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

Stefan Vandewalle; Wim Michiels; Koen Verheyden; Joris Vanbiervliet

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 3, page 478-493
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Vandewalle, Stefan, et al. "A nonsmooth optimisation approach for the stabilisation of time-delay systems." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 478-493. <http://eudml.org/doc/245876>.

@article{Vandewalle2008,
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.},
author = {Vandewalle, Stefan, Michiels, Wim, Verheyden, Koen, Vanbiervliet, Joris},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {stabilisation; delay differential equations; nonsmooth optimisation; bundle gradient methods},
language = {eng},
number = {3},
pages = {478-493},
publisher = {EDP-Sciences},
title = {A nonsmooth optimisation approach for the stabilisation of time-delay systems},
url = {http://eudml.org/doc/245876},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Vandewalle, Stefan
AU - Michiels, Wim
AU - Verheyden, Koen
AU - Vanbiervliet, Joris
TI - A nonsmooth optimisation approach for the stabilisation of time-delay systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2008
PB - EDP-Sciences
VL - 14
IS - 3
SP - 478
EP - 493
AB - 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.
LA - eng
KW - stabilisation; delay differential equations; nonsmooth optimisation; bundle gradient methods
UR - http://eudml.org/doc/245876
ER -

References

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