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Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

Julius P. J. Krebbekx, Roland Tóth, Amritam Das

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limit its applicability in analyzing practical nonlinear systems. We overcome this pitfall by modifying the SRG of a linear time invariant operator, combining the SRG with the Nyquist criterion, and apply our result to Lur'e systems. We thereby obtain a generalization of the celebrated circle criterion, which deals with a broader class of nonlinearities, and provides (incremental) $L_2$-gain performance bounds.

Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limit its applicability in analyzing practical nonlinear systems. We overcome this pitfall by modifying the SRG of a linear time invariant operator, combining the SRG with the Nyquist criterion, and apply our result to Lur'e systems. We thereby obtain a generalization of the celebrated circle criterion, which deals with a broader class of nonlinearities, and provides (incremental) -gain performance bounds.

Paper Structure

This paper contains 20 sections, 9 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Conventions
  4. Signals, Systems and Stability
  5. Scaled Relative Graphs
  6. Definitions
  7. Operations on SRGs
  8. Nonlinear System Analysis
  9. The Nyquist Criterion
  10. Nonlinear Feedback Systems
  11. System Analysis with Scaled Relative Graphs
  12. Pitfall of Stability Analysis with SRGs
  13. An apparent contradiction
  14. Understanding the Pitfall
  15. The SRG of an Unstable LTI System
  16. ...and 5 more sections

Key Result

Proposition 1

Let $0 \neq \alpha \in \mathbb{R}$ and let $R, S$ be arbitrary operators on the Hilbert space $\mathcal{L}$. Then, If the SRGs above contain $\infty$ or are the empty set, the above operations are slightly different, see ryuScaledRelativeGraphs2022.

Figures (5)

  • Figure 1: A simple linear feedback system.
  • Figure 2: Block diagram of a Lur'e system.
  • Figure 3: Block diagram of a general feedback interconnection.
  • Figure 4: SRGs and Nyquist diagram corresponding to $L(s) = \frac{-2}{s^2+s+1}$. The shaded area is the SRG and the bold line is the Nyquist diagram.
  • Figure 5: Figures for the SRG analysis of the example in Section \ref{['sec:examples']}: (a) graph of the nonlinearity $\phi$, (b) $\operatorname{SRG}(\phi)$, (c) Nyquist diagram of $G$, (d) $\operatorname{SRG}'(G)$, (e) visualization of the separation condition in Eq. \ref{['eq:separation_assumption']}.

Theorems & Definitions (18)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 4
  • Definition 1
  • Theorem 5
  • ...and 8 more