Itziar Baragaña

Column completion of a pair of matrices

Given a pair of matrices (A B) it is well known that its invariant factors and its controllability indices form a complete set of invariants for the Γ-equivalence [11] or block similarity [5]. How do they vary by adding columns to B? This problem was solved in [12] when B = 0; here we give a complete answer for this question.

Interlacing inequalities for regular pencils

Abstract We solve the problem of finding a necessary and sufficient condition for the existence of a regular pencil when its invariant homogeneous factors and a subpencil are prescribed. The solution obtained is an extension of the well-known Sa-Thompson interlacing inequalities for invariant factors of characteristic matrices.

Feedback invariants of supplementary pairs of matrices

Given a pair of matrices (A,B), with every (A,B)-invariant subspace and the corresponding supplementary sub-space we can associate pairs (A1,B1) and (A2, B2) which identify them up to feedback equivalence. A complete characterization of the controllability indices of (A,B) in terms of those of (A1,B1), (A2, B2) is provided under the assumption of complete controllability of (A1,B1). This problem is related to the feedback simulation problem.

Feedback invariants of restrictions and quotients: series connected systems

Abstract The problem of the feedback invariants of parallel and series connected systems is considered. It is shown that this problem is closely related to the characterization of the feedback invariants of a system with prescribed restriction and/or quotient to a controlled invariant subspace. Namely, under the assumption of complete controllability of the quotient, the relationship between the feedback invariants of ( A , B ) and those of its restriction and quotient is investigated.

Differentiable structure of realizable Markov parameters

Abstract The geometry of the finite sequences of p × m matrices that are realizable by systems of minimal order d is investigated. They can be stratified according to their partial row and column Kronecker indices. The case when the sum of the number of non-zero row and column indices is smaller than the number of elements in the sequence of parameters is considered. It is shown that, in this case, the whole set of Markov parameters and each stratum can be endowed with structures of differentiable manifolds, and their dimensions are computed.

On the geometry of the generalized partial realization problem

The geometry of the set of generalized partial realizations of a finite nice sequence of matrices is studied. It is proved that this set is a stratified manifold, the dimension of their strata is computed and its connection with the geometry of the cover problem is clarified. The results can be applied, as a particular case, to the classical partial realization problem.

Parametrising the solutions of the generalised partial realisation problem through an atlas of coordinates

Given a finite nice sequence of matrices ℒ and a positive integer d, the set of generalised partial realisations of ℒ of order d is a stratified smooth manifold. An atlas of coordinate charts for each stratum is provided that locally parametrises it. In particular, the set of all generalised partial realisations of order d of any given nice sequence ℒ will be described.

Versal deformation of realisable Markov parameters

ABSTRACT Let be the set of sequences (L 1,… , L n ), , admitting a minimal partial realisation of order d. To each , we associate two sequences of integers with r 1 ≥ r 2 ≥ … ≥ r β > 0 = r β+1 = … = r n and with s 1 ≥ s 2 ≥ … ≥ s α > 0 = s α+1 = … = s n called the partial Brunovsky column and row indices of L, respectively. Let be the subset of formed by the sequences L for which α + β ≤ n. Let Σ co be the set of matrix triples with (F, G) controllable and (H, F) observable. We denote by Σ co ≤ the subset of Σ co formed by the triples which are minimal partial realisations of the sequences . For every ξ ∈ Σ co ≤, we obtain a versal deformation of ξ corresponding to the action of the group , we show a method for obtaining a minimal partial realisation ξ of , and we derive a versal deformation of L from the obtained versal deformation of ξ.

Hermite indices and state feedback: generic case

The controllability indices are a complete system of invariants for the feedback equivalence relation of controllable matrix pairs. The Hermite indices are invariant for the similarity of matrix pairs but they are not invariant by changing basis on the input space and performing state feedback. The aim of this work is to partially characterize the Hermite indices of a controllable linear system, , when state feedback is performed on it. Namely, given a matrix pair (A, B), we study the problem of the existence of a matrix F such that (A+BF, B) has prescribed Hermite indices.

Hermite indices and equivalence relations

Abstract The Hermite indices are invariant for the right equivalence of non-singular polynomial matrices and for the similarity of controllable matrix pairs. Nevertheless, they do not form a complete system of invariants. The aim of this work is to define two equivalence relations, one in the set of non-singular polynomial matrices and the other one in the set of controllable matrix pairs for which the Hermite indices form a complete system of invariants.