Effect of delayed impulses on input-to-state stability of nonlinear systems

doi:10.1016/j.automatica.2016.08.009

Automatica

Volume 76, February 2017, Pages 378-382

https://doi.org/10.1016/j.automatica.2016.08.009 Get rights and content

Abstract

This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear systems with delayed impulses. By using Lyapunov method and the analysis technique proposed by Hespanha et al. (2005), some sufficient conditions ensuring ISS/iISS of the addressed systems are obtained. Those conditions establish the relationship between impulsive frequency and the time delay existing in impulses, and reveal the effect of delayed impulses on ISS/iISS. An example is provided to illustrate the efficiency of the obtained results.

Introduction

The theory of input-to-state stability (ISS) plays a central role in modern nonlinear control theory, in particular to robust stabilization of nonlinear systems, design of nonlinear observers, analysis of large-scale networks, etc. (Dashkovskiy et al., 2010, Freeman and Kokotovic, 2008, Jiang et al., 1994, Sontag, 2001). The concept of ISS was introduced by Sontag, 1989, Sontag, 1998. In last years, many interesting results on ISS properties of various systems such as discrete systems, switched systems and hybrid systems have been reported, see, (Cai and Teel, 2005, Mancilla-Aguilar and Garcia, 2001, Nesic and Teel, 2004a, Nesic and Teel, 2004b, Pepe and Jiang, 2006).

Impulsive systems serve as basic models to study the dynamics of processes that are subject to sudden changes at certain moments in their states. They have been extensively studied in the literature (Haddad et al., 2006, Lu et al., 2010, Naghshtabrizi et al., 2008). Especially, due to the facts that impulsive systems with external inputs arise naturally from a number of applications such as in control systems with communication constraints, control algorithms of uncertain systems and network control systems with scheduling protocol, it is important to guarantee the impulsive system to be input-to-state stable when it is affected by some external inputs (Chen and Zheng, 2009, Dashkovskiy and Mironchenko, 2012, Hespanha et al., 2008, Liu et al., 2011). Hence, it is of great practical significance to investigate the ISS property of impulsive systems and it has become one of the hot issues in control theory. The concepts of ISS and iISS of impulsive systems was proposed by Hespanha, Liberzon, and Teel (2005) and Hespanha et al. (2008). They developed the Lyapunov method to impulsive systems and established some criteria for ISS properties by controlling the frequency of impulse occurrence. Chen and Zheng (2009) studied the ISS and iISS of nonlinear impulsive systems with time delays and presented several Razumikhin-type criteria. In Liu et al. (2011), Liu and Xie considered the ISS properties of impulsive and switching hybrid systems with time delays using the method of multiple Lyapunov–Krasovskii functionals and it can be applied to impulsive systems with arbitrarily large delays. Recently, Dashkovskiy and Mironchenko (2012) and Dashkovskiy, Kosmykov, Mironchenko, and Naujok (2012) further studied the ISS and iISS of nonlinear impulsive systems with or without delays via the generalized average dwell-time method, and especially a Lyapunov–Krasovskii-type ISS theorem and a Lyapunov–Razumikhin-type ISS theorem for single impulsive time-delay systems were established in Dashkovskiy et al. (2012).

With the development of impulsive control theory, increasing attention has been paid to the study of dynamics and controller design of impulsive systems in which the impulses involve time delays which are sometimes called delayed impulses, see (Chen et al., 2013, Chen and Zheng, 2011, Khadra et al., 2009, Yang et al., 2014). Such kind of impulses describe a phenomenon where impulsive transients depend on not only their current but also historical states of the system. For instance, Khadra et al. (2009) studied the stability problem of autonomous impulsive differential systems with linear delayed impulses and then applied it to synchronization control of two coupled chaotic systems. Chen and Zheng (2011) and Chen et al. (2013) considered the effects of delayed impulses on nonlinear time-delay systems and Takagi–Sugeno fuzzy systems, respectively. Recently, Yang et al. (2014) considered the exponential synchronization of discontinuous chaotic systems via delayed impulsive control. However, it seems that there have been few results that consider the effect of delayed impulses on ISS property for nonlinear systems, which still remains as an important and open problem.

