A globally convergent frequency estimator

Online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications. In this paper we provide a solution to the long-standing fundamental open problem of ensuring a globally convergent estimation. More specifically, we propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features: (i) all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values; (ii) we obtain a simple tuning procedure for the estimator design parameters, which trades-off the adaptation tracking capabilities with noise sensitivity, ensuring (exponential) stability of the desired orbit; (iii) transient performance is considerably enhanced, even for small and large frequencies, as witnessed by extensive simulations. To reveal some of the stability-instability mechanisms of the existing algorithms and motivate our modifications we make appeal to a novel nonlinear (state-dependent) time scaling. The main advantage of working in the new time scale is that we remove the coupling,between the parameter update law and the filter itself, decomposing the system into a feedback form where the required modifications to ensure stability become apparent.