参数化LTI系统的不稳定性裕度分析及其在抑制子中的应用
Instability Margin Analysis for Parametrized LTI Systems with Application to Repressilator
AI总结：
本文针对动态扰动下的不稳定LTI系统定义鲁棒不稳定性半径，提出精确计算的理论与数值方法，并应用于合成生物学抑制子以验证振荡现象的全局持续性。
摘要：
本文关注动态扰动下单输入单输出不稳定线性时不变（LTI）系统的鲁棒不稳定性分析。标称系统本身可能受到不确定性静态增益的扰动，当非线性不确定系统在平衡点附近线性化时就会出现这种情况。我们将鲁棒不稳定性半径定义为使标称系统稳定的稳定线性扰动的最小$H_\infty$范数。主要有两个理论结果：一是对可精确计算鲁棒不稳定性半径的无扰动标称系统的部分刻画，二是对由扰动参数参数化的标称系统计算精确鲁棒不稳定性半径的数值可处理程序。这些结果应用于合成生物学中的抑制子，其中唯一平衡点的双曲不稳定性保证了全局意义上振荡现象的持续性，数值模拟证实了我们的线性鲁棒不稳定性分析的有效性。
英文摘要：
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 robust instability radius 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 robust instability radius 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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