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

TL;DR

This paper introduces a geometric, robust least-squares approach for data-driven predictive control, enhancing robustness and scalability by modeling data uncertainty on the Grassmannian manifold.

Contribution

It develops a novel geometric formulation of robust least-squares that extends classical methods and provides an efficient, interpretable algorithm for control applications.

Findings

Improves robustness in data-driven predictive control.

Achieves stronger robustness with better scaling under small uncertainty.

Provides a closed-form solution for the inner maximization problem.

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.