Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints

TL;DR

This paper introduces new methods for analyzing the local stability and estimating the region of attraction of neural network feedback systems, leveraging positivity constraints and Lyapunov functions to improve scalability and accuracy.

Contribution

It develops two novel approaches for ROA estimation using Lyapunov functions and tight sector bounds for neural networks within a localized Aizerman framework.

Findings

Improved ROA estimates over existing methods.

Enhanced scalability in stability analysis.

Effective certification of local exponential stability.

Abstract

We study the local stability of nonlinear systems in the Lur'e form with static nonlinear feedback realized by feedforward neural networks (FFNNs). By leveraging positivity system constraints, we employ a localized variant of the Aizerman conjecture, which provides sufficient conditions for exponential stability of trajectories confined to a compact set. Using this foundation, we develop two distinct methods for estimating the Region of Attraction (ROA): (i) a less conservative Lyapunov-based approach that constructs invariant sublevel sets of a quadratic function satisfying a linear matrix inequality (LMI), and (ii) a novel technique for computing tight local sector bounds for FFNNs via layer-wise propagation of linear relaxations. These bounds are integrated into the localized Aizerman framework to certify local exponential stability. Numerical results demonstrate substantial improvements over existing integral quadratic constraint-based approaches in both ROA size and scalability.