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Reliably-stabilizing piecewise-affine neural network controllers

Filippo Fabiani, Paul J. Goulart

TL;DR

This work tackles the stability challenge of neural-network approximations to model predictive control by introducing a rigorous offline framework that certifies closed-loop stability through two computable metrics: the worst-case approximation error $ar{e}_eta$ and the Lipschitz constant $\(e,X_)$. It shows how to exactly compute these quantities via mixed-integer linear programs (MILPs) and provides a stability guarantee: if these metrics satisfy certain bounds, the NN-based controller stabilizes the discretetime system with exponential convergence to a neighborhood of the origin. The approach leverages the piecewise-affine nature of ReLU networks and the explicit structure of MPC through KKT conditions, enabling a practical workflow to design minimum-complexity NN controllers that inherit MPC properties. Numerical examples with coupled oscillators demonstrate feasibility and the value of offline certification for embedded applications, where online computation must be minimal. The work thus offers a principled path toward reliable, certified NN controllers that retain MPC performance while reducing online computational burden.

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.

Reliably-stabilizing piecewise-affine neural network controllers

TL;DR

This work tackles the stability challenge of neural-network approximations to model predictive control by introducing a rigorous offline framework that certifies closed-loop stability through two computable metrics: the worst-case approximation error and the Lipschitz constant . It shows how to exactly compute these quantities via mixed-integer linear programs (MILPs) and provides a stability guarantee: if these metrics satisfy certain bounds, the NN-based controller stabilizes the discretetime system with exponential convergence to a neighborhood of the origin. The approach leverages the piecewise-affine nature of ReLU networks and the explicit structure of MPC through KKT conditions, enabling a practical workflow to design minimum-complexity NN controllers that inherit MPC properties. Numerical examples with coupled oscillators demonstrate feasibility and the value of offline certification for embedded applications, where online computation must be minimal. The work thus offers a principled path toward reliable, certified NN controllers that retain MPC performance while reducing online computational burden.

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.

Paper Structure

This paper contains 18 sections, 8 theorems, 35 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Summary of contributions
  4. Paper organization
  5. An approximation problem
  6. Stability analysis of piecewise-affine neural network controllers
  7. Mathematical background
  8. A family of neural networks
  9. and optimization
  10. Maximum gain computation as a mixed-integer linear program
  11. Quantifying the approximation quality of piecewise-affine neural networks
  12. Discussion of reliably-stabilizing controllers
  13. A user's guide for controllers
  14. Accommodating input constraints
  15. Numerical simulations
  16. ...and 3 more sections

Key Result

Lemma 3.2

There exists $\zeta > 0$ such that, if $\bar{e}_{\alpha} < \zeta$, the system in eq:LTI_sys with controller $u = u_{\textnormal{NN}}(x)$ converges exponentially to some neighbourhood of the origin $\Omega_b$, for all $x(0) \in \Omega_c \supset \Omega_b$, with $c \coloneqq \textnormal{max} \, \{a \ge

Figures (4)

  • Figure 1: Feedback loop with controller.
  • Figure 2: Two-dimensional schematic representation of the line of proof of Theorem \ref{['th:norm_comp']}.
  • Figure 3: Roadmap for using the proposed results.
  • Figure 4: Systems of 2, 3 and 4 oscillating masses each with one degree of freedom, connected each other through pairs of spring-damper blocks, and to walls (dark blocks on the sides). Control inputs $u_1$, $u_2$ and $u_3$ are either acting on each single mass at a time, or produce a joint action on multiple masses, according to the direction indicated by each arrow.

Theorems & Definitions (16)

  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Remark 3.5
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • Remark 4.6
  • Lemma 5.1
  • proof
  • ...and 6 more