Identification of the Mathieu equation parameters
Authors: Shcherbak V.F. 
Published: February 2025

Modern objects of automatic control are complex, non-linear multidimensional systems. One of the main problems in building mathematical models of such systems is the lack of “a priori” information about the state of the object and its parameters. Therefore, an important stage of their analysis is checking the adequacy of the model, which can be carried out with the help of inverse problems, which consist in determining the state of the object and its parameters. The paper proposes a method for obtaining asymptotic estimates of unknown components of a mathematical model based on the results of measuring output signals in real time. The method of invariant relations developed in analytical mechanics is used. To solve control theory problems related to determining the state and parameters of dynamic systems, a new method for synthesizing invariant relations is proposed, which are considered as additional algebraic equations for unknown components of the mathematical model. Based on the developed general approach, the problem of estimating the amplitude, and phase of a harmonic force acting on a mechanical system was considered. The problem have been studied over a long period of time, including recently. The reason for this is the possibility of using relevant methods in various theoretical and engineering fields, for example, in mechanical systems of conversion of kinetic energy of oscillations, in rotary mechanical processes, for vibration isolation of periodic components of noise through rotating mechanisms, for compensation of harmonic perturbations in automatic control algorithms, etc. A typical representative of such systems is the Mathieu equation. The Mathieu equation has been widely studied, and governs the forced motion of a swing, the stability of ships, Faraday surface wave patterns on water, and the behavior of parametric amplifiers based on electronic or superconducting devices. In principle, the method of least squares, Fourier analysis, Laplace transformation provide a potential solution to the problems of restoring the parameters of such oscillations. However, these methods cannot be directly applied in control algorithms with real-time data processing. From the point of view of control theory, there is the problem of identification, which is to determine the unknown parameters of the system based on information about its output, i.e., known information about motion. The possibility of solving the identification problem (the property of identifiability) significantly depends on the analytical structure of the right-hand sides of the dynamic equations and available information [3]. To solve the identification problem, we used the method of invariant relations, which was developed in analytical mechanics to find partial solutions (relationships between variables) of problems of the dynamics of a rigid body with a fixed point. The modification of this method to apply to observation problems made it possible to establish additional relationships between the known and unknown quantities of the original system that arise during the motion of its extended model. Our purpose here is to extend the method of synthesis of invariant relations to the problem of identifying the parameters of nonlinear oscillators.