Mapping back and forth between model predictive control and neural networks

TL;DR

This work establishes an exact correspondence between linear-quadratic MPC and implicit neural networks, showing that the MPC solution $\mathbf{u}^*(x)$ can be represented as the fixed point of an implicit NN $y(x)=W\phi(y(x))+Yx+b$ with $u^*(x)=W_f\phi(y(x))+Y_fx$. It then introduces an explicit unraveling of this implicit network into a feed-forward NN $w[j+1]=D\phi(w[j])+\zeta+f(w[j])$, where a designed feedback $f$ accelerates convergence, enabling accurate approximation with finite depth and bounding the error. The paper also provides a converse construction: given an implicit NN, one can recover MPC-like cost structures by piecewise-linear approximations and corresponding LMIs to ensure interior-region consistency. Together, these results bridge model-based MPC and data-driven NN controllers, offering a scalable, explainable framework for representing and approximating MPC through neural networks.

Abstract

Model predictive control (MPC) for linear systems with quadratic costs and linear constraints is shown to admit an exact representation as an implicit neural network. A method to "unravel" the implicit neural network of MPC into an explicit one is also introduced. As well as building links between model-based and data-driven control, these results emphasize the capability of implicit neural networks for representing solutions of optimisation problems, as such problems are themselves implicitly defined functions.

Figures (3)

• Figure 1: Linear quadratic model predictive control (MPC) is shown to be structured as an implicit neural network. This implicit neural network is then approximated by a feed-forward one. This neural network structure builds upon the results of valmorbida2023quadratic on the linear complementarity formulation of the MPC-QP problem.
• Figure 2: Comparison between the MPC controller and the implicit neural network defined by \ref{['lem:imp_NN']} for the numerical example. The input sequences $u[k]$, and states $x_1[k]$, $x_2[k]$ are shown for both policies. The equivalence between the two simulations validates the results of the paper. Note that the implicit neural network captures the input constraints imposed by the MPC policy, as predicted.
• Figure 3: (a) Residuals from unravelling the implicit neural network into an explicit one using the gains $K = -0.9I_m$, $K = -0.9I_m$ & $K = 0.2I_m$. (b) The control action from the implicit neural network of the numerical example as a function of the state-space. (b) The control action from the MPC of the numerical example as a function of the state-space. (d) Residuals of the implicit neural network during the simulation as $k$ evolves.

Theorems & Definitions (11)

• definition 1: MPC-QP
• definition 2
• definition 3
• remark 1
• definition 4
• theorem 1
• remark 2
• lemma 1
• proof
• remark 3