Improved Sum-of-Squares Stability Verification of Neural-Network-Based Controllers

TL;DR

This paper enhances the stability verification framework for neural-network-controlled nonlinear systems using semialgebraic sets and convex semidefinite programming, introducing new functions, proofs, and optimization problems for improved analysis.

Contribution

It introduces new semialgebraic functions compatible with RNNs, provides an alternative proof of stability analysis, and develops two optimization problems for better local stability and RoA analysis.

Findings

Expanded framework utility with new functions

Validated stability analysis with an alternative proof

Developed optimization problems for Lyapunov function parameterization

Abstract

This work presents several improvements to the closed-loop stability verification framework using semialgebraic sets and convex semidefinite programming to examine neural-network-based control systems regulating nonlinear dynamical systems. First, the utility of the framework is greatly expanded: two semialgebraic functions mimicking common, smooth activation functions are presented and compatibility with control systems incorporating Recurrent Equilibrium Networks (RENs) and thereby Recurrent Neural Networks (RNNs) is established. Second, the validity of the framework's state-of-the-art stability analyses is established via an alternate proof. Third, based on this proof, two new optimization problems simplifying the analysis of local stability properties are presented. To simplify the analysis of a closed-loop system's Region of Attraction (RoA), the first problem explicitly parameterizes a class of candidate Lyapunov functions larger than in previous works. The second problem utilizes the unique guarantees available under the condition of invariance to further expand the set of candidate Lyapunov functions and directly determine whether an invariant set forms part of the system's RoA. These contributions are successfully demonstrated in two numerical examples and suggestions for future research are provided.