Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems

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Abstract

This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear impulsive systems in the framework of event-triggered impulsive control (ETIC), where the stabilizing effect of time delays in impulses is fully considered. Some sufficient conditions which can avoid Zeno behavior and guarantee the ISS/iISS property of impulsive systems are proposed, where external inputs are considered in both the continuous dynamics and impulsive dynamics. A novel event-triggered delayed impulsive control (ETDIC) strategy which establishes a relationship among event-triggered parameters, impulse strength and time delays in impulses is presented. It is shown that time delays in impulses can contribute to the stabilization of impulsive systems in ISS/iISS sense. Finally, the effectiveness of the proposed theoretical results is illustrated by two numerical examples.

Introduction

The theory of ISS/iISS plays a key role in modern nonlinear control community, especially in analysis of large-scale networks, design of nonlinear observers and robust stabilization of nonlinear systems, see [1], [2], [3]. Briefly speaking, system with ISS/iISS property is bounded under bounded perturbation, and can be globally asymptotically stable (GAS) without disturbance. Due to the suitability in characterizing the effects of exogenous disturbances on systems, ISS/iISS has wide application in many fields (see [4], [5], [6]) and many valuable results have been reported about it (see [7], [8], [9], [10], [11], [12]). Initially, the concept of ISS/iISS is proposed by Sontag in [9], [10] for continuous-time systems, then it is extended to discrete-time systems [7], networked control systems [8], switched systems [11], impulsive systems [12], and so on (see [13], [14], [15]).

Especially, impulsive system as an important class of hybrid systems has received considerable attention in the past few years, see [16], [17], [18]. From the view of impulsive effects, the research on stability of impulsive systems includes two aspects: impulsive perturbation and impulsive control. The study of impulse perturbation usually considers the robustness of system including destabilizing impulses, while impulsive control focuses on the stabilization of system involving stabilizing impulses. As an effective discontinuous control approach, impulsive control has attracted increasing attention due to its advantages of simple structure, low control cost and wide application. Some interesting results on ISS/iISS property of impulsive systems based on impulsive control have been reported, see [12], [13], [19]. However, it can be observed that the impulsive time sequences in most of these results are determined artificially, which is essentially a time-triggered mechanism. It may leads to unnecessary control waste and resources consumption in the process of impulsive information transmission.

Recently, event-triggered control (ETC) has attracted more and more attention due to its superiority in avoiding unnecessary resources waste. Compared with the traditional time-triggered control, impulse signal under ETC is updated only when the well-designed event-triggered mechanism (ETM) is activated. It is actually a class of state-dependent impulses, which can effectively reduce control cost. Some related ETC strategies have been presented, see [20], [21], [22], [23]. However, ETC approach still has some limitations, for example, under such control strategies the state information of sampling instants should be transmitted all the time until it is updated by the next event-trigger. In fact, it is unnecessary and difficult to be achieved in real applications. More recently, event-triggered impulsive control (ETIC) as a combination of ETC and impulsive control has received increasing attention. Different from ETC method, control input in ETIC is only transmitted at triggering instants and there is no need for any information transmission between two consecutive triggering instants, which can reduce the communication resources further. Although ETIC method has obtained flourishing development in the past five years, the results on ISS/iISS under ETIC are still less, see [24], [25], [26], [27]. For instance, authors in [24] investigated ISS problem for a class of state unmeasurable nonlinear systems by designing observer-based ETIC strategies. Then, some Lyapunov-based criteria were established via ETIC in [25], [26], which can effectively avoid Zeno behavior and guarantee ISS/iISS property of nonlinear impulsive systems. However, in these literatures, the possibility of time delays in impulses is ignored, in which the impulsive jump of state depends on the current state information. In fact, the current state information is not always available in real problems since that time delays are unavoidable in sampling and transmission of the impulsive information, such as the sampling delays in communication security systems based on impulsive control and the time delays in animal husbandry or fishery industry involving impulsive harvesting and stocking, etc. Recently, some results taking time delays in impulses into account have been reported, for example, authors in [27] studied the stabilization to ISS for nonlinear system by delayed ETIC, where ETM involves three levels of events. Whereas, these results only focused on the negative effect of time delays in impulses, which was actually the robustness analysis of systems with respect to small delays in impulses. Moreover, the possibility of disturbance input in discrete dynamics was ignored. As we all know, time delays in impulses have double effects on system dynamics, that is, positive effect and negative effect. But until now, as far as we know, there are rare results about ISS/iISS problem based on ETIC involving stabilizing delays in impulses. It is still an open problem and a meaningful challenge.

