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Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Connections and Computations for Nonlinear Systems

Timo de Groot, Tom Oomen, W. P. M. H. Heemels, Sebastiaan van den Eijnden

Abstract

Scaled relative graphs (SRGs) enable graphical analysis and design of nonlinear systems. In this paper, we present a systematic approach for computing both soft and hard SRGs of nonlinear systems using dynamic integral quadratic constraints (IQCs). These constraints are exploited via application of the S-procedure to compute tractable SRG overbounds. In particular, we show that the multipliers associated with the IQCs define regions in the complex plane. Soft SRG computations are formulated through frequency-domain conditions, while hard SRGs are obtained via hard factorizations of multipliers and linear matrix inequalities. The overbounds are used to derive an SRG-based feedback stability result for Lur'e-type systems, providing a new graphical interpretation of classical IQC stability results with dynamic multipliers.

Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Connections and Computations for Nonlinear Systems

Abstract

Scaled relative graphs (SRGs) enable graphical analysis and design of nonlinear systems. In this paper, we present a systematic approach for computing both soft and hard SRGs of nonlinear systems using dynamic integral quadratic constraints (IQCs). These constraints are exploited via application of the S-procedure to compute tractable SRG overbounds. In particular, we show that the multipliers associated with the IQCs define regions in the complex plane. Soft SRG computations are formulated through frequency-domain conditions, while hard SRGs are obtained via hard factorizations of multipliers and linear matrix inequalities. The overbounds are used to derive an SRG-based feedback stability result for Lur'e-type systems, providing a new graphical interpretation of classical IQC stability results with dynamic multipliers.

Paper Structure

This paper contains 15 sections, 5 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on SRGs and IQCs
  3. Scaled Relative Graphs (SRGs)
  4. Integral Quadratic Constraints (IQCs)
  5. Connections between SRGs and static IQCs
  6. SRG overbounding via dynamic IQCs
  7. Overbounding soft SRGs
  8. Overbounding hard SRGs
  9. Overbound refinements
  10. Illustrative example
  11. Feedback stability analysis
  12. Lur'e system description
  13. An SRG--IQC stability result
  14. Popov multipliers
  15. Conclusions

Key Result

Lemma 1

Let $N=I$ and $M = \Pi \otimes I$ in eq:sIQC and eq:hIQC, where $\Pi = \Pi^\top \in \mathbb{R}^{2\times 2}$ is given. A causal system $H:\mathcal{L}_{2e}^n \to \mathcal{L}_{2e}^n$ satisfies the soft incremental IQC in eq:sIQC if and only if $\textup{SRG}(H) \subseteq \mathcal{S}(\Pi)$, with Furthermore, $H$ satisfies the hard incremental IQC in eq:hIQC if and only if $\textup{SRG}_e(H) \subseteq

Figures (2)

  • Figure C1: Overbounds on the soft SRG for systems satisfying the IQCs in \ref{['eq:SMN']} and \ref{['eq:HMN']} computed using: a) only the static IQC (hatched region), and b) both the static and dynamic IQCs. The SRG of the LTI system in \ref{['eq:LTI_example']} is shown in blue.
  • Figure D1: Lur'e feedback interconnection.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • ...and 3 more