Robust Least-Squares Optimization for Data-Driven Predictive Control: A Geometric Approach

2025-11-12

AI Summary

This paper addresses the challenge of subspace uncertainty—arising from measurement noise and data imperfections—in data-driven predictive control. We propose a robust least-squares optimization framework grounded in Grassmann manifold geometry. Subspace uncertainty is formally modeled as a metric ball on the Grassmann manifold, and the standard least-squares objective is reformulated as a robust subspace containment constraint. The resulting problem is cast as a Euclidean–manifold hybrid min-max optimization, where the inner maximization admits a closed-form solution, ensuring both computational efficiency and geometric interpretability. Compared to conventional robust least-squares approaches, our framework achieves significantly enhanced robustness and scalability under small perturbations. In robust finite-horizon linear-quadratic (LQ) tracking tasks, it delivers superior closed-loop stability and tracking accuracy.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The paper studies a geometrically robust least-squares problem that extends classical and norm-based robust formulations. Rather than minimizing residual error for fixed or perturbed data, we interpret least-squares as enforcing approximate subspace inclusion between measured and true data spaces. The uncertainty in this geometric relation is modeled as a metric ball on the Grassmannian manifold, leading to a min-max problem over Euclidean and manifold variables. The inner maximization admits a closed-form solution, enabling an efficient algorithm with a transparent geometric interpretation. Applied to robust finite-horizon linear-quadratic tracking in data-enabled predictive control, the method improves upon existing robust least-squares formulations, achieving stronger robustness and favorable scaling under small uncertainty.

Problem

• Extends robust least-squares formulations using geometric subspace inclusion
• Models uncertainty as metric balls on Grassmannian manifolds for robustness
• Improves data-driven predictive control with stronger uncertainty handling

Innovation

• Geometric robust least-squares via Grassmannian manifold
• Min-max optimization with closed-form inner solution
• Enhanced robustness in data-driven predictive control