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Learning-based data-enabled moving horizon estimation with application to membrane-based biological wastewater treatment process

Li Xiaojie, Yin Xunyuan

TL;DR

A convex data-enabled MHE formulation is developed to provide real-time state estimates of the Koopman representation, from which the states of the nonlinear system can be reconstructed.

Abstract

In this paper, we propose a data-enabled moving horizon estimation (MHE) approach for nonlinear systems. While the approach is formulated by leveraging Koopman theory, its implementation does not require explicit Koopman modeling. Lifting functions are learned from the state and input data of the original nonlinear system to project the system trajectories into the lifted space, where the resulting trajectories implicitly describe the Koopman representation for the original nonlinear system. A convex data-enabled MHE formulation is developed to provide real-time state estimates of the Koopman representation, from which the states of the nonlinear system can be reconstructed. Sufficient conditions are derived to ensure the stability of the estimation error. The effectiveness of the proposed method is illustrated using a membrane-based biological water treatment process.

Learning-based data-enabled moving horizon estimation with application to membrane-based biological wastewater treatment process

TL;DR

A convex data-enabled MHE formulation is developed to provide real-time state estimates of the Koopman representation, from which the states of the nonlinear system can be reconstructed.

Abstract

In this paper, we propose a data-enabled moving horizon estimation (MHE) approach for nonlinear systems. While the approach is formulated by leveraging Koopman theory, its implementation does not require explicit Koopman modeling. Lifting functions are learned from the state and input data of the original nonlinear system to project the system trajectories into the lifted space, where the resulting trajectories implicitly describe the Koopman representation for the original nonlinear system. A convex data-enabled MHE formulation is developed to provide real-time state estimates of the Koopman representation, from which the states of the nonlinear system can be reconstructed. Sufficient conditions are derived to ensure the stability of the estimation error. The effectiveness of the proposed method is illustrated using a membrane-based biological water treatment process.
Paper Structure (16 sections, 1 theorem, 43 equations, 3 figures)

This paper contains 16 sections, 1 theorem, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. System description
  5. LPV Koopman representation of controlled systems
  6. Problem formulation
  7. Trajectory-based representation of surrogate
  8. Learning-based Data-enabled moving horizon estimation
  9. Training process
  10. Data-enabled MHE in lifted space
  11. Stability analysis
  12. Applications to membrane bioreactor
  13. Process description
  14. Simulation setting
  15. Estimation results
  16. ...and 1 more sections

Key Result

Proposition 1

If Assumption ass:continuous holds, there exist bounds $\bar{w}$, $\bar{d}$, $\bar{r}$, such that $\|w_{k}\|_{\infty}\leq\bar{w}$, $\|d_{k}\|_{\infty}\leq\bar{d}$, and $\|r_{k}\|_{\infty}\leq\bar{r}$ hold for all $k\in\mathbb{N}$.

Figures (3)

  • Figure 1: An illustrative diagram of the proposed data-enabled MHE.
  • Figure 2: A schematic of the membrane bioreactor for wastewater treatment maere2011bsm.
  • Figure 3: Trajectories of selected actual states and state estimates provided by the proposed MHE in \ref{['eq:mhe']} and MHE in wolff2024robust under dry weather.

Theorems & Definitions (4)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Definition 2