Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error bounds for maxout neural network approximations of model predictive control

Dieter Teichrib, Moritz Schulze Darup

TL;DR

This work tackles the challenge of certifying stability for neural network approximations of model predictive control (MPC) when online OCP solutions are expensive. It extends prior ReLU-based results to maxout neural networks by deriving mixed-integer linear constraints that allow exact computation of the output $\boldsymbol{\Phi}(\boldsymbol{x})$ and the local gain $\boldsymbol{K}_{\text{NN}}(\boldsymbol{x})$, enabling exact evaluation of the maximum error $\overline{e}_\alpha$ and the Lipschitz bound $\mathcal{L}_\alpha(e,\mathcal{X})$ via MILP. The paper shows that maxout networks can exactly represent MPC laws under suitable architectures, and provides numerical examples (1D and 2D) where zero maximum error is achieved or where the error is vanishingly small, along with comparative analysis against ReLU nets. These results enable certified stability for NN-controlled MPC and suggest promising directions for integrating exact value-function descriptions in piecewise-quadratic MPC settings.

Abstract

Neural network (NN) approximations of model predictive control (MPC) are a versatile approach if the online solution of the underlying optimal control problem (OCP) is too demanding and if an exact computation of the explicit MPC law is intractable. The drawback of such approximations is that they typically do not preserve stability and performance guarantees of the original MPC. However, such guarantees can be recovered if the maximum error with respect to the optimal control law and the Lipschitz constant of that error are known. We show in this work how to compute both values exactly when the control law is approximated by a maxout NN. We build upon related results for ReLU NN approximations and derive mixed-integer (MI) linear constraints that allow a computation of the output and the local gain of a maxout NN by solving an MI feasibility problem. Furthermore, we show theoretically and experimentally that maxout NN exist for which the maximum error is zero.

Error bounds for maxout neural network approximations of model predictive control

TL;DR

This work tackles the challenge of certifying stability for neural network approximations of model predictive control (MPC) when online OCP solutions are expensive. It extends prior ReLU-based results to maxout neural networks by deriving mixed-integer linear constraints that allow exact computation of the output and the local gain , enabling exact evaluation of the maximum error and the Lipschitz bound via MILP. The paper shows that maxout networks can exactly represent MPC laws under suitable architectures, and provides numerical examples (1D and 2D) where zero maximum error is achieved or where the error is vanishingly small, along with comparative analysis against ReLU nets. These results enable certified stability for NN-controlled MPC and suggest promising directions for integrating exact value-function descriptions in piecewise-quadratic MPC settings.

Abstract

Neural network (NN) approximations of model predictive control (MPC) are a versatile approach if the online solution of the underlying optimal control problem (OCP) is too demanding and if an exact computation of the explicit MPC law is intractable. The drawback of such approximations is that they typically do not preserve stability and performance guarantees of the original MPC. However, such guarantees can be recovered if the maximum error with respect to the optimal control law and the Lipschitz constant of that error are known. We show in this work how to compute both values exactly when the control law is approximated by a maxout NN. We build upon related results for ReLU NN approximations and derive mixed-integer (MI) linear constraints that allow a computation of the output and the local gain of a maxout NN by solving an MI feasibility problem. Furthermore, we show theoretically and experimentally that maxout NN exist for which the maximum error is zero.
Paper Structure (13 sections, 4 theorems, 52 equations, 2 tables)

This paper contains 13 sections, 4 theorems, 52 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Fundamentals of MPC and NN
  4. Model predicitive control
  5. Neural networks with piecewise affine activations
  6. Approximate MPC
  7. Maxout neural networks for approximate MPC
  8. Exact maxout neural networks
  9. Maxout neural networks as MILP
  10. Numerical examples
  11. Example system with $n=1$
  12. Example system with $n=2$
  13. Conclusions and outlook

Key Result

Corollary 1

Let $F(\boldsymbol{x})$ be an arbitrary PWA function of the form eq:PWA_f with one dimensional output, i.e, $m=1$. Then for a maxout NN $\Phi(\boldsymbol{x})$ with $\ell=1$, $w_2=1$ and $\boldsymbol{b}^{(2)}=0$ there exist parameters and with $\Phi(\boldsymbol{x})=F(\boldsymbol{x})$.

Theorems & Definitions (8)

  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof