Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case
Nichols, N.K. and Kautsky, J. (2001) Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case. SIAM Journal on Matrix Analysis and Applications, 23 (1). pp. 77-102. ISSN 0895-4798
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To link to this article DOI: 10.1137/S0895479899362867
Feedback design for a second-order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second-order closed loop system but also that the system is robust, or insensitive to perturbations. We derive here new sensitivity measures, or condition numbers, for the eigenvalues of the quadratic matrix polynomial and define 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 structured perturbations. Numerically reliable methods for solving the structured generalized linear problem are developed that take advantage of the special 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 coefficient matrix in the quadratic polynomial is nonsingular, which ensures that the polynomial is regular. In a second part, we will examine the case where the open loop matrix polynomial is not necessarily regular.