Topological Linear System Identification via Moderate Deviations Theory

TL;DR

This paper introduces a method to classify the topological equivalence class of a stable dynamical system from limited observations, leveraging moderate deviations theory to quantify the probability of misclassification.

Contribution

It develops a novel approach using moderate deviations principles for system identification, providing exponential decay rates for misclassification probability based on the smallest singular value.

Findings

Misclassification probability decays exponentially with observations

The decay rate is proportional to the square of the smallest singular value of the system matrix

The method effectively classifies topological equivalence classes from single trajectories

Abstract

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 $\theta$.