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Koopman-Based Event-Triggered Control from Data

Zeyad M. Manaa, Ayman M. Abdallah, Mohamed Ismail, Samil El Ferik

TL;DR

The paper develops a data-driven Koopman-based event-triggered control (KOETC) framework for discrete-time nonlinear systems, enabling a lifted linear representation to design a state-feedback gain $K$ and an event-triggering policy from data. Stability is guaranteed in the Lyapunov sense with exponential convergence by solving LMIs and leveraging the S-procedure to bound the triggering threshold $\gamma$. The approach relies on persistence of excitation to identify the lifted operators and demonstrates substantial communication reductions across three illustrative examples, including nonlinear and linear cases. The work provides a scalable route to data-driven ETC for nonlinear dynamics and points to future extensions such as time-varying parameters, partial observations, and alternative lifting strategies.

Abstract

Event-triggered Control (ETC) presents a promising paradigm for efficient resource usage in networked and embedded control systems by reducing communication instances compared to traditional time-triggered strategies. This paper introduces a novel approach to ETC for discrete-time nonlinear systems using a data-driven framework. By leveraging Koopman operator theory, the nonlinear system dynamics are globally linearized (approximately in practical settings) in a higher-dimensional space. We design a state-feedback controller and an event-triggering policy directly from data, ensuring exponential stability in Lyapunov sense. The proposed method is validated through extensive simulation experiments, demonstrating significant resource savings.

Koopman-Based Event-Triggered Control from Data

TL;DR

The paper develops a data-driven Koopman-based event-triggered control (KOETC) framework for discrete-time nonlinear systems, enabling a lifted linear representation to design a state-feedback gain and an event-triggering policy from data. Stability is guaranteed in the Lyapunov sense with exponential convergence by solving LMIs and leveraging the S-procedure to bound the triggering threshold . The approach relies on persistence of excitation to identify the lifted operators and demonstrates substantial communication reductions across three illustrative examples, including nonlinear and linear cases. The work provides a scalable route to data-driven ETC for nonlinear dynamics and points to future extensions such as time-varying parameters, partial observations, and alternative lifting strategies.

Abstract

Event-triggered Control (ETC) presents a promising paradigm for efficient resource usage in networked and embedded control systems by reducing communication instances compared to traditional time-triggered strategies. This paper introduces a novel approach to ETC for discrete-time nonlinear systems using a data-driven framework. By leveraging Koopman operator theory, the nonlinear system dynamics are globally linearized (approximately in practical settings) in a higher-dimensional space. We design a state-feedback controller and an event-triggering policy directly from data, ensuring exponential stability in Lyapunov sense. The proposed method is validated through extensive simulation experiments, demonstrating significant resource savings.

Paper Structure

This paper contains 20 sections, 3 theorems, 40 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Notations and Basic Definitions
  5. Problem Overview
  6. Persistence of Excitation
  7. Framework
  8. Koopman operator theory
  9. Event-triggered Control for the Lifted Representation of the Non-linear Dynamics
  10. Learning Controller From Data
  11. Learning the Triggering Policy from Data
  12. Illustrative Simulations and Results
  13. Illustrative Example 1: Proof of Concept
  14. Illustrative Example 2: Polynomial System
  15. Illustrative Example 3: Linear Case Note
  16. ...and 5 more sections

Key Result

Lemma 1

The equivalent data-driven closed loop representation of the system (eqn:closed_loop) under satisfaction of assumption ass:PE and where holds, takes the following form where $L$ and $N$ are $T\times p$ matrices. $\Box$

Figures (8)

  • Figure 1: Block diagram visually providing representation that illustrates the core concept underlying . It showcases various components and their interconnections, highlighting the essential principles and operational dynamics of the framework.
  • Figure 2: Illustration of the Koopman Operator: The red panel represents the generic nonlinear state-space. Conversely, the green panel represents the linear space.
  • Figure 3: Results of the illustrative example 1. (a) Behaviour of state $x_1$ and $x_2$ for both ETC and TTC over the horizon. (b) Norms of the error $\|e_k\|$ and the threshold parameter $\gamma \|x_k\|$. (c) Inter-event times $k_{i+1} - k_i$ showing the intervals between successive events.
  • Figure 4: A simulation of ten random initial conditions drawn from a uniform distribution $X \sim \mathcal{U}(-5, 5)$. The figure shows the behaviour of $x_1$ (left), and $x_2$ (right).
  • Figure 5: The relationship between $\alpha$ and the Lyapunov function decay rate. Simulations confirm no violations in the decay rate, as all points lie below the boundary $\max (V(k+1)/V(k)) = \alpha$, ensuring system stability across the tested $\alpha$ range.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Definition 1: Koopman Operator (KO)
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1: Data-driven representation digge2022datade2019formulas
  • proof
  • Remark 4
  • Theorem 1: Direct Controller Design
  • proof
  • Theorem 2: Optimal Threshold
  • ...and 1 more