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On Driftless Systems with m controls and 2m or 2m-1 states that are Flat by Pure Prolongation

Jean Lévine, Jaume Franch

TL;DR

The paper addresses the problem of identifying when driftless nonlinear systems with $m$ inputs and $2m$ or $2m-1$ states are flat by pure prolongation, a restrictive yet algorithmically simple subclass of differential flatness. It leverages Levine's finite-prolongation algorithm to derive sufficient conditions for $P^2$-flatness in the three-input case, revealing two distinct minimal prolongations depending on the involutive structure of the input distributions, and then generalizes these results to arbitrary $m$-input systems with $2m-1$ or $2m$ states. The main contributions are twofold: (i) new, verifiable sufficient conditions that can differ from existing differential-flatness criteria (MR94), thus expanding the class of systems recognized as differentially flat; (ii) explicit procedures for computing flat outputs via annihilating appropriate involutive distributions and solving associated PDEs, including concrete PDE-based definitions of the flat outputs. The results provide practical criteria and constructive methods for trajectory planning and control design in a broad family of driftless systems, with clear directions for extending to more general drift or higher-dimensional input configurations.

Abstract

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems with two inputs have known necessary and sufficient conditions. For driftless systems with three or more inputs, the available conditions are only sufficient. This paper presents new findings on determining whether a system with m inputs and $2m$ or $2m-1$ states is flat by pure prolongation, a specific subclass of differential flatness. While this condition is more restrictive than general differential flatness, the algorithm for computing flat outputs remains remarkably simple, and the verification requirements are relatively lenient. Moreover, the conditions proposed in this work broaden the class of systems recognized as differentially flat, as our sufficient condition differs from existing criteria.

On Driftless Systems with m controls and 2m or 2m-1 states that are Flat by Pure Prolongation

TL;DR

The paper addresses the problem of identifying when driftless nonlinear systems with inputs and or states are flat by pure prolongation, a restrictive yet algorithmically simple subclass of differential flatness. It leverages Levine's finite-prolongation algorithm to derive sufficient conditions for -flatness in the three-input case, revealing two distinct minimal prolongations depending on the involutive structure of the input distributions, and then generalizes these results to arbitrary -input systems with or states. The main contributions are twofold: (i) new, verifiable sufficient conditions that can differ from existing differential-flatness criteria (MR94), thus expanding the class of systems recognized as differentially flat; (ii) explicit procedures for computing flat outputs via annihilating appropriate involutive distributions and solving associated PDEs, including concrete PDE-based definitions of the flat outputs. The results provide practical criteria and constructive methods for trajectory planning and control design in a broad family of driftless systems, with clear directions for extending to more general drift or higher-dimensional input configurations.

Abstract

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems with two inputs have known necessary and sufficient conditions. For driftless systems with three or more inputs, the available conditions are only sufficient. This paper presents new findings on determining whether a system with m inputs and or states is flat by pure prolongation, a specific subclass of differential flatness. While this condition is more restrictive than general differential flatness, the algorithm for computing flat outputs remains remarkably simple, and the verification requirements are relatively lenient. Moreover, the conditions proposed in this work broaden the class of systems recognized as differentially flat, as our sufficient condition differs from existing criteria.

Paper Structure

This paper contains 9 sections, 5 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Linearization by pure prolongations of driftless systems with 3 inputs
  4. Linearization by prolongations of driftless systems with $3$ inputs and $6$ states
  5. Linearization by prolongations of driftless systems with 3 inputs and 5 states
  6. Example
  7. Generalization to Driftless Systems with $m$ Inputs and $2m-1$ or $2m$ States
  8. Conclusions
  9. Appendix: Proof of Theorem \ref{['3-6theorem']}

Key Result

Theorem 1

A two input driftless system is differentially flat if, and only if, the ranks of the following distributions are $d_i=\mathrm{rank} D_i=i+2$, $\forall \, i=0,\dots,n-2$, with the usual notation $\{g_1,g_2\}$ for the distribution generated by $g_1$ and $g_2$ and with $[D,D] = \{ [\alpha, \beta] \mid \alpha, \beta \in D\}$, $[\alpha, \beta]$ denoting the Lie bracket of the vector fields $\alpha

Theorems & Definitions (9)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • proof