Observer of harmonic oscillator parameters

Abstract

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9], [10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I\&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].