Topological Linear System Identification via Moderate Deviations Theory

Two dynamical systems are topologically equivalent when their phase-portraits can be morphed into each other by a homeomorphic coordinate transformation on the state space. The induced equivalence classes capture qualitative properties such as stability or the oscillatory nature of the state trajectories, for example. In this paper we develop a method to learn the topological class of an unknown stable system from a single trajectory of finitely many state observations. Using a moderate deviations principle for the least squares estimator of the unknown system matrix $\theta$, we prove that the probability of misclassification decays exponentially with the number of observations at a rate that is proportional to the square of the smallest singular value of $\theta$.