Inmaculada de Hoyos

Perturbation of linear control systems

Abstract We study the variation of the controllability indices and the Jordan structure of a pair of matrices ( A , B ), under small perturbations.

Nearest pair with more nonconstant invariant factors and pseudospectrum

Abstract Let (A,B)∈ C n×n × C n×m . Suppose that the number of nonconstant (i.e., ≠1 ) invariant factors of the polynomial matrix λ[I n ,0]−[A,B] is less than k. For all complex number λ denote by σ n−(k−1) (λ[I n ,0]−[A,B]) the greatest (n−(k−1)) th singular value of the matrix λ[I n ,0]−[A,B] . The minimum absolute value of the real function of complex variable λ↦σ n−(k−1) (λ[I n ,0]−[A,B]) gives the distance from (A,B) to the set of pairs with more or equal number of nonconstant invariant factors. When k=1 , this specializes in the formula of Eising for the distance from a controllable pair (A,B) to the nearest uncontrollable pair. The complex numbers λ lying in the sublevel set {λ∈ C | σ n (λ[ I n , 0 ]−[ A , B ])⩽e} of the function λ↦σ n (λ[I n ,0]−[A,B]), are the uncontrollable modes of all the pairs that are within an e tolerance of (A,B) . All the results of this paper are an immediate consequence of the Singular Value Decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks.

A linear matrix equation: a criterion for block similarity

In this paper we study a homogeneous linear matrix equation related to the block similarity of rectangular matrices. We obtain the dimension of the vector space of its solutions and we describe these solutions. We give a characterization of the block similarity by rank tests. We extend Roths criterion to the corresponding non homogeneous equation.

The change of feedback invariants under column perturbations: particular cases

We study the variation of the feedback invariants of a complex rectangular n × (n + m) matrix when we make small additive perturbations to the elements of the last m columns. First of all, we obtain necessary conditions for the feedback invariants of all the matrices obtained by means of sufficiently small perturbations. Conversely, we prove that these conditions are also sufficient to find a matrix, as close as desired to the fixed matrix, with prescribed feedback invariants, in some particular cases: when the rectangular matrix is completely controllable, when the rectangular matrix is completely uncontrollable and when m = 1.