Estimating Quasiperiodic Disturbance With Unknown Frequency via Expectation–Maximization

This article is concerned with a quasiperiodic disturbance estimation problem for dynamic control systems without prior knowledge on frequency. As a major challenge of our work, the quasiperiodic disturbance to be treated is always submerged by untargeted waves, leading to complicated coupling between disturbance separation and frequency identification. Existing approaches on quasiperiodic disturbance rejection have circumvented, rather than overcome, this challenge by assuming either a known frequency or a measurable disturbance signal. In this work, an expectation–maximization (EM) framework is proposed where disturbance signal separation and frequency identification are carried out in an iterative manner. In the E-step, the expected log-likelihood function is evaluated via reconstruction of the quasiperiodic signal based on the latest frequency estimate; and in the M-step, the frequency estimate is updated by maximizing the log-likelihood function obtained in the E-step. To facilitate recursive frequency estimation, an online EM algorithm is also developed based on the forward-only smoothing techniques. Furthermore, we show that the proposed method can be easily extended to deal with nonlinear system models and time-varying frequencies.