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

TL;DR

The paper addresses robustness in data-driven predictive control by modeling uncertainty as a ball on the Grassmannian around the true finite-horizon subspace, transforming the problem into a min–max over subspaces. By leveraging the chordal distance, the inner maximization admits a closed-form solution, turning the overall problem into a convex optimization in $x$ that can be solved with a gradient-based method. The main contributions are the explicit solution to the inner maximization, a computationally efficient reformulation, and a receding-horizon controller that demonstrates favorable robustness and scaling compared to LMI-based approaches. The approach provides a principled geometric interpretation of robustness in terms of finite-horizon trajectories and shows strong potential for online data-driven estimation and control tasks.

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.

Figures (8)

• Figure 1: Performance comparison of proposed method (PROP) with DD--MPC and a nominal controller (NOM), of linear quadratic tracking for double-integrator system with reference $r=1$ (red dashed horizontal line). The first figure shows the trajectories for measurement noise level $\sigma = 0.1$, and the bottom figure is for noise level $\sigma = 0.2$.
• Figure 2: Performance comparison of proposed method (PROP) with DD--MPC, for linear quadratic tracking of the Laplacian system in Section \ref{['lapl']} with reference $r=0$. The first figure shows the tracking for measurement noise level $\sigma = 0.1$, and the bottom figure is for noise level $\sigma = 0.2$.
• Figure 3: Optimal control inputs for LQR of Laplacian system. The first figure shows the tracking for measurement noise level $\sigma = 0.1$, and the bottom figure is for noise level $\sigma = 0.2$.
• Figure 4: Cost $f(x_i,\mathcal{Y}^*(x_i))$ and gradnorm $\|\nabla_x f(x_i,\mathcal{Y}^*(x_i)) \|$ evolution with iteration index $i$, at $t=50$: First column corresponds to nominal (NOM) algorithm, second column corresponds to the (PROP) algorithm for $\sigma=0.1$ and third column is for $\sigma=0.2$.
• Figure 5: Cost $f(x_i,\mathcal{Y}^*(x_i))$ and gradnorm $\|\nabla_x f(x_i,\mathcal{Y}^*(x_i)) \|$ evolution with iteration index $i$, at $t=10$: First column corresponds to nominal (NOM) algorithm, second column corresponds to the (PROP) algorithm for $\sigma=0.1$ and third column is for $\sigma=0.2$.

Theorems & Definitions (9)

• lemma 1
• lemma 2
• lemma 3
• proof
• theorem 1
• proof
• remark 1: Characterization of $\lambda^*$
• proof
• proof