Estimation of oscillatory signals and asymptotic expansions

A tracking control problem is presented in which the signal to be tracked is a sine wave at a known frequency ??c with a random phase modeled as a Brownian motion process. The measurement process is the sum of this signal with corrupting additive white noise. This is an estimation problem for which an extended Kalman filter structure is assumed. The resuiting gain is characterized by a Riccati equation with periodic coefficients. By using multiple time scales, the solution to this Riccati equation can be obtained approximately as an expansion in the frequency ??c where 1/??c is used as an expansion parameter. This formulation serves as an alternate mechanization for phase-lock loop design from its classical couterpart if the amplitude of the signal is known. This more sophisticated formulation and solution allows better performance at low frequencies.