Invariant Set Estimation for Piecewise Affine Dynamical Systems Using Piecewise Affine Barrier Function

TL;DR

The paper addresses safety guarantees for systems controlled by ReLU networks or piecewise-affine dynamics by estimating forward invariant sets. It introduces a barrier-function framework represented as a piecewise-affine function and solves the resulting vertex-based linear program on a polyhedral partition, with a domain-refinement step to enlarge the invariant set. The approach enforces continuity across cell boundaries and Nagumo-type conditions to certify invariance, while a refinement loop expands the barrier's capacity when needed. Demonstrations on inverted-pendulum and MPC-based systems illustrate effective invariant-set estimation and reveal practical scalability for safety-critical neural-network–controlled applications.

Abstract

This paper introduces an algorithm for approximating the invariant set of closed-loop controlled dynamical systems identified using ReLU neural networks or piecewise affine PWA functions, particularly addressing the challenge of providing safety guarantees for ReLU networks commonly used in safety-critical applications. The invariant set of PWA dynamical system is estimated using ReLU networks or its equivalent PWA function. This method entails formulating the barrier function as a PWA function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of optimization is to maximize the invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of PWA dynamical systems.

Figures (7)

• Figure 1: A sample partition of a $\mathrm{PWA}$ dynamics. Cells $X_1$ ,$X_2$, $X_3$ and $X_4$ share points with the boundary, hence, $\{1,2,3,4\}\in I_{\partial \mathcal{D}}$. While cells $X_5$,$X_6$, and $X_7$ do not share any points with the boundary. As a result $\{5,6,7\}\notin I_{\partial \mathcal{D}}$. Furthermore, $v_1,v_2$ are vertices on the boundary, while $v_3$ and $v_4$ are not on the boundary. Hence, $\{(1,1),(1,2),(2,2)\}\in I_b$ and $\{(6,3),(6,4)\}\in I_{int}$.
• Figure 2: Vector fields for the Inverted Pendulum identified using ReLU NN and its estimated Invariant set
• Figure 3: $x_1$ and $x_2$ for a sample trajectory with initial condition $x_0=[-1,3]^T$
• Figure 4: $h(x)$ along the sample trajecty. Since the trajectory starts from outside the invariant set, the value of $h(x)$ is negative. As soon as they enter the invariant set, the value of $h(x)$ becomes positive.
• Figure 5: 2-D mpc vector fields, invariant set

Theorems & Definitions (15)

• Definition 1: Forward invariant set ames2019control
• Definition 2: Barrier function ames2019control
• Definition 3
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 1