Reachability analysis for piecewise affine systems with neural network-based controllers

TL;DR

This paper presents a method to analyze and certify the stability and safety of piecewise affine systems controlled by neural networks using reachability analysis and mixed-integer linear programming.

Contribution

It extends reachability analysis techniques to PWA systems with NN controllers, enabling stability certification without a baseline controller.

Findings

Over-approximate reachable sets for PWA systems with NN controllers.

Certify convergence to a small set containing the origin.

Modify NN controllers to ensure asymptotic stability.

Abstract

Neural networks (NN) have been successfully applied to approximate various types of complex control laws, resulting in low-complexity NN-based controllers that are fast to evaluate. However, when approximating control laws using NN, performance and stability guarantees of the original controller may not be preserved. Recently, it has been shown that it is possible to provide such guarantees for linear systems with NN-based controllers by analyzing the approximation error with respect to a stabilizing base-line controller or by computing reachable sets of the closed-loop system. The latter has the advantage of not requiring a base-line controller. In this paper, we show that similar ideas can be used to analyze the closed-loop behavior of piecewise affine (PWA) systems with an NN-based controller. Our approach builds on computing over-approximations of reachable sets using mixed-integer linear programming, which allows to certify that the closed-loop system converges to a small set containing the origin while satisfying input and state constraints. We also show how to modify a given NN-based controller to ensure asymptotic stability for the controlled PWA system.