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On phase in scaled graphs

Sebastiaan van den Eijnden, Chao Chen, Koen Scheres, Thomas Chaffey, Alexander Lanzon

TL;DR

This paper addresses the limitation of scaled graphs (SG) in nonlinear system phase analysis, where the phase is unsigned and cannot distinguish lead from lag. It introduces the signed scaled graph (SSG) by incorporating a Hilbert-transform-based signed phase and shows that $\mathrm{SSG}(H)\subset \mathrm{SG}(H)$ and $\mathrm{SSG}(H)\cup \mathrm{SSG}^*(H)=\mathrm{SG}(H)$; provides a stability interconnection result (Theorem SGH) based on graph separation, reducing conservatism. It also defines nonlinear positive real and negative imaginary concepts within the SSG framework, including a new SSG-NI notion via $\langle \hat{u},\mathcal{E}(y)\rangle$ and discusses its interconnection implications. The results are supported by motivating examples, linking to classical passivity and NI theory, and pointing to future work on computing and applying the SSG in nonlinear graphical analysis.

Abstract

The scaled graph has been introduced recently as a nonlinear extension of the classical Nyquist plot for linear time-invariant systems. In this paper, we introduce a modified definition for the scaled graph, termed the signed scaled graph (SSG), in which the phase component is characterized by making use of the Hilbert transform. Whereas the original definition of the scaled graph uses unsigned phase angles, the new definition has signed phase angles which ensures the possibility to differentiate between phase-lead and phase-lag properties in a system. Making such distinction is important from both an analysis and a synthesis perspective, and helps in providing tighter stability estimates of feedback interconnections. We show how the proposed SSG leads to intuitive characterizations of positive real and negative imaginary nonlinear systems, and present various interconnection results. We showcase the effectiveness of our results through several motivating examples.

On phase in scaled graphs

TL;DR

This paper addresses the limitation of scaled graphs (SG) in nonlinear system phase analysis, where the phase is unsigned and cannot distinguish lead from lag. It introduces the signed scaled graph (SSG) by incorporating a Hilbert-transform-based signed phase and shows that and ; provides a stability interconnection result (Theorem SGH) based on graph separation, reducing conservatism. It also defines nonlinear positive real and negative imaginary concepts within the SSG framework, including a new SSG-NI notion via and discusses its interconnection implications. The results are supported by motivating examples, linking to classical passivity and NI theory, and pointing to future work on computing and applying the SSG in nonlinear graphical analysis.

Abstract

The scaled graph has been introduced recently as a nonlinear extension of the classical Nyquist plot for linear time-invariant systems. In this paper, we introduce a modified definition for the scaled graph, termed the signed scaled graph (SSG), in which the phase component is characterized by making use of the Hilbert transform. Whereas the original definition of the scaled graph uses unsigned phase angles, the new definition has signed phase angles which ensures the possibility to differentiate between phase-lead and phase-lag properties in a system. Making such distinction is important from both an analysis and a synthesis perspective, and helps in providing tighter stability estimates of feedback interconnections. We show how the proposed SSG leads to intuitive characterizations of positive real and negative imaginary nonlinear systems, and present various interconnection results. We showcase the effectiveness of our results through several motivating examples.
Paper Structure (13 sections, 6 theorems, 54 equations, 1 figure)

This paper contains 13 sections, 6 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and motivation
  3. Scaled graphs
  4. Motivating example
  5. Signed scaled graphs
  6. The Hilbert transform
  7. Signed scaled graphs through the Hilbert transform
  8. An interconnection result
  9. nonlinear system characterizations
  10. Passive systems
  11. Negative imaginary systems
  12. Conclusions
  13. Proof of Theorem \ref{['th:SGH']}

Key Result

Theorem 1

Consider a pair of stable systems $H_1:\mathcal{L}_2 \to \mathcal{L}_2$ and $H_2:\mathcal{L}_2 \to \mathcal{L}_2$, and suppose that the feedback interconnection in Fig. fig:FB between $H_1$ and $\tau H_2$ is well-posed for all $\tau \in (0,1]$. If there exists $r >0$ such that for all $\tau \in (0,1 then the feedback interconnection is finite-gain stable.

Figures (1)

  • Figure 1: Negative feedback interconnection.

Theorems & Definitions (16)

  • Definition 1: Well-posedness
  • Definition 2: Feedback stability
  • Theorem 1
  • Example 1: Lead and lag filters
  • Lemma 1
  • Remark 1
  • Theorem 2
  • Example 2: Lead and lag filters revisited
  • Theorem 3
  • proof
  • ...and 6 more