Subspace tracking is an important problem in signal processing that finds applications in wireless communications, video surveillance, and source localization in radar and sonar. In recent years, it is recognized that a low-dimensional subspace can be estimated and tracked reliably even when the data vectors are partially observed with many missing entries, which is greatly desirable when processing high-dimensional and high-rate data to reduce the sampling requirement. This paper is motivated by the observation that the underlying low-dimensional subspace may possess additional structural properties induced by the physical model of data, which if harnessed properly, can greatly improve subspace tracking performance. As a case study, this paper investigates the problem of tracking direction-of-arrivals from subsampled observations in a unitary linear array, where the signals lie in a subspace spanned by columns of a Vandermonde matrix. We exploit the shift-invariant structure by mapping the data vector to a latent Hankel matrix, and then perform tracking over the Hankel matrices by exploiting their low-rank properties. Numerical simulations are conducted to validate the superiority of the proposed approach over existing subspace tracking methods that do not exploit the additional shift-invariant structure in terms of tracking speed and agility.