Tracking the state of pacemakers modeled by the van der Pol equation

Abstract

It is known that in many applications of physics, biology and other sciences as an approximate dynamic model of complex nonlinear oscillatory processes a model of one or more interconnected van der Pol oscillators or some of its modifications is used \cite{ShD01}. For this reason, nonlinear oscillators are studied as a method of modeling, analysis or even control in various fields, such as electronics \cite{ShD02}, control, robotics \cite{ShD03, ShD04}, biomedical research \cite{ShD05}, geology \cite{ShD06} and others. Naturally, in such modeling there are problems in determining the state and parameters of the models based on the results of measuring the output signals in real time. One of these problems, namely: the problem of determining the state of pacemaker models, which are obtained as certain modifications of the van der Pol oscillator equation, is considered in this paper. In the scientific literature, publications on the modeling of cardiovascular activity using oscillatory systems are widely represented. In recent years, among them there are works that are related to the solution of inverse problems for such models. In particular, in \cite{ShD07} using differential-geometric methods of control theory, a general scheme for constructing asymptotically accurate estimates of the state of a two-dimensional dynamical system is proposed. The obtained results are used to effectively solve the problem of observing two models of pacemakers. In our case in this observation problem we used the method of invariant relations \cite{ShD08}, which was developed in analytical mechanics to find partial solutions (dependencies between variables) in the problems of dynamics of a rigid body with a fixed point. Modification of this method to the problems of control theory, observation, identification allowed to synthesize between known and unknown values of the original system additional connections that arise during the motion of its extended model \cite{ShD09, ShD10, ShD11}. The corresponding technique is to expand the original system by introducing additional controlled differential equations and immersing the original system in a system of greater dimension, which due to its sufficiently free structure is more suitable for constructing an observer or identifier. Controls in an extended system are used to synthesize on its trajectories pre-proposed relations that define the unknown components of the mathematical model (phase vector, parameters) as functions of known quantities. The obtained theoretical results are illustrated by numerical simulations of the corresponding nonlinear observers in Section 5.