Input-to-state stabilization of time-delay systems: An event-triggered hybrid approach with delay-dependent impulses

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Abstract

This paper studies the input-to-state stabilization problem of nonlinear time-delay systems. A novel event-triggered hybrid controller is proposed, where feedback controller and distributed-delayed impulsive controller are taken into account. By using the Lyapunov-Krasovskii method, sufficient conditions for input-to-state stability are constructed under the designed event-triggered hybrid controller, the relation among control parameters, threshold parameter of the event-triggered mechanism and time delay in the impulsive signals is derived. Compared with the existing results, the obtained input-to-state stability criteria are applicable to time-delay systems with stabilizing delay-dependent impulsive effects and destabilizing ones. Numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Introduction

Impulsive systems, which consists of continuous dynamical evolution and discrete state changes, have attracted continued high research interests during the past decades because they provide a natural framework for mathematical modeling of many real world processes [1]. Such systems often arise from certain control techniques. In particular, a discrete control method known as impulsive control is proved to be a powerful tool in stabilization of a large class of dynamical systems [2], and has been successfully applied in a variety of applications, such as spacecraft maneuvers [3], multi-group formation tracking [4], pest control [5], etc. Recently, since inevitability of time delays in the sampling and transmission of state information [6], [7], [8], [9], lots of researchers have paid attention to stability analysis and control problems of nonlinear systems with delay-dependent impulses (see, e.g., [10], [11], [12], [13]).

Since its first introduction by Sontag [14], the concept of input-to-state stability (ISS), has been proved to be useful to the stability analysis and stabilizing control of dynamical systems. The ISS property characterizes the effects of external inputs to the dynamical behaviors of control systems, which means that no matter what size of the initial states are, the system states will remain bounded under the bounded external inputs, and tend to be asymptotically stable when the external inputs tend to zero. Recently, the ISS properties have been wildly investigated for various types of dynamical systems. Particularly, the ISS properties of impulsive systems has been extensively investigated in the literature (see [15], [16], [17], [18], [19] and their reference therein). In [15], the notions of ISS and integral ISS are first introduced for nonlinear (delay-free) impulsive systems. Extensions of the work of [15] are presented for time-delay impulsive systems in [16], where Razumikhin-type theorems are established for ISS properties of considered systems with an appropriate dwell-time condition. More recently in [18], a novel exponential ISS-Lyapunov function is constructed, Lyapunov-like sufficient conditions are provided for ISS of impulsive systems with multiple jump maps. The ISS and integral ISS for nonlinear (delay-free) systems with delayed impulse are studied in [17], where the stabilizing continuous dynamics are required. By use of Lyapunov functional method, the work of [19] further generalizes the results for ISS of time-delay systems subject to delay-dependent impulses, and sufficient conditions are constructed for three cases: ISS continuous dynamics with destabilizing impulses, unstable continuous dynamics with stabilizing impulses in sense of ISS, and ISS continuous dynamics with stabilizing impulses.

Recently, since the advantage of saving unnecessary usage of communication resource, lots of researchers have paid attention to the event-triggered control and its potential applications in various kinds of control problems [20], [21]. Different from the time-triggered control, the event-triggered control emphasizes that the execution of control updates are triggered by a predefined condition. Recently, a wealth of interesting event-triggered control strategies have been reported in the literature, such as distributed event-triggered control [22], self-triggered control [23], dynamic event-triggered control [24], etc., and various of control strategies are cooperated with these event-triggered mechanisms for the stability analysis of dynamical systems. Among them, the event-triggered impulsive control, which integrates the event-triggered mechanism with the impulsive control, is a powerful method to improve the resource usage for the involving systems with limited bandwidth resource. Along this line, some related results are reported in the literature [25], [26], [27], [28]. However, these results are only applicable for some specific systems and some drawbacks exist in these results. In [25], a novel event-triggered impulsive control scheme with three level of events is designed for ISS property of dynamical systems, where the upper bound of two event triggers are restricted by a forced triggered period Δ. In [26], consensus problems of leader-following multi-agent systems are solved based on distributed event-triggered impulsive control. However, the sufficient conditions depending on the whole triggered interval distributions, which is very strict and hard to implement. Synchronization problems of memristive neural networks are studied by event-triggered impulsive control in [27], where the results are only applicable for systems with stable continuous dynamics with delay-free impulses. Although these interesting results for event-triggered impulsive control have been reported, the event-triggered impulsive control with delay-dependent impulses, is seldom considered. In fact, since the inevitability of time delays in the sampling and transmission of impulsive information, it is reasonable to consider time delays in the impulsive controller. Especially, in many practical applications, distributed-delayed impulsive control is perceived as a better way to deal with the practical problems. For example, in financial analysis such as advertising input, the impulsive advertising input is not depend on the advertising effect of a certain moment, but the effect during a history time period, i.e., the distributed-delayed impulsive control should be applied in this application. Therefore, it is highly desirable to investigate the event-triggered distributed-delayed impulsive control for dynamical systems.

