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Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems

Pouya Samanipour, Hasan Poonawala

TL;DR

This work tackles safety guarantees for dynamical systems identified by ReLU networks or their PWA equivalents by replacing the conventional nonlinear class K_infinity barrier function with a practical Leaky ReLU alpha and by introducing the Union of Invariant Sets (UIS) to fuse information from multiple barrier candidates. It develops a systematic PWA barrier-function design on a shared partition, and constructs a UIS by taking the maximum of several BF candidates obtained with different linear alphas, producing a potentially larger forward-invariant set. The key contributions are (i) a Leaky ReLU-based alpha that ensures barrier conditions for bounded barrier functions, (ii) a PWA BF formulation with tractable linear optimizations, and (iii) the UIS framework that combines multiple invariant sets without increasing computational cost. The method is validated on ReLU-based dynamics, including inverted pendulum scenarios, demonstrating that UIS can enlarge invariant sets while preserving safety guarantees, with open-source code provided for reproducibility and practical adoption.

Abstract

In this paper, a systematic framework is presented for determining piecewise affine PWA barrier functions and their corresponding invariant sets for dynamical systems identified via Rectified Linear Unit (ReLU) neural networks or their equivalent PWA representations. A common approach to determining the invariant set is to use Nagumo's condition, or to utilize the barrier function with a class K-infinity function. It may be challenging to find a suitable class K-infinity function in some cases. We propose leaky ReLU as an efficient substitute for the complex nonlinear K-infinity function in our formulation. Moreover, we propose the Union of Invariant Sets (UIS) method, which combines information from multiple invariant sets in order to compute the largest possible PWA invariant set. The proposed framework is validated through multiple examples, showcasing its potential to enhance the analysis of invariant sets in ReLU-based dynamical systems. Our code is available at: https://github.com/PouyaSamanipour/UIS.git.

Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems

TL;DR

This work tackles safety guarantees for dynamical systems identified by ReLU networks or their PWA equivalents by replacing the conventional nonlinear class K_infinity barrier function with a practical Leaky ReLU alpha and by introducing the Union of Invariant Sets (UIS) to fuse information from multiple barrier candidates. It develops a systematic PWA barrier-function design on a shared partition, and constructs a UIS by taking the maximum of several BF candidates obtained with different linear alphas, producing a potentially larger forward-invariant set. The key contributions are (i) a Leaky ReLU-based alpha that ensures barrier conditions for bounded barrier functions, (ii) a PWA BF formulation with tractable linear optimizations, and (iii) the UIS framework that combines multiple invariant sets without increasing computational cost. The method is validated on ReLU-based dynamics, including inverted pendulum scenarios, demonstrating that UIS can enlarge invariant sets while preserving safety guarantees, with open-source code provided for reproducibility and practical adoption.

Abstract

In this paper, a systematic framework is presented for determining piecewise affine PWA barrier functions and their corresponding invariant sets for dynamical systems identified via Rectified Linear Unit (ReLU) neural networks or their equivalent PWA representations. A common approach to determining the invariant set is to use Nagumo's condition, or to utilize the barrier function with a class K-infinity function. It may be challenging to find a suitable class K-infinity function in some cases. We propose leaky ReLU as an efficient substitute for the complex nonlinear K-infinity function in our formulation. Moreover, we propose the Union of Invariant Sets (UIS) method, which combines information from multiple invariant sets in order to compute the largest possible PWA invariant set. The proposed framework is validated through multiple examples, showcasing its potential to enhance the analysis of invariant sets in ReLU-based dynamical systems. Our code is available at: https://github.com/PouyaSamanipour/UIS.git.

Paper Structure

This paper contains 10 sections, 3 theorems, 26 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Piecewise Affine Functions
  4. Invariant Set Estimation
  5. Forward Invariance
  6. Leaky ReLU: A practical choice for alpha(x) in BF design
  7. PWA barrier function
  8. Union of Invariant Sets(UIS)
  9. Results and Simulations
  10. Conclusion

Key Result

Theorem 1

Let $S \subset \mathcal{D} \subseteq \mathbb{R}^n$ represent the super-level set, as defined in Equation eq:BF Domain, of a continuously differentiable function $h(x)$ with respect to the closed-loop dynamic eq:general nl. If $h(x)$ is a bounded function such that $h_{\min} \leq h(x) \leq h_{\max}$

Figures (3)

  • Figure 1: The new invariant set $\overline{S}$ shown with a red dashed line is the union of four invariant sets obtained from optimization problem \ref{['eq:opt UIS']} with different $\alpha$. As can be seen $S(\mathcal{P}_4,\alpha_4)$ does not contribute to the $\overline{S}$.
  • Figure 2: The final invariant set will be the Union of invariant sets with 3 different $\alpha$ as described in UIS \ref{['sec:UIS']}.
  • Figure 3: The final invariant set will be the Union of invariant sets with 2 different $\alpha$ as described in UIS \ref{['sec:UIS']} for the Example \ref{['comparison example']}.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2: Forward invariant set ames2019control
  • Definition 3: Barrier function ames2019control
  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Corollary 1
  • ...and 1 more