Synchronization of Goodwin’s oscillators under boundedness and nonnegativeness constraints for solutions

Proskurnikov, A. V. & Cao, M., 2-Jan-2017, In : IEEE Transactions on Automatic Control. 62, 1, p. 372-378 7 p.

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In the recent paper by Hamadeh et al. (2012) an elegant analytic criterion for incremental output feedback passivity (iOFP) of cyclic feedback systems (CFS) has been reported, assuming that the constituent subsystems are incrementally output strictly passive (iOSP). This criterion was used to prove that a network of identical CFS can be synchronized under sufficiently strong linear diffusive coupling. A very important class of CFS consists of biological oscillators, named after Brian Goodwin and describing self-regulated chains of enzymatic reactions, where the product of each reaction catalyzes the next reaction in the chain, however, the last product inhibits the first reaction in the chain. Goodwin’s oscillators are used to model the dynamics of genetic circadian pacemakers, hormonal cycles and some metabolic pathways. In this paper we show that for Goodwin’s oscillators, where the individual reactions have nonlinear (e.g. Mikhaelis-Menten) kinetics, the synchronization criterion, obtained by Hamadeh et al., cannot be directly applied. This criterion relies on the implicit assumption of the solution boundedness, dictated also by the chemical feasibility (the state variables stand for the concentrations of chemicals). Furthermore, to test the synchronization condition one needs to know an explicit bound for a solution, which generally cannot be guaranteed under linear coupling. At the same time, we show that these restrictions can be avoided for a nonlinear coupling protocol, where the control inputs are saturated by a special nonlinear function (belonging to a wide class), which guarantees nonnegativity of the solutions and allows to get explicit ultimate bounds for them. We prove that oscillators synchronize under such a protocol, provided that the couplings are sufficiently strong.
Original languageEnglish
Pages (from-to)372-378
Number of pages7
JournalIEEE Transactions on Automatic Control
Issue number1
Publication statusPublished - 2-Jan-2017

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