Generalized exponential input-to-state stability of nonlinear systems with time delay

doi:10.1016/j.cnsns.2016.08.016

Communications in Nonlinear Science and Numerical Simulation

Volume 44, March 2017, Pages 352-359

https://doi.org/10.1016/j.cnsns.2016.08.016 Get rights and content

Highlights

•
Provide several concepts of general input-to-state stability for the nonlinear delay systems. That is generalized globally exponential integral input-to-state stability (GGE-iISS), generalized globally integral exponential integral input-to-state stability (GGIE-iISS), and e^λt -weighted generalized globally integral exponential integral input-to-state stability ( e^λt -weighted GGIE-iISS).
•
Give out the stability conditions of the general input-to-state stability for the nonlinear delay systems. Moreover, strict theoretical derivation is given to prove the sufficiency and correctness of the given conditions.
•
An example is given to illustrate the correctness of the obtained theoretical results.

Abstract

This paper studies the general input-to-state stability problem of the nonlinear delay systems. By employing Lypaunov–Razumikhin technique, several general input-to-state stability concepts, that is generalized globally exponential integral input-to-state stability (GGE-iISS), generalized globally integral exponential integral input-to-state stability (GGIE-iISS), and e^λt-weighted generalized globally integral exponential integral input-to-state stability (e^λt-weighted GGIE-iISS) are studied. An example is given to illustrate the correctness of the obtained theoretical results.

Introduction

Stability problem of nonlinear systems has become more and more important because of its widely applications. Usually, systems in the real world are affected by the external interference, such as disturbance in control or errors on observation. Time delay usually is inevitable in dynamic systems due to the external environment or internal factors, such as parameter variability, measurement, transmission or transport lags, and computational delays [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. External disturbance inputs and time delay often lead to the loss of stability for an otherwise stable system [5], [6], [7], [8], [9], [10], [11], [12], [13]. When a control system is affected by external input, it is important to guarantee the control system to be input-to-state stable. The notion of input-to-state stability (ISS), which characterizes the continuity of state trajectories on the initial states and the external inputs, is first introduced by Sontag [14]. Roughly speaking, ISS means that, no matter what the initial state is, if the external input is small, then the state must be eventually small. The analysis of ISS aims to investigate how external inputs affect the stability of dynamic systems, and it has been proven very useful in the analysis and designing controllers of nonlinear systems [13], [14], [15], [16], [17], [18], [19]. Integral input-to-state stability (iISS), which is a natural generalization of ISS, is a weaker concept introduced in [20]. ISS was originally defined for continuous-time systems. However, some other ISS properties have been proposed for different types of dynamical systems, such as discrete-time systems [21], [22], [23], impulsive systems [12], [24], [25], [26], switched systems [27], [28], hybrid systems [29], [30], [31], stochastic systems [32], [33], [34], [35], and so on. Zhu et al. [12] study an issue of input-to-state stability for a class of impulsive stochastic Cohen–Grossberg neural networks with mixed delays. Some new sufficient conditions are given to ensure the considered system with/without impulse control is mean-square exponentially input-to-state stable. Vu et al. [27] provided that, when the switching signals have large enough average dwell time, the switched system is ISS, e^λt-weighted ISS, and e^λt-weighted iISS, respectively, if the individual subsystems are ISS. Pepe et al. [36] studied the integral input-to-state globally uniformly stable (iIS-GUS) and integral input-to-state globally uniform asymptotical stable (iIS-GUAS). They gave some sufficient conditions for the stability of dynamic systems.

Since the ISS is very useful for the analysis and designing controllers for dynamics, especially for nonlinear systems and complex networked control systems, and time delay is inevitable in the actual system, therefore it is necessary to study the stability of the nonlinear control systems with external inputs and time delay. To promote the application of the method, this work proposes several more general concepts about the input-to-state stability, such as GGE-iISS, GGIE-iISS and e^λt-weighted GGIE-iISS. By adopting the Lypaunov–Razumikhin technique, some sufficient conditions of the generalized ISS for nonlinear delay systems are established. As far as we know, there is no result on the related topics till now.

The rest of this paper is organized as follows. Section 2 presents the problem formulation, related notions and definitions. The main results are given in Section 3. An example is presented in Section 4 to illustrate the correctness of the obtained theoretical results. Conclusions are collected in Section 5.

Section snippets

Preliminaries and problem formulation

Throughout this paper, R denotes the real number set, $R^{+}$ is subset of R, which is defined by $R^{+} = [0, + \infty),$ and Rⁿ denotes the n-dimensional real column vectors with the usual Euclidean norm | · |. A continuous function $γ : R^{+} \to R^{+}$ is of class- $K (γ \in K),$ if it is continuous, zero at zero, and strictly increasing; γ is of class $K_{\infty} (γ \in K_{\infty}),$ if it is of class- $K$ and γ(s) → ∞ as s → ∞.

