Robust Eigenstructure Assignment in Quadratic Matrix Polynomials: Nonsingular Case

• Published in: SIAM journal on matrix analysis and applications Vol. 23; no. 1; pp. 77 - 102
• Main Authors: Nichols, N. K., Kautsky, J.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2001
• Subjects: Applied sciences; Assignment problem; Closed loop systems; Computer science; control theory; systems; Control theory. Systems; Controllers; Eigenvalues; Eigenvectors; Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; System theory; Closed loop; Second order; Feedback design; Control system; Feedback regulation; Quadratic polynomial; Perturbation; Inverse problem; Assignment; Polynomial function; Time invariant; Eigenvalue problem; Robustness
• ISSN: 0895-4798; 1095-7162
• 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.