A Razumikhin-type lemma for functional differential equations with application to adaptive control
type: Item
instance of: scholarly article
title: A Razumikhin-type lemma for functional differential equations with application to adaptive control
zbMATH Open document ID: 0934.93038
DOI: 10.1016/S0005-1098(98)00227-1
author: E. P. Ryan
author: Franco Blanchini
published in: Automatica
publication date: 1999-12-05T00:00:00Z
review text: The authors consider a class of \(m\)-input, \(m\)-output controlled functional differential equations of the form \[ \dot z(t)=g(t,z_{t}^{a})+B(t,u(t)), \quad y(t)=Cz(t), \] where \(z(t),u(t),y(t)\in\mathbb{R}^{m}\), \(z_{t}^{a}(\theta)=z(t+\theta)\), \(-a\leq\theta\leq 0\), \(g\) and \(B\) are of Carathéodory class, \(\det C\neq 0\), \(a\geq 0\). Under some restrictions on \(g\) and \(B\), an adaptive strategy -- universal for the class -- is developed for approximate tracking of any reference signal \(r(t)\) of Sobolev class \(W^{1,\infty}\), i.e., the tracking error \(e(t)=y(t)-r(t)\) should tend to the closed ball of radius \(\lambda\) centered at the origin in \(\mathbb{R}^{n}\) for any prescribed \(\lambda>0\). The result is based on a Razumikhin-type lemma on boundedness of solutions of retarded functional differential equations. The approach is also extended to provide approximate tracking for a class of dynamically perturbed functional differential equations. An application in the control of a thermal process is presented. The paper is very well written and all statements are proved.
Mathematics Subject Classification ID: 93C40
Mathematics Subject Classification ID: 93C23
Mathematics Subject Classification ID: 34K35
zbMATH DE Number: 1325751
zbMATH Keywords: adaptive control
zbMATH Keywords: tracking
zbMATH Keywords: functional differential equations
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: nonlinear systems
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
Wikidata QID: Q59341891
DBLP publication ID: journals/automatica/BlanchiniR99
description: scientific article; zbMATH DE number 1325751