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Learning Piecewise Residuals of Control Barrier Functions for Safety of Switching Systems using Multi-Output Gaussian Processes

Mohammad Aali, Jun Liu

TL;DR

This work addresses safety guarantees for switching (hybrid) systems under model uncertainty by learning piecewise residuals on CLF and CBF constraints. A batch multi-output Gaussian process (MOGP) with region-specific kernels is used to estimate residuals $d_r^V$ and $d_r^h$ for each switching surface, turning uncertainty-augmented constraints into a convex second-order cone program (SOCP). The authors provide probabilistic confidence bounds and a feasibility analysis, establishing necessary and sufficient conditions for SOCP feasibility. An episodic data-collection and retraining scheme iteratively improves the model, and simulations on switching adaptive cruise control show that the MOGP-SOCP approach maintains safety where baselines fail. The approach offers a tractable, real-time framework for safety-critical control in uncertain, hybrid environments, with potential extensions to impulse effects and overlapping regions.

Abstract

Control barrier functions (CBFs) have recently been introduced as a systematic tool to ensure safety by establishing set invariance. When combined with a control Lyapunov function (CLF), they form a safety-critical control mechanism. However, the effectiveness of CBFs and CLFs is closely tied to the system model. In practice, model uncertainty can jeopardize safety and stability guarantees and may lead to undesirable performance. In this paper, we develop a safe learning-based control strategy for switching systems in the face of uncertainty. We focus on the case that a nominal model is available for a true underlying switching system. This uncertainty results in piecewise residuals for each switching surface, impacting the CLF and CBF constraints. We introduce a batch multi-output Gaussian process (MOGP) framework to approximate these piecewise residuals, thereby mitigating the adverse effects of uncertainty. A particular structure of the covariance function enables us to convert the MOGP-based chance constraints CLF and CBF into second-order cone constraints, which leads to a convex optimization. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through a simulation of a switching adaptive cruise control system.

Learning Piecewise Residuals of Control Barrier Functions for Safety of Switching Systems using Multi-Output Gaussian Processes

TL;DR

This work addresses safety guarantees for switching (hybrid) systems under model uncertainty by learning piecewise residuals on CLF and CBF constraints. A batch multi-output Gaussian process (MOGP) with region-specific kernels is used to estimate residuals and for each switching surface, turning uncertainty-augmented constraints into a convex second-order cone program (SOCP). The authors provide probabilistic confidence bounds and a feasibility analysis, establishing necessary and sufficient conditions for SOCP feasibility. An episodic data-collection and retraining scheme iteratively improves the model, and simulations on switching adaptive cruise control show that the MOGP-SOCP approach maintains safety where baselines fail. The approach offers a tractable, real-time framework for safety-critical control in uncertain, hybrid environments, with potential extensions to impulse effects and overlapping regions.

Abstract

Control barrier functions (CBFs) have recently been introduced as a systematic tool to ensure safety by establishing set invariance. When combined with a control Lyapunov function (CLF), they form a safety-critical control mechanism. However, the effectiveness of CBFs and CLFs is closely tied to the system model. In practice, model uncertainty can jeopardize safety and stability guarantees and may lead to undesirable performance. In this paper, we develop a safe learning-based control strategy for switching systems in the face of uncertainty. We focus on the case that a nominal model is available for a true underlying switching system. This uncertainty results in piecewise residuals for each switching surface, impacting the CLF and CBF constraints. We introduce a batch multi-output Gaussian process (MOGP) framework to approximate these piecewise residuals, thereby mitigating the adverse effects of uncertainty. A particular structure of the covariance function enables us to convert the MOGP-based chance constraints CLF and CBF into second-order cone constraints, which leads to a convex optimization. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through a simulation of a switching adaptive cruise control system.
Paper Structure (13 sections, 7 theorems, 42 equations, 2 figures, 1 table)

This paper contains 13 sections, 7 theorems, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setup and statement
  3. Control Lyapunov Functions for Switching Systems
  4. Control Barrier Functions
  5. Impact of uncertainty in the switching systems
  6. Proposed batch mogp-based design for piecewise residuals
  7. Single-output GP for real-valued functions
  8. Batch multi-output GP for piecewise residuals
  9. Confidence bounds for the estimation of the effect of uncertainty
  10. Feasibility analysis
  11. Data collection and model fitting
  12. Simulation results
  13. Conclusion

Key Result

Corollary 1

Given CBF $h:\mathbb{R}^n \rightarrow \mathbb{R}$ with the associated set $\mathcal{C}$ in (eq3), if $\nabla h(\bm x) \neq 0$ for all $\bm x \in \partial \mathcal{C}$, any Lipschitz continuous controller $\boldsymbol{u}(\boldsymbol{x})$ satisfying (eq5), guarantees that the set $\mathcal{C}$ is forw

Figures (2)

  • Figure 1: Snapshots of the simulation to show the unsafe (red color) and safe (green color) distance between cars in different implementations. The nominal QP controller (left) and single GP-SOCP controller (middle) violate the safety distance. MOGP-based controller (right) ensures safety. See https://youtu.be/8nmcIIJSGJE for the simulation video.
  • Figure 2: Comparison of the MOGP-SOCP (yellow), single GP-SOCP (cyan), nominal QP (blue), and true design (red dashed). System state $x_2$ and distance between two cars $z$ (left column). The CBF and control input $u$ (right column).

Theorems & Definitions (15)

  • Definition 1: Class $\mathcal{K}$ function khalil2002nonlinear
  • Definition 2: Control Lyapunov functions
  • Definition 3: Control barrier function ames2019control
  • Corollary 1
  • Proposition 1
  • proof
  • Lemma 1: srinivas2009gaussian
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 5 more