Time-varying frequency estimation and periodic disturbance cancellation

A structure and algorithm to identify and cancel a periodic signal and/or disturbance with unknown frequency is presented. This approach is based on the behavior of an internal model with an adaptive parameter, which is employed to identify and cancel a sinusoidal signal or disturbance in parallel with a stabilizing controller, in an error feedback system. In steady state, the time-varying states of the internal model can be mapped to two variables: the magnitude and the difference between the nominal frequency of the model and the true frequency of the input periodic signal or disturbance. An integral controller, a least-squares estimator or Kalman filter can be used to drive this error to zero. The stability of the whole feedback system with this algorithm and convergence of the algorithm to the correct frequency with exact disturbance cancellation have been proven by integral manifold (slow manifold) and averaging theories. The algorithm is locally exponentially stable, rather than asymptotically stable. Signals or disturbances including multiple frequency components have also been studied. For the purpose of a direct estimation of the instantaneous frequency of a signal, an approach which has as design parameters a fictitious plant and feedback controller is presented. A strategy for choosing the transfer functions of the fictitious plant and controller to incorporate a desired bandpass filter with an adjustable notch in the scheme is also proposed. A bandpass filter characteristic enhances the ability of the algorithm to reject noise. A noise analysis for the 'measurement' of the frequency of the periodic signal in the presence of white noise is given. Some formulae to calculate the means and variances of the measured difference between the true frequency and nominal frequency for high SNR and low SNR are derived. When an integral controller is used to eliminate this difference, we prove that this frequency estimation is unbiased. The formulae to calculate the mean and variance are also given for the output of the integral controller. In the multiple harmonic case, an optimal estimation method based on Gauss-Markov theorem is suggested to achieve minimum variance estimation by using the derived mean and variance for the measured signal. A least squares estimator and Kalman filter are also presented based on the derived mean and variance for the measured signal. The simulations confirm all the approximations in the analysis.