Robust Stability Analysis of Positive Lure System with Neural Network Feedback

TL;DR

This paper presents a scalable method for robustness analysis of positive Lur'e systems with neural network feedback, deriving explicit stability radius formulas and refining neural network sector bounds.

Contribution

It introduces a novel approach leveraging positivity properties to analyze robustness and stability of uncertain Lur'e systems with neural network controllers.

Findings

Derived explicit stability radius formulas for positive Lur'e systems.

Extended analysis to systems with neural network feedback loops.

Proposed a refinement method for neural network sector bounds.

Abstract

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and 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 NNs. 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.