Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical Discretization Schemes that Preserve Flatness

Ashutosh Jindal, Florentina Nicolau, David Martin Diego, Ravi Banavar

TL;DR

The work tackles preserving differential and difference flatness under discretization for nonlinear control systems. It introduces discretization maps on manifolds to construct geometry-preserving discretizations and leverages dynamic endogenous feedback to linearize, enabling flatness preservation in the discrete domain. The main theorem constructs flatness-preserving discretizations by discretizing the linearized form and lifting via a local diffeomorphism to the original system, ensuring that flat outputs remain intact. A concrete illustrative example demonstrates the method and simulations validate first-order accuracy and practical applicability for discrete-time, flatness-based control.

Abstract

Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems. Although many control systems evolve in continuous time, control implementation is performed digitally, requiring discretization. It is well known in the literature that discretization does not necessarily preserve structural properties, and it has been established that, in general, flatness is not preserved under discretization (whether exact or approximate). In this paper, inspired by our previous work [1] and based on the notion of discretization maps, we construct numerical schemes that preserve flatness.

Numerical Discretization Schemes that Preserve Flatness

TL;DR

The work tackles preserving differential and difference flatness under discretization for nonlinear control systems. It introduces discretization maps on manifolds to construct geometry-preserving discretizations and leverages dynamic endogenous feedback to linearize, enabling flatness preservation in the discrete domain. The main theorem constructs flatness-preserving discretizations by discretizing the linearized form and lifting via a local diffeomorphism to the original system, ensuring that flat outputs remain intact. A concrete illustrative example demonstrates the method and simulations validate first-order accuracy and practical applicability for discrete-time, flatness-based control.

Abstract

Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems. Although many control systems evolve in continuous time, control implementation is performed digitally, requiring discretization. It is well known in the literature that discretization does not necessarily preserve structural properties, and it has been established that, in general, flatness is not preserved under discretization (whether exact or approximate). In this paper, inspired by our previous work [1] and based on the notion of discretization maps, we construct numerical schemes that preserve flatness.

Paper Structure

This paper contains 12 sections, 3 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Differential and Difference Flatness
  3. Differential Flatness (Continuous Time)
  4. Difference Flatness (Discrete time)
  5. Flatness Preserving Discretization
  6. Discretization Maps
  7. Main Result: Flatness Preserving Discretizations
  8. An Illustrative Example
  9. Flatness preserving discretization
  10. Trajectory tracking using difference flatness
  11. Simulation Results
  12. Conclusion

Key Result

Proposition III.3

Let $\mathcal{D}$ be any arbitrary discretization map on $\mathrm{T}\mathcal{X}$ and $\mathcal{D}^{-1}$ be its inverse. Let $\dot{x} = f(x,u)$, $x\in\mathcal{X}$, $u\in\mathcal{U}$, be a given control system. For a given stepsize $h>0$, and a given control sequence $k\longmapsto u[k]$, define a sequ with $\mathrm{T}\mathcal{X}\ni(x,\nu)\longmapsto\tau_{\mathcal{X}}(x,\nu) = x$ as the canonical pro

Figures (5)

  • Figure 3.1: Discretization map $\mathcal{D}$ mapping points from $\mathrm{T}\mathcal{M}$ on to $\mathcal{M}\times\mathcal{M}$.
  • Figure 3.2: Discretization scheme for $\dot x = f(x,u)$, from a given discretization map $\mathcal{D}$.
  • Figure 3.3: $\mathcal{D}$ and $\mathcal{D}'$ commute.
  • Figure 3.4: Discrete time implementation of \ref{['ctssys']} via a flatness preserving discretization scheme, with flat outputs $\varphi_i$, $1\leq i\leq m$.
  • Figure 5.1: Simulation Results for $h=50$ milliseconds: (a) system trajectories $x[k],x(t)$, (b) Control input $u$, (c) Relative error $\left\lVert x[k]-x(kh) \right\rVert/\left\lVert x(kh) \right\rVert$, the accumulated error is reset every 1 second by setting $x[k]=x(kh)$ and (d) Flat outputs $(z_1,z_4)$ tracking reference $(z^*_1,z^*_4)$

Theorems & Definitions (8)

  • Example III.1
  • Proposition III.3: Discretization of a vector field 21MBLDMdD
  • Example III.4
  • Proposition III.5: retr_disc_map
  • Theorem III.6: Flatness Preserving Discretization
  • proof
  • Remark III.7
  • Remark IV.1