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ReLU Barrier Functions for Nonlinear Systems with Constrained Control: A Union of Invariant Sets Approach

Pouya Samanipour, Hasan A. Poonawala

Abstract

Certifying safety for nonlinear systems with polytopic input constraints is challenging because CBF synthesis must ensure control admissibility under saturation. We propose an approximation--verification pipeline that performs convex barrier synthesis on piecewise-affine (PWA) surrogates and certifies safety for the original nonlinear system via facet-wise verification. To reduce conservatism while preserving tractability, we use a two-slope Leaky ReLU surrogate for the extended class-$\mathcal{K}$ function $α(\cdot)$ and combine multiple certificates using a Union of Invariant Sets (UIS). Counterexamples are handled through local uncertainty updates. Simulations on pendulum and cart-pole systems with input saturation show larger certified invariant sets than linear-$α$ designs with tractable computation time.

ReLU Barrier Functions for Nonlinear Systems with Constrained Control: A Union of Invariant Sets Approach

Abstract

Certifying safety for nonlinear systems with polytopic input constraints is challenging because CBF synthesis must ensure control admissibility under saturation. We propose an approximation--verification pipeline that performs convex barrier synthesis on piecewise-affine (PWA) surrogates and certifies safety for the original nonlinear system via facet-wise verification. To reduce conservatism while preserving tractability, we use a two-slope Leaky ReLU surrogate for the extended class- function and combine multiple certificates using a Union of Invariant Sets (UIS). Counterexamples are handled through local uncertainty updates. Simulations on pendulum and cart-pole systems with input saturation show larger certified invariant sets than linear- designs with tractable computation time.
Paper Structure (13 sections, 3 theorems, 32 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 13 sections, 3 theorems, 32 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Piecewise Affine Functions
  4. Forward Invariance and Barrier Functions
  5. Problem Description
  6. PWA Barrier Functions for Surrogate Dynamics
  7. Leaky ReLU: A Practical Choice for alpha(·) in BF Design
  8. Union of Invariant Sets (UIS)
  9. Constructing CBFs for Nonlinear Dynamics
  10. Cell-local robustification via uncertainty update
  11. Search algorithm
  12. Results and Simulations
  13. Conclusion

Key Result

Theorem 1

Let $S \subset \mathcal{D} \subseteq \mathbb{R}^n$ denote the super-level set eq:safeset of a locally Lipschitz function $h:\mathcal{D}\!\to\!\mathbb{R}$ under the closed-loop dynamics eq:cl-system. Assume $h(\mathcal{D})\subset [h_{\min},h_{\max}]$ with $h_{\min}<0<h_{\max}$, where these bounds are

Figures (3)

  • Figure 1: Geometric interpretation of the Union of Invariant Sets (UIS). Each solid curve shows an invariant set obtained with a different linear $\alpha(x)$. Instead of selecting one candidate set, UIS certifies the union (red dashed line) via max-composition of the corresponding barrier functions. This typically yields a larger certified safe region and never smaller than the best single-$\alpha$ choice, without incurring additional optimization cost.
  • Figure 2: The final invariant set is obtained by verifying a CBF synthesized through the UIS method with three different choices of $\alpha$, as described in Sec. \ref{['sec:UIS']}.
  • Figure 3: The final invariant set is obtained by UIS with two different $\alpha$ values (see Sec. \ref{['sec:UIS']}) for Example \ref{['example:comparison example']}. The certified region is compared against baselines from sum-of-squares (SOS) programming and SyntheBC, showing that UIS achieves a larger invariant set.

Theorems & Definitions (13)

  • Definition 1: Forward invariance ames2019control
  • Definition 2: Control barrier function (CBF) ames2019control
  • Definition 3: Barrier function for closed-loop dynamics
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • ...and 3 more