Nonregular Feedback Linearization of a Class of Multi-Input Nonlinear Control Systems

Abstract

In this paper we study feedback linearization of multi-input control-affine systems via a particular class of nonregular feedback transformations, namely by reducing the number of controls by one. A complete geometric characterization of systems of that class requires special properties of the linearizability distributions $mathcal{D}^0 subset mathcal{D}^1 subsetmathcal{D}^2 subset cdots$. Contrary to the case of regular feedback linearization, they need not be involutive but the first noninvolutive one has to contain a sufficiently large involutive subdistribution. Recently, we solved the problem of $mathcal{D}^0$ being noninvolutive. In the present paper, we study the case of $mathcal{D}^0$ involutive but $mathcal{D}^1$ noninvolutive. This case is specially interesting because it covers a big class of mechanical control systems. We will provide geometric necessary and sufficient conditions describing our class of systems that can be verified by differentiation and algebraic operations only. We illustrate our results by several examples and discuss relations with other linearizability problems (static invertible feedback linearization or dynamic linearization).