Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions
Jean Lévine
TL;DR
The paper addresses structural conditions for differential flatness achieved by pure prolongation in nonlinear multi-input systems. It introduces a two-filtration framework on prolonged distributions, proving that flatness by pure prolongation holds iff Δ_k^(𝚥) is involutive with invariant, locally constant dimension and saturated controllability, with G_k^(𝚥) decomposing as Γ_k^(𝚥) ⊕ Δ_k^(𝚥); it then provides a finite algorithm to compute minimal prolongation orders and corresponding flat outputs. Through multiple two-input and three-input examples, including a pendulum that is differentially flat but not flat by pure prolongation, the method demonstrates both its constructive power and its limitations. The results yield a computationally tractable pathway to synthesize flat outputs via pure prolongations, offering practical tools for motion planning and trajectory tracking in constrained nonlinear systems.
Abstract
In this article, we introduce the notion of differential flatness by pure prolongation: loosely speaking, a system admits this property if, and only if, there exists a pure prolongation of finite order such that the prolonged system is feedback linearizable. We obtain Lie-algebraic necessary and sufficient conditions for a general nonlinear multi-input system to satisfy this property. These conditions are comprised of the involutivity and relative invariance of a pair of filtrations of distributions of vector fields. An algorithm computing the minimal prolongation lengths of the input channels that achieve the system linearization, yielding the associated flat outputs, is deduced. Examples that show the efficiency and computational tractability of the approach are then presented.