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Safe Event-triggered Gaussian Process Learning for Barrier-Constrained Control

Armin Lederer, Azra Begzadić, Sandra Hirche, Jorge Cortés, Sylvia Herbert

TL;DR

The paper tackles safety for unknown control-affine systems by integrating control barrier functions (CBFs) with safe, event-triggered Gaussian process (GP) learning. It derives a robust, data-efficient control law that preserves CBF feasibility with probability $1-\delta$ while adaptively collecting informative data via a safe excitation mechanism; Zeno behavior is ruled out through a lower-bounded inter-event time. The contributions include closed-form control updates that maintain feasibility during GP updates, probabilistic safety guarantees, and a mechanism to trade off sampling rate with excitation magnitude. Demonstrations in adaptive cruise control show the method closely matches exact-model safety and outperforms naive time-triggered or Zeno-prone schemes, highlighting practical impact for safety-critical, data-driven control.

Abstract

While control barrier functions (CBFs) are employed in addressing safety, control synthesis methods based on them generally rely on accurate system dynamics. This is a critical limitation, since the dynamics of complex systems are often not fully known. Supervised machine learning techniques hold great promise for alleviating this weakness by inferring models from data. We propose a novel \revision{approach for safe event-triggered learning of Gaussian process models in CBF-based continuous-time control for unknown control-affine systems. By applying a finite excitation at triggering times, our approach ensures a sufficient information gain to maintain the feasibility of the CBF-based safety condition with high probability. Our approach probabilistically guarantees safety based on a suitable GP prior and rules out} Zeno behavior in the triggering scheme. The effectiveness of the proposed approach and theory is demonstrated in simulations.

Safe Event-triggered Gaussian Process Learning for Barrier-Constrained Control

TL;DR

The paper tackles safety for unknown control-affine systems by integrating control barrier functions (CBFs) with safe, event-triggered Gaussian process (GP) learning. It derives a robust, data-efficient control law that preserves CBF feasibility with probability while adaptively collecting informative data via a safe excitation mechanism; Zeno behavior is ruled out through a lower-bounded inter-event time. The contributions include closed-form control updates that maintain feasibility during GP updates, probabilistic safety guarantees, and a mechanism to trade off sampling rate with excitation magnitude. Demonstrations in adaptive cruise control show the method closely matches exact-model safety and outperforms naive time-triggered or Zeno-prone schemes, highlighting practical impact for safety-critical, data-driven control.

Abstract

While control barrier functions (CBFs) are employed in addressing safety, control synthesis methods based on them generally rely on accurate system dynamics. This is a critical limitation, since the dynamics of complex systems are often not fully known. Supervised machine learning techniques hold great promise for alleviating this weakness by inferring models from data. We propose a novel \revision{approach for safe event-triggered learning of Gaussian process models in CBF-based continuous-time control for unknown control-affine systems. By applying a finite excitation at triggering times, our approach ensures a sufficient information gain to maintain the feasibility of the CBF-based safety condition with high probability. Our approach probabilistically guarantees safety based on a suitable GP prior and rules out} Zeno behavior in the triggering scheme. The effectiveness of the proposed approach and theory is demonstrated in simulations.
Paper Structure (18 sections, 5 theorems, 29 equations, 3 figures, 1 algorithm)

This paper contains 18 sections, 5 theorems, 29 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Notation
  4. Problem Setting
  5. Safe Control through Event-Triggered Learning
  6. Gaussian Process Regression
  7. Learning Models of Control-Affine Systems
  8. Safe Learning-Based Control Synthesis
  9. Excitation Requirement for Guaranteed Accuracy
  10. Safe Event-Tiggered Gaussian Process Learning
  11. Ruling out Zeno Behavior
  12. Numerical Evaluation in Adaptive Cruise Control
  13. Conclusions
  14. Proof of \ref{['lem:GPbound']}
  15. Proof of \ref{['prop:safety']}
  16. ...and 3 more sections

Key Result

Lemma 1

Consider a dynamical system eq:si_sys and a set $\mathcal{C}$ defined by a continuously differentiable function $\psi: \mathcal{X} \rightarrow \mathbb{R}$. If there exists an extended class $\mathcal{K}_{\infty}$ function $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ such that with $c_0(\bm{x})=\nabla_{\bm{x}}^T \psi (\bm{x})\bm{f}(\bm{x})+\alpha(\psi(\bm{x}))$ and $\bm{c}^T(\bm{x})=[c_1(\bm{x}), \ldo

Figures (3)

  • Figure 1: The continuous-time control approach in \ref{['alg:control_algorithm']} with event-triggered online learning results in almost identical velocity (top) and distance (bottom) trajectories as a CBF-based controller with exact model knowledge in contrast to an analogous approach with periodically updated GP model, which becomes infeasible after $\approx 6\,$s.
  • Figure 2: The event-triggered update scheme in \ref{['alg:control_algorithm']}probabilistically ensures the necessary model accuracy for guaranteeing safety by sampling data until the system state has converged. The excitation filter ensures that data is generated using sufficiently large control amplitudes $\bar{u}_{\mathrm{GP}}(\bm{x})$ with high probability.When the approach from castaneda2023 is applied, Zeno behavior occurs and essentially stops the simulation after $\approx 3s$.
  • Figure 3: The parameter $\gamma$ allows to trade-off the admissible peak control magnitude $\max_t|u(t)|$ and the minimum inter-event time $\min_N T_{N+1}-t_N$.

Theorems & Definitions (6)

  • Definition 1: Safety cohen_safety-critical_2024
  • Lemma 1: Control Barrier Functions cohen_safety-critical_2024
  • Lemma 2
  • Proposition 1
  • Theorem 1
  • Proposition 3