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Minimum-Energy Control For Control-Affine Systems

Cyprien Tamekue, Zongxi Yu, ShiNung Ching

Abstract

In this letter, we derive minimum-energy controls for a broad class of control-affine systems using a Lagrange multiplier fixed-point equation and a generally non-symmetric Gramian-like matrix. In feasible coercivity classes, this fixed point is unique and can be computed by standard Picard iteration. These iterates converge with factorial decay, yielding an implementable, highly scalable synthesis with an intrinsic energy bound. As a demonstration of concept, we use uniform complete controllability results for linear time-varying systems to derive a bracket-generating condition ensuring complete controllability for time-dependent planar control-affine systems with scalar inputs. Special treatment for the unicycle kinematic model is also provided, and numerical examples illustrate the approach's effectiveness.

Minimum-Energy Control For Control-Affine Systems

Abstract

In this letter, we derive minimum-energy controls for a broad class of control-affine systems using a Lagrange multiplier fixed-point equation and a generally non-symmetric Gramian-like matrix. In feasible coercivity classes, this fixed point is unique and can be computed by standard Picard iteration. These iterates converge with factorial decay, yielding an implementable, highly scalable synthesis with an intrinsic energy bound. As a demonstration of concept, we use uniform complete controllability results for linear time-varying systems to derive a bracket-generating condition ensuring complete controllability for time-dependent planar control-affine systems with scalar inputs. Special treatment for the unicycle kinematic model is also provided, and numerical examples illustrate the approach's effectiveness.
Paper Structure (17 sections, 10 theorems, 66 equations, 1 figure)

This paper contains 17 sections, 10 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General assumptions and flow representation
  3. Representation of solutions and Gramians
  4. Main Results: Feasible coercivity classes and minimum-energy synthesis
  5. Examples
  6. Models considered and verification of control feasibility
  7. Planar control-affine systems with scalar inputs
  8. Fully-actuated control-affine systems
  9. Unicycle kinematic model
  10. Numerical illustrations
  11. Torque-controlled pendulum with varying length
  12. Fully-actuated 3D recurrent neural network (RNN)
  13. CONCLUSIONS
  14. Additional Proofs
  15. Proof of Lemma \ref{['lem:minimizer-is-fp']}
  16. ...and 2 more sections

Key Result

Lemma III.1

For all $(x^0,u)\in{\mathbb R}^d\times {\mathcal{Y}}$, system eq:state-dependent-input admits a unique absolutely continuous solution $x_u\in C^0([t_0, T]; {\mathbb R}^d)$ that can be represented by

Figures (1)

  • Figure 1: Comparison of the steering controls $\bar{u}_2$, $u_2$, and $u_{\mathrm{fl}}$ over six transfer problems: pendulum (P1--P2), 3D RNN (3D-1--3D-2), and unicycle (U1--U2). Top: control energy $\frac{1}{2}\|u\|_{L^2}^2$. Bottom: peak amplitude $\|u\|_{L^\infty}$. In all cases, the terminal error $\|x_u(T)-x^1\|_2$ is negligible, ranging from $10^{-8}$ to $10^{-15}$. We denote by P1--P2 the two pendulum transfers, by 3D-1--3D-2 the two 3D RNN transfers, and by U1--U2 the two unicycle transfers.

Theorems & Definitions (16)

  • Remark II.3
  • Lemma III.1
  • Lemma III.2
  • Definition 2
  • Definition 3: tamekue2025controlanalysis
  • Lemma IV.1
  • Lemma IV.2
  • Theorem IV.3: Main Theorem
  • Lemma IV.4
  • proof : Proof of Theorem \ref{['thm:main controllability result nonlinear']}
  • ...and 6 more