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On input-to-state stability verification of identified models obtained by Koopman operator

Wenjie Mei, Dongzhe Zheng, Yu Zhou, Ahmad Taha, Chengyan Zhao

Abstract

This paper proposes a class of basis functions for realizing the input-to-state stability verification of identified models obtained from the true system (assumed to be input-to-state stable) using the Koopman operator. The formulated input-to-state stability conditions are in the form of linear matrix inequalities. Two extensions are presented to relax the imposed restrictions on the basis functions. Several numerical examples are provided to demonstrate the efficacy of the proposed results.

On input-to-state stability verification of identified models obtained by Koopman operator

Abstract

This paper proposes a class of basis functions for realizing the input-to-state stability verification of identified models obtained from the true system (assumed to be input-to-state stable) using the Koopman operator. The formulated input-to-state stability conditions are in the form of linear matrix inequalities. Two extensions are presented to relax the imposed restrictions on the basis functions. Several numerical examples are provided to demonstrate the efficacy of the proposed results.
Paper Structure (30 sections, 3 theorems, 33 equations, 2 figures, 2 tables)

This paper contains 30 sections, 3 theorems, 33 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representing Koopman operator via a matrix
  4. Connecting Koopman operator and system identification
  5. Approximating Koopman operator
  6. Problem statement
  7. Main results: Basis functions for ISS analysis of identified models
  8. A class of basis functions
  9. ISS property of the identified model
  10. Extensions of basis functions
  11. Numerical Case Studies
  12. Preliminary Experiments
  13. Traffic Flow Model
  14. Generalized Persidskii System Model
  15. Experimental Setup
  16. ...and 15 more sections

Key Result

Theorem 1

Let Assumption main_assum_sector_condition be satisfied with $\phi \in \mathbb{N} \backslash \{0\}$. If there exist $0\leq P=P^{\top}\in\mathbb{R}^{n\times n}$; $\left\{\Lambda^{j}=\mathop{\mathrm{diag}}(\Lambda_{1}^{j},\dots,\Lambda_{n}^{j}) \right\}_{j=1}^{M}$, $\left\{\Xi^{k} \right\}_{k=1}^{M}$, where and $\phi$ is defined in appendix:gps, then a forward complete system F_x_approximation is I

Figures (2)

  • Figure 1: Three-dimensional representation of true trajectories (blue solid lines) and identified trajectories (red dashed lines) under four disturbance scenarios.
  • Figure 2: Comparison of Prediction Results from Different Models with Real Data

Theorems & Definitions (6)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Corollary 1
  • Definition 2
  • Theorem 2