Stochastic input-to-state stability of random impulsive nonlinear systems

https://doi.org/10.1016/j.jfranklin.2018.11.035Get rights and content

Abstract

In this paper, the stochastic input-to-state stability is investigated for random impulsive nonlinear systems, in which impulses happen at random moments. Employing Lyapunov-based approach, sufficient conditions for the stochastic input-to-state stability are established based on the connection between the properties of system and impulsive intervals. Two classes of impulsive systems are considered: (1) the systems with single jump map; (2) the systems with multiple jump maps. Finally, some examples are provided to illustrate the effectiveness of the proposed results.

Introduction

Impulsive systems are a special class of hybrid systems, which combine continuous-time dynamics with instantaneous state jumps [1], [2]. In recent years, impulsive systems have attracted increasing research attention because of their potential applications in practical systems from a variety of scenarios such as sampled-data control systems [3], complex networks [4], multi-agent systems [5], networked control systems [6], [7] and neural network systems [8], [9].

It is well known that stability of a system is always an important subject in the research of control theory. A large amount of salient results of stability on impulsive systems are provided in the literature [3], [8], [9], [10], [11], [12], [13], [14], [15]. For instance, the exponential stability conditions of nonlinear time-varying impulsive systems were established in [3] by using Lyapunov functions with discontinuity at the impulse times. In [9], the exponential stability in the mean square was discussed for the impulsive Markovian jump stochastic bidirectional associative memory neural networks with time-varying delay and distributed delay. The noise-to-state(NSS) stability and globally asymptotic stability were investigated in [10] for a class of nonlinear systems with random disturbances and impulses. The criteria of stability and asymptotic stability for nonlinear impulsive systems were presented in [11]. The finite-time stability was addressed in [13] for a class of nonlinear impulsive stochastic systems with time-varying delays based on the average impulsive interval condition.

For a control system in reality, it is important to characterize the effects of external signals (e.g., external disturbances and input). The central concept in this theory is input-to-state stability (ISS), originally proposed for deterministic continuous-time systems in [16]. For the systems perturbed by non-white noise, the concepts of NSS and eλt-weighted (integral) NSS are introduced in [10], [17], [18] to characterize effects of stochastic disturbances. To study ISS of impulsive nonlinear systems, we should concentrate on two ingredients, namely, the stability of systems and the impulsive interval. An intuitive idea is that when the continuous dynamics is ISS but the impulses are destabilizing, the impulses should not occur too frequently, and when the impulses are stabilizing but the continuous dynamics is not ISS, there must not be overly long intervals between impulses. This idea is formalized in terms of impulsive interval (or dwell-time) conditions, such as fixed impulsive interval(i.e., each impulsive interval has a uniform upper bound or lower bound) [19], [20], average impulsive interval [21], [22], [23], [24], and constant impulsive interval (also called periodic impulse) [25], etc.

However, the aforementioned impulsive nonlinear systems require the impulses to happen at determined moments. In practice, impulses may be affected by external environmental factors, and do not always occur at fixed points but usually at random ones [28]. In modeling such processes, it is essential to consider the nonlinear system with random impulses occurring at random moments(i.e., impulsive moments are random variables). Differing from deterministic impulsive nonlinear systems [11], [12], [19], [21] and impulsive stochastic nonlinear systems [13], [20], [22], [23], [24], [34], the impulsive intervals of random impulsive systems are random variables. Hence, the impulsive interval conditions mentioned above cannot be applied to random impulsive systems directly. The fundamental difficulty is how to characterize the property of random impulsive intervals and connect it with the key properties of continuous-time dynamics and instantaneous state jumps. Thus, random impulsive systems are more realistic than deterministic impulsive systems and the study of random impulsive nonlinear systems is a new area of research [26], [27], [28], [29], [30], [31], [32], [33]. In [26], the p-moment exponential stability of random impulsive differential equations was studied by employing appropriate Lyapunov functions. By means of the Lyapunov–Razumikhin functional, some sufficient conditions for p-moment stability of functional differential equations with random impulses were presented in [28]. The p-moment exponential stability of Caputo fractional differential equations with impulses occurring at random moments was studied in [31]. Nevertheless, to the best of our knowledge, there is no work on ISS of random impulsive nonlinear systems. Furthermore, in real applications, the nonlinear systems may be affected by different types of impulsive jumps in different manners. In other words, some impulsive jumps can contribute towards stability and others can potentially destroy stability. For example, in [4], two kinds of impulses, namely, desynchronizing and synchronizing impulses, can make different effects on the synchronization of complex dynamical networks.

