Inference of dominant modes for linear stochastic processes

For dynamical systems that can be modelled as asymptotically stable linear systems forced by Gaussian noise, this paper develops methods to infer (estimate) their dominant modes from observations in real time. The modes can be real or complex. For a real mode (monotone decay), the goal is to infer its damping rate and mode shape. For a complex mode (oscillatory decay), the goal is to infer its frequency, damping rate and (complex) mode shape. Their amplitudes and correlations are encoded in a mode covariance matrix that is also to be inferred. The work is motivated and illustrated by the problem of detection of oscillations in power flow in AC electrical networks. Suggestions of some other applications are given.