GREAT: Grassmannian Recursive Algorithm for Tracking & Online System Identification

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 representatio...

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Published in:IEEE transactions on automatic control pp. 1 - 16
Main Authors: Sasfi, Andras, Padoan, Alberto, Markovsky, Ivan, Dorfler, Florian
Format: Journal Article
Language:English
Published: IEEE 2025
Subjects:
Behavioral systems
Convergence
Dynamical systems
Linear systems
manifold optimization
Manifolds
Measurement
Prediction algorithms
Signal processing algorithms
subspace methods
System identification
Time-varying systems
Vectors
ISSN:0018-9286, 1558-2523
Online Access:Get full text
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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.
AbstractList 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.
Author Padoan, Alberto
Dorfler, Florian
Sasfi, Andras
Markovsky, Ivan
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  organization: Department of Information Technology and Electrical Engineering, ETH Zürich, Zürich, Switzerland
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SubjectTerms Behavioral systems
Convergence
Dynamical systems
Linear systems
manifold optimization
Manifolds
Measurement
Prediction algorithms
Signal processing algorithms
subspace methods
System identification
Time-varying systems
Vectors
Title GREAT: Grassmannian Recursive Algorithm for Tracking & Online System Identification
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