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Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification

Hendrik Alsmeier, Lukas Theiner, Anton Savchenko, Ali Mesbah, Rolf Findeisen

TL;DR

A framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks is presented and a training adjustment is introduced, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds.

Abstract

This paper presents a framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks. Leveraging the Lipschitz properties of these neural networks, we derive a bound that guides dataset design to ensure the approximation error remains at chosen limits. We discuss how this method can be used to design a stable neural network controller with performance guarantees employing existing robust model predictive control approaches for data generation. Additionally, we introduce a training adjustment, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds. We verify that the proposed augmentation results in improvements to the network's predictive capabilities and a reduction of the Lipschitz constant. Moreover, on a simulated inverted pendulum problem, we show that the approach results in a closer match of the closed-loop behavior between the imitation and the original model predictive controller.

Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification

TL;DR

A framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks is presented and a training adjustment is introduced, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds.

Abstract

This paper presents a framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks. Leveraging the Lipschitz properties of these neural networks, we derive a bound that guides dataset design to ensure the approximation error remains at chosen limits. We discuss how this method can be used to design a stable neural network controller with performance guarantees employing existing robust model predictive control approaches for data generation. Additionally, we introduce a training adjustment, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds. We verify that the proposed augmentation results in improvements to the network's predictive capabilities and a reduction of the Lipschitz constant. Moreover, on a simulated inverted pendulum problem, we show that the approach results in a closer match of the closed-loop behavior between the imitation and the original model predictive controller.
Paper Structure (9 sections, 2 theorems, 24 equations, 2 figures, 2 tables)

This paper contains 9 sections, 2 theorems, 24 equations, 2 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM SETUP
  3. PERFORMANCE GUARANTEES FOR NN APPROXIMATED CONTROLLERS
  4. SENSITIVITY-REGULARIZED TRAINING
  5. Sensitivity of Parametric Nonlinear Programs
  6. Loss Adaptation
  7. Simulation Experiment
  8. Discussion
  9. CONCLUSIONS

Key Result

Theorem 1

If Assumptions a:1 and a:2 hold, then for any $\epsilon > 0$ the approximation error is bounded by if the following conditions for the dataset $\mathcal{D}$ hold: for $L_{\mathrm{NN}}$ and $L_\mathrm{MPC}$ being Lipschitz constants of $\kappa_{\mathrm{NN}}$ and $\kappa$ correspondingly, such that

Figures (2)

  • Figure 1: Closed-loop simulation for $4.5~\mathrm{s}$ of the real MPC, Norminal NN, and Sensitivity-Regularized NN. Shown are the first (top) and second state (center) and the input (bottom).
  • Figure 2: Approximation error for the nominal NN and the sensitivity-regularized NN (top), divergence of the input values along the closed-loop trajectory (middle), and magnitude of constraint violation in closed-loop (bottom). $\epsilon_\mathrm{D}$ denoted as the maximal error according to \ref{['eq:explespd']}. In all box plots, the boxes show the $25\,\%$ to $75\,\%$ quartile, and the whiskers extend to include the majority of the data.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Corollary 1