Research Article  Open Access
Guoqi Ma, Linlin Qin, Xinghua Liu, Gang Wu, "ObserverBased EventTriggered Control for Switched Systems with TimeVarying Delay and NormBounded Disturbance", Mathematical Problems in Engineering, vol. 2016, Article ID 7920394, 14 pages, 2016. https://doi.org/10.1155/2016/7920394
ObserverBased EventTriggered Control for Switched Systems with TimeVarying Delay and NormBounded Disturbance
Abstract
This paper is concerned with the problem of observedbased eventtriggered control for switched linear systems with timevarying delay and exogenous disturbance. First by employing a state observer, an observerbased eventtriggered controller is designed to guarantee the finitetime boundedness and finitetime stabilization of the resulting dynamic augmented closedloop system. Then based on the Lyapunovlike function method and the average dwell time technique, some sufficient conditions are given to ensure the finitetime boundedness and finitetime 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 continuoustime dynamics and discretetime 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 finitetime stability was proposed in [20]. To be specific, a system is said to be finitetime 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 finitetime stability is more practically meaningful compared with Lyapunov asymptotic stability. For more related results on finitetime stability, interested readers can be referred to [6, 14, 21–23] and references cited therein. Moreover, timedelay 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 realtime 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 welldeveloped sampleddata 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 sampleddata control paradigm, eventtriggered control (ETC), also called eventbased control or eventdriven control in the literatures, has been proposed; see, for example, [34–36]. In the eventtriggered control framework, the control task will not get updated until an external event occurs, generated in light of some prescribed eventtriggered mechanism, rather than according to the elapse of a certain fixed time period as utilized in the conventional periodic sampleddata 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 closedloop performance [37]. Since the early seminal works on eventtriggered control [34–36], several different eventtriggered mechanisms and control schemes have been proposed, and quite a few theoretical results have appeared that investigate the eventtriggered 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 eventtriggered model predictive control for discretetime linear systems are presented in [40]; later in [41], the problem of eventtriggered model predictive control for continuoustime nonlinear systems subject to bounded disturbances is dealt with. The contribution of [35] is that a novel eventtriggered 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 eventtriggered PID controller presented in [35], whereas it should be pointed out that most of the previous results are concerned with the statefeedback 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 eventtriggered output feedback control strategies, some results of which can be found in [42].
Moreover, it should be noted that, among the existing literatures on eventtriggered control, most results are focused on linear systems, while the problem of eventtriggered control for switched delay systems with exogenous disturbance has not been yet addressed, which motivates the current study. In this paper, the problem of observerbased eventtriggered control for continuoustime switched linear systems with timevarying delay and normbounded disturbance is investigated, and we opt for a continuoustime eventtrigger 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 observerbased eventtriggered controller is designed to guarantee the finitetime boundedness and finitetime stabilization of the resulting closedloop system. The main contributions of this paper lie in the following. (i) The eventtriggered control scheme is firstly applied to switched systems subject to timevarying delay and normbounded exogenous disturbance. (ii) Sufficient conditions for finitetime boundedness and finitetime 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 eventtriggering 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 finitetime boundedness, finitetime 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 (positivesemi, negative, negativesemi) 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 eventtriggered control system considered in this paper can be divided into the following three modules: (i) the physical plant and observer; (ii) the eventtrigger; (iii) the eventtriggered controller.
2.1. Physical Plant and Observer
Consider a continuoustime switched linear system with timevarying delay and exogenous disturbance described bywhere is the system state vector, represents the control input vector, and denotes the output vector. is the timevarying exogenous disturbance input satisfying . denotes the timevarying delay satisfying . is a continuous vectorvalued initial function on which specifies the initial values of states of system (1). are all wellknown constant system matrices with appropriate dimensions, and the pairs and are controllable and observable, respectively. is the switching signal which is a piecewise constant function and specifies which subsystem functions at a certain time instant. Corresponding to the switching signal , we have the following switching sequence:where represent the initial time instant and state, respectively, and . When , the subsystem is activated.
For the switched linear system (1), the purpose of this paper is to design the Luenberger observer, which can generate the accurate state estimates of system (1) by using the measured output . Then, an eventtriggered controller will be synthesized based on the estimates to stabilize the resulting closedloop system.
The Luenberger observer for system (1) is introduced as follows: where is the observer state and is the observer gain to be appropriately designed.
Assumption 1. The state trajectories of system (1) do not jump at switching instants; that is, the trajectories of are everywhere continuous.
2.2. EventTrigger and EventTriggered Controller
The eventtrigger detects the eventtriggered condition to determine whether an “event” is generated or not. Once an event happens, the eventtrigger will transmit the latest state estimates to the eventtriggered controller. Specifically, we have the following eventtrigger instant sequence:with . Without loss of generality, we assume that the first event happens at time instant and . With the estimates of system states sampled at time instant , the next sampling instant can be determined by the eventtrigger, and in this paper, we focus on the continuoustime eventtrigger which is given bywhere and is the exponentially decreasing event threshold with given parameters , , and .
In this paper, the controller is eventtriggered, and the control inputs are determined on the basis of the sampled observer states, which can be given by where is the feedback gain with appropriate dimension. Note that, in the eventtriggered control scenario, at sampling time instant , the controller in (6) will receive the sampled estimated states , which will be held constant until next event is generated at time instant . Thereby, the control input in (6) is only updated at the sampling instants. Hence, a zeroorder holder (ZOH) is equipped in the system in order to keep the control signal continuous.
Before proceeding with the main results, we will first recall the following definitions and lemmas, which play an important role in the proof of the main results.
