Speaker
Description
A rectangular polynomial matrix with more columns than rows generically has no eigenvalues. We aim to find a smallest perturbation (with respect to the 2-norm of the concatenated coefficients of the polynomial matrix) so that the perturbed polynomial matrix has an eigenvalue, that is prescribed. This problem is motivated by locating a nearest uncontrollable system for a first-order, as well as a higher-order continuous-time control system. We consider the cases when the polynomial matrix is subject to complex and real perturbations. For both, we derive singular value optimization characterizations that facilitate locating a nearest polynomial matrix with the prescribed eigenvalue. The real perturbation case, inspired by the work of Qiu et al. [Automatica, Vol. 31, pp. 879-890] for the distance to instability of a linear continuous-time system under real perturbations, is much more involved, and leads to a more complicated singular value formula.
We exploit the derived singular value optimization characterizations to compute a nearest rectangular polynomial matrix with an eigenvalue under complex perturbations and under real perturbations. The approaches that we devise are based on level-set methods that date back to Byers, Boyd-Balakrishnan, Bruinsma-Steinbuch for the distance to instability, as well as Lipschitz-continuity based global optimization techniques. They seem to work extremely effectively. For instance, we are able to compute a nearest uncontrollable system for a first-order or higher-order system of medium size in a few seconds.