Instability Margin Analysis for Parametrized LTI Systems with Application to Repressilator

TL;DR

This paper develops a method to analyze the stability margins of parametrized LTI systems under perturbations, with applications to biological oscillators like the repressilator, providing exact and numerical tools for stability assessment.

Contribution

It introduces a novel approach for calculating the robust instability radius of parametrized LTI systems, including a tractable procedure and partial characterization results.

Findings

Exact robust instability radius can be computed for certain systems.

Numerical procedure effectively calculates stability margins.

Application confirms relevance to biological oscillators like the repressilator.

Abstract

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.