GREAT: Grassmannian REcursive Algorithm for Tracking & Online System Identification

TL;DR

This work addresses online identification of time-varying subspaces that encapsulate the behavior of linear dynamical systems. It leverages optimization on the Grassmann manifold, formulating subspace tracking as gradient descent on Gr(n,d) with a data-window-based cost, and provides non-asymptotic guarantees including an exponential convergence rate in the noise-free setting and explicit uncertainty bounds under bounded noise and subspace drift. The GREAT algorithm combines a practical update rule (Exp map) with rigorous analysis to yield an invariant tube around the true subspace, enabling robust online system identification and reliable prediction. Numerical studies on synthetic geodesics and an airplane model demonstrate competitive performance against standard subspace and parametric online methods, while highlighting improved robustness to measurement errors. The approach offers a principled, coordinate-free, non-parametric alternative for tracking evolving linear behaviors in a variety of data-driven control contexts, with clear avenues for future extensions in forgetting factors and adaptive subspace dimension.

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.

Figures (8)

• Figure 1: Illustration of the signal-to-noise ratio properties on a two dimensional subspace $\mathbf{U}$ in $\mathbb{R}^3$ (shaded). Two data samples, $u_1$ and $u_2$ are depicted along with their noise components (dotted lines). The signal component is $P_{\mathbf{U}_{}}W$ with $W=[u_1~u_2]$ and singular values $\sigma_1$ and $\sigma_2$. The image and the 2-norm condition of the matrix $P_{\mathbf{U}_{}}W$ are illustrated by the dashed ellipse. Note that, since $\sigma_2>0$, the matrix $P_{\mathbf{U}_{}}W$ spans the whole subspace $\mathbf{U}$, i.e., the ellipse lies in the plane $\mathbf{U}$.
• Figure 2: Illustration of the convergence rate $\rho$ as a function of the step size $\alpha$ in Lemma \ref{['lemma:one-step-improvement-noisy']}. The convergence rate is positive for any $\alpha$ from the interval $\left(0,\underline{\sigma}^2/(2\overline{\sigma}^2)\right)$. The maximal value of $\rho$ is $\underline{\sigma}^4/(8\overline{\sigma}^2)$, which is achieved with $\alpha=\underline{\sigma}^2/(4\overline{\sigma}^2)$.
• Figure 3: Illustration of Assumption \ref{['ass:sufficient_decrease']} and Lemma \ref{['lemma:grad_dom']}. If the estimate $\hat{\mathbf{U}}_{t-1}$ is in the metric ball $\mathbb{B}_{r_\mathrm{b}-c}(\mathbf{U}_{t-1})$ (solid yellow circle), it is also in $\mathbb{B}_{r_\mathrm{b}}(\mathbf{U}_{t})$ (blue circle), since $\mathbf{U}_{t} \in \mathbb{B}_c(\mathbf{U}_{t-1})$ (red circle) holds by Assumption \ref{['ass:LTV-Lipschitz']}. Furthermore, under Assumption \ref{['ass:sufficient_decrease']}, the gradient descent updates in the inner loop of the algorithm guarantee that the distance between the estimate and $\mathbf{U}_{t}$ reduces from $r_\mathrm{b}$ to $r_\mathrm{b}-c$, i.e., $\hat{\mathbf{U}}_{t}\in\mathbb{B}_{r_\mathrm{b}-c}(\mathbf{U}_{t})$ (dashed yellow circle). Therefore, the estimates are always contained in the ball $\mathbb{B}_{r_\mathrm{b}}$ centered around the current true subspace.
• Figure 4: Illustration of the bound in Theorem \ref{['thm:main']}. The blue tube illustrates a sequence of metric balls with varying radius centered around the true sequence of subspaces (dashed line). The estimates from the algorithm are guaranteed to remain in the tube. The evolution of the tube's radius is a function of the bound on the true subspace's rate of change $c$, the measurement error $\epsilon$, and the signal $\overline{\sigma},\underline{\sigma}$.
• Figure 5: Illustration of the subspace tracking algorithm with the derived theoretical bounds. For different step sizes $\alpha$, the evolution of the distance between the estimates and the true subspace (yellow, solid) is shown with the bound from \ref{['eq:main_thm_bound']} (blue, dotted) and the ultimate bound from \ref{['eq:main_thm_ultimate_bound']} (red, dashed) in Theorem \ref{['thm:main']}. The step size $\alpha^\mathrm{cvg}$ (left) maximizes the convergence rate, and $\alpha^\mathrm{ub}$ (right) minimizes the ultimate bound (c.f., Remark \ref{['rem:step-size']}).

Theorems & Definitions (22)

• Lemma 1
• proof
• Lemma 2: Noise bound
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 3: Single step decay bound
• Remark 5
• Remark 6