Semi-Global Direct Estimation of Multiple Frequencies with an Adaptive Observer having Minimal Parametrization

This paper deals with the problem of estimating the frequencies of the n sinusoidal components of a multisinusoidal signal. A distinctive feature of the proposed method is that the frequencies are directly adapted, thus not requiring further steps of eigenvalue extraction or polynomial rootfinding to retrieve the frequencies from the characteristic polynomial, as typically done in the literature. The frequency estimation problem is approached by formulating a new statespace realization of the signal generator (oscillatory internal model) which is characterized by a minimal parameterization in the sense that only n parameters are used to assign the spectrum of the generator, i.e., the n frequencies of the components. In contrast with other existing adaptive observer-based methods that provide direct estimates of the frequencies, the proposed technique does not require state augmentation, making the overall dynamic order (internal model’s state+parameters) equal to 3n.