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

Joris Vanbiervliet; Koen Verheyden; Wim Michiels; Stefan Vandewalle

ESAIM: Control, Optimisation and Calculus of Variations (2007)

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

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topVanbiervliet, Joris, et al. "A nonsmooth optimisation approach for the stabilisation of time-delay systems." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 478-493. <http://eudml.org/doc/90879>.

@article{Vanbiervliet2007,

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 = {Vanbiervliet, Joris, Verheyden, Koen, Michiels, Wim, Vandewalle, Stefan},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Stabilisation; delay differential equations; nonsmooth optimisation; bundle gradient methods; stabilisation},

language = {eng},

month = {11},

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/90879},

volume = {14},

year = {2007},

}

TY - JOUR

AU - Vanbiervliet, Joris

AU - Verheyden, Koen

AU - Michiels, Wim

AU - Vandewalle, Stefan

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/11//

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; stabilisation

UR - http://eudml.org/doc/90879

ER -

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