On estimation of rates of convergence in Lyapunov–Razumikhin approach

doi:10.1016/j.automatica.2020.108928

Automatica

Volume 116, June 2020, 108928

https://doi.org/10.1016/j.automatica.2020.108928 Get rights and content

Abstract

Considering a retarded nonlinear system, this note proposes several modifications of the Lyapunov–Razumikhin approach guaranteeing the existence of an upper estimate on convergence rate of the system solutions. The cases of exponential, finite-time and fixed-time (with respect to a ball) convergences are studied. The proposed approach is illustrated by simulation of academic examples.

Introduction

There exist two generic frameworks assessing asymptotic stability of time-delay systems, which are based on analysis of a Lyapunov–Razumikhin function or a Lyapunov–Krasovskii functional (Fridman, 2014, Kolmanovskii and Myshkis, 1999). The latter method has been proven to be equivalent to the asymptotic stability property for some particular classes of the time-delay systems (Efimov and Fridman, 2019, Pepe and Karafyllis, 2013, Pepe et al., 2017), and it can also be used to establish finite-time stability (Moulay, Dambrine, Yeganefar, & Perruquetti, 2008). The former approach is only sufficient for the asymptotic stability (Fridman, 2014, Kolmanovskii and Myshkis, 1999), and it is less intuitive while obtaining the rate of solution convergence (Aleksandrov and Zhabko, 2012, Efimov, Polyakov et al., 2014, Myshkis, 1995). An advantage of Lyapunov–Razumikhin approach with respect to Lyapunov–Krasovskii one is that in many nonlinear cases it is more simple to find a Lyapunov–Razumikhin function than a Lyapunov–Krasovskii functional (Efimov, Perruquetti et al., 2014, Efimov et al., 2016) (e.g., a Lyapunov function for the delay-free case can be tested).

The objective of this work is to overcome one of the main drawbacks of the Lyapunov–Razumikhin function approach, and to propose its several extensions, which allow the rate of solution convergence to be estimated using the method. Three cases will be considered: the systems with asymptotic rate of convergence of solutions to the origin (the expansion of Myshkis, 1995 is given), with a finite time of convergence and a fixed-time one with respect to a ball (the definitions of these kinds of comportment are given below). The obtained results are used to formulate examples of differential inequalities for Lyapunov–Razumikhin function providing the studied convergence rates.

The paper is organized as follows. Preliminaries are given in Section 2. The problem statement is presented in Section 3, and the main results are formulated in Section 4. The performed simulations are described in Section 5. The final remarks and discussion are presented in Section 6.

Section snippets

Preliminaries

The real numbers are denoted by $R$ , $R_{+} = {τ \in R : τ \geq 0}$ . Euclidean norm for a vector $x \in R^{n}$ is denoted as $| x |$ . We denote by $C_{[a, b]}$ , $- \infty < a < b < + \infty$ the Banach space of continuous functions $ϕ : [a, b] \to R^{n}$ with the uniform norm $‖ ϕ ‖ = {sup}_{a \leq ς \leq b} | ϕ (ς) |$ .

Consider an autonomous functional differential equation of retarded type (Kolmanovsky & Nosov, 1986): $d x (t) ∕ d t = f (x_{t}), t \geq 0,$ where $x (t) \in R^{n}$ and $x_{t} \in C_{[- τ, 0]}$ is the state function, $x_{t} (s) = x (t + s)$ , $- τ \leq s \leq 0$ and $τ > 0$ is a finite delay; $f : C_{[- τ, 0]} \to R^{n}$ is a continuous function, $f (0) = 0$ , and is

Problem statement

As we mentioned above, there exist two methods evaluating asymptotic stability of the system (1) based on a Lyapunov–Razumikhin function or a Lyapunov–Krasovskii functional. The Lyapunov–Krasovskii approach is used in Proposition 1, Proposition 2 to establish finite-time/fixed-time to a ball stability of (1), and there are also converse results for asymptotic stability in Efimov and Fridman, 2019, Pepe and Karafyllis, 2013 and Pepe et al. (2017), while the Lyapunov–Razumikhin method can be

Main result

The previous evaluations by the Lyapunov–Razumikhin approach of the convergence rate for an asymptotically stable system have been presented in Aleksandrov and Zhabko, 2012, Mao, 1996 and Myshkis (1995).

