IDENTIFICATION OF DISSIPATION CHARACTERISTICS FOR VAN DER POL OSCILLATORS

Abstract

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems gives the key to understanding and exploring a huge number of different phenomena. With this approach it is a priori assumed that the entire physical manifold can be represented in the form of fairly simple model equations. It is contributes to a qualitative study of complex systems for various physical nature since basic models individually are well studied, and their parameters have a physical interpretation.

In particular, it is well known that oscillatory motion of various systems with a
stable limit cycle can be modeled by a system consisting of one or more coupled van der Pol oscillators. This systems are widely represented, for example, in the study and modeling of some biological functions of the body, such as cardiac activity, respiration, locomotor activity, etc. Therefore, the tasks of determining in real time the state and parameters of such systems based on the results of measuring the output signals are relevant. One of these problems, namely, the problem of identification some parameters of an oscillatory system, is considered in this article. In the paper it is proposed Aa method is proposed for obtaining asymptotic estimates of parameters that determine the nonlinear nature of oscillations for a system of interconnected van der Pol oscillators by information about their motion. On the first step the corresponding identification problem was solved for one van der Pol oscillator; further, the results obtained results are extended to a system of interconnected oscillators. 

The unknown parameter characterizes the nonlinear term in the van der Pol
equation and determines the threshold value for deviations at which the damping value in the system changes sign. The method of invariant relations is used for identification scheme design. Such approach allows us to synthesize additional relations arising between known and unknown quantities during the observed motion of the system considered. The constructed identifier provides an asymptotic estimation of unknown parameters for oscillatory networks of arbitrary structure. The numerical simulation confirms the operability of the proposed scheme for solving the identification problem.