Instability of an equilibrium point often provides a basis for an oscillatory mode of operation in certain nonlinear dynamical systems. This paper is concerned with a fundamental robustness analysis for instability of an equilibrium point in the presence of a dynamic perturbation. Our analysis is based on the linearized system around the equilibrium point, which may move due to the perturbation . We define the instability margin as the infimum of the H_\infty norm of the stable linear perturbation that exponentially stabilizes the corresponding equilibrium point . There are two main theoretical results ; one is on characterizations of linear systems for which the robust instability radius can be calculated exactly  and the other is on the exact instability margin for a class of nonlinear systems by properly taking the change of the equilibrium point into account . The results are applied to the Repressilator in synthetic biology, where hyperbolic instability of a unique equilibrium guarantees the persistence of oscillation phenomena in the global sense, and the effectiveness  is confirmed by numerical simulations.
 This paper is concerned with a robust instability analysis for the single-input-single-output unstable linear time-invariant (LTI) system under dynamic perturbations. The nominal system itself is possibly perturbed by the static gain of the uncertainty, which would be the case when a nonlinear uncertain system is linearized around an equilibrium point . We define the instability margin as the smallest H_\infty norm of the stable linear perturbation that stabilizes the nominal system . There are two main theoretical results : one is on a partial characterization of unperturbed nominal systems for which the robust instability radius can be calculated exactly , and the other is a numerically tractable procedure for calculating the exact instability margin for nominal systems parametrized by a perturbation parameter . The results are applied to the Repressilator in synthetic biology, where hyperbolic instability of a unique equilibrium guarantees the persistence of oscillation phenomena in the global sense, and the effectiveness of our linear robust instability analysis is confirmed by numerical simulations.
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Instability of an equilibrium point often provides a basis for an oscillatory mode of operation in certain nonlinear dynamical systems.
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fundamental robustness analysis for instability of an equilibrium point in the presence of a dynamic perturbation. Our analysis is based on the linearized system around the equilibrium point, which may move due to the perturbation
robust instability analysis for the single-input-single-output unstable linear time-invariant (LTI) system under dynamic perturbations. The nominal system itself is possibly perturbed by the static gain of the uncertainty, which would be the case when a nonlinear uncertain system is linearized around an equilibrium point
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infimum of the
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exponentially stabilizes the corresponding equilibrium point
stabilizes the nominal system
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characterizations of linear
a partial characterization of unperturbed nominal
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a numerically tractable procedure for calculating
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a class of nonlinear systems by properly taking the change of the equilibrium point into account
nominal systems parametrized by a perturbation parameter
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of our linear robust instability analysis