Theorem 2

Let there exist a locally Lipschitz continuous Lyapunov–Razumikhin function $V : R^{n} \to R_{+}$ such that

(i) for some $α_{1}, α_{2} \in K_{\infty}$ and all $x \in R^{n}$ : $α_{1} (| x |) \leq V (x) \leq α_{2} (| x |);$

(ii) for some $γ^{'} > 1$ , $α^{'} > 0$ and all $φ \in C_{[- τ, 0]}$ : $max_{θ \in [- τ, 0]} V (φ (θ)) \leq γ^{'} V (φ (0)) \Rightarrow$ $D^{+} V (φ (0)) f (φ) \leq - α^{'} V (φ (0)) .$ Then the origin is globally asymptotically stable for

Simulations

Let us demonstrate by simulations that under the conditions introduced in Theorem 2, Theorem 3, Theorem 4, the solutions of the system (1) have the corresponding convergence rates. For simplicity of illustration we will consider several examples for a scalar variable $V (t) \in R_{+}$ , which represents a possible behavior of a Lyapunov–Razumikhin function, and we will restrict ourselves to the scenarios with pointwise constant delay $τ > 0$ .

First, assume that for all $t \geq 0$ and $V_{0} \in C_{[- τ, 0]}$ : $\dot{V} (t) \leq - a V (t) + b V (t - τ),$

Conclusions

Three extensions of the Lyapunov–Razumikhin function approach are proposed, which allow an upper estimate on the rate of decreasing of solutions to be obtained for a time-delay system. The case of the exponential convergence just adds an additional parametric restriction to the conventional method. While the cases of finite-time and fixed-time to a ball stability need more severe modifications of the approach. The latter property is also introduced in this work together with a

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Denis Efimov received Ph.D. degree in Automatic Control from the Saint-Petersburg State Electrical Engineering University (Russia) in 2001, and Dr.Sc. degree in Automatic control in 2006 from the Institute for Problems of Mechanical Engineering RAS (Saint-Petersburg, Russia). From 2006 to 2011 he was working in the L2S CNRS (Supelec, France), the Montefiore Institute (University of Liege, Belgium) and IMS CNRS lab (University of Bordeaux, France). In 2011, he joined Inria (Lille — Nord Europe center). Starting from 2018 he is the scientific head of Valse team. He is a member of several IFAC TCs and a Senior Member of IEEE. He is also serving as an Associate Editor for IEEE Transactions on Automatic Control, IFAC Journal on Nonlinear Analysis: Hybrid Systems and Asian Journal of Control.

Alexander Aleksandrov received the M.S. degree in Applied Mathematics, and the Ph.D. degree in Automatic Control in 1985 and 1989, respectively, both from the Leningrad State University, Russia. He received the Doctor of Sciences (higher doctorate) degree in Mathematical Modeling in 2000 from the St. Petersburg Institute for Informatics and Automation of the Academy of Science of Russia. Currently, he is a Professor of the faculty of Applied Mathematics and Control Processes of St. Petersburg State University. He is an author of more than 200 scientific articles and books. His current research interests include stability of nonlinear time-varying systems, switched systems, time-delay systems, nonlinear and robust control.

^☆: This work is partially supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, by Government of Russian Federation (Grant 08-08) and by the Russian Foundation for Basic Research (grant no. 19-01-00146-a). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nikolaos Bekiaris-Liberis under the direction of Editor Miroslav Krstic.

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Brief paperOn estimation of rates of convergence in Lyapunov–Razumikhin approach☆

Abstract

Introduction

Section snippets

Preliminaries

Problem statement

Main result

Simulations

Conclusions

Automatica

Stochastic Processes and their Applications

Systems & Control Letters

Systems & Control Letters

Systems & Control Letters

Asymptotic stability conditions and estimates of solutions for nonlinear multiconnected time-delay systems

Circuits, Systems, and Signal Processing

Stability analysis and estimation of the convergence rate of solutions for nonlinear time-delay systems

On the asymptotic stability of solutions of nonlinear systems with delay

Siberian Mathematical Journal

Brief paper
On estimation of rates of convergence in Lyapunov–Razumikhin approach☆