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Computing the Hard Scaled Relative Graph of LTI Systems

Julius P. J. Krebbekx, Eder Baron-Prada, Roland Tóth, Amritam Das

TL;DR

This work develops an exact, non-approximate method to compute the hard SRG of square LTI systems that may be unstable, using circle-bound radii $r_\alpha$ and $R_\alpha$ derived from $G_\alpha$ to express $\\operatorname{SRG}_\mathrm{e}(G)=\\mathcal{G}(\\{r_\alpha\\},\\{R_\alpha\\})$. It proves that in the SISO case the hard SRG coincides with the extended SRG, thereby linking the graphical SRG framework to Nyquist encirclement concepts while maintaining a robust well-posedness perspective. The authors show that the hard SRG provides an alternative to the MIMO Nyquist criterion and the Generalized Nyquist Criterion, offering stability margins and a scalable design view via gain-tuning in the loop. A bounding algorithm is presented that yields exact results for square LTI systems and is demonstrated on both SISO and MIMO examples, including integrators and unstable plants. Overall, the results extend SRG analysis to unstable LTI plants and position the hard SRG as a practical, exact tool for nonlinear SRG-based stability and performance analysis.

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems, where Linear Time-Invariant (LTI) systems are the fundamental building block. To analyze feedback loops with unstable LTI components, the hard SRG is required, since it aptly captures the input/output behavior on the extended $L_2$ space. In this paper, we develop a systematic computational method to exactly compute the hard SRG of LTI systems, which may be unstable and contain integrators. We also study its connection to the Nyquist criterion, including the multivariable case, and demonstrate our method on several examples.

Computing the Hard Scaled Relative Graph of LTI Systems

TL;DR

This work develops an exact, non-approximate method to compute the hard SRG of square LTI systems that may be unstable, using circle-bound radii and derived from to express . It proves that in the SISO case the hard SRG coincides with the extended SRG, thereby linking the graphical SRG framework to Nyquist encirclement concepts while maintaining a robust well-posedness perspective. The authors show that the hard SRG provides an alternative to the MIMO Nyquist criterion and the Generalized Nyquist Criterion, offering stability margins and a scalable design view via gain-tuning in the loop. A bounding algorithm is presented that yields exact results for square LTI systems and is demonstrated on both SISO and MIMO examples, including integrators and unstable plants. Overall, the results extend SRG analysis to unstable LTI plants and position the hard SRG as a practical, exact tool for nonlinear SRG-based stability and performance analysis.

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems, where Linear Time-Invariant (LTI) systems are the fundamental building block. To analyze feedback loops with unstable LTI components, the hard SRG is required, since it aptly captures the input/output behavior on the extended space. In this paper, we develop a systematic computational method to exactly compute the hard SRG of LTI systems, which may be unstable and contain integrators. We also study its connection to the Nyquist criterion, including the multivariable case, and demonstrate our method on several examples.

Paper Structure

This paper contains 34 sections, 7 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Conventions
  4. Signals and Systems
  5. Signal spaces
  6. Systems
  7. LTI Systems
  8. Transfer functions, poles and zeros
  9. Inverting LTI systems
  10. Scaled Relative Graphs
  11. Hyperbolic convexity
  12. Soft SRGs
  13. Soft SRG of LTI systems
  14. The extended SRG
  15. Hard SRGs
  16. ...and 19 more sections

Key Result

Proposition 1

Let $G \in RH_\infty^{p \times p}$, then $\operatorname{SRG}(G) = \mathcal{G}(\{\underline{\sigma}(G_\alpha)\}, \{\overline{\sigma}(G_\alpha)\})$.

Figures (2)

  • Figure 1: Feedback interconnection with LTI loop transfer $L(s)$.
  • Figure 2: The hard SRG bound of the systems in Section \ref{['sec:examples']} in gray, obtained from \ref{['eq:hard_srg_G_circle_bound_thm']}, where the soft SRG bound is depicted in dark gray and is always contained in the hard SRG bound. Note that for SISO systems, the hard/soft SRG bound is exact.

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Theorem 1: Ch. 4.5 desoerFeedbackSystemsInputoutput1975
  • Definition 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 10 more