Eigenvalue range determination for interval and parametric matrices

Abstract

SUMMARYStudy 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.

A new approach to hull consistency

“…A method for determining x * has been proposed in [6] which is based on the linear LP-solution (25). An improved version that employs the quadratic p-solution (26) is suggested here.…”

Determining the stability margin in linear interval parameter electric circuits via a DAE model

“…Let µ′ denote the maximum real eigenvalue of (29). We assume that µ′ is a simple eigenvalue (this assumption and its physical meaning, the so‐called structural stability , have already been discussed earlier in 15, 17, 28). Let x ′ denote the corresponding right eigenvector (that is unique up to normalization).…”

“…Step 2 . Solve (3) for the range , related to the first (real) eigenvalue λ 1 , using an exact method 17, 28.…”

Necessary and sufficient stability condition of fractional-order interval linear systems

“…Since the set of vertex matrices needs to be stable and it also provides a sufficient condition (i.e., the existence of common quadratic Lyapunov function), it looks like that the vertex matrices still might play a key role in establishing the exact robust stability condition for some classes of fractional-order interval systems. Related to this argument, better results might be developed for fractional-order cases using the results in integer-order systems recently developed in [5]- [7].…”