Stereographic Projection of Probabilistic Frequency-Domain Uncertainty

TL;DR

This paper introduces a probabilistic framework using stereographic projection to analyze frequency-domain uncertainty in control systems, providing a new way to quantify stochastic distances between uncertain and nominal models.

Contribution

It develops the CDF of the chordal distance in the stereographic projection of Nyquist plots, advancing probabilistic robust control analysis.

Findings

CDF of the stochastic distance is derived.

Framework enables probabilistic robustness assessment.

Opens new research directions in control theory.

Abstract

This paper investigates the stereographic projection of points along the Nyquist plots of single input single output (SISO) linear time invariant (LTI) systems subject to probabilistic uncertainty. At each frequency, there corresponds a complex-valued random variable with given probability distribution in the complex plane. The chordal distance between the stereographic projections of this complex value and the corresponding value for a nominal model, as per the well-known Nu-Gap metric of Vinnicombe, is also a random quantity. The main result provides the cumulative density function (CDF) of the chordal distance at a given frequency. Such a stochastic distance framework opens up a fresh and a fertile research direction on probabilistic robust control theory.