M. I. García-Planas

On matrix inverses modulo a subspace

Given a pair of matrices (A,B)∈Rn×n×Rn×m with coefficients in a commutative ring we study the problem of finding a matrix F such that A+BF becomes invertible. We point out some relations between the problem of finding feedback inverses of A modulo B and the pole-shifting problem for the pair (A,B). In fact we give feedback invertibility results over a well known large class of rings related with the pole-shifting: the class of PA rings. On the other hand we also give a pointwise-global characterization. The ring R of rational integers and the coordinate ring of the real unit circle R[x,y]/(x2+y2−1) are studied in some detail. Finally the problem of derivative feedback standardization of generalized linear systems is reviewed as a particular case.

Stratification of linear systems. Bifurcation diagrams for families of linear systems

Abstract A dynamical system can be represented by x =Ax+Bu, y=Cx, where A is a square matrix and B, C are rectangular matrices. The question of uncertain parameters e in the entries of the matrices A, B, C is particularly important when using the Kronecker form of the triple of matrices (A,B,C): the eigenstructure may depend discontinuously on the parameters when the matrices A(e),B(e),C(e) depend smoothly on those parameters. It is of great interest to know which different structures can arise from small perturbations of a dynamical system, and discuss the generic behaviour of smooth few-parameter families of linear systems. A fundamental way of dealing with these problems is, in a first step, to stratify the space of triples of matrices defining the systems. Here an important role is played by the miniversal deformations. A second step is to induce a partition in the space of parameters parametrizing the family of linear systems. We need to consider transversal families in order to ensure that the induced partition (called the bifurcation diagram) is also a stratification. In this case the induced partition is called a bifurcation diagram.

Miniversal deformations of linear systems under the full group action

Abstract We consider the equivalence relation in the space of time-invariant linear dynamical systems of the form x =Ax+Bu, y=Cx+Du, under the full group action of state feedback/output injection transformations. As in the case of the reduction of a matrix to its Jordan canonical form, the reduction of a quadruple (A,B,C,D) defining a system as above to its canonical reduced form is an unstable operation. Following V.I. Arnold’s techniques, the starting point for the study of local perturbations is to obtain a miniversal deformation of a differentiable family of quadruples. In this paper, a “real” miniversal deformation and some applications are shown.

Rigid systems of second-order linear differential equations

Abstract We say that a system of differential equations x ¨ ( t ) = A x ˙ ( t ) + Bx ( t ) + Cu ( t ) , A , B ∈ C m × m , C ∈ C m × n , is rigid if it can be reduced by substitutions x ( t ) = Sy ( t ) , u ( t ) = U y ˙ ( t ) + Vy ( t ) + Pv ( t ) , with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A, B, C. We prove that there exists a rigid system for given m and n if and only if m n ( 1 + 5 ) / 2 , and describe all rigid systems.

A generalized sylvester equation: a criterion for structural staility of triples of matrices

A generalized Sylvester matrix equation is considered. This equation is related with the stabilizer of a triple of matrices under the Lie group action on the space of triples of matrices which corresponds to the equivalence relation generalizing, in a natural way, the similarity between square matrices.criterion for the structural stability of a triple of matrices t is deduced in terms of therank of a matrix M(t). Also a differentiable family of triples of matrices is considered and we give conditions for differentiablity of the related family of tangent spaces to stabilizers T(ϕ(x)), deducing from it another criterion for structural stability of triples of matrices.