Motivated by above discussions, this paper is to investigate the ISS/iISS property of nonlinear impulsive systems based on ETIC strategy involving stabilizing delays in impulses, i.e., event-triggered delayed impulsive control (ETDIC) strategy. With the help of Lyapunov method and impulsive control theory, some sufficient conditions are proposed to guarantee non-Zeno behavior and the ISS/iISS property via ETDIC. Compared with the existing results, our contribution is twofold:

  • In the framework of ETDIC, several Lyapunov-based conditions for non-Zeno behavior and ISS/iISS are provided, where exogenous inputs are considered in both the continuous dynamics and impulsive dynamics;

  • The information of time delays in impulses is fully fetched and integrated into the dynamic analysis of the system. A relationship among event-triggered parameters, impulse strength and time delays in impulses is established, which fully illustrates the stabilizing effect of time delays in impulses on ISS/iISS.

The remainder of this paper is organized as follows. In Section 2, the problem is formulated and some preliminary concepts are given. The main results are presented in Section 3. In Section 4, an application is provided. Then two numerical examples are given in Section 5, and conclusions are given in Section 6.

Notation

Let Z+, R+ and R denote the set of positive integers, positive real numbers, and real numbers, respectively. Rn is the n-dimensional real spaces equipped with the Euclidean norm ||, and ||H denotes the supremum norm over the interval H. The notation is used to denote the symmetric block in one symmetric matrix. I is an identity matrix of appropriate dimension. The notation AT and A1 denote the transpose and the inverse of A, respectively. A continuous function α:RR+ is said to be of class K if it is strictly increasing and α(0)=0. Especially, if α is unbounded, it is of class K. A function β:R+×R+R+ is said to be of class KL if for each fixed t, the mapping β(s,t) is of class K, and for each fixed s, it is decreasing to zero as t. ab and ab are the maximum and minimum of a and b, respectively.

Section snippets

Preliminaries

Consider the following impulsive system ẋ(t)=f(x(t),w(t)),ttk,tt0,x(t)=gk(x(r),w(r)),t=tk,r=rk{tk}θ,where kZ+, x(t)Rn is the system state, ẋ denotes the right-hand derivative of x(t), w(t)Rm is a measurable bounded exogenous disturbance input, continuous function f and g:Rn×RmRn satisfy some suitable conditions such that system (1) exists unique solution when tt0 (see [28], [29]), and assume that f(0,0)=gk(0,0)=0 such that system (1) can admit a zero solution. The time sequence {tk,

Main results

In this section, considering the effect of time delays in impulses, some sufficient conditions are established for non-Zeno behavior, ISS and iISS properties of system (1) in the framework of ETDIC approach, respectively. First, consider the following ETM: tk=min{tk,tk1+Δk},tk=inf{ttk1:V(x(t))exp(ak)V(x(tk1))+exp(bk)φ(|w|[t0,t])}, where kZ+, φK, V(x(t)) denotes the Lyapunov function depending on the state trajectory x(t) of system (1) at time t. The forced impulse parameter ΔkR+ and

Application

As an application, we consider the following nonlinear control system with exogenous disturbance input: ẋ(t)=Ax(t)+Bf(t)+Cw(t),tt0,subjects to impulses x(t)=Dx(r),t=tk,r=rk{tk}θ,where x(t)Rn, f():RnRn is a global Lipschitz continuous function with Lipschitz matrix L, w(t)Rm is locally bounded exogenous disturbance input, A, BRn×n, CRn×m are pre-given real matrices, {tk} is the triggered impulse sequence, r=rk{tk}θ with θ(0,1], and DRn×n is the impulse gain matrix. Our purpose is

Examples

Example 1

Consider the following delayed impulsive system: ẋ(t)=0.1x(t)cos(x(t))+w(t),ttk,t0,x(t)=exp(0.2)x(r),t=tk,r=rk{tk}θ,where delayed parameter θ=0.01, w(t)=0.1|sin(t)| is the bounded exogenous input, and {tk} denotes the impulse sequence to be designed later. As is shown in Fig. 1 (red curve), when there is no control input (i.e., u(t)=0), the corresponding system is non-GAS without exogenous disturbance (i.e., w(t)=0). It implies that system (23) without control input is not ISS. In the

Conclusion

This paper investigated the ISS/iISS property of nonlinear impulsive systems based on ETDIC strategy, where external inputs are considered in both the continuous dynamics and impulsive dynamics. The proposed results in this paper include a relationship among ETM, impulse strength and time delays in impulses, which fully reflects the positive effect of time delays in impulses. Future work would be further exploration for ETIC approaches involving discrete delays in impulses. Moreover,

CRediT authorship contribution statement

Mingzhu Wang: Data curation, Writing – original draft, Simulation. Peng Li: Software, Visualization, Validation. Xiaodi Li: Methodology, Supervision.

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.

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This work was supported by National Natural Science Foundation of China (62173215), Major Basic Research Program of the Natural Science Foundation of Shandong Province in China (ZR2021ZD04, ZR2020ZD24), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (2019KJI008).

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