Stability Analysis of Complementarity Systems with Neural Network Controllers

TL;DR

This paper presents a novel method for analyzing the stability of complementarity systems controlled by neural networks, by representing ReLU networks as linear complementarity problems and verifying stability via LMIs.

Contribution

It introduces a new approach to model ReLU neural network controllers as linear complementarity systems for stability analysis.

Findings

The method successfully verifies stability in multi-contact robotic systems.

ReLU networks can be represented as solutions to linear complementarity problems.

The approach handles systems with non-unique solutions in friction models.

Abstract

Complementarity problems, a class of mathematical optimization problems with orthogonality constraints, are widely used in many robotics tasks, such as locomotion and manipulation, due to their ability to model non-smooth phenomena (e.g., contact dynamics). In this paper, we propose a method to analyze the stability of complementarity systems with neural network controllers. First, we introduce a method to represent neural networks with rectified linear unit (ReLU) activations as the solution to a linear complementarity problem. Then, we show that systems with ReLU network controllers have an equivalent linear complementarity system (LCS) description. Using the LCS representation, we turn the stability verification problem into a linear matrix inequality (LMI) feasibility problem. We demonstrate the approach on several examples, including multi-contact problems and friction models with non-unique solutions.