On Stochastic Continuous-Time Linear System Identification for Adaptive Stabilization

Learning to stabilize an unknown continuous-time stochastic linear system is a central problem in control theory and reinforcement learning. This paper numerically studies data-driven identification of such systems with possibly unstable dynamics for the purpose of stabilization. We present a method that learns a stabilizing gain from observed trajectories, and examine how the failure probability, namely the probability that the learned gain does not stabilize the system, depends on the amount of data, dimension, noise intensity, excitation, and controllability. Numerical experiments on an unstable batch-reactor benchmark show that the squared identification error decays inversely with the aggregate observation time, that is, the combined length of all trajectories, while the failure probability decays exponentially in the same quantity. Hence, despite instability, a single long trajectory can achieve similar stabilization performance to that of many short trajectories of the same total length. We also observe \emph{signal saturation}: stronger excitation improves stabilization only logarithmically with the excitation strength. Finally, we show experimentally that accurate identification of the full dynamics is \emph{not} necessary for successful stabilization.