Let p, m, n, d be positive integers and let Ln(d)Ln(d) denote the set of sequences L=(L1,…,Ln)L=(L1,…,Ln) of p×mp×m real or complex matrices which are realizable by systems of minimal order d. It was shown in ; that Ln(d)Ln(d) can be endowed with a structure of differentiable manifold when p=m=1p=m=1; that is, when the sequences are realizable by Single Input/Single Output (SISO) systems. In this paper a similar result is obtained for more general sequences. Specifically, we will consider the set View the MathML sourceLn(r_,s_) of sequences L which are realizable by systems of minimal order d and having View the MathML sourcer_ and View the MathML sources_ as Brunovsky indices of controllability and observability, respectively. It is shown in this paper that when one of the two collections of indices View the MathML sourcer_ or View the MathML sources_ is constant, then View the MathML sourceLn(r_,s_) can be provided with a structure of differentiable manifold and a formula of its dimension is given. The special cases View the MathML sourcer_=(1,…,1) or View the MathML sources_=(1,…,1) correspond to sequences realizable, respectively, by Single Input/Multi Output (SIMO) or Multi Input/Single Output (MISO) systems.
The aim of this paper is the study of local perturbations of a bimodal system which consists of two linear
control systems on each side of a given hyperplane. We follow Arnold’s technique based on obtaining
a miniversal deformation corresponding to the action of a group associated to a simultaneous feedback
equivalence. An application to the study of the controllability of local perturbations of such a system is
Nuestra intencion es a partir de la caracterizacion de los sistemas controlables que se encuentra en  y de las deformaciones versales que calculamos y aplicamos sobre la formas reducidas obtenidas en , hacer un estudio del comportamiento de la controlabilidad en tales sistemas cuando se aplican pequeñas
perturbaciones sobre ellos.
En particular, se demuestra que el conjunto de sistemas bimodales controlables es un conjunto abierto pero que no es denso; es decir, contrariamente a lo que pasa en los sistemas lineales ordinarios, en los sistemas bimodales la controlabilidad no es una condicion generica. Ademas se muestran ejemplos de sistemas no controlables que contienen puntos interiores y de sistemas no controlables con entornos controlables y entornos no controlables.
 M.K. Camlibel, W.P.M.H. Heemels and J.M. Shumacher. On the controllability of Bimodal Piecewise Linear Systems. Hybrid Systems: Computation and Control. Lecture Notes in Computer and Science, Vol 2993 (2004), pp 250-264.
 X. Puerta. Feedback reduced and canonical forms for switched and bimodal linear systems. Preprint
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.
Given a bimodal system de¯ned by the equations
x_ (t) = A1x(t) + Bu(t) if ctx(t) · 0
x_ (t) = A2x(t) + Bu(t) if ctx(t) ¸ 0 (1)
where B 2Mn;m and Ai 2Mn, i = 1; 2, are such that A1;A2 coincide on the hyper-
plane V =Kerct. We consider in the set of matrices de¯ning the above systems the
simultaneous feedback equivalence de¯ned by ([A1;B]; [A2;B]) » ([A0
i B0] = S¡1[Ai B]
i = 1; 2 with S(V) = V
This equivalent relation corresponds to the action of a Lie group. Under this action we
obtain, in the case m · 1, the semiuniversal deformation, following Arnold's technique.
Then the problem of structural stability is studied.
Given a pair of matrices (A,B) we study the Lipschitz stability
of its controlled invariant subspaces. A sufficient condition is derived from
the geometry of the set formed by the quadruples (A,B, F, S) where S is an
(A,B)-invariant subspace and F a corresponding feedback.
Given a pair of matrices (A;B) we study the stability of their invariant subspaces from the geometry of the manifold of quadruples
(A;B; S; F) where S is an (A;B)-invariant subspace and F is such that (A + BF)S ½ S. In particular, we derive a su±cient computable condition of stability.