In this paper, motivated by the discussion above, we aim to investigate the stochastic input-to-state stability(SISS) for random impulsive nonlinear systems. Based on the Lyapunov function and the probability distribution of impulsive interval, the sufficient conditions for SISS are established. The contributions of this paper can be summarized as follows: (1) differing from nonlinear systems with impulses occurring at deterministic moments [10], [11], [12], [13], [19], [25], the impulsive time sequence in this paper is considered as a stochastic process, which is more practical; (2) compared with [26] in which the continuous dynamics is required to be stable, two cases for the random impulsive nonlinear systems with single jump map are considered in this paper. The first case is that the continuous dynamics is stable; the second case is that the discrete impulses are stabilizing; (3) SISS is investigated for random impulsive nonlinear system with multiple jump maps which depend on probability, that is, the nonlinear system may be affected by both stabilizing and destabilizing impulsive jumps. In Section 2, the random impulsive nonlinear system with single jump map is presented, together with some definitions, assumptions and lemmas. The conditions for SISS of corresponding system is provided. SISS of the random impulsive nonlinear system with multiple jump maps is analyzed in Section 3. Some examples are provided in Section 4 to illustrate the effectiveness of the proposed results.

Notation: Rn denotes the n-dimensional Euclidean space, R+ stands for the sets of the nonnegative real numbers and Rt0+={tR+|tt0}. N+ represents the set of positive integers. | · | represents the Euclidean vector norm; P denotes the probability measure; E denotes the mathematical expectation. C2,1 stands for the space of the functions that are continuously twice differentiable on the first augment and continuously differentiable on the second augment. A function α(t):R+R+ is said to belong to class K if it is continuous, strictly increasing and α(0)=0; α(t) belongs to class K if α(t)K and α(t) → ∞ as t → ∞; α(t) belongs to class L if it is continuous, strictly decreasing and α(t) → 0 as t → ∞. A function β(s,t):R+×R+R+ is said to belong to class KL if β(s,t)K for each fixed t ≥ 0 and β(s, t) decreases to zero as t → ∞ for each fixed s ≥ 0. I(·) denotes the indicator function. Ln denotes the set of all the measurable and locally essentially bounded signal xRn on Rt0+ with norm xsuptt0inf{AΩ,P{A}=0}sup{|x(t,ω)|:ωΩA}

Section snippets

SISS of random impulsive nonlinear system with single jump map

In this section, we will consider the random impulsive nonlinear system with the effects of single impulsive jump map:{x˙(t)=f(t,x,u),tTk,x(t)=g(x(t),u(t)),t=Tk,kN+ where x(t)XRn is the system state, u(t)ULm is the external input. The function f:Rt0+×X×UX is assumed to be continuous with respect to t, x, u and uniformly locally Lipschitz with respect to x, u; f(·,0,0)=0. The function g:X×UX is continuous with respect to x, u; g(0,0)=0. The random impulsive time sequence T={Tk}k=0 is

SISS of random impulsive nonlinear system with multiple jump maps

In this section, we focus on the impulsive nonlinear systems with multiple jump maps which depend on probability. Firstly, an example will be given to show our motivation.

Example 1

Let the number x ≥ 0 of goods in a storage be continuously decreasing proportionally to the number of items with rate coefficient 0.2. Meanwhile, the decreasing rate may be affected by the factor of market ω(t). On some days (Tk), a delivery truck double the number of items(with probability 0.4 and denoted by r(k)=1) or takes

Numerical examples

In this section, some examples are given to illustrate the results in the previous section.

Example 2

In this example, we consider the following random impulsive nonlinear system with stable continuous dynamics and destabilizing impulses of the form:{x˙1(t)=x1+sin2x2·u,tTk,x1(t)=3x1(t),t=Tk,k=1,2,3,x˙2(t)=x2/2,tTk,x2(t)=3x2(t),t=Tk,k=1,2,3,where {Tk}k=0 is random impulsive time sequence defined in Section 2. Choose Lyapunov function V(x(t),t)=(x12(t)+x22(t))/2 and the function ρ(v)=5|v|/2. It

Conclusion

In this paper, the stochastic input-to-state stability is investigated for random impulsive nonlinear systems. Both the situation that the random impulsive nonlinear systems with single jump map and multiple jump maps are considered. The sufficient conditions for the stochastic input-to-state stability are established based on the frequency of impulses occurring condition and Lyapunov-based approach. Finally, some examples are given to illustrate the developed theory. In our future research, we

Acknowledgments

This work is supported by the National Natural Science Foundation of China [grant number 11571322]. The authors highly appreciate the above financial support.

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      Citation Excerpt :

      Simply stated, under any bounded initial condition, the system state will finally converge to an origin’s neighborhood whose size is proportional to the input magnitude [26]. The concepts of ISS and iISS have been extended to different types of dynamical systems, including stochastic systems [27,28], discrete-time systems [29,30], switched systems [31,32], impulsive systems [33,34], etc. Random impulses are used to describe the external impulsive stochastic disturbances that dynamical systems may encounter.

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