying Delay and Norm-Bounded Disturbance

Department of Automation, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China

Received 31 August 2015; Accepted 2 March 2016

Academic Editor: Naohisa Otsuka

Copyright © 2016 Guoqi Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the problem of observed-based event-triggered control for switched linear systems with time-varying delay and exogenous disturbance. First by employing a state observer, an observer-based event-triggered controller is designed to guarantee the finite-time boundedness and finite-time stabilization of the resulting dynamic augmented closed-loop system. Then based on the Lyapunov-like function method and the average dwell time technique, some sufficient conditions are given to ensure the finite-time boundedness and finite-time stabilization, respectively. Furthermore, the lower bound of the minimum interevent interval is proved to be positive, which thus excludes the Zeno behavior of sampling. A numerical example is finally exploited to verify the effectiveness and potential of the achieved control scheme.

#### 1. Introduction

Broadly speaking, hybrid systems are such a class of systems where continuous-time dynamics and discrete-time dynamics interact. Switched systems, consisting of a finite number of subsystems described by differential or difference equations and a logical rule orchestrating the switching order among these subsystems, can be regarded as a special category of hybrid systems. In recent decades, numerous efforts have been devoted to the study of switched systems due to their intrinsic characteristic and their practical applications in a wide range of areas, for example, power electronics [1], networked control systems [2], robot control systems [3], air traffic control systems [4], and modern agriculture systems [5], to list a few. In addition, along the line of theoretical research, a great deal of valuable results on switched systems have been presented, such as stability and stabilization [6–14], controllability and reachability [15], and observability [16, 17]. In general, the stability and stabilization problems are the principal concerns for switched systems. Currently, most of the existing literatures on stability and stabilization of switched systems are focused on Lyapunov asymptotic stability, which is defined over an infinite time interval. Nevertheless, in practice, one may be only interested in a bound of system trajectories over a fixed short time interval, as there may exist such a case that a system is Lyapunov stable but completely of no practical use if it possesses undesirable transient performances, such as the systems with saturation elements [18, 19]. In order to study the transient performances of a system, the concept of finite-time stability was proposed in [20]. To be specific, a system is said to be finite-time stable if, given a bound on the initial state condition, the system state trajectories stay within a prescribed range during a fixed time interval. Hence, the finite-time stability is more practically meaningful compared with Lyapunov asymptotic stability. For more related results on finite-time stability, interested readers can be referred to [6, 14, 21–23] and references cited therein. Moreover, time-delay is a common phenomenon arising in various practical applications, for example, networked control systems, chemical engineering systems, and power systems [24–28]. And time delays are the inherent characteristics of a large number of physical plants and the big sources of instability and poor performances for switched systems [29] as well. Therefore, it is nontrivial to investigate the control problem for switched systems with time delays.

On the other hand, in quite a lot of modern industrial control applications, in order to run real-time operating systems, the controllers are usually implemented on digital platforms which are furnished with microprocessors [30]. In such an implementation, the control task comprises sampling the plant outputs, computing and implementing new control signals. Traditionally, the control task is executed periodically on the basis of the well-developed sampled-data system theory from an analysis and design point of view [31, 32]. Nonetheless, it is well worth pointing out that the aforementioned control strategy is conservative from a resource utilization point of view: that is, sampling at a constant rate regardless of whether it is really necessary or not will result in a waste of communication resource when no disturbances are influencing the system and the system is approaching its desired equilibrium [33]. To address the problems arising from the periodic control scenario, an alternative to sampled-data control paradigm, event-triggered control (ETC), also called event-based control or event-driven control in the literatures, has been proposed; see, for example, [34–36]. In the event-triggered control framework, the control task will not get updated until an external event occurs, generated in light of some prescribed event-triggered mechanism, rather than according to the elapse of a certain fixed time period as utilized in the conventional periodic sampled-data control strategy. Consequently, the number of control task executions and the communication frequency between the sensors and the controllers can be dramatically reduced while guaranteeing a satisfactory closed-loop performance [37]. Since the early seminal works on event-triggered control [34–36], several different event-triggered mechanisms and control schemes have been proposed, and quite a few theoretical results have appeared that investigate the event-triggered control systems; see, for example, [38, 39] and references cited therein. Over the past few decades, a wealth of constructive complementary contributions have been made toward this interesting topic; to mention a few, the approaches of event-triggered model predictive control for discrete-time linear systems are presented in [40]; later in [41], the problem of event-triggered model predictive control for continuous-time nonlinear systems subject to bounded disturbances is dealt with. The contribution of [35] is that a novel event-triggered PID controller is designed, and the PID controller does not compute the control input until the change of the measurement signal is large enough. The work of [42] proposes some improvements of the event-triggered PID controller presented in [35], whereas it should be pointed out that most of the previous results are concerned with the state-feedback control schemes, which is on the presupposition that all states of the plant can be measured. However, in many control applications full state measurements are not always available for feedback; therefore, in such cases, it is of great significance to investigate the event-triggered output feedback control strategies, some results of which can be found in [42].

Moreover, it should be noted that, among the existing literatures on event-triggered control, most results are focused on linear systems, while the problem of event-triggered control for switched delay systems with exogenous disturbance has not been yet addressed, which motivates the current study. In this paper, the problem of observer-based event-triggered control for continuous-time switched linear systems with time-varying delay and norm-bounded disturbance is investigated, and we opt for a continuous-time event-trigger to “observe” the “event,” defined as some error signals exceeding a given threshold, to determine the updating of the controller. Besides, we utilize the state observer to generate the state estimates; then the observer-based event-triggered controller is designed to guarantee the finite-time boundedness and finite-time stabilization of the resulting closed-loop system. The main contributions of this paper lie in the following. (i) The event-triggered control scheme is firstly applied to switched systems subject to time-varying delay and norm-bounded exogenous disturbance. (ii) Sufficient conditions for finite-time boundedness and finite-time stabilization of switched delay systems are given. Besides, the analysis on minimum interevent interval is performed, and the lower bound of the minimum interevent interval is obtained. (iii) Combining event-triggering signal and subsystem switching signal together, a time interval partition algorithm is presented.

The rest of this paper is organized as follows: Section 2 contains the problem statement and preliminaries; Section 3 presents finite-time boundedness, finite-time stabilization, and minimum interevent interval performance for switched delay systems; Section 4 provides a numerical example to verify the effectiveness of the proposed results; concluding remarks are given in Section 5.

*Notations.* The following notations are used throughout the paper. represents the set of natural numbers, stands for positive integer, denotes the dimensional Euclidean space, and is the set of all matrices. For any real number , denotes the integer part of . (, , ), where and are both symmetric matrices, meaning that is positive (positive-semi, negative, negative-semi) definite. The identity matrix of order is denoted as (or, simply, if no confusion arises). The superscript “” is used to stand for matrix transposition. For a symmetric block matrix, we use to denote the terms introduced by symmetry. For any symmetric matrix , and denote the maximum and minimum eigenvalues of matrix , respectively. is the Euclidean norm of vector , , while is spectral norm of matrix , . Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

#### 2. Problem Statement and Preliminaries

As shown in Figure 1, the event-triggered control system considered in this paper can be divided into the following three modules: (i) the physical plant and observer; (ii) the event-trigger; (iii) the event-triggered controller.