Semi-global direct estimation of multiple frequencies with an adaptive observer having minimal parameterization

This paper deals with the problem of estimating the frequencies of the n sinusoidal components of a multi-sinusoidal 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 root-finding to retrieve the frequencies from the characteristic polynomial, as typically done in the literature. The frequency estimation problem is approached by formulating a new state-space 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.