On the Effect of Instability on Learning Continuous-Time Linear Control Systems
This addresses the challenge of learning unstable dynamics in control systems, which is incremental but provides new theoretical tools.
The paper tackles the problem of system identification for stochastic continuous-time linear control systems from a single trajectory, showing that estimation error decays with trajectory length, excitability, and signal-to-noise ratio while growing with dimension.
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.