Technical Report CIT / CDS 94 - 006 California Institute of Technology DIFFERENTIAL FLATNESS AND ABSOLUTE EQUIVALENCEOF

This paper presents a formulation of diierential atness|a concept originally introduced by Fliess, Levine, Martin, and Rouchon|in terms of absolute equivalence between exterior diierential systems. Systems which are diierentially at have several useful properties which can be exploited to generate eeective control strategies for nonlinear systems. The original deenition of atness was given in the context of diierential algebra, and required that all mappings be meromorphic functions. The formulation of atness presented here does not require any algebraic structure and allows one to use tools from exterior diierential systems to help characterize diierentially at systems. In particular, it is shown that, under regularity assumptions, in the case of single input control systems (i.e., codimen-sion 2 Pfaaan systems), a system is diierentially at if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability about an equilibrium point and atness are not equivalent: one must be careful with the role of time as well as the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of diierential atness to nonlinear control systems and open questions are also discussed. 1. Introduction The problem of equivalence of nonlinear systems (in particular to linear systems, that is, feedback linearization) is traditionally approached in the context of diieren-tial geometry 15, 20]. A complete characterization of static feedback linearizability in the multi-input case is available, and for single input systems it has been shown that static and dynamic feedback linearizability are equivalent 5]. Some special results have been obtained for dynamic feedback linearizability of multi-input systems , but the general problem remains unsolved. Typically, the conditions for