On Necessary and Sufficient Conditions for Differential Flatness

Abstract

This paper is devoted to the characterization of differentially flat nonlinear systems. Implicit representations of nonlinear systems, where the input variables are eliminated, are studied in the differential geometric framework of jets of infinite order. In this context, flatness may be seen as a generalization of the property of uniformization of Hilbert’s 22nd problem. The notion of Lie-Bäcklund equivalence, introduced in [16], is adapted to this implicit setting. Lie-Bäcklund isomorphisms associated to a flat system, called trivializations, can be locally characterized in terms of polynomial matrices of the indeterminate d dt. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal. Various examples and consequences are presented. Keywords. Nonlinear system, implicit system, manifold of jets of infinite order, Hilbert’s 22nd problem, polynomial matrices, ideals, differential forms, differential flatness, flat output.