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Partition-based distributed extended Kalman filter for large-scale nonlinear processes with application to chemical and wastewater treatment processes

Xiaojie Li, Adrian Wing-Keung Law, Xunyuan Yin

Abstract

In this paper, we address a partition-based distributed state estimation problem for large-scale general nonlinear processes by proposing a Kalman-based approach. First, we formulate a linear full-information estimation design within a distributed framework as the basis for developing our approach. Second, the analytical solution to the local optimization problems associated with the formulated distributed full-information design is established, in the form of a recursive distributed Kalman filter algorithm. Then, the linear distributed Kalman filter is extended to the nonlinear context by incorporating successive linearization of nonlinear subsystem models, and the proposed distributed extended Kalman filter approach is formulated. We conduct rigorous analysis and prove the stability of the estimation error dynamics provided by the proposed method for general nonlinear processes consisting of interconnected subsystems. A chemical process example is used to illustrate the effectiveness of the proposed method and to justify the validity of the theoretical findings. In addition, the proposed method is applied to a wastewater treatment process for estimating the full state of the process with 145 state variables.

Partition-based distributed extended Kalman filter for large-scale nonlinear processes with application to chemical and wastewater treatment processes

Abstract

In this paper, we address a partition-based distributed state estimation problem for large-scale general nonlinear processes by proposing a Kalman-based approach. First, we formulate a linear full-information estimation design within a distributed framework as the basis for developing our approach. Second, the analytical solution to the local optimization problems associated with the formulated distributed full-information design is established, in the form of a recursive distributed Kalman filter algorithm. Then, the linear distributed Kalman filter is extended to the nonlinear context by incorporating successive linearization of nonlinear subsystem models, and the proposed distributed extended Kalman filter approach is formulated. We conduct rigorous analysis and prove the stability of the estimation error dynamics provided by the proposed method for general nonlinear processes consisting of interconnected subsystems. A chemical process example is used to illustrate the effectiveness of the proposed method and to justify the validity of the theoretical findings. In addition, the proposed method is applied to a wastewater treatment process for estimating the full state of the process with 145 state variables.
Paper Structure (24 sections, 7 theorems, 79 equations, 10 figures, 3 tables, 1 algorithm)

This paper contains 24 sections, 7 theorems, 79 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributed full-information estimation
  4. Distributed Kalman filter algorithm
  5. Proposed distributed extended Kalman filter
  6. Error dynamics for the proposed DEKF
  7. Stability analysis
  8. Applications
  9. Data Availability and Reproducibility Statement
  10. Conclusions
  11. Acknowledgment

Key Result

Proposition 1

Let $\hat{x}_{k|k}=\mathrm{col}\{\hat{x}_{k|k}^{1}, \hat{x}_{k|k}^{2}, \ldots, \hat{x}_{k|k}^{n}\}$. If the Assumption assume:continuous holds, then the estimation error $e_{k|k}=x_{k}-\hat{x}_{k|k}$, $k\geq 0$, for the entire process model_nonlinear_centralized given by the proposed DEKF algorithm In phi_varphi$, \phi$ and $\varphi$ are high-order terms of the first-order Taylor-series expansion

Figures (10)

  • Figure 1: A schematic diagram of proposed partition-based distributed (extended) Kalman filter scheme
  • Figure 2: The trajectories of the actual states and the state estimates for the numerical example.
  • Figure 3: The trajectories of the RMSEs for the three designs: 1) proposed DKF; 2) DMHE in Li et al li2023iterative with same hyper-parameters as DKF; 3) DMHE in Li et al li2023iterative with fine-tuned hyper-parameters. The shadowed region shows the possible range of the RMSE over 500 simulations.
  • Figure 4: A schematic of the four-CSTR process.
  • Figure 5: The trajectories of the actual states and the state estimates for four reactors.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Proof
  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Proof
  • Lemma 3
  • Lemma 4
  • ...and 4 more