M. Dolors Magret

An alternative system of structural invariants of quadruples of matrices

Abstract The equivalence relation between time-invariant linear dynamical systems deduced from the action of the group consisting of all basis changes, state feedback and output injection transformations is considered. An alternative complete system of structural invariants is presented. Discrete invariants in this system are the ranks of suitable matrices, generalizing the numerical invariants obtained for pairs of matrices under block-similarity. The continuous invariants are characterized in terms of the rank of a matrix.

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

Switched singular linear systems

We consider switched singular linear systems and determine the set of reachable/controllable states. We derive necessary and sufficient conditions for such a system to be reachable/controllable when an “equisingularity condition” holds.

Differentiable Families of Planar Bimodal Linear Control Systems

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along it. For a differentiable family of planar bimodal linear control systems, we obtain its stratification diagram and, if controllability holds for each value of the parameters, we construct a differentiable family of feedbacks which stabilizes both subsystems for each value of the parameters.

Associating matrix pencils to generalized linear multivariable systems

We consider quadruples of matrices (E,A,B,C) representing generalized linear multivariable systems Ex(t)=Ax(t)+Bu(t),y(t)=Cx(t), with E, A square matrices and B, C rectangular matrices. We characterize equivalent quadruples, by associating matrix pencils to them, with respect to the equivalence relation corresponding to standard transformations: basis changes (for the state, control and output spaces), state feedback, derivative feedback and output injection. Equivalent quadruples are those whose associated matrix pencils are “simultaneously equivalent”.

Isometries of the hamming space and equivalence relations of linear codes over a finite field

Detection and error capabilities are preserved when applying to a linear code an isomorphism which preserves Hamming distance. We study here two such isomorphisms: permutation isometries and monomial isometries.

Upper bounds for the distance between a controllable switched linear system and the set of uncontrollable ones

The set of controllable switched linear systems is an open and dense set in the space of all switched linear systems. Therefore it makes sense to compute the distance from a controllable system to the nearest uncontrollable one. In the case of a standard system, , R. Eising, D. Boley, and W. S. Lu obtain some results for this distance, both in the complex and real cases. In this work we explore this distance, for switched linear systems in the real case, obtaining upper bounds for it. The main contribution of the paper is to prove that a natural generalization of the upper bound obtained by D. Boley and W. S. Lu is true in the case of switched linear systems.

Differentiable families of stabilizers for planar bimodal linear control systems

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which point wise stabilizes both subsystems.

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.

Perturbation of quadrics

Abstract The aim of this paper is to study what happens when a slight perturbation affects the coefficients of a quadratic equation defining a variety (a quadric) in R n . Structurally stable quadrics are those such that a small perturbation on the coefficients of the equation defining them does not give rise to a “different” (in some sense) set of points. In particular, we characterize structurally stable quadrics and give the “bifurcation diagrams” of the non-stable ones (showing which quadrics meet all of their neighbourhoods), when dealing with the “affine” and “metric” equivalence relations. This study can be applied to the case where a set of points, which constitute the set of solutions of a problem, is defined by a quadratic equation whose coefficients are given with parameter uncertainty.