Motivated by the aforementioned discussion, the objective of this paper focuses on the ISS problem of nonlinear time-delay systems by event-triggered delay-dependent impulsive control. By using the method of Lyapunov-Krasovskii functionals, sufficient conditions for ISS of nonlinear time-delay systems are derived. The main contributions of this paper are given in the following four aspects: (1) A novel event-triggered hybrid control algorithm, which consists of the feedback controller and the distributed-delayed impulsive controller, is proposed. The hybrid controller is more efficient and robust with regard to the disturbance signals. Especially, if one controller is unserviceable or fails to work effectively, the other one would still be effective in the control of systems; (2) The designed discrete event-triggered mechanism only needs monitor the system states and corresponding measurements at discrete instants, under which the Zeno behavior is ruled out, and the communication resources could be further saved for ISS property of considered systems by the event-triggered impulsive control algorithm; (3) The relation among control gains, bounds of triggering intervals, threshold parameter and delay size of impulses is derived in the obtained results, which is beneficial to the analysis and co-design of controller and event-triggered mechanism; (4) Based on the method of Lyapunov-Krasovskii functionals and the idea of hybrid controller, the ISS criteria are constructed for nonlinear time-delay systems with stabilizing delay-dependent impulsive effects and destabilizing ones, respectively. The remainder of this paper is as follows. Section 2 formulates the ISS problem of time-delay systems. In Section 3, the main results are presented. Numerical examples are given in Section 4, and conclusions are given in Section 5.

Section snippets

Problem formulation and preliminaries

Let N+ be the set of positive integers, R the set of real numbers, R+ the set of nonnegative real numbers. Rn denote the n-dimensional real space equipped with the Euclidean norm denoted by ·. Rn×n denotes the n×n real matrices. P0(P0) means that matrix P is symmetric and semi-positive (semi-negative) definite. In denotes the n×n real identity matrix. The notation T denotes the transpose of a matrix or a vector. λmin(·) and λmax(·) denote the minimum and the maximum eigenvalue of the

Main results

In this section, the ISS property for time-delay system (3) will be studied. First, by employing the Lyapunov-Krasovskii method, two necessary propositions are proposed to show the continuous dynamics and the impulsive effects of the system, respectively. After that, sufficient conditions are presented in Theorem 1 to ensure ISS property of system (3) via the designed event-triggered hybrid control algorithm. Moreover, two corollaries are derived for the case that the feedback controller fails

Numerical example

In this section, two representative examples will be presented to demonstrate our main results.

Example 1

Consider the time-delay system (1) with n=2, t0=0, τ(t)=1+0.3sin(2t), f(x)=g(x)=tanh(x), A=I,B=(1.620.10.212.3),F=(2.210.144.73.2),D=(0.8000.5),H=(1.51.25).

So we can obtain τ1=0.7, τ2=1.3, η=0.6, and f(x) is Lipschitz continuous with K1=K2=I. Set the initial value φ(s)=[2,0.5]T for s[1.3,0]. Under the given parameters, the time-delay system (1) exhibits chaotic behaviors with the external

Conclusion

This paper has analyzed the ISS problem of nonlinear time-delay systems by event-triggered control. The event-triggered hybrid controller that takes into account both feedback controller and distributed-delayed impulsive controller has been constructed. By utilizing the Lyapunov-Krasovskii functional method and the event-triggered hybrid control algorithms, some ISS criteria are derived for time-delay systems with stabilizing delay-dependent impulsive effects and destabilizing ones. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank Yuan Shen, with the Department of Applied Mathematics, University of Waterloo, for his support, and Dr. Kexue Zhang, with the Department of Mathematics and Statistics, University of Calgary, for his helpful comments. This work is supported by the National Natural Science Foundation of China (61673247), the Research Fund for Distinguished Young Scholars of Shandong Province (JQ201719), the Australian Research Council (DP160102819), and the NSERC Canada.

References (33)

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    The validity and superiority of the proposed stability criteria have been displayed by discussing two example simulations in details. In future work, the fuzzy quantized sampled-data control strategy will be taken into account for nonlinear time-delay systems [45,46], and the improved inequality technique will be extended to stability analysis of nonlinear impulsive systems with event-triggered control [47–50]. Furthermore, since the underlying descriptor Markov jump systems [51–53] have extensively applications in the real world, applying the theoretical results proposed and the approaches utilized in this paper to further discuss the HMM-based asynchronous control issue for descriptor Markov jump systems is also part of our future research efforts.

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