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.
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,x˙t=Axt+But, 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.
Abstract:It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field F and hyperinvariant subspaces of F n under an appropriate linear transformation.Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.
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.
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