Matrix polynomials with completely prescribed eigenstructure

The proceeding at: Joint ALAMA-GAMM/ANLA 2014 Meeting, took place 2014, July 14-16, in Barcelona (Spain).We present necessary and su cient conditions for the existence of a matrix polynomial when its degree, its nite and in nite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary in nite elds and are determined mainly by the \index sum theorem", which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to x the problem of the existence of (l)-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures.This research was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-32542 and by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.Publicad

A Complete Parametric Solutions of Eigenstructure Assignment by State-Derivative Feedback for Linear Control Systems

In this paper we introduce a complete parametric approach for solving the problem of eigenstructure assignment via state-derivative feedback for linear systems. This problem is always solvable for any controllable systems iff the open-loop system matrix is nonsingular. In this work, two parametric solutions to the feedback gain matrix are introduced that describe the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve robustness of the closed-loop system. Accordingly, the sensitivity of the assigned eigenvalues to perturbations in the system and gain matrix is minimized. Numerical examples are included to show the effectiveness of the proposed approach.

Robustness and perturbations of minimal bases II: The case with given row degrees

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.Comment: arXiv admin note: text overlap with arXiv:1612.0379

Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data

The structural data of any rational matrix R(\lambda ), i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nonespaces, is known to satisfy a simple condition involving certain sums of these indices. This fundamental constraint was first proved by Van Dooren in 1978; here we refer to this result as the rational index sum theorem. An analogous result for polynomial matrices has been independently discovered (and rediscovered) several times in the past three decades. In this paper we clarify the connection between these two seemingly different index sum theorems, describe a little bit of the history of their development, and discuss their curious apparent unawareness of each other. Finally, we use the connection between these results to solve a fundamental inverse problem for rational matrices---for which lists \scrL of prescribed structural data does there exist some rational matrix R(\lambda ) that realizes exactly the list \scrL ? We show that Van Dooren's condition is the only constraint on rational realizability; that is, a list \scrL is the structural data of some rational matrix R(\lambda ) if and only if \scrL satisfies the rational index sum condition