On the optimal pole assignment for time-delay systems

Abstract

The well-known fact about linear time-invariant timedelay systems (LTI-TDS) is that these systems have an infinite spectrum. Not only plants themselves but also the whole control feedbacks then have this undesirable feature in most cases. The aim of this contribution is to present algebraic controller design in a special ring of proper and stable meromorphic functions followed by an optimal pole assignment minimizing the spectral abscissa. The main problem is how to place feedback poles to the prescribed positions exactly by a finite number of free (controller) parameters. Clearly, it is not possible to place all poles but the idea is to push the rightmost ones as left as possible, which gives rise to the task of the spectral abscissa minimization. The spectral abscissa is a nonsmooth nonconvex function of free controller parameters in general. Moreover, there is a problem of its sensitivity to infinitesimally small delay changes. Four advanced iterative algorithms; namely, Quasi- Continuous Shifting Algorithm, Nelder-Mead algorithm, Extended Gradient Sampling Algorithm and Self-Organizing Migration Algorithm, are described as a possible numerical tools when minimization. Only two of them have already been used for the spectral abscissa minimization and none of them with the combination with algebraic controller design.