GREAT: Grassmannian REcursive Algorithm for Tracking & Online System Identification

TL;DR

This paper presents GREAT, an online algorithm for tracking time-varying subspaces in linear dynamical systems using Grassmannian optimization, with proven convergence guarantees and practical demonstrations.

Contribution

The paper introduces a novel Grassmannian gradient descent method for online subspace tracking with theoretical guarantees and uncertainty quantification.

Findings

Proven exponential convergence rate of the algorithm.

Guaranteed bounds on the estimation error.

Successful numerical demonstrations of the method.

Abstract

This paper introduces an online approach for identifying time-varying subspaces defined by linear dynamical systems. The approach of representing linear systems by non-parametric subspace models has received significant interest in the field of data-driven control recently. This system representation enables us to provide rigorous guarantees for linear time-varying systems, which are difficult to obtain for parametric system models. The proposed method leverages optimization on the Grassmann manifold leading to the Grassmannian Recursive Algorithm for Tracking (GREAT). We view subspaces as points on the Grassmann manifold and adapt the estimate based on online data by performing optimization on the manifold. At each time step, a single measurement from the current subspace corrupted by a bounded error is available. The subspace estimate is updated online using Grassmannian gradient descent on a cost function incorporating a window of the most recent data. Under suitable assumptions on the signal-to-noise ratio of the online data and the subspace's rate of change, we establish theoretical guarantees for the resulting algorithm. More specifically, we prove an exponential convergence rate and provide an uncertainty quantification of the estimates in terms of an upper bound on their distance to the true subspace. The applicability of the proposed algorithm is demonstrated by means of numerical examples.