Robust Stability Analysis of Positive Lure System with Neural Network Feedback

2025-05-25

AI Summary

This paper addresses the robust stability problem of positive Lur’e systems subject to parametric uncertainties in the linear subsystem and unknown sector-bounded nonlinearities. To overcome the conservatism of conventional methods and their inability to handle neural network (NN)-based nonlinear feedback, we integrate positive systems theory with Metzler matrix robustness analysis and introduce the positive Aizerman conjecture, yielding an analytical expression for the stability radius. We propose a sector-bound adaptive refinement method tailored to feedforward NNs, relaxing the restrictive fixed-sector assumption. Furthermore, we establish a scalable stability criterion for positive nonlinear control systems. Numerical experiments demonstrate that the proposed approach significantly reduces conservatism and effectively supports robustness verification of NN-based feedback systems.

Technology Category

Intelligent Robots: State EstimationNatural Language Processing: Safety and RobustnessMachine Learning: Adversarial Learning & Robustness

Application Category

Security and Privacy: Large-scale security measurementsSocial Networks and Social Media: Generative AI / large language models and their impact on social systemsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and the robustness properties of Metzler matrices. Specifically, we consider a control system of Lur'e type in which not only the linear part includes parametric uncertainty but also the nonlinear sector bound is unknown. We investigate tools from positive linear systems to effectively solve the problems in complicated and uncertain nonlinear systems. By leveraging the positivity characteristic of the system, we derive an explicit formula for the stability radius of Lur'e systems. Furthermore, we extend our analysis to systems with neural network (NN) feedback loops. Building on this approach, we also propose a refinement method for sector bounds of feedforward neural networks (FFNNs). This study introduces a scalable and efficient approach for robustness analysis of both Lur'e and NN-controlled systems. Finally, the proposed results are supported by illustrative examples.

Problem

• Analyzes robustness of Lur'e systems under positivity constraints
• Extends stability analysis to neural network feedback systems
• Proposes refined sector bounds for feedforward neural networks

Innovation

• Positive linear systems for uncertain nonlinear systems
• Stability radius formula for Lur'e systems
• Refinement method for neural network sector bounds