Taming Heterogeneity and Complexity of Embedded Control

Generic Families and Generic Bifurcations of Control-Affine Systems

Marek Wojciech Rupniewski^a1 Witold Respondek^b

 

^aPolish Academy of Sciences, Institute of Mathematics ul. Sniadeckich 8 02-956 Warsaw 10, Poland

^aWarsaw University of Technology, Institute of Electronic Systems 00-665 Warsaw, Poland [e-mail]

^bInstitute National des Sciences Appliquées de Rouen Laboratoire de Mathématiques de l’INSA Pl. Emile Blondel, 76 131 Mont Saint Aignan, Cedex, France [e-mail]

 

ABSTRACT. We consider control-affine systems with n - 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank. We provide a complete classification of such generic systems and their oneparameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback and orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.

KEYWORDS: control system, feedback equivalence, involutive distribution, 1-parameter family, bifurcation

1. Introduction

In this paper we deal with nonlinear control systems of the form

(Σ)

on an n-dimensional manifold M, and with 1-parameter families of such systems, in which case we replace the vector ...