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Learning Control Barrier Functions with Deterministic Safety Guarantees

Amy K. Strong, Ali Kashani, Claus Danielson, Leila Bridgeman

TL;DR

This work tackles safe set synthesis with deterministic guarantees for Lipschitz, discrete-time dynamics from data by learning a continuous piecewise affine barrier function $W$ over a triangulation and a data-driven classifier $\gamma$ to enforce the BF condition $W(g(\mathbf{x},\mathbf{u}))-\gamma(W(\mathbf{x}),\mathbf{x})\le0$. The nonconvex learning problem is solved via Iterative Convex Overbounding (ICO), which alternates convex updates to maximize the safe-set volume while ensuring the BF condition holds on each simplex; this yields a provably safe, contractive sublevel set ${\mathcal{S}}=\{\mathbf{x}\mid W(\mathbf{x})\le0\}$ contained in $\mathcal{X}$. The CPA/triangle-based representation affords hard BF constraints and data efficiency, with convergence to a local minimum and the option to adaptively add data in regions where the constraint is tight. Experiments on 2D autonomous and non-autonomous systems show the learned safe sets can outperform model-based baselines in volume and closely approach maximal invariant sets, validating the approach's practicality for data-driven safety guarantees.

Abstract

Barrier functions (BFs) characterize safe sets of dynamical systems, where hard constraints are never violated as the system evolves over time. Computing a valid safe set and BF for a nonlinear (and potentially unmodeled), non-autonomous dynamical system is a difficult task. This work explores the design of BFs using data to obtain safe sets with deterministic assurances of control invariance. We leverage ReLU neural networks (NNs) to create continuous piecewise affine (CPA) BFs with deterministic safety guarantees for Lipschitz continuous, discrete-time dynamical system using sampled one-step trajectories. The CPA structure admits a novel classifier term to create a relaxed \ac{bf} condition and construction via a data driven constrained optimization. We use iterative convex overbounding (ICO) to solve this nonconvex optimization problem through a series of convex optimization steps. We then demonstrate our method's efficacy on two-dimensional autonomous and non-autonomous dynamical systems.

Learning Control Barrier Functions with Deterministic Safety Guarantees

TL;DR

This work tackles safe set synthesis with deterministic guarantees for Lipschitz, discrete-time dynamics from data by learning a continuous piecewise affine barrier function over a triangulation and a data-driven classifier to enforce the BF condition . The nonconvex learning problem is solved via Iterative Convex Overbounding (ICO), which alternates convex updates to maximize the safe-set volume while ensuring the BF condition holds on each simplex; this yields a provably safe, contractive sublevel set contained in . The CPA/triangle-based representation affords hard BF constraints and data efficiency, with convergence to a local minimum and the option to adaptively add data in regions where the constraint is tight. Experiments on 2D autonomous and non-autonomous systems show the learned safe sets can outperform model-based baselines in volume and closely approach maximal invariant sets, validating the approach's practicality for data-driven safety guarantees.

Abstract

Barrier functions (BFs) characterize safe sets of dynamical systems, where hard constraints are never violated as the system evolves over time. Computing a valid safe set and BF for a nonlinear (and potentially unmodeled), non-autonomous dynamical system is a difficult task. This work explores the design of BFs using data to obtain safe sets with deterministic assurances of control invariance. We leverage ReLU neural networks (NNs) to create continuous piecewise affine (CPA) BFs with deterministic safety guarantees for Lipschitz continuous, discrete-time dynamical system using sampled one-step trajectories. The CPA structure admits a novel classifier term to create a relaxed \ac{bf} condition and construction via a data driven constrained optimization. We use iterative convex overbounding (ICO) to solve this nonconvex optimization problem through a series of convex optimization steps. We then demonstrate our method's efficacy on two-dimensional autonomous and non-autonomous dynamical systems.

Paper Structure

This paper contains 20 sections, 7 theorems, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Problem Statement and Preliminaries
  5. Barrier Functions and Safe Sets
  6. CPA Functions
  7. CPA Barrier Function
  8. Implementation via Iterative Convex Overbounding
  9. Computation Complexity
  10. Numerical Experiments
  11. Iterative Convex Overbounding
  12. Proof of Lemma \ref{['lem:icoSoln']}
  13. Initialization of ICO
  14. Convergence properties of Algorithm \ref{['alg:ico']}
  15. Searching for additional data
  16. ...and 5 more sections

Key Result

lemma 1

Consider the eq:dynSys and let Assumption assum:Lipschitz hold. Define the function $\gamma:\mathbb R \times \mathbb R^n\rightarrow\mathbb R$, where $\gamma(0,\mathbf{x}) = 0$ and $\langle \gamma (y,\mathbf{x}), y \rangle \geq 0$ for all $y\in \mathbb{R}^{}, \mathbf{x}\in \mathbb{R}^{n}.$ For $W:\ma

Figures (5)

  • Figure 1: The resulting safe sets found for the nonlinear autonomous system \ref{['fig:nonlinSys']} and the linear non-autonomous system \ref{['fig:linearSys_nonAut']}. The safe set boundary is in blue, while safe sets found using model based methods are shown as green polygons. The values of $\gamma$ across the triangulation $\mathcal{T} \subseteq \mathcal{X}$ are shown for the nonlinear autonomous system \ref{['fig:nonlinSys_gamma']} and linear non-autonomous system \ref{['fig:linearSys_nonAut_gamma']}. Black denotes a minimal value of $\gamma$ (near $0.1$) and bright yellow denotes the maximal value of $\gamma$ (ranging from 12.4 (\ref{['fig:nonlinSys_gamma']}) to 4.456 (\ref{['fig:linearSys_nonAut_gamma']})).
  • Figure 2: The resulting safe sets found for the linear autonomous system \ref{['fig:linearSys']} The safe set boundary is in blue, while the maximal safe set for the system as found by MPT3 is in green. In \ref{['fig:linearSysAlpha']}, the value of $\gamma$ are shown across the simplices of the triangualtion.
  • Figure 3: The resulting CPA barrier function found for the linear autonomous system \ref{['fig:linearSys_bf']} and nonlinear autonomous system \ref{['fig:nonlinearSys_bf']} are shown. The high data density for the nonlinear system results in very small simplices, while the linear system's bf was generated with much fewer data points.
  • Figure 4: In \ref{['fig:linearSys_addPoints']}, the safe set found for the linear autonomous system when using adaptive sampling during Algorithm \ref{['alg:ico']} is shown. The safe set boundary is in blue, which approaches the maximal safe set of the system (find via model based geometric methods MPT3), which is shown as the green polygon. In \ref{['fig:linearSys_addPoints_alpha']}, the corresponding $\gamma$ values across the triangulation are shown -- with black representing a smaller value of $\gamma$ near the minimum of 0.01 and yellow representing a higher value of $\gamma$ near the maximum of 14.45.
  • Figure 5: The cpa bf found for the linear, non-autonomous system is shown, where the function is affine on each simplex of the triangualtion.

Theorems & Definitions (17)

  • definition 1
  • lemma 1
  • proof
  • definition 2
  • definition 3
  • definition 4
  • lemma 2
  • lemma 3
  • theorem 1
  • proof
  • ...and 7 more