Frequency estimation of periodic signals

This paper addresses the problem of estimating on-line the unknown period of a periodic signal: this is a crucial problem in the design of learning and synchronizing controls, in fault detection and for the attenuation of periodic disturbances. Given a measurable continuous, bounded periodic signal, with non-zero first harmonic in its Fourier series expansion, a dynamic algorithm is proposed which provides an on-line globally exponentially convergent estimate of the unknown period. The period estimate exponentially converges from any initial condition to a neighborhood of the true period whose size is explicitly characterized in terms of the higher order harmonics contained in the signal. It is shown that the converging period estimate can be used to initialize a locally exponentially convergent estimator for the unknown period. Existing results on local frequency estimation of periodic signals are extended in two ways: any initial frequency estimate is allowed without imposing any restrictions on the algorithm design parameters; the exact value of the period is exponentially obtained, provided that the initial conditions for the period estimate are sufficiently close to the true value. When the periodic signal is a biased sinusoid, the unknown frequency is exactly estimated, along with its bias, amplitude and phase from any initial condition, thus recovering a well-known result.