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Geometric Data-Driven Dimensionality Reduction in MPC with Guarantees

Roland Schurig, Andreas Himmel, Rolf Findeisen

TL;DR

The paper tackles online dimensionality reduction in MPC by designing a low-dimensional subspace through a geometric, data-driven framework on Grassmann manifolds, ensuring stability, feasibility, and an initial-feasibility condition. By integrating a state-dependent offset and an admissible-guess mechanism, the reduced-order MPC preserves recursive feasibility and achieves comparable closed-loop performance to a full-order controller with a longer horizon. The key contribution is a Riemannian optimization formulation that enforces subspace-admissibility constraints and demonstrates improved interpretability and feasibility guarantees over Euclidean alternatives. The practical impact is a computationally lighter MPC with provable guarantees, suitable for fast-sampling or hardware-limited control applications, with clear avenues for extending the approach to learning-driven parameter tuning.

Abstract

We address the challenge of dimension reduction in the discrete-time optimal control problem which is solved repeatedly online within the framework of model predictive control. Our study demonstrates that a reduced-order approach, aimed at identifying a suboptimal solution within a low-dimensional subspace, retains the stability and recursive feasibility characteristics of the original problem. We present a necessary and sufficient condition for ensuring initial feasibility, which is seamlessly integrated into the subspace design process. Additionally, we employ techniques from optimization on Riemannian manifolds to develop a subspace that efficiently represents a collection of pre-specified high-dimensional data points, all while adhering to the initial admissibility constraint.

Geometric Data-Driven Dimensionality Reduction in MPC with Guarantees

TL;DR

The paper tackles online dimensionality reduction in MPC by designing a low-dimensional subspace through a geometric, data-driven framework on Grassmann manifolds, ensuring stability, feasibility, and an initial-feasibility condition. By integrating a state-dependent offset and an admissible-guess mechanism, the reduced-order MPC preserves recursive feasibility and achieves comparable closed-loop performance to a full-order controller with a longer horizon. The key contribution is a Riemannian optimization formulation that enforces subspace-admissibility constraints and demonstrates improved interpretability and feasibility guarantees over Euclidean alternatives. The practical impact is a computationally lighter MPC with provable guarantees, suitable for fast-sampling or hardware-limited control applications, with clear avenues for extending the approach to learning-driven parameter tuning.

Abstract

We address the challenge of dimension reduction in the discrete-time optimal control problem which is solved repeatedly online within the framework of model predictive control. Our study demonstrates that a reduced-order approach, aimed at identifying a suboptimal solution within a low-dimensional subspace, retains the stability and recursive feasibility characteristics of the original problem. We present a necessary and sufficient condition for ensuring initial feasibility, which is seamlessly integrated into the subspace design process. Additionally, we employ techniques from optimization on Riemannian manifolds to develop a subspace that efficiently represents a collection of pre-specified high-dimensional data points, all while adhering to the initial admissibility constraint.
Paper Structure (9 sections, 7 theorems, 20 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 7 theorems, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. MPC FORMULATION
  3. DIMENSIONALITY REDUCTION IN MPC
  4. DATA-DRIVEN SUBSPACE DESIGN
  5. Design of the optimal subspace
  6. Riemannian approach
  7. Comparison to a Euclidean approach
  8. SIMULATION RESULTS
  9. CONCLUSIONS

Key Result

Proposition 1

Given a state $x \in \mathcal{X}_N$ and an input sequence $z \in \mathbb{U}^N(x)$, we create an input sequence $\mathantt{s}_\mathrm{f}(x,z) \in \mathbb{U}^N(\mathantt{x}_z(1,x))$. If $\mathantt{x}_z(1,x) = 0$, then we set $\mathantt{s}_\mathrm{f}(x,z) \mathop{\mathrm{\coloneqq}}\nolimits 0$. Otherw with $x_{N} = \mathantt{x}_z(N,x)$. We call $\mathantt{s}_\mathrm{f}(x,z)$ the admissibly shifted i

Figures (3)

  • Figure 1: Illustration of the subspace approach for $d=2$, $r=1$.
  • Figure 2: Illustration of the admissibility problem.
  • Figure 3: Initial set and data points for pendulum example.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Definition 4
  • Remark 3
  • Lemma 1
  • ...and 16 more