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Robustifying Event-Triggered Control to Measurement Noise

Koen J. A. Scheres, Romain Postoyan, W. P. Maurice H. Heemels

TL;DR

This work addresses the practical challenge of measurement noise in event-triggered control (ETC) by introducing a general hybrid-model framework that guarantees input-to-state practical stability (ISpS) of a networked ETC system while enforcing a strictly positive minimum inter-event time (MIET) for all triggering mechanisms. The authors develop dynamic and static triggering rules based on space-regularization, ensuring Zeno-freeness without requiring differentiability of noise, and provide a prescriptive Lyapunov-based set of conditions (Assumptions) that ensure IS(p)S under network-induced errors. The framework is validated through case studies on nonlinear single-system stabilization and distributed consensus, including robustifications of Garcia2013, Dolk2019, and Berneburg schemes, with theoretical guarantees of SGiMIET and ISpS in various graph topologies. Numerical simulations with measurement noise illustrate that space-regularization yields larger average inter-event times while maintaining convergence to or near the desired sets, highlighting the practical impact for communication-constrained networks. Overall, the paper offers a versatile, scalable approach to designing noise-robust triggering conditions for distributed ETC in both single-plant and multi-agent settings, with rigorous stability and inter-event-time guarantees.

Abstract

While many event-triggered control strategies are available in the literature, most of them are designed ignoring the presence of measurement noise. As measurement noise is omnipresent in practice and can have detrimental effects, for instance, by inducing Zeno behavior in the closed-loop system and with that the lack of a positive lower bound on the inter-event times, rendering the event-triggered control design practically useless, it is of great importance to address this gap in the literature. To do so, we present a general framework for set stabilization of (distributed) event-triggered control systems affected by additive measurement noise. It is shown that, under general conditions, Zeno-free static as well as dynamic triggering rules can be designed such that the closed-loop system satisfies an input-to-state practical set stability property. We ensure Zeno-freeness by proving the existence of a uniform strictly positive lower-bound on the minimum inter-event time. The general framework is applied to point stabilization and consensus problems as particular cases, where we show that, under similar assumptions as the original work, existing schemes can be redesigned to robustify them to measurement noise. Consequently, using this framework, noise-robust triggering conditions can be designed both from the ground up and by simple redesign of several important existing schemes. Simulation results are provided that illustrate the strengths of this novel approach.

Robustifying Event-Triggered Control to Measurement Noise

TL;DR

This work addresses the practical challenge of measurement noise in event-triggered control (ETC) by introducing a general hybrid-model framework that guarantees input-to-state practical stability (ISpS) of a networked ETC system while enforcing a strictly positive minimum inter-event time (MIET) for all triggering mechanisms. The authors develop dynamic and static triggering rules based on space-regularization, ensuring Zeno-freeness without requiring differentiability of noise, and provide a prescriptive Lyapunov-based set of conditions (Assumptions) that ensure IS(p)S under network-induced errors. The framework is validated through case studies on nonlinear single-system stabilization and distributed consensus, including robustifications of Garcia2013, Dolk2019, and Berneburg schemes, with theoretical guarantees of SGiMIET and ISpS in various graph topologies. Numerical simulations with measurement noise illustrate that space-regularization yields larger average inter-event times while maintaining convergence to or near the desired sets, highlighting the practical impact for communication-constrained networks. Overall, the paper offers a versatile, scalable approach to designing noise-robust triggering conditions for distributed ETC in both single-plant and multi-agent settings, with rigorous stability and inter-event-time guarantees.

Abstract

While many event-triggered control strategies are available in the literature, most of them are designed ignoring the presence of measurement noise. As measurement noise is omnipresent in practice and can have detrimental effects, for instance, by inducing Zeno behavior in the closed-loop system and with that the lack of a positive lower bound on the inter-event times, rendering the event-triggered control design practically useless, it is of great importance to address this gap in the literature. To do so, we present a general framework for set stabilization of (distributed) event-triggered control systems affected by additive measurement noise. It is shown that, under general conditions, Zeno-free static as well as dynamic triggering rules can be designed such that the closed-loop system satisfies an input-to-state practical set stability property. We ensure Zeno-freeness by proving the existence of a uniform strictly positive lower-bound on the minimum inter-event time. The general framework is applied to point stabilization and consensus problems as particular cases, where we show that, under similar assumptions as the original work, existing schemes can be redesigned to robustify them to measurement noise. Consequently, using this framework, noise-robust triggering conditions can be designed both from the ground up and by simple redesign of several important existing schemes. Simulation results are provided that illustrate the strengths of this novel approach.
Paper Structure (26 sections, 11 theorems, 81 equations, 7 figures, 1 table)

This paper contains 26 sections, 11 theorems, 81 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Graph theory
  5. Hybrid systems
  6. Problem formulation
  7. Hybrid model
  8. Main result
  9. Main Assumption
  10. Dynamic triggering rules
  11. Static triggering rules
  12. Boundedness of $\xi$
  13. Case studies
  14. The nonlinear single-system case
  15. Consensus for multi-agent systems
  16. ...and 11 more sections

Key Result

Proposition 1

Suppose $\mathcal{H}$ is persistently flowing and let $\mathcal{A}\subset\mathbb{R}^{n_\xi}$ be a non-empty closed set. If there exist $V:\mathop{\mathrm{dom}}\limits V\to\mathbb{R}_{\geqslant0}$, $\alpha,\underline{\alpha},\overline{\alpha}\in\mathcal{K}_\infty$, $\gamma\in\mathcal{K}$ and $c\in\ma then $\mathcal{A}$ is ISpS. If, moreover, $c=0$ in item iii), then $\mathcal{A}$ is ISS.

Figures (7)

  • Figure 1: Networked control setup with Event-Triggering Mechanism (ETM). ETM $i$ determines when the current noisy output $\widetilde{y}_i$ is transmitted over the network.
  • Figure 2: The undirected communication topology used in the numerical examples.
  • Figure 3: Evolution of the states (top) and inter-event times (bottom) of the MAS using the dynamic trigger obtained by applying Corollary \ref{['cor:statictrigger']} to Proposition \ref{['prop:garcia']} with $c_i=0$ and initial condition $x(0,0)=(8,6,4,2,-2,-4,-8)$.
  • Figure 4: Evolution of the states (top) and inter-event times (bottom) of the MAS using the dynamic trigger obtained by applying Corollary \ref{['cor:statictrigger']} to Proposition \ref{['prop:garcia']} with $c_i=2\cdot10^{-6}$ and initial condition $x(0,0)=(8,6,4,2,-2,-4,-8)$.
  • Figure 5: Evolution of the states (top) and inter-event times (bottom) of the MAS using the dynamic trigger obtained by applying Theorem \ref{['thm:dynamictrigger']} to Proposition \ref{['prop:dolk']} with $c_i=0$ and initial condition $x(0,0)=(8,6,4,2,-2,-4,-8)$.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1: Heemels2021
  • Definition 2
  • Remark 1
  • Definition 3
  • Proposition 1
  • Definition 4
  • Theorem 1
  • Corollary 1
  • Remark 2
  • Remark 3
  • ...and 19 more