Nonlinear averaging theorems, and the determination of parameter convergence rates in adaptive control
type: Item
instance of: scholarly article
title: Nonlinear averaging theorems, and the determination of parameter convergence rates in adaptive control
zbMATH Open document ID: 0612.93038
DOI: 10.1016/0167-6911(86)90108-8
author: Marc Bodson
author: Brian D. O. Anderson
author: S. Shankar Sastry
author: Iven Mareels
author: Robert R. Bitmead
published in: Systems \& Control Letters
publication date: 1986-01-01T00:00:00Z
review text: The system of differential equations \[ (1)\quad \dot x=\epsilon f(t,x,y),\quad (2)\quad \dot y=A(x)+\epsilon g(t,x,y),\quad x(0)=x_ 0,\quad y(0)=y_ 0 \] is considered. The existence of a function \(f_{av}(x)\) is assumed such that for all t, \(T\geq 0\), \[ | \frac{1}{T}\int^{t+T}_{t}f(\tau,x,0)d\tau -f_{av}(x)| \leq \gamma (T) \] where \(\gamma\) (T)\(\to 0\) as \(T\to \infty\). The matrix A(x) is uniformly exponentially stable for all \(x\in B_ n\), and some other assumptions are made. There exists a strictly increasing function \(\psi\) (\(\epsilon)\), \(\psi (0)=0\), such that, given \(T\geq 0\), \[ | x(t)- x_{av}(t)| \leq \psi (\epsilon)b_ T \] for some \(b_ T\), \(\epsilon_ T>0\) and for all \(t\in [0,T/\epsilon]\), \(0<\epsilon <\epsilon_ T\). Here \(x_{av}(t)\) is the solution of the averaged system \[ (3)\quad \dot x_{av}=f_{av}(x_{av}),\quad x_{av}(0)=x_ 0. \] If (3) is an exponentially stable system, then the original system (1), (2) is exponentially stable for \(\epsilon >0\) sufficiently small. These results are applied, to the Narendra-Valavani model reference adaptive control algorithm [\textit{K. S. Narendra} and \textit{L. S. Valavani}, IEEE Trans. Autom. Control AC-23, 570-583 (1978; Zbl 0389.93025)]. The results of some computer experiments are given.
Mathematics Subject Classification ID: 93C40
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93D05
zbMATH DE Number: 3991371
zbMATH Keywords: parameter convergence rates
zbMATH Keywords: averaged system
zbMATH Keywords: model reference adaptive control
reviewed by: V. A. Yakubovich
MaRDI profile type: Publication
full work available at URL: 0167-6911(86)90108-8
OpenAlex ID: W2050978482
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description: scientific article; zbMATH DE number 3991371