Exact Instability Radius of Discrete-Time LTI Systems
Chung-Yao Kao, Sei Zhen Khong, Shinji Hara, Yu-Jen Lin
TL;DR
The paper addresses robust instability analysis for nominally unstable discrete-time LTI systems by introducing and characterizing the exact robust instability radius (RIR). It develops a discrete-time phase change rate (PCR) maximization framework, proving that optimal worst-case perturbations are first-order all-pass functions and deriving necessary and sufficient marginal-stability and exact-RIR conditions for systems with a unique peak gain frequency. The results yield explicit criteria for when an unstable plant has exact RIR, and provide constructive perturbations and controller structures, complemented by two practical applications in magnetic levitation and neural dynamics. Together, these contributions advance a concrete, technically grounded theory linking robust instability, single-mode marginal stabilization, and strong stabilization, with clear pathways for extension to broader unstable networks.
Abstract
The robust instability of an unstable plant subject to stable perturbations is of significant importance and arises in the study of sustained oscillatory phenomena in nonlinear systems. This paper analyzes the robust instability of linear discrete-time systems against stable perturbations via the notion of robust instability radius (RIR) as a measure of instability. We determine the exact RIR for certain unstable systems using small-gain type conditions by formulating the problem in terms of a phase change rate maximization subject to appropriate constraints at unique peak-gain frequencies, for which stable first-order all-pass functions are shown to be optimal. Two real-world applications -- minimum-effort sampled-data control of magnetic levitation systems and neural spike generations in the FitzHugh--Nagumo model subject to perturbations -- are provided to illustrate the utility of our results.