Online Subspace Learning on Flag Manifolds for System Identification

Data-driven control methods based on subspace representations are powerful but are often limited to linear time-invariant systems where the model order is known. A key challenge is developing online data-driven control algorithms for time-varying systems, especially when the system's complexity is unknown or changes over time. To address this, we propose a novel online subspace learning framework that operates on flag manifolds. Our algorithm leverages streaming data to recursively track an ensemble of nested subspaces, allowing it to adapt to varying system dimensions without prior knowledge of the true model order. We show that our algorithm is a generalization of the Grassmannian Recursive Algorithm for Tracking. The learned subspace models are then integrated into a data-driven simulation framework to perform prediction for unknown dynamical systems. The effectiveness of this approach is demonstrated through a case study where the proposed adaptive predictor successfully handles abrupt changes in system dynamics and outperforms several baselines.