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Set-based and Dynamical Feedback-augmented Hands-off Control

Andrei Sperilă, Sorin Olaru, Stéphane Drobot

TL;DR

This work reframes hands-off control from a purely time-minimization perspective to a set-based, spatial approach for FD LTI systems. It introduces a hands-off region in the state space and uses dynamical feedback to keep the system's state inside this region, switching off control whenever possible. A joint, S-procedure–based synthesis procedure designs both a dynamical controller and lifted invariant sets that guarantee state and input constraints while enabling hands-off behavior with recursive feasibility and convergence guarantees. A numerical example with a discretized vehicle model demonstrates efficient, sub-millisecond computation and clear switching between open- and closed-loop modes, illustrating practicality and robustness of the approach. The results offer a principled, scalable path toward distributed or multi-agent implementations in contexts where centralized control is impractical.

Abstract

A novel set-theoretical approach to hands-off control is proposed, focusing on spatial arguments for command limitation rather than temporal ones. By employing dynamical feedback alongside invariant set-based constraints, actuation is employed only to drive the system's state within a "hands-off region" of its state-space, where the plant can freely evolve in open-loop configuration. A computationally-efficient procedure with strong theoretical guarantees is devised, and its effectiveness is showcased via an intuitive practical example.

Set-based and Dynamical Feedback-augmented Hands-off Control

TL;DR

This work reframes hands-off control from a purely time-minimization perspective to a set-based, spatial approach for FD LTI systems. It introduces a hands-off region in the state space and uses dynamical feedback to keep the system's state inside this region, switching off control whenever possible. A joint, S-procedure–based synthesis procedure designs both a dynamical controller and lifted invariant sets that guarantee state and input constraints while enabling hands-off behavior with recursive feasibility and convergence guarantees. A numerical example with a discretized vehicle model demonstrates efficient, sub-millisecond computation and clear switching between open- and closed-loop modes, illustrating practicality and robustness of the approach. The results offer a principled, scalable path toward distributed or multi-agent implementations in contexts where centralized control is impractical.

Abstract

A novel set-theoretical approach to hands-off control is proposed, focusing on spatial arguments for command limitation rather than temporal ones. By employing dynamical feedback alongside invariant set-based constraints, actuation is employed only to drive the system's state within a "hands-off region" of its state-space, where the plant can freely evolve in open-loop configuration. A computationally-efficient procedure with strong theoretical guarantees is devised, and its effectiveness is showcased via an intuitive practical example.

Paper Structure

This paper contains 9 sections, 4 theorems, 54 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem Statement
  5. Main Results
  6. Theoretical Aspects
  7. Synthesis Procedure
  8. Numerical Example
  9. Conclusions and Future Work

Key Result

Theorem III.1

Consider a system of type eq:ss_sys and a controller of type eq:ss_ctl, for which $A_{CL}$ is a Schur matrix and for which $\Omega_I$ along with $\Omega_O$ satisfy I1)-I3) and O1)-O4), respectively. If the system's initialization satisfies $x[k_0]\in\mathcal{S}_c\,$, then applying Algorithm alg:ho_p

Figures (4)

  • Figure 1: The greyscale shading inside the sets indicates the connection between states and their corresponding control action: dark grey indicates admissible states that will never be reached (due to the applied commands), medium grey marks full "hands-on control" space, light grey designates the transitional space, and white is assigned to full "hands-off control" space.
  • Figure 2: The external acceleration profiles that act upon the vehicle.
  • Figure 3: State evolution of the vehicle, under the action of Algorithm \ref{['alg:ho_periodic']}.
  • Figure 4: The feedback actuation and switching computed by Algorithm \ref{['alg:ho_periodic']}.

Theorems & Definitions (13)

  • Remark II.1
  • Remark III.1
  • Theorem III.1
  • proof
  • Corollary III.1
  • proof
  • Remark III.2
  • Theorem III.2
  • proof
  • Theorem III.3
  • ...and 3 more