Robust eigenstructure assignment in quadratic matrix polynomials: Nonsingular case

Feedback design for a second-order control system leads to an eigenstructur e assignment problem for a quadratic matrix polynomial. It is desirable tha t the feedback controller not only assigns specified eigenvalues to the sec ond-order closed loop system but also that the system is robust, or insensi tive to perturbations. We derive here new sensitivity measures, or conditio n numbers, for the eigenvalues of the quadratic matrix polynomial and de ne a measure of the robustness of the corresponding system. We then show that the robustness of the quadratic inverse eigenvalue problem can be achieved by solving a generalized linear eigenvalue assignment problem subject to s tructured perturbations. Numerically reliable methods for solving the struc tured generalized linear problem are developed that take advantage of the s pecial properties of the system in order to minimize the computational work required. In this part of the work we treat the case where the leading coe fficient matrix in the quadratic polynomial is nonsingular, which ensures t hat the polynomial is regular. In a second part, we will examine the case w here the open loop matrix polynomial is not necessarily regular.