Definition 2 (see [43]). Given time instants and such that , let denote the switching number of over , ifholds for and an integer ; then is called an average dwell time.
Definition 3 (see [44]). Given four positive constants , and such that , a matrix , and a switching signal . Ifholds, then system (1) with is said to be finitetime bounded with respect to . If (8) holds for any switching signal , system (1) with is said to be uniformly finitetime bounded with respect to .
Definition 4 (see [44]). Given four positive constants , and , such that , a matrix , and a switching signal . Ifholds, then system (1) with observerbased control (6) is said to be finitetime stabilizable with respect to .
Lemma 5 (see [45]). Let be a symmetric matrix, and let ; then the following inequality holds:
Lemma 6 ((Schur complement), see [46]). Given symmetric matrix , where , then the following three conditions are equivalent: ; ; .
Lemma 7 (see [47]). Assume is Hurwitz; then there exists a positive scalar such thatwhere .
3. Main Results
3.1. Modeling of ClosedLoop System
Define the estimation error . For the time period , the plant and error systems can be rewritten as follows:where , andDefine ; then we have the following augmented closedloop system:where
3.2. Time Interval Partition
From (2) and (4), we know that there exist two different “switching signals” which partition the time axis into different consecutive tiny intervals, which makes the problem of eventtriggered control for switched systems more interesting and challenging. To address this difficulty, it is desirable to partition the time axis afresh to obtain the “switching instants” denoted by . As depicted in Figure 2, when , that is, the th subsystem is activated, and it is assumed that is the latest eventtrigger instant before ; obviously, there are two different cases of the location of , as illustrated in Figures 2(a) and 2(b), respectively. Figure 2(a) shows that ; consequently, for the interval , the control input ; hence, for this interval, we can utilize the welldeveloped analysis approach of eventtriggered control for linear systems. Then we can set and . Accordingly, we have and for Figure 2(b). The procedure of determining is given in Algorithm 1.

(a)
(b)
Remark 8. On the basis of the above discuss, we can conclude that the set of the “switching instants” , if the subsystem switching instants and eventtrigger instants do not occur simultaneously. In the sequel, we denote by the value of “switching instants” at time instant .
3.3. FiniteTime Boundedness Analysis
The following theorem indicates that the finitetime boundedness of the closedloop system (14) can be guaranteed by using the abovementioned eventtriggered control scheme.
Theorem 9. For any , suppose that there exist matrices , let , and there exist constants such thatIf the average dwell time of the “switching signal” satisfiesthen system (14) is finitetime bounded with respect to , where , , , and
Proof. Choose the following Lyapunovlike function:Taking the derivative of with respect to along the trajectory of system (14) readsAccording to (16), we havewhich is equivalent toto continueIntegrating (25) from to givesSuppose at the instant . Therefore, if the condition (17) holds, thenHence, we can further obtainwhere denotes the “switching” number of over , which means ; therefore, it can be obtained thatOn the other hand,Combining (29), (30), and (31), it can be deduced thatIf , then according to (18), we haveIf , according to (19), one can getSubstituting (34) to (32) yieldsThen according to Definition 3, we can conclude that the finitetime boundedness of the resulting closedloop system (14) can be guaranteed; thus, this proof is completed.
Remark 10. In Theorem 9, the meaning of average dwell time is somewhat different from the commonly used, due to the complexity of the problem considered here. Theorem 9 shows that, in order to guarantee the finitetime bounded quality of eventtriggered switched control systems, the switching frequency of subsystems and eventtrigger should have an upper bound over a finitetime interval.
Now, based on Theorem 9, the following corollary is obtained to give some conditions which guarantee uniform finitetime boundedness of system (14).
Corollary 11. For any , let , suppose there exist matrices , and constants such thatand then according to Definition 3, system (14) is uniformly finitetime bounded with respect to .
Proof. Choose a Lyapunovlike function:Substituting with into the proof procedure of Theorem 9, it is easy to get the conclusion.
3.4. FiniteTime Stabilization Analysis
Next, let us consider the finitetime stabilization problem for system (14) based on the above results. The following theorem gives the conditions that can guarantee system (14) is finitetime stabilizable.
Theorem 12. Given positive constants , a positive definite matrix , system (14) is finitetime stabilizable with respect to , if the average dwell time of the “switching signal” satisfies (19), and the feedback gain is selected according to , the observer gain is selected according to , and for any , there exist matrices , , , , , and constant such thatwhere
Proof. If condition (38) holds, then substituting , into (16) and in view of Lemma 6, we can getwhereThe rest proof procedure is similar to that of Theorem 9, and due to limited space, we will not cover that again; thus, this proof is completed.
Remark 13. In the proofs of Theorems 9 and 12, it is not required that , which is required for asymptotical stability of switched systems via a common Lyapunov function. However, conditions (18) and (19), which are essential for finitetime stabilization of switched systems, are not needed for asymptotical stability. Moreover, the switched system concerned in this paper is a certain dynamical system, whereas some valuable results for uncertain dynamical systems are presented in chapter 9 of [48], where the problem of absolute parameter stability for uncertain singular perturbation systems is thoroughly investigated and several different sorts of Lyapunov functions are constructed.
3.5. The Minimum Interevent Interval
To exclude the Zeno behavior of sampling, it is needed to show there always exists a nonzero lower bound of the minimum interevent interval. The following theorem shows that there exists a positive lower bound of the minimum interevent interval.
Theorem 14. With the sampling instants determined by (5), the minimum interevent interval is lower bounded by a positive scalar.
Proof. For any , let be a sampling instant. Then, in the time interval , is constant. In light of the definition of in (12), for , we havewhich yields thatMoreover, noticing (13), we havewhich leads to