Discrete-time Frequency-Locked-Loop filters for exact asymptotic rejection of sinusoidal disturbances

The paper proposes a discrete-time adaptive control scheme for the asymptotical rejection of unknown and possibly time-varying sinusoidal disturbances acting on a plant. The approach is based on a recently proposed discrete-time Frequency-Locked-Loop (FLL) nonlinear filter which is used in the scheme as an adaptive frequency tracker. The properties of the scheme are fully analyzed and it is shown that standard discretization techniques fail to provide discrete-time FLL filters that exhibit the same good properties of their continuous-time counterparts. In particular, unlike continuous-time FLL filters, biased estimates are usually produced by these discrete-time FLL filters and the corresponding rejection properties of the scheme are poor. In order to overcome this drawback, a Tustin discretization technique equipped with an adaptive pre-warping mechanism consisting of properly adjusting the pre-warping frequency on-line is shown to be sufficient to solve the problem and recover the rejection performance of the continuous-time schemes. Several final examples are provided for assessment purposes.