A simple positive state observer for multidimensional Goodwin's oscillator

Periodic phenomena and oscillations are fundamental characteristics of the dynamics of living systems at all levels of organization, from a single cell to complex organisms. In spite of the recent progress in understanding biological oscillators and clocks, most of the aspects of their control, observation, and identification still remain nearly unexplored. In this paper, we address the problem of observer design for Goodwin's oscillator that stands as a prototypic model of a biological rhythm and has been used to portray e.g. genetic oscillators, metabolic pathways, and hormonal axes. We show that, despite its nonlinear dynamics, Goodwin's oscillator admits a simple Luenberger-type observer that preserves positivity of solutions and is free of many flaws of the standard high-gain state reconstruction, such as the peaking phenomenon and noise amplification. These improvements are achieved through exploiting the properties of the plant model rather than canceling the nonlinear dynamics by means of a high observer gain. The results are illustrated by numerical simulations for the third-order Goodwin model with a Hill nonlinearity.

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