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Published December 1994 | Published
Book Section - Chapter Open

Differential flatness and absolute equivalence


In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which 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. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we show that 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. However, in higher codimensions feedback linearizability and flatness are not equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed.

Additional Information

© 1994 IEEE. Research supported in part by NASA. Research supported in part by the Powell Foundation. The authors would like to thank Willem Sluis for many fruitful and inspiring discussions and for introducing us to Cartan's work and its applications to control theory. We also thank Shankar Sastry for valuable comments on this paper, and Philippe Martin for several useful discussions which led to a more complete understanding of the relationship between endogenous feedback and differential flatness.

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August 20, 2023