On Stochastic Continuous-Time Linear System Identification for Adaptive Stabilization

Abstract

System identification studies the canonical problem of reliably learning system-dynamics from state and input trajectories. One of its important applications consists of estimating a feedback gain matrix that can stabilize an unknown system. This paper focuses on the analysis of such adaptive stabilization schemes for multivariate linear time-invariant state-space models. The system instability practically renders trajectories short, i.e., the observed data is notably limited. So, it is crucial to tightly consider effects of \emph{all} involved quantities including length and number of trajectories, dimension, controllability Gramian, and signal-to-noise ratio. We put forward a simple, practical, and reliable relationship between the chance that adaptive stabilization (based on system identification) fails to stabilize the system, in terms of all the above-mentioned quantities. Our findings show that despite exponential effects of all other quantities on failure probability, such effects do not hold for signal-to-noise ratio. More specifically, this study shows that the strength of the excitation signal helps stabilization only \emph{logarithmically}; a very important phenomenon that we explain under the term \emph{signal saturation}.