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On the Local Input-Output Stability of Event-Triggered Control Systems

Mohsen Ghodrat, Horacio J Marquez

TL;DR

The paper addresses preserving finite local $L_2$-gain performance for nonlinear event-triggered control systems subject to disturbances bounded by a Lipschitz function of the state. It develops a triggering framework grounded in an ISS/Lyapunov formulation that guarantees ${\|\mathscr{G}_e\|}_{\mathcal{L}_2}\le \Gamma$ when implementing a pre-designed continuous-time controller, and disturbances lie in $\mathscr{W}_Q$. To further reduce communication, it introduces an exponentially decaying triggering threshold and a discrete variant, proving that inter-event times remain uniformly nonzero and, in the absence of disturbances, the zero-input system is globally asymptotically stable; illustrative examples corroborate the theory. The work advances the design of event-triggered controllers by enabling quantitative inter-event-time guarantees and extending stability/performance guarantees to a broader class of nonlinear, disturbance-affected systems. This facilitates more reliable and efficient networked control where data transmission is constrained.

Abstract

This paper studies performance preserving event design in nonlinear event-based control systems based on a local L2-type performance criterion. Considering a finite gain local L2-stable disturbance driven continuous-time system, we propose a triggering mechanism so that the resulting sampled-data system preserves similar disturbance attenuation local L2-gain property. The results are applicable to nonlinear systems with exogenous disturbances bounded by some Lipschitz-continuous function of state. It is shown that an exponentially decaying function of time, combined with the proposed triggering condition, extends the inter-event periods. Compared to the existing works, this paper analytically estimates the increase in intersampling periods at least for an arbitrary period of time. We also propose a so-called discrete triggering condition to quantitatively find the improvement in inter-event times at least for an arbitrary number of triggering iterations. Illustrative examples support the analytically derived results.

On the Local Input-Output Stability of Event-Triggered Control Systems

TL;DR

The paper addresses preserving finite local -gain performance for nonlinear event-triggered control systems subject to disturbances bounded by a Lipschitz function of the state. It develops a triggering framework grounded in an ISS/Lyapunov formulation that guarantees when implementing a pre-designed continuous-time controller, and disturbances lie in . To further reduce communication, it introduces an exponentially decaying triggering threshold and a discrete variant, proving that inter-event times remain uniformly nonzero and, in the absence of disturbances, the zero-input system is globally asymptotically stable; illustrative examples corroborate the theory. The work advances the design of event-triggered controllers by enabling quantitative inter-event-time guarantees and extending stability/performance guarantees to a broader class of nonlinear, disturbance-affected systems. This facilitates more reliable and efficient networked control where data transmission is constrained.

Abstract

This paper studies performance preserving event design in nonlinear event-based control systems based on a local L2-type performance criterion. Considering a finite gain local L2-stable disturbance driven continuous-time system, we propose a triggering mechanism so that the resulting sampled-data system preserves similar disturbance attenuation local L2-gain property. The results are applicable to nonlinear systems with exogenous disturbances bounded by some Lipschitz-continuous function of state. It is shown that an exponentially decaying function of time, combined with the proposed triggering condition, extends the inter-event periods. Compared to the existing works, this paper analytically estimates the increase in intersampling periods at least for an arbitrary period of time. We also propose a so-called discrete triggering condition to quantitatively find the improvement in inter-event times at least for an arbitrary number of triggering iterations. Illustrative examples support the analytically derived results.

Paper Structure

This paper contains 13 sections, 15 theorems, 71 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Notation and Definitions
  4. Problem Setup
  5. $\mathcal{L}_2$-gain Performance of Event Triggered Nonlinear Systems
  6. Improving Average Sampling Frequency
  7. Continuous Triggering Condition Scenario
  8. Stability Analysis
  9. Inter-Event Lower Bound Comparison
  10. Discrete Triggering Condition Scenario
  11. Illustrative Examples
  12. Conclusion
  13. Appendix

Key Result

Theorem 2.1

The nonlinear system $\mathscr{G}$ is finite gain locally $\mathcal{L}_2$-stable with zero bias and has ${\|\mathscr{G}\|}_{\mathcal{L}_2}\leqslant {\Gamma}$, provided there exist a positive definite ${\bf{C}}^1$ function $V$ and a control input $u\in\mathscr{U}$ such that for all $w \in \mathscr{W}

Figures (6)

  • Figure 1: Verification of $\mathcal{L}_2$-gain.
  • Figure 2: System's trajectory (Left). Actuator signal (Right).
  • Figure 3: Verification of $\mathcal{L}_2$-gain (Left). Actuator signal (Right).
  • Figure 4: System's trajectories.
  • Figure 5: Verification of $\mathcal{L}_2$-gain (Left). Actuator signal (Right).
  • ...and 1 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • proof
  • Remark 2.1
  • proof
  • Definition 2.6
  • ...and 31 more