On the stability of continuous-time adaptive controllers for pure delay systems
type: Item
instance of: scholarly article
title: On the stability of continuous-time adaptive controllers for pure delay systems
zbMATH Open document ID: 0609.93050
DOI: 10.1016/0167-6911(86)90025-3
published in: Systems \& Control Letters
publication date: 1986-01-01T00:00:00Z
review text: The paper addresses the problem of analyzing the stability properties of the one-dimensional retarded differential-difference equation (RDDE) that arises when an adaptive controller is applied to a system described by a pure delay, i.e. \(y(t)=ku(t-h)\), \(h>0\) and k an unknown positive gain. The RDDE in question is \[ {\tilde \theta}\dot {\;}=-\sigma {\tilde \theta}(t)-\gamma B(t){\tilde \theta}(t-h)+f(t) \] where \(\gamma >0\), \(\sigma >0\) and B(t), f(t) are uniformly bounded functions of t. Three different results are given. Firstly, under the assumption that \(B(t+T)=B(t)\) with \(h=mT\) for an integer n, necessary and sufficient conditions for stability are established. The technical device used for the proof is Floquet theory. Secondly, using averaging theory, a stability-instability boundary is derived for sufficiently small values of \(\gamma\). The result applies to B(t) almost periodic. Finally, we introduce a modified adaptation and control laws which gives a B(t) of the form \[ B(t)=K(t)y^ 2[t-h(t)]/\{1+y^ 2[t-h(t)]\}, \] with K,h: \({\mathbb{R}}_+\to {\mathbb{R}}\). It is shown that the equilibrium of the RDDE in this case is globally stable. Two alternative proofs, one based on Lyapunov functionals and the second one using Razumikhin-type theorems, are given.
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 34K35
Mathematics Subject Classification ID: 93C40
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C30
zbMATH DE Number: 3987166
zbMATH Keywords: one-dimensional retarded differential-difference equation
zbMATH Keywords: adaptive controller
zbMATH Keywords: Floquet theory
zbMATH Keywords: averaging
zbMATH Keywords: stability-instability boundary
author: Romeo Ortega
MaRDI profile type: Publication
full work available at URL: 0167-6911(86)90025-3
OpenAlex ID: W1985412422
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description: scientific article; zbMATH DE number 3987166