Asymptotic evaluation of the state and stiffness of the van der Paul oscillator

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 diverseness 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, 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. Such systems are widely represented in various technical devices and in the study and modeling of some biological functions of the body, such as cardiac activity, respiration, locomotor activity, etc. It is considered a typical situation for many practical applications of control theory when the complete state vector of the system is unknown and only some of the functions of the state variables -- the outputs of the system are accessible to measurement. Therefore, the problem 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 inverse control problems, namely, the problem of observability and parameter identification of an model oscillatory system is considered in this article. For observation and identification scheme design the method of invariant relations developed in analytical mechanics is used. Its modification in control problems allows us to synthesize additional relationships between known and unknown quantities of a dynamical system that arise during the observed motion. The method does not involve linearization of the original system and is essentially non-linear. The constructed nonlinear observer provides an asymptotic estimation of unknown parameter and velocity of oscillations.