Unobservable Planar Bimodal Linear Systems: Miniversal Deformations, Controllability and Stabilization

TL;DR

This paper characterizes unobservable planar bimodal linear systems, providing explicit controllability conditions, miniversal deformations, bifurcation diagrams, and proving their stabilizability.

Contribution

It offers a simple explicit controllability characterization and constructs miniversal deformations for unobservable planar bimodal systems, advancing understanding of their bifurcations and stabilization.

Findings

Explicit controllability conditions for unobservable planar bimodal systems

Construction of miniversal deformations based on canonical forms

Proof that unobservable controllable systems are stabilizable

Abstract

We consider the set of bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing on the unobservable planar ones, we obtain a simple explicit characterization of controllability. Moreover, we apply the canonical forms of these systems depending on two state variables to obtain explicitly miniversal deformations, to illustrate bifurcation diagrams and to prove that the unobservable controllable systems are stabilizable. Preprint of an article submitted for consideration in IJBC \copyright 2011 copyright World Scientific Publishing Company http://www.worldscinet.com/ijbc/