Robust stabilisation of linear time-delay systems with uncertainties in the system matrices and in the delay terms

摘要

The pseudospectral abscissa is a powerful tool in the evaluation of the robustness of a dynamical system's stability, as it provides a worst-case analysis of the stability of a system when this includes potential uncertainties on parameters. In this paper we consider systems of DDAEs (Delay Differential Algebraic Equations) that incorporate real-valued structured uncertainties on the system matrices and on the delay terms. We propose a robust stabilisation approach for a system of DDAEs, which is based on the optimization of the pseudospectral abscissa with respect to a fixed set of tunable design or controller parameters. Given the non-smoothness nature of the pseudospectral abscissa, standard optimization methods are not efficient and we make use of bundle gradient methods. This approach proves to be very effective, and particularly useful in the design of robust static or dynamic fixed-order controllers for a system of DDEs (Delay Differential Equations) of retarded type. The approach fully exploits the structure on the uncertainty and the property that in applications perturbations are usually real valued, i.e., the uncertainty is not over-bounded and the employed robust stability criterion is necessary and sufficient.