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

2025-05-28

AI Summary

This paper addresses the local stability analysis and region-of-attraction (ROA) estimation for Lur’e-type nonlinear systems under positivity constraints, where the static nonlinear feedback is implemented via a feedforward neural network (FFNN). To exploit the FFNN’s local sector-bounded property, we propose a stability analysis framework grounded in a localized Aizerman conjecture. We introduce a novel hierarchical linear relaxation propagation method to compute tight local sector bounds precisely. Furthermore, we design a low-conservatism, LMI-based ROA estimation scheme using quadratic Lyapunov sublevel sets. By integrating positivity system theory, sector-boundedness analysis, and Lyapunov methods, our approach significantly enlarges the estimated ROA across multiple benchmark systems while improving computational scalability—outperforming state-of-the-art integral quadratic constraint (IQC) methods.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Local Search

Application Category

• Social Networks and Social Media: Social media analysis through the lenses of networks
• Search and Retrieval-Augmented AI: Large language models for search
• Security and Privacy: Large-scale security measurements

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.

Problem

• Analyzing local stability of neural network feedback systems
• Estimating Region of Attraction using Lyapunov and novel methods
• Improving stability analysis via tight local sector bounds

Innovation

• Leverages positivity constraints for stability analysis
• Uses Lyapunov-based LMI approach for ROA estimation
• Computes tight sector bounds via linear relaxations