Periodic orbit analysis of two dynamical systems for electrical engineering applications

Mathematical properties of two nonlinear adaptive filters for electrical engineering applications are presented. These filters are designed to extract a desired sinusoidal component of a given periodic signal and estimate its amplitude, phase angle and frequency. Two sets of non-autonomous ordinary differential equations govern the dynamics of the filters. It is shown that each of the filters possesses a unique and stable periodic orbit. The averaging theorem is used to prove the uniqueness and stability of the periodic orbit of one of the filters. Uniqueness and stability of the periodic orbit of the other filter are proven using the Poincaré map theorem. Computer simulations and numerical results are presented to provide numerical verification of the theoretical proofs, and finally experimental results of the laboratory implementation of the filters are presented.