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Computing Scaled Relative Graphs of Discrete-time LTI Systems from Data

Talitha Nauta, Richard Pates

TL;DR

A robust version of the Scaled Relative Graph is introduced, which can be computed from noisy data trajectories and contains the SRG of the actual system.

Abstract

Graphical methods for system analysis have played a central role in control theory. A recently emerging tool in this field is the Scaled Relative Graph (SRG). In this paper, we further extend its applicability by showing how the SRG of discrete-time linear-time-invariant (LTI) systems can be computed exactly from its state-space representation using linear matrix inequalities. We additionally propose a fully data-driven approach where we demonstrate how to compute the SRG exclusively from input-output data. Furthermore, we introduce a robust version of the SRG, which can be computed from noisy data trajectories and contains the SRG of the actual system.

Computing Scaled Relative Graphs of Discrete-time LTI Systems from Data

TL;DR

A robust version of the Scaled Relative Graph is introduced, which can be computed from noisy data trajectories and contains the SRG of the actual system.

Abstract

Graphical methods for system analysis have played a central role in control theory. A recently emerging tool in this field is the Scaled Relative Graph (SRG). In this paper, we further extend its applicability by showing how the SRG of discrete-time linear-time-invariant (LTI) systems can be computed exactly from its state-space representation using linear matrix inequalities. We additionally propose a fully data-driven approach where we demonstrate how to compute the SRG exclusively from input-output data. Furthermore, we introduce a robust version of the SRG, which can be computed from noisy data trajectories and contains the SRG of the actual system.
Paper Structure (11 sections, 4 theorems, 45 equations, 3 figures)

This paper contains 11 sections, 4 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Scaled Relative Graphs
  5. SRGs of discrete-time state-space models
  6. SRGs from data trajectories
  7. Robust SRGs from noisy data trajectories
  8. Examples
  9. Unstable MIMO
  10. Low- and High-pass filters
  11. Conclusions

Key Result

Corollary 1

Consider the operators $\mathbf{T}^\tau$ and $\mathbf{T}$ as defined in Definition def:Ttau and def:Tinf. Then $\lim_{\tau \rightarrow \infty} \mathrm{SRG}\left(\mathbf{T}^\tau\right)$ is equal to and $\mathrm{cl}\;\mathrm{SRG}(\mathbf{T})$ is equal to

Figures (3)

  • Figure 1: The figure shows how Corollary \ref{['cor:circle_thm']} can be used to compute the . The grey area shows the approximation given by the intersection of the gain bounds obtained for $\{ \alpha_1, \alpha_2, \alpha_3\}$. The orange area is the .
  • Figure 2: The for the operators $\mathbf{T}$, to the left, and $\lim_{\tau \rightarrow \infty} \mathbf{T}^\tau$, to the right, of the system in \ref{['eq:ss_MIMO_ex']}. The orange area is the obtained from state-space representation or data trajectories, while the blue area shows the robust extension obtained from noisy data trajectories. Note that the areas go to infinity for $\lim_{\tau \rightarrow \infty} \mathbf{T}^\tau$.
  • Figure 3: The orange area shows the of the operator $\mathbf{T}$ associated with the low-pass filter in \ref{['eq:lowpass']}, to the left, and a high-pass filter in \ref{['eq:highpass']}, to the right, obtained from state-space representation or data trajectories. The extended blue area shows the robust version obtained from noisy data trajectories.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Corollary 1
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4
  • Theorem 2
  • proof
  • Remark 1
  • ...and 3 more