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Discrete-Time Event-Triggered Extremum Seeking

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstić, Frank Allgöwer

Abstract

This paper proposes a discrete-time event-triggered extremum seeking control scheme for real-time optimization of nonlinear systems. Unlike conventional discrete-time implementations relying on periodic updates, the proposed approach updates the control input only when a state-dependent triggering condition is satisfied, reducing unnecessary actuation and communication. The resulting closed-loop system combines extremum seeking with an event-triggering mechanism that adaptively determines the input update instants. Using discrete-time averaging and Lyapunov analysis, we establish practical convergence of the trajectories to a neighborhood of the unknown extremum point and show exponential stability of the associated average dynamics. The proposed method preserves the optimization capability of classical extremum seeking while significantly reducing the number of input updates. Simulation results illustrate the effectiveness of the approach for resource-aware real-time optimization.

Discrete-Time Event-Triggered Extremum Seeking

Abstract

This paper proposes a discrete-time event-triggered extremum seeking control scheme for real-time optimization of nonlinear systems. Unlike conventional discrete-time implementations relying on periodic updates, the proposed approach updates the control input only when a state-dependent triggering condition is satisfied, reducing unnecessary actuation and communication. The resulting closed-loop system combines extremum seeking with an event-triggering mechanism that adaptively determines the input update instants. Using discrete-time averaging and Lyapunov analysis, we establish practical convergence of the trajectories to a neighborhood of the unknown extremum point and show exponential stability of the associated average dynamics. The proposed method preserves the optimization capability of classical extremum seeking while significantly reducing the number of input updates. Simulation results illustrate the effectiveness of the approach for resource-aware real-time optimization.

Paper Structure

This paper contains 9 sections, 1 theorem, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Event-Triggered of the Discrete-Time Gradient-Based Extremum Seeking Tuning Law
  5. Event-Triggering Mechanism
  6. Average Closed-Loop System
  7. Stability Analysis
  8. Simulation results
  9. Conclusions

Key Result

Theorem 1

Consider the closed-loop average dynamics of the gradient estimate (eq:hatG_av_k+1_1), the average error vector (eq:e_av_k_1), under Assumption assumption_lyapunovEq, and the average static event-triggering mechanism given by Definition def:staticEvent_discrete_av. For $\epsilon>0$ sufficiently smal In addition, there exists a lower bound $k^{\ast}$ for the inter-execution interval $k_{l+1}-k_{l}$

Figures (2)

  • Figure 1: Event-Triggered for Discrete-Time Gradient-based Extremum Seeking Control.
  • Figure 2: Simulations of the Discrete-time Event-triggered Extremum Seeking Control System.

Theorems & Definitions (5)

  • Definition 1: Shift Operator, AW:1997
  • Definition 2: Event-Triggering Condition
  • Definition 3: Average Event-Triggering Condition
  • Theorem 1
  • proof