Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers

TL;DR

This paper presents a new stability analysis method for systems with neural network controllers, ensuring positivity and exponential stability by establishing sector bounds for fully connected feedforward neural networks without biases.

Contribution

It introduces a novel stability theorem for positive feedback systems with FFNN controllers, leveraging sector bounds and positive Lur'e system principles to guarantee global exponential stability.

Findings

Proves stability of linear systems with FFNN controllers using sector bounds.

Demonstrates practical application on a linear system with trained FFNN controller.

Addresses stability challenges in highly nonlinear systems with neural network controllers.

Abstract

This paper introduces a novel method for the stability analysis of positive feedback systems with a class of fully connected feedforward neural networks (FFNN) controllers. By establishing sector bounds for fully connected FFNNs without biases, we present a stability theorem that demonstrates the global exponential stability of linear systems under fully connected FFNN control. Utilizing principles from positive Lur'e systems and the positive Aizerman conjecture, our approach effectively addresses the challenge of ensuring stability in highly nonlinear systems. The crux of our method lies in maintaining sector bounds that preserve the positivity and Hurwitz property of the overall Lur'e system. We showcase the practical applicability of our methodology through its implementation in a linear system managed by a FFNN trained on output feedback controller data, highlighting its potential for enhancing stability in dynamic systems.