Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

EDMD-Based Robust Observer Synthesis for Nonlinear Systems

Xiuzhen Ye, Wentao Tang

TL;DR

The paper addresses robust nonlinear state observation from data by combining EDMD-based Koopman surrogates with a linear fractional representation to account for modeling errors. It proves probabilistic error bounds linking data size to a conic uncertainty, and formulates an SDP with LMIs that guarantees exponential convergence of the observer with high probability. The key contributions are the data-driven LFR framework, data-size dependent convergence certification, and validation on systems with and without invariant Koopman lifting. The approach offers a scalable, certifiable pathway for nonlinear observation in practical settings such as chemical processes.

Abstract

This paper presents a data-driven Koopman operator-based approach for designing robust state observers for nonlinear systems. Based on a finite-dimensional surrogate of the Koopman generator, identified via an extended dynamic mode decomposition (EDMD) procedure, a tractable formulation of the observer design problem is enabled on the data-driven model with conic uncertainties. The resulting problem is cast as a semidefinite program (SDP) with linear matrix inequalities (LMIs), guaranteeing exponential convergence of the observer with a predetermined rate in a probabilistic sense. The approach bridges the gap between statistical error tolerance and observer convergence certification, and enables an explicit use of linear systems theory for nonlinear observation in a data-driven framework. Numerical studies demonstrate the effectiveness and flexibility of the proposed method.

EDMD-Based Robust Observer Synthesis for Nonlinear Systems

TL;DR

The paper addresses robust nonlinear state observation from data by combining EDMD-based Koopman surrogates with a linear fractional representation to account for modeling errors. It proves probabilistic error bounds linking data size to a conic uncertainty, and formulates an SDP with LMIs that guarantees exponential convergence of the observer with high probability. The key contributions are the data-driven LFR framework, data-size dependent convergence certification, and validation on systems with and without invariant Koopman lifting. The approach offers a scalable, certifiable pathway for nonlinear observation in practical settings such as chemical processes.

Abstract

This paper presents a data-driven Koopman operator-based approach for designing robust state observers for nonlinear systems. Based on a finite-dimensional surrogate of the Koopman generator, identified via an extended dynamic mode decomposition (EDMD) procedure, a tractable formulation of the observer design problem is enabled on the data-driven model with conic uncertainties. The resulting problem is cast as a semidefinite program (SDP) with linear matrix inequalities (LMIs), guaranteeing exponential convergence of the observer with a predetermined rate in a probabilistic sense. The approach bridges the gap between statistical error tolerance and observer convergence certification, and enables an explicit use of linear systems theory for nonlinear observation in a data-driven framework. Numerical studies demonstrate the effectiveness and flexibility of the proposed method.

Paper Structure

This paper contains 12 sections, 4 theorems, 42 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Data-Driven Dynamic Modeling
  3. Problem Setting
  4. Data-Driven System Representation via Koopman Operator
  5. Error Bound for Koopman Approximation
  6. Koopman-based Robust Observer Design
  7. LFR of the EDMD Model
  8. Observer Synthesis: SDP Formulation
  9. Numerical Examples
  10. A Nonlinear System with Invariant Koopman Lifting
  11. A Nonlinear System without Invariant Koopman Lifting
  12. Conclusions

Key Result

Proposition 1

(data_bound, data_bound, Thm. 3) Suppose that Assumption Assump1 holds and the data samples are independent and identically distributed, and let an error bound $c_r > 0$ and a probability tolerance $\delta \in (0, 1)$ be given. Then, there is an amount of data $d_0 \in \mathbb{N}$ such that for all with probability $1-\delta$.

Figures (2)

  • Figure 1: Observer performance for the system with invariant Koopman lifting.
  • Figure 2: Observer performance for the system without invariant Koopman lifting.

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Remark 2
  • Theorem 1
  • proof