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Stereographic Projection of Probabilistic Frequency-Domain Uncertainty

Anton Nystrom, Venkatraman Renganathan, Michael Cantoni

TL;DR

The paper addresses probabilistic uncertainty in SISO LTI systems by projecting Nyquist plots onto the Riemann sphere via stereographic projection and measuring distances with the chordal metric $\kappa$. It derives an explicit expression for the CDF $\mathbb{F}_{\mathsf{K}}(d)$ of the point-wise chordal distance $K=\kappa(\bar{P},P)$ at a fixed frequency, showing that $K=\frac{\sqrt{Q}}{c}$ with $Q=z/w$ and $c=\sqrt{1+|\bar{P}|^2}$, and expressing the result through the joint density $f_{zw}$ and the corresponding $f_{xy}$. A numerical example with Gaussian uncertainty around $\bar{P}=1+j$ validates the monotonic CDF $\mathbb{F}_{\mathsf{K}}(d)$ and demonstrates the method. The work lays a foundation for probabilistic robust control by linking distributions over complex Nyquist values to distributional statements about stability-related distances, and it outlines future extensions to the $\nu$-gap and MIMO settings. The approach enables quantifiable probabilistic guarantees for closed-loop performance under model uncertainty.

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.

Stereographic Projection of Probabilistic Frequency-Domain Uncertainty

TL;DR

The paper addresses probabilistic uncertainty in SISO LTI systems by projecting Nyquist plots onto the Riemann sphere via stereographic projection and measuring distances with the chordal metric . It derives an explicit expression for the CDF of the point-wise chordal distance at a fixed frequency, showing that with and , and expressing the result through the joint density and the corresponding . A numerical example with Gaussian uncertainty around validates the monotonic CDF and demonstrates the method. The work lays a foundation for probabilistic robust control by linking distributions over complex Nyquist values to distributional statements about stability-related distances, and it outlines future extensions to the -gap and MIMO settings. The approach enables quantifiable probabilistic guarantees for closed-loop performance under model uncertainty.

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.
Paper Structure (9 sections, 4 theorems, 33 equations, 5 figures)

This paper contains 9 sections, 4 theorems, 33 equations, 5 figures.

Table of Contents

  1. Research Motivation
  2. Motivating Problem $1$
  3. Motivating Problem $2$
  4. Contribution and Paper Organization
  5. Notation and Preliminaries
  6. Stereographic Projection
  7. Formulating Chordal Distance as a Random Quantity
  8. Numerical Simulations
  9. Conclusion & Future Outlook

Key Result

Proposition 1

(From vinnicombe_tac_1993) Given a nominal continuous time LTI plant $\bar{P}$, and nominal feedback compensator $\bar{C}$, let and $\|\cdot\|_\infty$ denotes the $\mathcal{H}_\infty$ norm. Then, any controller $\bar{C}$ that achieves $b_{\bar{P}, \bar{C}} > \alpha$ stabilises the set of plants $\{ P : \delta_{\nu}(P, \bar{P}) \leq \alpha\}$ and where $\delta_\nu(P,\bar{P})$ denotes the $\nu$-ga

Figures (5)

  • Figure 1: Nyquist plot of the identified system \ref{['eq:model_set']} along with uncertainty regions corresponding to $5$ standard deviations on a frequency grid of interval length $5\,\mathrm{rad/s}$ is shown here.
  • Figure 2: The histogram of the $\nu$-gap between the nominal model and $1000$ independent trials is shown here. Clearly, more systems with very small distance measure from the nominal model got realised during the trials.
  • Figure 3: Realizations of chordal distance from the nominal model across a range of frequencies during $1000$ trials are plotted here.
  • Figure 4: Stereographic projection on $\mathbb{R}$ is illustrated here. The uncertainty in point $P$ is depicted as an orange interval $\mathbf{S} \subset \mathbb{R}$. Both the nominal point $\bar{P}$ and its projection $\phi^{-1}(\bar{P})$ are shown in blue color. A realization of the random point $P \in \mathbf{S}$ and its projection $\phi^{-1}(P)$ are shown in red color. Since $P$ is random, the angle $\alpha := \measuredangle P\mathfrak{N}\bar{P}$ comes random and hence the corresponding chordal distance line in green color are random as well.
  • Figure 5: The CDF of $\kappa(\bar{P}, P)$ following Theorem \ref{['thm_pdf_kappa']} when the underlying distribution $f_{\mathsf{P}}$ is Gaussian is depicted here.

Theorems & Definitions (9)

  • Proposition 1
  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • proof