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Event-Triggered Newton Extremum Seeking for Multivariable Optimization

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstic, Paulo Tabuada

TL;DR

The paper addresses the slow and curvature-dependent convergence of multivariable gradient-based extremum seeking by introducing a static event-triggered Newton-based approach. It uses a Riccati-based estimator to dynamically compute the inverse Hessian $H^{*-1}$ via the update law $\dot{\Gamma}=\omega_r\Gamma-\omega_r\widehat{H}^{\!*}(t)\Gamma$, yielding a Newton-like controller $u(t)=-K\Gamma(t)\hat{G}(t)$ whose convergence rate can be tuned independently of the unknown curvature. Stability is established through averaging theory for discontinuous systems, proving local exponential practical stability and precluding Zeno behavior, while the event-triggered mechanism significantly reduces control updates. Numerical results demonstrate faster convergence and up to an 8x reduction in actuation frequency compared with gradient-based ES, highlighting the practical benefits for real-time, resource-constrained optimization in networked or embedded systems.

Abstract

This paper presents a static event-triggered control strategy for multivariable Newton-based extremum seeking. The proposed method integrates event-triggered actuation into the Newton-based optimization framework to reduce control updates while maintaining rapid convergence to the extremum. Unlike traditional gradient-based extremum seeking, where the convergence rate depends on the unknown Hessian of the cost function, the proposed approach employs a dynamic estimator of the Hessian inverse, formulated as a Riccati equation, enabling user-assignable convergence rates. The event-triggering mechanism is designed to minimize unnecessary actuation updates while preserving stability and performance. Using averaging theory, we establish local stability results and exponential convergence to a neighborhood of the unknown extremum point. Additionally, numerical simulations illustrate the benefits of the proposed approach over gradient-based and continuously actuated Newton-based extremum seeking, showing improved convergence rates and reduced control update frequency, leading to more efficient implementation in real-time optimization scenarios.

Event-Triggered Newton Extremum Seeking for Multivariable Optimization

TL;DR

The paper addresses the slow and curvature-dependent convergence of multivariable gradient-based extremum seeking by introducing a static event-triggered Newton-based approach. It uses a Riccati-based estimator to dynamically compute the inverse Hessian via the update law , yielding a Newton-like controller whose convergence rate can be tuned independently of the unknown curvature. Stability is established through averaging theory for discontinuous systems, proving local exponential practical stability and precluding Zeno behavior, while the event-triggered mechanism significantly reduces control updates. Numerical results demonstrate faster convergence and up to an 8x reduction in actuation frequency compared with gradient-based ES, highlighting the practical benefits for real-time, resource-constrained optimization in networked or embedded systems.

Abstract

This paper presents a static event-triggered control strategy for multivariable Newton-based extremum seeking. The proposed method integrates event-triggered actuation into the Newton-based optimization framework to reduce control updates while maintaining rapid convergence to the extremum. Unlike traditional gradient-based extremum seeking, where the convergence rate depends on the unknown Hessian of the cost function, the proposed approach employs a dynamic estimator of the Hessian inverse, formulated as a Riccati equation, enabling user-assignable convergence rates. The event-triggering mechanism is designed to minimize unnecessary actuation updates while preserving stability and performance. Using averaging theory, we establish local stability results and exponential convergence to a neighborhood of the unknown extremum point. Additionally, numerical simulations illustrate the benefits of the proposed approach over gradient-based and continuously actuated Newton-based extremum seeking, showing improved convergence rates and reduced control update frequency, leading to more efficient implementation in real-time optimization scenarios.
Paper Structure (17 sections, 4 theorems, 104 equations, 5 figures)

This paper contains 17 sections, 4 theorems, 104 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review of the Event-Triggered Multivariable Gradient-based Extremum Seeking
  3. Classical Gradient-based Extremum Seeking
  4. Static Gradient-Feedback Tuning Law
  5. Event-Triggered Multivariable Gradient-Based Extremum Seeking
  6. Event-Triggered Multivariable Newton-based Extremum Seeking
  7. Continuous-time Newton-based Extremum Seeking: Hessian Estimate and its Inverse
  8. Continuous-Time Tuning Law
  9. Emulation of the Continuous-Time Tuning Law
  10. Event-Triggering Mechanism
  11. Rescaling of Time
  12. Average Closed-Loop System
  13. Linearized Average Closed-Loop System
  14. Stability Analysis
  15. Simulation Results
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the dynamics of the gradient estimate given by equation (eq:dhatGdt_20250206_1) with control law (eq:u) emulated by $u(t)=K\hat{G}(t_{k})$, for all $t \in [t_k, t_{k+1})$, where $t_k$ denotes the most recent update time. The next update instant $t_{k+1}$ is determined by the event-triggerin where $a=\sqrt{\sum_{i=1}^{n}a_{i}^{2}}$, with $a_i$ defined in (eq:S_v1) and (eq:M_v1), and the co

Figures (5)

  • Figure 1: Multivariable Gradient-based Extremum Seeking.
  • Figure 2: Event-Triggered Multivariable Gradient-based Extremum Seeking.
  • Figure 3: Multivariable Newton-based Extremum Seeking.
  • Figure 4: Event-Triggered Multivariable Newton-based Extremum Seeking.
  • Figure 5: Static and Dynamic Event-triggered Multivariable Extremum Seeking Systems.

Theorems & Definitions (6)

  • Theorem 1
  • Definition 1: Event-Triggering Condition
  • Definition 2: Average Event-Triggering Condition
  • Theorem 2
  • Theorem 3
  • Theorem 4