On the exponential stability of a class of nonlinear systems including delayed perturbations.
type: Item
instance of: scholarly article
title: On the exponential stability of a class of nonlinear systems including delayed perturbations.
zbMATH Open document ID: 1033.93055
DOI: 10.1016/S0377-0427(03)00550-8
author: D. Massart
published in: Journal of Computational and Applied Mathematics
publication date: 2003-11-20T00:00:00Z
review text: The authors consider the system \[ \dot x= F(x,t)+ G(x, t)u(t), \] whose equilibrium at the origin for \(u(t)\equiv 0\) is exponentially stable, this property being ensured by a \({\mathcal C}^1\) Lyapunov function satisfying \[ \lambda^2_1| x|^2\leq V(x, t)\leq \lambda^2_2| x|^2,\quad {\partial V\over\partial t}+ (\text{grad}_x V)F(x,t)\leq -\lambda_3 V(x,t). \] This system is perturbed by a delayed state dependent disturbance satisfying \(| H(x,t)|\leq \beta| x|\). It is shown that the control law \[ u(t)=- {G^T(x, t)(\text{grad}_x V(x,t))^T \beta^2\chi^2(t)\over |\text{grad}_x V(x,t) G(x,t)|_\beta\chi(t)+ \varepsilon e^{-\alpha t}} \] exponentially stabilizes the system \[ \dot x= F(x(t), t)+ G(x(t), t)[H(x(t- h(t), t)+ u(t)],\quad 0\leq h(t)\leq\overline h. \] Here \(\alpha> 0\), \(\varepsilon> 0\) and \(\chi(t)= \sup_{t-\overline h\leq\theta\leq\overline h}| x(\theta)|\).
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 93D21
zbMATH DE Number: 2006409
zbMATH Keywords: nonlinear system
zbMATH Keywords: exponential stabilization
zbMATH Keywords: time delay
zbMATH Keywords: Lyapunov function
reviewed by: Vladimir Răsvan
MaRDI profile type: Publication
cites work: A complicated version of a robust control scheme
cites work: Stabilization of uncertain dynamic systems including state delay
cites work: Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems
cites work: Bounded controllers for robust exponential convergence
cites work: On the state observation and output feedback problems for nonlinear uncertain dynamic systems
cites work: Uncertain dynamical systems--A Lyapunov min-max approach
cites work: Design of robust controllers for time-delay systems
cites work: Sufficient conditions for robust stability of LQG optimal control systems including delayed perturbations
cites work: Exponential stability of a class of nonlinear dynamical systems with uncertainties
cites work: Stability robustness analysis of linear systems with delayed perturbations
description: scientific article; zbMATH DE number 2006409