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Event-triggered control of nonlinear systems from data

Hailong Chen, Claudio De Persis, Andrea Bisoffi, Pietro Tesi

TL;DR

This work develops a data-driven framework for event-triggered control of unknown nonlinear systems by combining two controller design philosophies—nonlinearity cancellation and contraction—with two data-derived triggering policies (error-state and error-library). The emulation-based approach uses offline data and a library of basis functions to certify Lyapunov-based stability and to guarantee a positive minimum inter-event time, avoiding reliance on ISS assumptions. The authors provide convex optimization formulations to obtain stabilizing controllers from data and derive triggering conditions with provable inter-event-time bounds and invariant sets, demonstrated through a polynomial system and an inverted pendulum. The results offer a practical path for stabilizing nonlinear systems directly from data while efficiently coordinating control updates over networks. The framework also discusses robustness considerations and potential extensions to noisy data and derivative-free implementations.

Abstract

In a recent paper [8], we introduced a data-based approach to design event-triggered controllers for linear systems directly from data. Here, we extend the results in [8] to a class of nonlinear systems. We provide two data-based designs certified by a (classical) Lyapunov function. For these two designs, we devise event-triggered policies that rely on the previously found Lyapunov function, have parameters tuned from data, ensure a positive minimum inter-event time, and act based either on the state error or on the library error. These two different policies, and their respective advantages, are illustrated numerically.

Event-triggered control of nonlinear systems from data

TL;DR

This work develops a data-driven framework for event-triggered control of unknown nonlinear systems by combining two controller design philosophies—nonlinearity cancellation and contraction—with two data-derived triggering policies (error-state and error-library). The emulation-based approach uses offline data and a library of basis functions to certify Lyapunov-based stability and to guarantee a positive minimum inter-event time, avoiding reliance on ISS assumptions. The authors provide convex optimization formulations to obtain stabilizing controllers from data and derive triggering conditions with provable inter-event-time bounds and invariant sets, demonstrated through a polynomial system and an inverted pendulum. The results offer a practical path for stabilizing nonlinear systems directly from data while efficiently coordinating control updates over networks. The framework also discusses robustness considerations and potential extensions to noisy data and derivative-free implementations.

Abstract

In a recent paper [8], we introduced a data-based approach to design event-triggered controllers for linear systems directly from data. Here, we extend the results in [8] to a class of nonlinear systems. We provide two data-based designs certified by a (classical) Lyapunov function. For these two designs, we devise event-triggered policies that rely on the previously found Lyapunov function, have parameters tuned from data, ensure a positive minimum inter-event time, and act based either on the state error or on the library error. These two different policies, and their respective advantages, are illustrated numerically.

Paper Structure

This paper contains 19 sections, 5 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. System and control objective
  4. Scenario of interest
  5. Learning a feedback controller from data
  6. Linearization method
  7. Contractive design
  8. Learning a triggering policy from data
  9. Error-state threshold
  10. Error-library threshold
  11. Discussion
  12. Numerical examples
  13. A polynomial system
  14. The triggering policy with error-state threshold
  15. The triggering policy with error-library threshold
  16. ...and 4 more sections

Key Result

Lemma 1

Let Assumptions ass:library-ass:rich hold. Consider any matrix $K \in \mathbb R^{m \times s}$. Then, systemZ with the control law $u=K\zeta(x)$ results in the closed-loop dynamics where $G \in \mathbb R^{T \times s}$ is any solution to $\left[\right]=\left[\right] G$.

Figures (4)

  • Figure 1: The set $\mathcal{Z}$ and the largest estimate sub-level set $\mathcal{R}_{\gamma^\star}$ included in $\mathcal{Z}$.
  • Figure 2: Left: state solutions. Middle: evolution of $|e(\cdot)|$ and $\sigma|x(\cdot)|$. Right: triggering inter-event times.
  • Figure 3: The set $\mathcal{V}$ and the largest estimate sub-level set $\mathcal{R}_{\gamma^\star}$ included in $\mathcal{V}$.
  • Figure 4: Left: state solutions. Middle: evolution of $|e(\cdot)|$ and $\sigma|x(\cdot)|$. Right: triggering inter-event times.

Theorems & Definitions (10)

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