A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems

Abstract

This paper investigates the dwell-time dependent stability for impulsive systems by employing a new Lyapunov-like functional that is of the second order in time t. In contrary to those built on [t_k, t], a part of the impulsive interval $[t_k,t_{k+1}]$, the Lyapunov-like functional is two-sided in the sense of employing the system information on $[t,t_{k+1}]$ as well as [t_k, t]. To deal with the derivative of the two-sided Lyapunov-like functional, which involves integrals of the state and integrals coupled by $[t,t_{k+1}]$ and [t_k, t], integral equations of the impulsive systems are introduced and an advanced inequality is employed. By the Lyapunov-like functional theory, new dwell-time dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results turn out to be less conservative than some existing ones, which is illustrated by numerical examples.

Introduction

An impulsive system is modeled after many real world processes that undergo abrupt state changes, and it is comprised of three parts: a continuous-time dynamical equation which governs the evolution of the system between two successive impulsive instants; a difference equation which describes how the system state is changed at impulsive instants; and finally a criterion for determining the impulsive instant sequence [1], [2], [3], [4], [5], [6], [7]. Stability with minimal dwell-time or maximal dwell-time for impulsive systems was defined and discussed based on the characters of the system matrices in [4]. The stability of impulsive systems has been paid attention due to their applications in many fields such as epidemiology [8], impulsive control [9,10], power electronics [11], networked control systems [12,13], sampled-data systems [14–18]. For example, in [16], the sampled-data systems were formulated as impulsive systems, and robust stability of sampled-data systems was investigated. The discrete time method is a basic approach to the stability of impulsive systems, by which the impulsive system is transformed into discrete time systems, and then the eigenvalue analysis is conducted. This method, however, encounters difficulties when impulsive systems involve uncertainties or aperiodic impulses. To deal with this problem, a Lyapunov functional method was proposed in [16] to address the stability for impulsive systems, and the stability results were applied to sampled-data systems with uncertainties. To improve the Lyapunov functional method further, there are two methods to deal with the stability for impulsive systems. One is the Lyapunov-like functional approach where the Lyapunov-like functional is not required positive definite or continuous [17,18]. In [17] stability results were derived for periodic impulsive systems with or without polytope uncertainties. Less conservative stability results were obtained by constructing a different Lyapunov-like functional in [18]. The other approach to improve the Lyapunov functional method is the looped-functional method [19], [20], [21], [22], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. Convex dwell-time stability characterizations were given for the linear impulsive systems with uncertainties in [19,20], where the stability analysis was formulated as an infinite dimensional feasibility problem, which was hard to solve. By contrast, the stability results obtained in [21,22] were in the form of linear matrix inequalities (LMIs) and could be checked easily. Recently the stability for impulsive systems was studied by using the Lyapunov-like functional approach [23], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. The stability results obtained in [23] have improved over those in [21,22]. However, the Lyapunov-like functional in [23] as well as [17], and the looped-functional in [21] or [22] only used the system information on the interval(t_k, t], failed to make use of that on $[t,t_{k+1}]$. Therefore, the existing results can be expected to further improve if the Lyapunov-like functional or the looped-functional is expanded by making full use of the system information on the whole impulsive interval $[t_k,t_{k+1}]$. It is challenging how to employ the information on $[t,t_{k+1}]$ and (t_k, t] to construct a two-sided Lyapunov functional and develop new techniques to deal with the derivative of the two-sided Lyapunov functional.

It is worth noting that recently some scholars have turned towards impulsive networks [30,31] and impulsive stochastic functional differential systems [32,33]. For instance, in [33] the stability problem was addressed for time-varying stochastic functional differential systems with distributed-delay dependent impulsive effects. Based on stochastic theory, some stability results were derived by employing a Lyapunov approach. Nevertheless, in this paper we will further study stability for the linear impulsive systems in [17–23] by a new Lyapunov-like functional. The study features:

• The Lyapunov-like functional is second order in time t, and two-sided in the sense of its including the system information on both $[t,t_{k+1}]$ and (t_k, t].

• An advanced inequality and integral equations of the impulsive system are employed when estimating the derivative of the two-sided Lyapunov-like functional.

• Improved dwell-time dependent stability results, including those with ranged dwell-time, maximal dwell-time and minimal dwell-time for impulsive systems with periodic or aperiodic impulses, are derived. Numerical examples are given to illustrate the less conservatism of the dwell-time dependent stability results.

Notations: Throughout this paper, X^T and $X^{-1}$denote the transposition and the inverse of the matrixX, respectively.* in the symmetric matrix stands for the symmetric terms. For a symmetric matrix X, X > 0  ( < 0) denotes X is positive definite (negative definite). I denotes the identity matrix with appropriate dimensions. | · | is the Euclidean norm for a vector and ‖ · ‖ is the induced matrix norm. ${\mathbb{R}}^n$, ${\mathbb{R}}^{n\times m}$, $\mathbb{k}$ refer to the set of n dimension vectors, n × m matrices and non-negative integers, respectively. λ_min( · ), λ_max( · ) and ρ( · ) denote the smallest eigenvalue, largest eigenvalue and the spectral radius of a real symmetric matrix. For a given matrix A, He{A} stands for $A+A^T$. For the impulsive instant t_k, $x\left(t_k^{+}\right)={\lim }_{s\downarrow t_k}x\left(s\right)$, $x\left(t_k\right)={\lim }_{s\uparrow t_k}x\left(s\right)$. ${\mathbb{S}}_n$ and ${\mathbb{S}}_n^{+}$ are the set of symmetric and symmetric positive definite matrices of size n × n respectively.