Reliably-stabilizing piecewise-affine neural network controllers

TL;DR

This paper introduces a method to certify the stability of neural network controllers approximating MPC policies by quantifying errors and Lipschitz constants, ensuring reliable closed-loop system performance.

Contribution

It provides a novel, optimization-based approach to verify stability and performance of ReLU neural network controllers approximating MPC schemes.

Findings

Exact computation of worst-case approximation error and Lipschitz constant.

Conditions established for stability certification of NN-based controllers.

Method applicable to design minimal complexity neural networks with guaranteed stability.

Abstract

A common problem affecting neural network (NN) approximations of model predictive control (MPC) policies is the lack of analytical tools to assess the stability of the closed-loop system under the action of the NN-based controller. We present a general procedure to quantify the performance of such a controller, or to design minimum complexity NNs with rectified linear units (ReLUs) that preserve the desirable properties of a given MPC scheme. By quantifying the approximation error between NN-based and MPC-based state-to-input mappings, we first establish suitable conditions involving two key quantities, the worst-case error and the Lipschitz constant, guaranteeing the stability of the closed-loop system. We then develop an offline, mixed-integer optimization-based method to compute those quantities exactly. Together these techniques provide conditions sufficient to certify the stability and performance of a ReLU-based approximation of an MPC control law.