State Observer for the Pacemaker Model Based on the Van der Pol Equation

Abstract

A Dutch physiologist and a founder of electrocardiography V. Einthoven [10] proposed the first known model of the cardiac electrical activity. Later, van der Pol and van der Mark [11] developed a model of the heart, where the heartbeat is considered as a relaxation oscillation. From this point of view, to model the operation of pacemakers, the van der Pol equation [14,15,19] can be useful. The paper offers modeling of only one heart node that is the S-A (sinoatrial) node, which is the main heart pacemaker [20].Many control algorithms for dynamic systems are based on feedback, which involves the full state vector of a dynamic system. However, in practice, the full state vector is not always known. So, in the case of cardiac electrical activity, the potentials of the nodes rather than their changing rates are measured. To restore the full state vector from existing measurements, state observers are often used.In this paper, we solve the task of constructing an observer with linear error dynamics [22.25]. A necessary condition for the existence of such an observer is the system observability. The sufficient conditions can be formulated in the framework of the differential-geometric approach [25] using the ideas of duplicity [25,26]. Within this approach, an algorithm for observer construction can be developed. In the paper, a general problem to construct an observer for two-dimensional systems is solved and the results obtained are applied to the pacemaker model based on the Van der Pol oscillator. The numerical simulation enables us to illustrate operation of the observer developed.

Keywords

• pacemaker
• state observer
• observer error
• canonical observability form
• van der pol equation