Stabilizability preserving quotients and partial feedback linearization

Abstract

This paper presents the problem of stabilizability preserving quotients. Given a control system and its quotient the problem of stabilizability preserving quotients seeks to characterize all the quotients for which the system is stabilizable if and only if the quotient is stabilizable. This paper presents a mathematical formulation of this problem using the language of differential geometry. Taking partial feedback linearization as an instance of quotienting of a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability preserving if the “zero dynamics” are stable. In addition to this sufficient condition another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that the “zero dynamics” can be defined as a vertical vector field that is related to the horizontal lift of the linear system.