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Exact Learning of Linear Model Predictive Control Laws using Oblique Decision Trees with Linear Predictions

Jiayang Ren, Qiangqiang Mao, Tianwei Zhao, Yankai Cao

TL;DR

This work introduces an exact-learning framework that maps the Linear MPC control law to oblique decision trees with linear predictions (ODT-LP). It proves CPWA equivalence between Linear MPC and finite-depth ODT-LP, and develops a gradient-based offline training pipeline using grid-generated state-action data with smooth indicator relaxations. The approach yields controllers that match MPC performance while delivering orders-of-magnitude faster online evaluation and enabling verifiable, interpretable control logic. Robustness and ISS analyses show the method remains stable under bounded approximation errors and disturbances, making it suitable for safety-critical applications.

Abstract

Model Predictive Control (MPC) is a powerful strategy for constrained multivariable systems but faces computational challenges in real-time deployment due to its online optimization requirements. While explicit MPC and neural network approximations mitigate this burden, they suffer from scalability issues or lack interpretability, limiting their applicability in safety-critical systems. This work introduces a data-driven framework that directly learns the Linear MPC control law from sampled state-action pairs using Oblique Decision Trees with Linear Predictions (ODT-LP), achieving both computational efficiency and interpretability. By leveraging the piecewise affine structure of Linear MPC, we prove that the Linear MPC control law can be replicated by finite-depth ODT-LP models. We develop a gradient-based training algorithm using smooth approximations of tree routing functions to learn this structure from grid-sampled Linear MPC solutions, enabling end-to-end optimization. Input-to-state stability is established under bounded approximation errors, with explicit error decomposition into learning inaccuracies and sampling errors to inform model design. Numerical experiments demonstrate that ODT-LP controllers match MPC's closed-loop performance while reducing online evaluation time by orders of magnitude compared to MPC, explicit MPC, neural network, and random forest counterparts. The transparent tree structure enables formal verification of control logic, bridging the gap between computational efficiency and certifiable reliability for safety-critical systems.

Exact Learning of Linear Model Predictive Control Laws using Oblique Decision Trees with Linear Predictions

TL;DR

This work introduces an exact-learning framework that maps the Linear MPC control law to oblique decision trees with linear predictions (ODT-LP). It proves CPWA equivalence between Linear MPC and finite-depth ODT-LP, and develops a gradient-based offline training pipeline using grid-generated state-action data with smooth indicator relaxations. The approach yields controllers that match MPC performance while delivering orders-of-magnitude faster online evaluation and enabling verifiable, interpretable control logic. Robustness and ISS analyses show the method remains stable under bounded approximation errors and disturbances, making it suitable for safety-critical applications.

Abstract

Model Predictive Control (MPC) is a powerful strategy for constrained multivariable systems but faces computational challenges in real-time deployment due to its online optimization requirements. While explicit MPC and neural network approximations mitigate this burden, they suffer from scalability issues or lack interpretability, limiting their applicability in safety-critical systems. This work introduces a data-driven framework that directly learns the Linear MPC control law from sampled state-action pairs using Oblique Decision Trees with Linear Predictions (ODT-LP), achieving both computational efficiency and interpretability. By leveraging the piecewise affine structure of Linear MPC, we prove that the Linear MPC control law can be replicated by finite-depth ODT-LP models. We develop a gradient-based training algorithm using smooth approximations of tree routing functions to learn this structure from grid-sampled Linear MPC solutions, enabling end-to-end optimization. Input-to-state stability is established under bounded approximation errors, with explicit error decomposition into learning inaccuracies and sampling errors to inform model design. Numerical experiments demonstrate that ODT-LP controllers match MPC's closed-loop performance while reducing online evaluation time by orders of magnitude compared to MPC, explicit MPC, neural network, and random forest counterparts. The transparent tree structure enables formal verification of control logic, bridging the gap between computational efficiency and certifiable reliability for safety-critical systems.

Paper Structure

This paper contains 16 sections, 6 theorems, 16 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Oblique Decision Tree with Linear Predictions
  4. Linear Model Predictive Control
  5. Exact Learning of MPC law using Oblique Decision Tree with Linear Predictions
  6. Equivalence of Linear MPC and Oblique Decision Trees
  7. Procedure of the ODT-LP-based MPC
  8. Data generation
  9. Decision Tree Training
  10. Robustness of ODT-LP based MPC
  11. Error Analysis of the ODT-LP Policy
  12. Input-to-State Stability Analysis
  13. Numerical Experiments
  14. Case Study 1: A Two-State System
  15. Case Study 2: A Four-State System
  16. ...and 1 more sections

Key Result

Corollary 1

(Adapted from bemporad_explicit_2002, Corollary 2) The MPC control law $\kappa_N(x_0) = f(x_0), \ f:\mathbb{R}^n \to \mathbb{R}^m$, defined by the optimization problem eqn:mpc, is continuous and piecewise affine: where the polyhedral sets $\{H^i x_0 \leq l^i\}, \ i = 1, \ldots, N_{\text{mpc}}$, form a partition of the given set of states.

Figures (4)

  • Figure 1: Examples of ODT-LP, NN, and Explicit MPC control laws. (Red paths show the activated paths needed to obtain actions. ODT-LP visits one path, NN visits all paths, and explicit MPC visits all regions in worst cases.)
  • Figure 2: Procedure of the ODT-LP-based MPC
  • Figure 3: Trained ODT-LP control law of case study 1.
  • Figure 4: Closed-loop response of ODT-LP-based MPC controller.

Theorems & Definitions (12)

  • Corollary 1
  • Theorem 1
  • proof
  • Lemma 1: Policy Approximation Error Bound
  • proof
  • Definition 1: Robust Positive Invariance
  • Definition 2: Input-to-State Stability (ISS)
  • Definition 3: ISS-Lyapunov Function
  • Proposition 1: Bounded Function Differences
  • Proposition 2: Subadditivity of $\mathcal{K}$-Functions
  • ...and 2 more