Eigenvalue range determination for interval and parametric matrices

摘要

Study of the dynamic behaviour of linear limped parameter circuits or systems under parametric uncertainties is a well-established research area. The present paper addresses the problem of determining the exact (within rounding errors) ranges for the real and imaginary parts of an eigenvalue of real matrices whose elements are either independent intervals or linear (affine) functions of independent interval parameters. A unified method for solving the above problem is suggested in the paper. It is iterative and reduces to forming and solving, at each iteration, two real (incomplete) quadratic systems to find outer interval bounds on the set of the right and left eigenvectors associated with the eigenvalue considered. The method is applicable if: (i) the solutions of both quadratic systems involved are positive and (ii) the outer interval bounds on the eigenvectors satisfy certain constant sign conditions. It is shown that its numerical complexity is polynomial in the size n of the matrix studied. Numerical examples illustrating the applicability of the new method are provided.