Flatness necessary and sufficient conditions for nonlinear fractional systems using fractional differential forms

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders: a fractional exterior derivative is defined [1]. This definition is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. A characterization of differentially flat nonlinear systems in implicit representation, where the input variables are eliminated. Lie-Bäcklund isomorphisms associated to a flat system, called trivializations, can be locally characterized in terms of polynomial matrices. These enable to compute the ideal of differential forms generated by the differentials of all possible trivializations. After introducing the notion of strongly closed ideal, flatness is equivalent to the strong closedness of the differential form ideal.