Differential Flatness and Absolute Equivalence of Nonlinear Control Systems

This paper presents a formulation of differential flatness—a concept originally introduced by Fliess, Levine, Martin, and Rouchon—in terms of absolute equivalence between exterior differential systems. Systems that are differentially flat have several useful properties that can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra and required that all mappings be meromorphic functions. The formulation of flatness presented here does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, it is shown that, under regularity assumptions and in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. In higher codimensions our approach does not allow one to prove that feedback linearizability about an equilibrium point and flatness are equivalent: one must be careful with the role of time as well as the use of prolongations that may not be realizable as dynamic feedback in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed.