High-gain robust adaptive controllers for multivariable systems
type: Item
instance of: scholarly article
title: High-gain robust adaptive controllers for multivariable systems
zbMATH Open document ID: 0632.93046
DOI: 10.1016/0167-6911(87)90077-6
author: David H. Owens
author: Dieter Prätzel-Wolters
author: Achim Ilchmann
published in: Systems \& Control Letters
publication date: 1987-01-01T00:00:00Z
full work available at URL: SCL1987 IOP.pdf
review text: The time-varying linear system \[ (1)\quad dx/dt=({\mathcal A}-k(t){\mathcal D})x(t),\quad t\geq 0 \] is considered. Here \({\mathcal A}\), \({\mathcal D}\in {\mathbb{R}}^{n^ 2}\), \(k(t)\in{\mathcal K}\), where \({\mathcal K}\) is a set of all piecewise continuous functions monotonically non-decreasing, \(k(t)\to +\infty\) as \(t\to \infty\). The following conditions are equivalent: (i) Re \(\lambda_ j({\mathcal D})>0\), \(j=1,...,n\), where \(\lambda_ j({\mathcal D})\) are eigenvalues of \({\mathcal D},\) (ii) Re \(\lambda_ j[{\mathcal A}-k(t){\mathcal D}]\to -\infty\), \(j=1,...,n,\) (iii) For some \(M>0\), \(\omega\) (\(\cdot)\in {\mathcal K}:\| \Phi (t,t_ 0)\| \leq Me^{-\omega (t_ 0)(t-t_ 0)}\), for all \(t\geq t_ 0\geq 0\), where \(\Phi (t,t_ 0)\) is the evolution matrix of (1). Let the following conditions be satisfied: (a) There exist \(k^*\), \(\epsilon >0\) such that \(\lambda_ j[{\mathcal A}-k{\mathcal D}]<-\epsilon\) for all \(k>k^*,\) (b) If \({\mathfrak O}\in \{\lambda_ j({\mathcal D})\}\) then \({\mathfrak O}\) is semisimple. Then (1) is exponentially stable for every k(\(\cdot)\in {\mathcal K}\). Moreover, there exists \(M'>0\) such that for all \(\epsilon >0\) and all k(\(\cdot)\in {\mathcal K}\) we can find \(T=T(\epsilon,k(\cdot))\) such that for all \(t\geq t_ 0\geq T:\| \Phi (t,t_ 0)\| \leq M'e^{-\epsilon (t- t_ 0)}.\) This result can be used for the construction of the adaptive regulator \(u(\cdot)=\psi [\sigma (\cdot)]\sigma (t)\) of the system \[ (2)\quad dx/dt=Ax+Bu,\quad \sigma =Cx. \] The matrices A,B,C are unknown, \(A,D=BC\) satisfy the conditions (a),(b); u(t), \(\sigma (t)\in {\mathbb{R}}^ 1\). The robustness of this adaptive controller with respect to small nonlinear perturbations of (2) is proved. Reviewer's remark. For stabilization problems of minimum phase objects, the adaptive regulators of the above type were widely used in the Russian literature, see, e.g., the book [\textit{V. N. Fomin}, \textit{A. L. Fradkov}, and \textit{V. …
Mathematics Subject Classification ID: 93C40
Mathematics Subject Classification ID: 93C35
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 4029396
zbMATH Keywords: time-dependent
MaRDI profile type: Publication
OpenAlex ID: W2081016457
Wikidata QID: Q123302314
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description: scientific article; zbMATH DE number 4029396