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.