Hybrid Observer for an Intrinsic Impulsive Feedback System * *A. Medvedev and D. Yamalova are in part financed by the Swedish Research Council, Grant 2015-05256. A. Churilov was partly supported by the Russian Foundation for Basic Research, Grant 17-01-00102-a, and by Saint Petersburg State University, Grant 6.38.230.2015

Abstract

Abstract This paper deals with continuous plants subject to intrinsic pulse-modulated feedback, thus exhibiting hybrid closed-loop dynamics. The system structure implements a hybrid oscillator and arises in living organisms, e.g. when episodically firing neurons control the production of hormones in endocrine glands. Hybrid observers reconstructing both the continuous and discrete states of the hybrid plant from only continuous measured outputs are considered. They excel over the existing solutions through the introduction of two co-ordinated feedbacks of the output estimation error: one correcting the continuous state estimates and another adjusting the discrete ones. Different types of the feedback operator to the discrete estimates are analyzed. The observer design problem is reduced to synchronization of the observer solution with that of the plant. The synchronous mode of the observer is rendered locally stable by the selection of the feedback gains. Numerical illustration of the design procedure and observer performance with respect to a pulse-modulated model of testosterone regulation is provided.