Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Observer-Based Performance-Barrier Event-Triggered Control of $2\times2$ Linear Hyperbolic PDEs

Eranda Somathilake, Mamadou Diagne

Abstract

Performance-barrier event-triggered control (P-ETC) is a methodology implemented to increase the dwell-times between events while still preserving a prescribed performance of the system under event-triggered control (ETC). This is achieved by considering the performance residual of the system, which is a measure of the system performance with respect to the prescribed performance. This allows the Lyapunov function candidate to deviate from decreasing monotonically. In order to determine the performance residual, it is required to know the full-state information, leading to all work related to P-ETC to be under full-state feedback. In this article, we propose a novel dynamic performance-barrier under output feedback with an exponentially convergent observer. We consider event-triggered boundary control of a class of $2\times2$ linear hyperbolic PDEs with anti-collocated measurements with the control input. Under the proposed P-ETC mechanism, we prove the existence of a minimum dwell-time, and show the global exponential stability of the spatial $L^2$ norm of the solution of the system. Simulation results are presented to validate the theoretical claims.

Observer-Based Performance-Barrier Event-Triggered Control of $2\times2$ Linear Hyperbolic PDEs

Abstract

Performance-barrier event-triggered control (P-ETC) is a methodology implemented to increase the dwell-times between events while still preserving a prescribed performance of the system under event-triggered control (ETC). This is achieved by considering the performance residual of the system, which is a measure of the system performance with respect to the prescribed performance. This allows the Lyapunov function candidate to deviate from decreasing monotonically. In order to determine the performance residual, it is required to know the full-state information, leading to all work related to P-ETC to be under full-state feedback. In this article, we propose a novel dynamic performance-barrier under output feedback with an exponentially convergent observer. We consider event-triggered boundary control of a class of linear hyperbolic PDEs with anti-collocated measurements with the control input. Under the proposed P-ETC mechanism, we prove the existence of a minimum dwell-time, and show the global exponential stability of the spatial norm of the solution of the system. Simulation results are presented to validate the theoretical claims.

Paper Structure

This paper contains 7 sections, 4 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries and Problem statement
  4. Continuous-Time Output Feedback Control and its Emulation
  5. Performance-Barrier Event-Triggered Control
  6. Numerical Simulations
  7. Conclusion

Key Result

Proposition 1

Let $j\in\mathbb{N}$, and $U(t)\in\mathbb{R}$ be constant for $t\in[t_j,t_{j+1})$. For a given $(u[t_j], v[t_j])^T \in L^2((0,1);\mathbb{R}^2)$ and $(\hat{u}[t_j], \hat{v}[t_j])^T \in L^2((0,1);\mathbb{R}^2)$, there exist unique solutions such that $(u, v)^T \in \mathcal{C}^0([t_j, t_{j+1}]; L^2((0

Figures (7)

  • Figure C1: Schematic of the proposed P-ETC mechanism
  • Figure C2: Schematic of the dynamics of $m(t)$
  • Figure D1: Variation of the spatial $L^2$ norm of the states with time
  • Figure D2: Variation of the control input with time
  • Figure D3: Dwell-times under the proposed triggering condition
  • ...and 2 more figures

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Lemma 1
  • Definition 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof