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Asymptotic Behavior of Inter-Event Times in Planar Systems under Event-Triggered Control

Anusree Rajan, Pavankumar Tallapragada

TL;DR

This work addresses predicting the asymptotic inter-event times in planar LTI systems under scale-invariant event-triggering by formulating the problem through an angle map $\phi$ that evolves the state angle between successive events. The core method expresses inter-event times as $\tau_e(x)=\tau_s(\theta)$ and studies fixed points of $\phi$ to determine whether inter-event times converge to a constant, become periodic, or fail to converge, with ergodic theory providing a route to the asymptotic average $\tau_{avg}$. Key contributions include sufficient conditions for convergence or non-convergence of inter-event times to a constant, a framework for analyzing $\tau_{avg}$ via ergodicity and rotation theory, and detailed results for the special case $M(\cdot)=M_2(\cdot)$, complemented by numerical demonstrations. The findings offer principled guidance for scheduling and resource planning in networked control systems under broad scale-invariant triggering rules, and lay the groundwork for extending the angle-map approach to higher-dimensional or nonlinear settings.

Abstract

This paper analyzes the asymptotic behavior of inter-event times in planar linear systems, under event-triggered control with a general class of scale-invariant event triggering rules. In this setting, the inter-event time is a function of the angle of the state at an event. This viewpoint allows us to analyze the inter-event times by studying the fixed points of the angle map, which represents the evolution of the angle of the state from one event to the next. We provide a sufficient condition for the convergence or non-convergence of inter-event times to a steady state value under a scale-invariant event-triggering rule. Following up on this, we further analyze the inter-event time behavior in the special case of threshold based event-triggering rule and we provide various conditions for convergence or non-convergence of inter-event times to a constant. We also analyze the asymptotic average inter-event time as a function of the angle of the initial state of the system. With the help of ergodic theory, we provide a sufficient condition for the asymptotic average inter-event time to be a constant for all non-zero initial states of the system. Then, we consider a special case where the angle map is an orientation-preserving homeomorphism. Using rotation theory, we comment on the asymptotic behavior of the inter-event times, including on whether the inter-event times converge to a periodic sequence. We illustrate the proposed results through numerical simulations.

Asymptotic Behavior of Inter-Event Times in Planar Systems under Event-Triggered Control

TL;DR

This work addresses predicting the asymptotic inter-event times in planar LTI systems under scale-invariant event-triggering by formulating the problem through an angle map that evolves the state angle between successive events. The core method expresses inter-event times as and studies fixed points of to determine whether inter-event times converge to a constant, become periodic, or fail to converge, with ergodic theory providing a route to the asymptotic average . Key contributions include sufficient conditions for convergence or non-convergence of inter-event times to a constant, a framework for analyzing via ergodicity and rotation theory, and detailed results for the special case , complemented by numerical demonstrations. The findings offer principled guidance for scheduling and resource planning in networked control systems under broad scale-invariant triggering rules, and lay the groundwork for extending the angle-map approach to higher-dimensional or nonlinear settings.

Abstract

This paper analyzes the asymptotic behavior of inter-event times in planar linear systems, under event-triggered control with a general class of scale-invariant event triggering rules. In this setting, the inter-event time is a function of the angle of the state at an event. This viewpoint allows us to analyze the inter-event times by studying the fixed points of the angle map, which represents the evolution of the angle of the state from one event to the next. We provide a sufficient condition for the convergence or non-convergence of inter-event times to a steady state value under a scale-invariant event-triggering rule. Following up on this, we further analyze the inter-event time behavior in the special case of threshold based event-triggering rule and we provide various conditions for convergence or non-convergence of inter-event times to a constant. We also analyze the asymptotic average inter-event time as a function of the angle of the initial state of the system. With the help of ergodic theory, we provide a sufficient condition for the asymptotic average inter-event time to be a constant for all non-zero initial states of the system. Then, we consider a special case where the angle map is an orientation-preserving homeomorphism. Using rotation theory, we comment on the asymptotic behavior of the inter-event times, including on whether the inter-event times converge to a periodic sequence. We illustrate the proposed results through numerical simulations.
Paper Structure (20 sections, 19 theorems, 33 equations, 4 figures)

This paper contains 20 sections, 19 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contributions
  4. Organization
  5. Notation
  6. Problem Setup
  7. System Dynamics
  8. Triggering Rules
  9. Objective
  10. Inter-event time as a function of the state
  11. Properties of the inter-event time function
  12. Continuity of the inter-event time function $\tau_s(\theta)$
  13. Evolution of the inter-event time
  14. Stability of the Fixed Points of the Angle map
  15. Analysis of fixed points of $\phi(.)$ with $M(.)=M_2(.)$
  16. ...and 5 more sections

Key Result

Lemma 3

(For any fixed $\tau$, $f_s(\theta,\tau)$ is a sinusoidal function with a shift in phase and mean). Let $m_{ij}(\tau)$ be the $(ij)^{th}$ element of $M(\tau) \in \mathbb{R}^{2 \times 2}$. For any fixed $\tau \in \mathbb{R}_{>0}$,

Figures (4)

  • Figure 1: Simulation results of Case 1 when $A_c$ has real eigenvalues at $\{-0.5528,-1.4472\}$.
  • Figure 2: Simulation results of Case 2 when $A_c$ has complex conjugate eigenvalues at $\{-1+i,-1-i\}$.
  • Figure 3: Simulation results of Case 3 when $A_c$ has complex conjugate eigenvalues at $[-1+0.2i,-1-0.2i]$.
  • Figure 4: Simulation results of Case 4 with discontinuous inter-event time function.

Theorems & Definitions (47)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • proof
  • ...and 37 more