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A nonsmooth optimisation approach for the stabilisation of time-delay systems

Published online by Cambridge University Press:  21 November 2007

Joris Vanbiervliet ,
Koen Verheyden ,
Wim Michiels  and
Stefan Vandewalle
Joris Vanbiervliet
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; joris.vanbiervliet@cs.kuleuven.be
Koen Verheyden
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; joris.vanbiervliet@cs.kuleuven.be
Wim Michiels
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; joris.vanbiervliet@cs.kuleuven.be
Stefan Vandewalle
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; joris.vanbiervliet@cs.kuleuven.be
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Abstract

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm to infinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms to efficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.

Keywords

Stabilisationdelay differential equationsnonsmooth optimisationbundle gradient methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 478 - 493
Copyright
© EDP Sciences, SMAI, 2007

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