Bounding the distance of a controllable and observable system to an uncontrollable or unobservable one

Let $(A,B,C)$ be a triple of matrices representing a time-invariant linear system \left .\aligned \dot x(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\endaligned \right \} under similarity equivalence, corresponding to a realization of a prescribed transfer function matrix. In this paper we measure the distance between a irreducible realization, that is to say a controllable and observable triple of matrices $(A,B,C)$ and the nearest reducible one that is to say uncontrollable or unobservable one. Different upper bounds are obtained in terms of singular values of the controllability matrix $C(A,B,C)$, observability matrix $O(A,B,C)$ and controllability and observability matrix $CO(A,B,C)$ associated to the triple

Second order generalized linear systems. A geometric approach

Let (E;A1;A2;B) be a quadruple of matrices representing a two-order generalized time-invariant linear system, E¨x = A1 ˙ x + A2x + Bu. We study the controllability character under an algebraic point of view

Controllability of second order linear systems

Let (A1;A2;B) be a triple of matrices representing two-order time-invariant linear systems, ¨x = A1 ˙ x+A2x+Bu. Using linearization process we study the controllability of second order linear systems. We obtain su±cient conditions for controllability and we analyze the kind of systems verifying these conditions

Bounding the distance of a controllable system to an uncontrollable one

Let $(A,B)$ be a pair of matrices representing a time-invariant linear system $\dot x(t)=Ax(t)+Bu(t)$ under block-similarity equivalence. In this paper we measure the distance between a controllable pair of matrices $(A,B)$ and the nearest uncontrollable one. A bound is obtained in terms of singular values of the controllability matrix $C(A,B)$ associated to the pair. This bound is not simply based on the smallest singular value of $C(A,B)$ contrary to what one may expect. Also a lower bound is obtained using geometrical techniques expressed in terms of the singular values of a matrix representing the tangent space of the orbit of the pair $(A,B)$

Orbits of controllable and observable systems

Let a time-invariant linear system \left .\aligned \dot x(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\endaligned \right \} corresponding to a realization of a prescribed transfer function matrix can be represented by triples of matrices $(A,B,C)$. The permitted transformations of basis changes in the space state on the systems can be seen in the space of triples of matrices as similarity equivalence. In this paper we give a geometric characteriaztion of controllable and observable systems as orbits under a Lie group action. As a corollary we obtain a lower bound of the distance between a controllable and observable triple and the nearest uncontrollable one

Some considerations about reachability of switched linear singular systems

No necessary and sufficient condition for reachability of switched linear singular systems has been found, exceptuating the case of the so-called “equisingular systems”. Such a condition is not valid in the general case, as examples show.Peer ReviewedPostprint (published version

Distance from a controllable switched linear system to an uncontrollable one

The set of controllable switched linear systems is an open set in the space of all switched linear systems. Then it makes sense to compute the distance from a controllable switched linear system to the set of uncontrollable systems. In this work we obtain an upper bound for such distance.Postprint (published version

Perturbation of quadrics

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 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 deffined by a quadratic equation whose coefficients are given with parameter uncertainty

Familias diferenciables de inversas de Drazin

Las inversas de Drazin son una clase de inversa generalizada entre cuyas aplicaciones podemos mencionar la resoluci´on de ecuaciones diferenciales, ecuaciones en diferencias, estudio de cadenas de Markov, etc. En esta presentaci´on se caracterizan las matrices cuyas inversas de Drazin tienen misma forma reducida de Jordan y se estudia la partici´on del espacio de matrices cuadradas correspondiente a la relaci´on de equivalencia derivada. Finalmente, se encuentra una condici´on sufi- ciente para que una familia de inversas de Drazin de una familia diferenciable de matrices sea a su vez una familia diferenciable