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Observer-based Periodic Event-triggered and Self-triggered Boundary Control of a Class of Parabolic PDEs

Bhathiya Rathnayake, Mamadou Diagne

TL;DR

This work addresses the challenge of implementing boundary control for a class of parabolic PDEs with digital platforms by developing observer-based PETC and STC schemes grounded in PDE backstepping. It transforms a continuous-time triggering mechanism into periodic and self-triggered variants, guaranteeing Zeno-freeness, providing explicit sampling bounds, and preserving exponential stabilization in the spatial $L^2$ norm (global for PETC, local for STC). The contributions cover anti-collocated and collocated sensing/actuation configurations, with rigorous well-posedness proofs and numerical simulations validating the theoretical results. The proposed methods enable efficient, event-driven boundary control of PDEs in practical settings where continuous monitoring is infeasible.

Abstract

This paper introduces the first observer-based periodic event-triggered control (PETC) and self-triggered control (STC) for boundary control of a class of parabolic PDEs using PDE backstepping control. We introduce techniques to convert a certain class of continuous-time event-triggered control into PETC and STC, eliminating the need for continuous monitoring of the event-triggering function. For the PETC, the event-triggering function requires only periodic evaluations to detect events, while the STC proactively computes the time of the next event right at the current event time using the system model and the continuously available measurements. For both strategies, the control input is updated exclusively at events and is maintained using a zero-order hold between events. We demonstrate that the closed-loop system is Zeno-free. We offer criteria for selecting an appropriate sampling period for the PETC and for determining the time until the next event under the STC. We prove the system's global exponential convergence to zero in the spatial $L^2$ norm for both anti-collocated and collocated sensing and actuation under the PETC. For the STC, local exponential convergence to zero in the spatial $L^2$ norm for collocated sensing and actuation is proven. Simulations are provided to illustrate the theoretical claims.

Observer-based Periodic Event-triggered and Self-triggered Boundary Control of a Class of Parabolic PDEs

TL;DR

This work addresses the challenge of implementing boundary control for a class of parabolic PDEs with digital platforms by developing observer-based PETC and STC schemes grounded in PDE backstepping. It transforms a continuous-time triggering mechanism into periodic and self-triggered variants, guaranteeing Zeno-freeness, providing explicit sampling bounds, and preserving exponential stabilization in the spatial norm (global for PETC, local for STC). The contributions cover anti-collocated and collocated sensing/actuation configurations, with rigorous well-posedness proofs and numerical simulations validating the theoretical results. The proposed methods enable efficient, event-driven boundary control of PDEs in practical settings where continuous monitoring is infeasible.

Abstract

This paper introduces the first observer-based periodic event-triggered control (PETC) and self-triggered control (STC) for boundary control of a class of parabolic PDEs using PDE backstepping control. We introduce techniques to convert a certain class of continuous-time event-triggered control into PETC and STC, eliminating the need for continuous monitoring of the event-triggering function. For the PETC, the event-triggering function requires only periodic evaluations to detect events, while the STC proactively computes the time of the next event right at the current event time using the system model and the continuously available measurements. For both strategies, the control input is updated exclusively at events and is maintained using a zero-order hold between events. We demonstrate that the closed-loop system is Zeno-free. We offer criteria for selecting an appropriate sampling period for the PETC and for determining the time until the next event under the STC. We prove the system's global exponential convergence to zero in the spatial norm for both anti-collocated and collocated sensing and actuation under the PETC. For the STC, local exponential convergence to zero in the spatial norm for collocated sensing and actuation is proven. Simulations are provided to illustrate the theoretical claims.
Paper Structure (7 sections, 7 theorems, 57 equations, 4 figures)

This paper contains 7 sections, 7 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Continuous-time Event-triggered Control (CETC)
  3. Periodic Event-triggered Control (PETC) and Self-triggered Control (STC)
  4. Periodic Event-triggered Control (PETC)
  5. Self-triggered Control (STC)
  6. Numerical Simulations
  7. Conclusions

Key Result

Proposition 1

For every $u[t_{j}^\omega],\hat{u}[t_{j}^\omega]\in L^{2}(0,1)$, there exist unique solutions $u,\hat{u}:[t_j^\omega,t_{j+1}^\omega]\times[0,1]\rightarrow\mathbb{R}$ between two time instants $t_{j}^\omega$ and $t_{j+1}^\omega$ such that $u,\hat{u}\in C^{0}([t_{j}^\omega,t_{j+1}^\omega];L^{2}(0,1))\

Figures (4)

  • Figure 1: Evolution of $\Vert u[t]\Vert+\Vert\hat{u}[t]\Vert$
  • Figure 2: Boundary control inputs.
  • Figure 3: Dwell-times under CETC and PETC.
  • Figure 4: Dwell-times under STC.

Theorems & Definitions (9)

  • Proposition 1: rathnayake2021observer
  • Remark 1
  • Theorem 1: Results under CETC rathnayake2021observerrathnayake2022sampled
  • Theorem 2: Results under PETC
  • Theorem 3: Results under STC
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3