Geometric Theory of State Feedback: Tools

Abstract

In the last two Chapters of these notes, we discuss problems similar to the ones analyzed in Chapters 4 and 5, for more general classes of multivariable nonlinear systems. We will not assume anymore that the systems we deal with have some vector relative degree at the point of interest (i.e., in particular, that the matrix (5.1.2) is nonsingular at x°), but we will replace this hypothesis by some milder regularity assumptions, namely the constancy of the dimensions of certain distributions (and/or of the ranks of certain mappings) around a given point. For every nonlinear system of the class we consider in these notes, these assumptions are satisfied at each point of an open and dense set in the state space. In addition to these more “technical” hypotheses, we will assume sometimes that the system—viewed as a mapping between inputs and outputs—is “invertibile”, in a sense that will be precised later on.