Exact Instability Radius of Discrete-Time LTI Systems

TL;DR

This paper precisely calculates the robust instability radius for certain unstable discrete-time systems, providing exact measures of how much stable perturbation they can withstand before becoming unstable, with applications in control and neuroscience.

Contribution

It introduces a method to determine the exact robust instability radius for specific unstable systems using phase change rate maximization and all-pass functions.

Findings

Exact RIR computed for certain unstable systems

Stable all-pass functions are optimal at peak-gain frequencies

Applications demonstrated in magnetic levitation control and neural models

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.