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Instability Margin Analysis for Parametrized LTI Systems with Application to Repressilator

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Document pages: 17 pages

Abstract: This paper is concerned with a robust instability analysis for thesingle-input-single-output unstable linear time-invariant (LTI) system underdynamic perturbations. The nominal system itself is possibly perturbed by thestatic gain of the uncertainty, which would be the case when a nonlinearuncertain system is linearized around an equilibrium point. We define theinstability margin as the smallest $H infty$ norm of the stable linearperturbation that stabilizes the nominal system. There are two main theoreticalresults: one is on a partial characterization of unperturbed nominal systemsfor which the robust instability radius can be calculated exactly, and theother is a numerically tractable procedure for calculating the exactinstability margin for nominal systems parametrized by a perturbationparameter. The results are applied to the Repressilator in synthetic biology,where hyperbolic instability of a unique equilibrium guarantees the persistenceof oscillation phenomena in the global sense, and the effectiveness of ourlinear robust instability analysis is confirmed by numerical simulations.

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