On estimation of rates of convergence in Lyapunov–Razumikhin approach

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 fixedtime (with respect to a ball) convergences are studied. The proposed approach is illustrated by simulation of academic examples.

On finite-time stabilization of a class of nonlinear time-delay systems: Implicit Lyapunov-Razumikhin approach

“…Let us formulate a Lyapunov-Razumikhin theorem on asymptotic stability of time-delay systems [9] and its mild modifications [6] allowing one to estimate the exponential, finite-time and nearly fixed-time convergence rates. For brevity only the global stability will be considered in the sequel.…”

“…Theorem 3 (Theorem 8 [6]). Let there exists a locally Lipschitz continuous Lyapunov-Razumikhin function V : R n → R, satisfying condition 1) of Theorem 1, such that 2c) for some µ ∈ (−1, 0), c > 0, α > 0 and all…”

“…Hence, it gives mainly a qualitative conclusion. Recently several extensions of Lyapunov-Razumikhin method have been proposed [6], allowing the rate of solution convergence to be estimated.…”

“…Therefore, the goal of this work is to propose the implicit versions of Lyapunov-Razumikhin theorems on asymptotic and non-asymptotic stability developed in [6]. Differently from [8], the proposed approach allows one to use "simple" ILF (e.g.…”

Fixed-Time Stabilization of Linear Delay Systems by Smooth Periodic Delayed Feedback

“…In this vein, we note that previous finite-time/fixed-time stabilization approaches for delay systems typically consist of two steps: In the first step, by applying the Artstein transformation, the delayed linear system is transformed into a linear system without delay, and next a nonlinear controller is designed for the resultant linear system so as to achieve finite-time/fixed-time stability [5], [32]. As a result, a nonlinear control design is generally required, which in turn requires the construction of Lyapunov-Razumikhin (LR) functions or the Lyapunov-Krasovskii (LK) functionals to ascertain finite-time stability [10], [27], [32], [35]. The FTS design method developed here remains a linear control design, and as such appears conceptually more straightforward and intuitively more appealing.…”