Feedback linearization and driftless systems ∗ Philippe MARTIN † Pierre ROUCHON ‡ November 1994, to appear in MCSS R´ esum´ e The problem of dynamic feedback linearization is recast using the notion of dy- namic immersion. We investigate here a “generic” property which holds at every point of a dense open subset, but may fail at some points of interest, such as equi- librium points. Linearizable systems are then systems that can be immersed into linear controllable ones. This setting is used to study the linearization of driftless systems : a geometric sufficient condition in terms of Lie brackets is given; this condition is shown to be also necessary when the number of inputs equals two. Though non invertible feedbacks are not a priori excluded, it turns out that li- nearizable driftless systems with two inputs can be linearized using only invertible feedbacks, and can also be put into chained form by (invertible) static feedback. Most of the developments are done within the framework of differential forms and Pfaffian systems. Key words : nonlinear systems, feedback linearization, dynamic immersion, driftless systems, Pfaffian systems. 1 Introduction The problem of feedback linearization (see e.g., [CLM89]) of a (smooth) control system ˙ x = f (x, u) defined on an open subset X × U of R n × R m consists in finding a (smooth) dynamic feedback ˙ z = a(x,z,v) u = σ(x,z,v) * This work was partially supported by INRIA, NSF grant ECS-9203491, GR “Automatique” (CNRS) and DRED (Minist` ere de l’ ´ Education Nationale). Part of it was done while the first author was visiting the Center for Control Engineering and Computation, University of California at Santa Barbara. † Centre Automatique et Syst` emes, ´ Ecole des Mines de Paris, 35, rue Saint-Honor´ e, 77305 Fontainebleau Cedex, FRANCE. Tel: 33 (1) 64 69 48 57. E-mail: martin@cas.ensmp.fr ‡ Centre Automatique et Syst` emes, ´ Ecole des Mines de Paris, 60, Bd Saint-Michel, 75272 Paris Cedex 06, FRANCE. Tel: 33 (1) 40 51 91 15. E-mail: rouchon@cas.ensmp.fr 1