In this paper, we shall study the ISS property of nonlinear systems with delayed impulses and external input affecting both the continuous dynamics and the state impulse map. We extend the analysis technique developed in Hespanha et al., 2008, Hespanha et al., 2005 to the systems with delayed impulses. Some sufficient conditions ensuring the ISS/iISS are obtained. The rest of this paper is organized as follows. In Section 2, the problem is formulated and some notations and definitions are given. In Section 3, we present the main results. An example is given in Section 4, and conclusions follow in Section 5.

Section snippets

Preliminaries

Notations. Let $R$ denote the set of real numbers, $R_{+}$ denote the set of nonnegative real numbers, $R^{n}$ and $R^{n \times m}$ the $n$ -dimensional and $n \times m$ -dimensional real spaces, respectively, $Z_{+}$ the set of positive integer numbers, $| \cdot |$ the Euclidean norm, and ${‖ \cdot ‖}_{J}$ the supremum norm on an interval $J \in R$ . Let $α \lor β$ and $α \land β$ denote the maximum and minimum value of $α$ and $β$ , respectively. Let $K_{\infty} = {α \in C (R_{+}, R_{+}) | α (0) = 0, α (r) is strictly increasing in r, and α (r) \to + \infty as r \to + \infty}$ , $KL = {β \in C (R_{+} \times R_{+}, R_{+}) | β (r, t) is in class K w.r.t.$

Main results

In this section, we will establish some criteria which provide sufficient conditions for ISS/iISS of system (1). The idea is inspired by the work of Hespanha, Liberzon and Teel in Hespanha et al., 2008, Hespanha et al., 2005.

Theorem 1 Uniformly ISS

Let $V$ be a candidate exponential ISS-Lyapunov function of system (1) with rate coefficients $c \in R_{+}$ and $d \in (0, τ c)$ . For any constants $σ, η \in R_{+}$ and function $μ \in U_{σ}^{η, d}$ , let $F (μ)$ denote the class of impulse time sequences ${t_{k}}_{k \in Z_{+}}$ satisfying $(τ c - d) N (t, s) - c (t - s) \leq μ (t - s) \forall t \geq s \geq t_{0} .$ Then

An example

In this section, an example is given to show the effectiveness of our obtained results.

Example 1

Consider the following impulsive system $\dot{x} (t) = - sat (x (t)) + b sat (u (t)), t \neq t_{k},$ $x (t) = ϱ x (t^{-} - τ) + β sat (u (t^{-} - τ)), t = t_{k},$ where sat( $\cdot$ ) denotes the saturation function limited at $\pm 1$ and with unit slope on $[- 1, 1]$ ; $b, ϱ$ and $β$ are scalars satisfying $| b | < 1$ and $| ϱ | + | β | < 1$ .

Choose LF $V (x) = {\begin{matrix} x^{2}, & | x | \leq 1 \\ e^{2 (| x | - 1)}, & | x | > 1 . \end{matrix}$ . When $| x | \leq 1$ , it is easy to check that $▽ V (x) \cdot f \leq - (2 - | b |) V (x) + | b | {sat}^{2} (u)$ ; While $| x | > 1$ , $▽ V (x) \cdot f \leq - 2 (1 - | b |) V$ . It then can be

Conclusion

In this paper, we have presented some sufficient conditions for ISS/iISS properties of nonlinear systems with delayed impulses. Those conditions established the relationship between impulsive frequency and the time delay existing in impulses, and showed the effect of delayed impulses on ISS/iISS. An example has been given to illustrate the efficiency of the obtained results. However, due to the presence of time delay on the impulses, our results apply only when the continuous dynamics are

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^☆: This work was jointly supported by National Natural Science Foundation of China (11301308, 61673247, 41427806), China PSFF (2014M561956, 2015T80737) and Research Fund for Excellent Youth Scholars of Shandong Province (ZR2016JL024). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor A. Pedro Aguiar under the direction of Editor André L. Tits.

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Technical communiqueEffect of delayed impulses on input-to-state stability of nonlinear systems☆

Abstract

Introduction

Section snippets

Preliminaries

Main results

An example

Conclusion

Automatica

Automatica

Nonlinear Analysis. Hybrid Systems

Automatica

Automatica

Automatica

Systems & Control Letters

Systems & Control Letters

Automatica

Systems & Control Letters

Systems & Control Letters

Communications in Nonlinear Science and Numerical Simulation

Technical communique
Effect of delayed impulses on input-to-state stability of nonlinear systems☆