On the Effect of Instability on Learning Continuous-Time Linear Control Systems

TL;DR

This paper investigates the challenges of identifying parameters in unstable continuous-time linear systems from limited data, proposing a randomized control approach and providing theoretical guarantees on estimation accuracy.

Contribution

It introduces a novel method for estimating unstable system matrices using randomized inputs and offers new theoretical tools for analyzing non-stationary stochastic processes.

Findings

Estimation error decreases with longer trajectories and better signal-to-noise ratio.

The method effectively estimates unstable system matrices from finite data.

New stochastic bounds improve understanding of non-stationary martingales.

Abstract

We study the problem of system identification for stochastic continuous-time dynamics, based on a single finite-length state trajectory. We present a method for estimating the possibly unstable open-loop matrix by employing properly randomized control inputs. Then, we establish theoretical performance guarantees showing that the estimation error decays with trajectory length, a measure of excitability, and the signal-to-noise ratio, while it grows with dimension. Numerical illustrations that showcase the rates of learning the dynamics, will be provided as well. To perform the theoretical analysis, we develop new technical tools that are of independent interest. That includes non-asymptotic stochastic bounds for highly non-stationary martingales and generalized laws of iterated logarithms, among others.