Consider the following nonlinear dynamic system $\begin{matrix} \dot{x} (t) = f (t, x (s), u (t); σ \leq s \leq t), \forall t \geq 0, \end{matrix}$ where x(t) ∈ Rⁿ is the state, $\dot{x} (t)$ denotes the

Main results

In this section, some sufficient conditions for GGE-iISS, GGIE-iISS, and e^λt-weighted GGIE-iISS of system (2.1) are provided through several theorems. For convenience, suppose that u ∈ Φ_γ, and denote $x (t) = x (t, ϕ, u)), V (t) = V (t, x (ϕ, u))$ throughout all the paper.

Theorem 3.1

Suppose that there exists a Lyapunov function $V : R^{+} \times R^{+} \to R^{+},$ functions $α (\cdot), β (\cdot) \in K_{\infty}$ and $γ (\cdot) \in K,$ such that for some λ > 0, $α (| x |) \leq V (t, x (t)) \leq β (| x |),$ and $V (t, x (t)) \geq V (τ, x (τ)), σ \leq τ \leq t,$ implies that $\begin{matrix} {[(V (t, x (t)) - \int_{0}^{t} γ (| u (s) |) d s) e^{λ t}]}^{'} \leq 0, \forall t \geq 0 . \end{matrix}$ Then with the

Examples

In this part, an example is given to illustrate the correctness of the obtained theoretical results.

Example 4.1

Consider the following dynamic system $\begin{matrix} \dot{x} (t) = - x (t) + \frac{1}{4} x (t - τ) u (t) + u (t), t \geq 0, \end{matrix}$ where x(t) is the state of system (4.1), u(t) is the control input, and τ > 0 is the time delay. Suppose that the initial state is zero, the control input u satisfies $\int_{0}^{+ \infty} | u (s) | d s < + \infty .$ In the following, the generalized exponential input-to-state stability of system (4.1) is analyzed.

Define $V (t, x) = | x | e^{- \int_{0}^{t} | u (s) | d s} .$ Then along

Conclusions

Several new input-to-state conceptions, that is generalized globally exponential integral input-to-state stability (GGE-iISS), generalized globally integral exponential integral input-to-state stability (GGIE-iISS), and e^λt-weighted generalized globally integral exponential integral input-to-state stability (e^λt-weighted GGIE-iISS) of nonlinear systems with time delay are proposed in this work. By employing Lypaunov–Razumikhin technique, some sufficient conditions for the generalized ISS

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^☆: This work is supported by National Natural Science Foundation (NNSF) of China (Grant Nos. 61503053, 61472374, 61304197and 61673080), the Natural Science Foundation Project of CQ CSTC, China (Grant No. cstc2013jcyjA40018), the Natural Science Foundation of CQJW, China (Grant Nos. KJ130506 and KJ1400435), the Youth Science Research Project of CQUPT, China (Grant Nos. A2012-78 and A2012-82), and the Training Programme Foundation for the Talents of Higher Education by Chongqing Education Commission.

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Research paperGeneralized exponential input-to-state stability of nonlinear systems with time delay☆

Highlights

Abstract

Introduction

Section snippets

Preliminaries and problem formulation

Main results

Examples

Conclusions

Commun Nonlinear Sci Numer Simul

Neurocomputing

Acta Autom. Sin.

Syst Control Lett

Automatica

Syst Control Lett

Commun Nonlinear Sci Numer Simul

Automatica

Appl Math Comput

Appl Math Comput

J Franklin Inst Eng Appl Math

Syst Control Lett

Lyapunov–Krasovskii functionals for input-to-state stability of switched non-linear systems with time-varying input delay

IET Control Theory Appl.

Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control

IEEE Trans Circuits Syst I Regul Pap

A remark on exponential stability of timed-delayed burgers equation

Discrete Continuous Dyn Syst Ser B

On design of robust reliable h∞ control and input-to-state stabilization of uncertain stochastic systems with state delay

Commun Nonlinear Sci Numer Simul

Global exponential stability for stochastic networks of coupled oscillators with variable delay

Commun Nonlinear Sci Numer Simul

Input-to-state stability for nonlinear systems with large delay periods based on switching techniques

IEEE Trans Circuits Syst I Regul Pap

Research paper
Generalized exponential input-to-state stability of nonlinear systems with time delay☆

On design of robust reliable h_∞ control and input-to-state stabilization of uncertain stochastic systems with state delay