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Event-Based Control via Sparsity-Promoting Regularization: A Rollout Approach with Performance Guarantees

Shumpei Nishida, Kunihisa Okano

TL;DR

The work tackles sparse intermittent actuation in linear systems by balancing infinite-horizon performance with actuation rate through a rollout-based co-design. It constructs a discounted-cost surrogate with a base periodic policy, performs $h$-step lookahead to compute online control and triggering decisions, and proves a performance bound relative to the optimal periodic strategy plus mean-square stability. Theoretical guarantees include a finite-gap bound $J^a(\mu^{u,\text{ro}},\mu^{\delta,\text{ro}})\le J^a(\mu^{u,\text{per}},\mu^{\delta,\text{per}})+1/h}$ and mean-square stability, supported by ergodicity results for the induced Markov chain. A numerical example demonstrates favorable trade-offs against periodic control and an $\ell_1$-relaxed MPC baseline, highlighting the practical impact for networked and resource-constrained control systems.

Abstract

This paper presents a controller design framework aiming to balance control performance and actuation rate. Control performance is evaluated by an infinite-horizon average cost, and the number of control actions is penalized via sparsity-promoting regularization. Since the formulated optimal control problem has a combinatorial nature, we employ a rollout algorithm to obtain a tractable suboptimal solution. In the proposed scheme, actuation timings are determined through a multistage minimization procedure based on a receding-horizon approach, and the corresponding control inputs are computed online. We establish theoretical performance guarantees with respect to periodic control and prove the stability of the closed-loop system. The effectiveness of the proposed method is demonstrated through a numerical example.

Event-Based Control via Sparsity-Promoting Regularization: A Rollout Approach with Performance Guarantees

TL;DR

The work tackles sparse intermittent actuation in linear systems by balancing infinite-horizon performance with actuation rate through a rollout-based co-design. It constructs a discounted-cost surrogate with a base periodic policy, performs -step lookahead to compute online control and triggering decisions, and proves a performance bound relative to the optimal periodic strategy plus mean-square stability. Theoretical guarantees include a finite-gap bound and mean-square stability, supported by ergodicity results for the induced Markov chain. A numerical example demonstrates favorable trade-offs against periodic control and an -relaxed MPC baseline, highlighting the practical impact for networked and resource-constrained control systems.

Abstract

This paper presents a controller design framework aiming to balance control performance and actuation rate. Control performance is evaluated by an infinite-horizon average cost, and the number of control actions is penalized via sparsity-promoting regularization. Since the formulated optimal control problem has a combinatorial nature, we employ a rollout algorithm to obtain a tractable suboptimal solution. In the proposed scheme, actuation timings are determined through a multistage minimization procedure based on a receding-horizon approach, and the corresponding control inputs are computed online. We establish theoretical performance guarantees with respect to periodic control and prove the stability of the closed-loop system. The effectiveness of the proposed method is demonstrated through a numerical example.

Paper Structure

This paper contains 16 sections, 12 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proposed Rollout-Based algorithm
  4. Preliminaries for the Rollout Algorithm
  5. Optimal Periodic Policy and The Corresponding Cost
  6. Algorithm Design
  7. Performance and Stability Guarantees of the Proposed Algorithm
  8. Illustrative Example
  9. Conclusion
  10. Preliminaries for the Proofs
  11. Preliminary Lemmas
  12. Proof of Theorem \ref{['thm:performance_guarantee']}
  13. Proof of Lemma \ref{['lemma:foster_lyapunov']}
  14. Proof of Lemma \ref{['lemma:markov_chain']}
  15. Proof of Theorem \ref{['thm:ergodicity']}
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that Assumptions assump:eigenvalue_condition and assump:performance_and_stability_guarantees hold. Then, the following inequality holds:

Figures (3)

  • Figure 1: Considered feedback system.
  • Figure 2: Plot of the average control cost versus the average actuation rate.
  • Figure 3: Average control cost (top) and average actuation rate (bottom) as functions of $\theta$, with error bars indicating the standard deviation from the mean.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 11 more