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Harnessing Data for Accelerating Model Predictive Control by Constraint Removal

Zhinan Hou, Feiran Zhao, Keyou You

TL;DR

This work tackles the computational burden of real-time MPC under thousands of constraints by introducing constraint-adaptive MPC (ca-MPC) that uses the Lipschitz continuity of the MPC policy and historical data to perform constraint removal without changing the policy. A key contribution is an explicit, offline-computable Lipschitz constant κ_max derived from the model parameters, with an improved bound via a transformation Φ to tighten the estimate. The method constructs an outer approximation M(x) as a sphere around the previous optimal solution z^*(\tilde{x}) and defines a sphere-halfspace-based removal rule to safely discard constraints, preserving exact optimality. Simulations on a double integrator demonstrate substantial constraint reduction (over 80%) and 10–100× speedups, while maintaining identical closed-loop trajectories to the original MPC policy, highlighting strong practical efficiency gains for high-constraint MPC problems.

Abstract

Model predictive control (MPC) solves a receding-horizon optimization problem in real-time, which can be computationally demanding when there are thousands of constraints. To accelerate online computation of MPC, we utilize data to adaptively remove the constraints while maintaining the MPC policy unchanged. Specifically, we design the removal rule based on the Lipschitz continuity of the MPC policy. This removal rule can use the information of historical data according to the Lipschitz constant and the distance between the current state and historical states. In particular, we provide the explicit expression for calculating the Lipschitz constant by the model parameters. Finally, simulations are performed to validate the effectiveness of the proposed method.

Harnessing Data for Accelerating Model Predictive Control by Constraint Removal

TL;DR

This work tackles the computational burden of real-time MPC under thousands of constraints by introducing constraint-adaptive MPC (ca-MPC) that uses the Lipschitz continuity of the MPC policy and historical data to perform constraint removal without changing the policy. A key contribution is an explicit, offline-computable Lipschitz constant κ_max derived from the model parameters, with an improved bound via a transformation Φ to tighten the estimate. The method constructs an outer approximation M(x) as a sphere around the previous optimal solution z^*(\tilde{x}) and defines a sphere-halfspace-based removal rule to safely discard constraints, preserving exact optimality. Simulations on a double integrator demonstrate substantial constraint reduction (over 80%) and 10–100× speedups, while maintaining identical closed-loop trajectories to the original MPC policy, highlighting strong practical efficiency gains for high-constraint MPC problems.

Abstract

Model predictive control (MPC) solves a receding-horizon optimization problem in real-time, which can be computationally demanding when there are thousands of constraints. To accelerate online computation of MPC, we utilize data to adaptively remove the constraints while maintaining the MPC policy unchanged. Specifically, we design the removal rule based on the Lipschitz continuity of the MPC policy. This removal rule can use the information of historical data according to the Lipschitz constant and the distance between the current state and historical states. In particular, we provide the explicit expression for calculating the Lipschitz constant by the model parameters. Finally, simulations are performed to validate the effectiveness of the proposed method.
Paper Structure (10 sections, 4 theorems, 46 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 4 theorems, 46 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation and background
  3. Linear MPC
  4. Constraint-adaptive MPC
  5. Exploiting historical data for ca-MPC
  6. The Lipschitz constant of reduced mp-QP
  7. The outer approximation $\mathcal{M}(x)$
  8. Simulations
  9. Conclusions
  10. Proof of Theorem \ref{['continuity']}

Key Result

Lemma 1

Consider the original and reduced MPC optimization problem 2_7 and 2_8. Let the set-valued mapping $\mathbb{C}: \mathbb{R}^n \rightrightarrows \mathbb{N}_{[1,n_c]}$ denote the set of indices of removed constraints, i.e., $\mathbb{C}(x) = \mathbb{N}_{[1,n_c]} - \mathbb{I}(x)$. If there exists a mappi Then, it holds that $z^*(x, \mathbb{I}(x)) = z^*(x)$.

Figures (3)

  • Figure 1: The illustrations of Lemma \ref{['outer']}. The black dashed lines denote level sets of the cost function. The orange region denotes $\mathcal{M}(x)$. The green solid and dashed lines denote the constraints specified by $\mathbb{I}(x)$ and $\mathbb{C}(x)$, respectively. The left half-space of the green dashed line is the set $\mathbb{Z}(x,\mathbb{C}(x))$.
  • Figure 2: The closed-loop state trajectory using the original MPC policy and the proposed ca-MPC policy.
  • Figure 3: The percentage of constraints and the computation time with respect to the original MPC solution over time.

Theorems & Definitions (7)

  • Lemma 1: Theorem 1, nouwens2023constraint
  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof