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On the circle criterionfor boundarycontrol systemsin factor form: Lyapunov stability and Lur'e equations

Published online by Cambridge University Press:  15 December 2005

Piotr Grabowski and
Frank M. Callier
Piotr Grabowski
Affiliation:
Institute of Automatics, AGH University of Science and Technology, avenue A. Mickiewicz 30, B1, rm.314, 30-059 Cracow, Poland; pgrab@ia.agh.edu.pl
Frank M. Callier
Affiliation:
University of Namur (FUNDP), Department of Mathematics, Rempart de la Vierge 8, 5000 Namur, Belgium; frank.callier@fundp.ac.be
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Abstract

A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found,which is based on results by Oostveen and Curtain [Automatica34 (1998) 953–967]. All theresults are illustrated in detail by an electrical transmission line example of thedistortionless loaded $\mathfrak{RLCG}$ -type. The paper uses extensively thephilosophy of reciprocal systems with bounded generating operators as recentlystudied and used by Curtain in (2003) [Syst. Control Lett.49 (2003) 81–89; SIAM J. Control Optim.42 (2003) 1671–1702].

Keywords

Infinite-dimensional control systemssemigroupsLyapunovfunctionalscircle criterion.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006, pp. 169 - 197
Copyright
© EDP Sciences, SMAI, 2006

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