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Amplitude Dependent Bode Diagrams via Scaled Relative Graphs

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

Abstract

Scaled Relative Graphs (SRGs) provide an intuitive graphical frequency-domain method for the analysis of Nonlinear (NL) systems, generalizing the Nyquist diagram. In this paper, we develop a method for computing $L_2$-gain bounds for Lur'e systems over bounded frequency and amplitude ranges. We do this by restricting the input space of the SRG both in frequency and energy content, and combining with methods from Sobolev theory. The resulting gain bounds over restricted sets of inputs are less conservative than bounds computed over all of $L_2$, and yield three-dimensional NL generalization of the Bode diagram, plotting $L_2$-gain as function of both input frequency and energy content. In the zero-energy limit, the Linear Time-Invariant (LTI) Bode diagram is recovered, and at the infinite-energy zero-frequency limit, we recover the $L_2$-gain. The effectiveness of our method is demonstrated on an example that resembles Phase-Locked Loop dynamics.

Amplitude Dependent Bode Diagrams via Scaled Relative Graphs

Abstract

Scaled Relative Graphs (SRGs) provide an intuitive graphical frequency-domain method for the analysis of Nonlinear (NL) systems, generalizing the Nyquist diagram. In this paper, we develop a method for computing -gain bounds for Lur'e systems over bounded frequency and amplitude ranges. We do this by restricting the input space of the SRG both in frequency and energy content, and combining with methods from Sobolev theory. The resulting gain bounds over restricted sets of inputs are less conservative than bounds computed over all of , and yield three-dimensional NL generalization of the Bode diagram, plotting -gain as function of both input frequency and energy content. In the zero-energy limit, the Linear Time-Invariant (LTI) Bode diagram is recovered, and at the infinite-energy zero-frequency limit, we recover the -gain. The effectiveness of our method is demonstrated on an example that resembles Phase-Locked Loop dynamics.
Paper Structure (23 sections, 7 theorems, 29 equations, 2 figures)

This paper contains 23 sections, 7 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signals
  4. Systems
  5. Scaled Relative Graphs
  6. Amplitude and Frequency Dependent Gains
  7. Frequency-Dependent Gain
  8. Amplitude-Dependent Gain
  9. Bounding Amplitude using Sobolev Methods
  10. Computing Frequency- and Amplitude-Dependent Gains
  11. Gain Computation using SRG Methods
  12. SRG of LTI Systems
  13. Input to output
  14. Derivative map
  15. SRG of Nonlinearities
  16. ...and 8 more sections

Key Result

Proposition 1

Suppose that Condition condition:lure holds, $\operatorname{SG}(\phi)$ satisfies the chord propertySee ryuScaledRelativeGraphs2022 for the definition of the chord property. and there exist $r, r_\tau$ such that then $\gamma([G,\phi]) \leq 1/r_1$.

Figures (2)

  • Figure 1: Lur'e system with LTI $G$ and NL $\phi : \mathbb{R} \to \mathbb{R}$.
  • Figure 2: Gain and amplitude for the example in \ref{['eq:example_lure']}.

Theorems & Definitions (15)

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