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Iterative distributed moving horizon estimation of linear systems with penalties on both system disturbances and noise

Xiaojie Li, Song Bo, Yan Qin, Xunyuan Yin

TL;DR

This work tackles scalable distributed state estimation for large-scale interconnected linear systems by partitioning the centralized moving horizon estimation (MHE) objective into local DMHE objectives that penalize subsystem disturbances and measurement noise with horizon length $N$. It introduces two DMHE formulations: DMHE-1 (unconstrained) and DMHE-2 (constrained), both executed iteratively within each sampling period to converge toward the centralized MHE solution. Theoretical results establish convergence conditions, including $ ho(M_d^{-1}M_r)<1$ for DMHE-1 and a scaled gradient projection criterion $0<\gamma<2\alpha/K$ for DMHE-2, ensuring asymptotic stability of the estimation error under suitable tuning of $P$, $Q$, and $R$. A case study on a reactor-separator benchmark demonstrates improved estimation accuracy and scalability over existing partition-based approaches, validating the practical relevance of the proposed method.

Abstract

In this paper, partition-based distributed state estimation of general linear systems is considered. A distributed moving horizon state estimation scheme is developed via decomposing the entire system model into subsystem models and partitioning the global objective function of centralized moving horizon estimation (MHE) into local objective functions. The subsystem estimators of the distributed scheme that are required to be executed iteratively within each sampling period are designed based on MHE. Two distributed MHE algorithms are proposed to handle the unconstrained case and the case when hard constraints on states and disturbances, respectively. Sufficient conditions on the convergence of the estimates and the stability of the estimation error dynamics for the entire system are derived for both cases. A benchmark reactor-separator process example is introduced to illustrate the proposed distributed state estimation approach.

Iterative distributed moving horizon estimation of linear systems with penalties on both system disturbances and noise

TL;DR

This work tackles scalable distributed state estimation for large-scale interconnected linear systems by partitioning the centralized moving horizon estimation (MHE) objective into local DMHE objectives that penalize subsystem disturbances and measurement noise with horizon length . It introduces two DMHE formulations: DMHE-1 (unconstrained) and DMHE-2 (constrained), both executed iteratively within each sampling period to converge toward the centralized MHE solution. Theoretical results establish convergence conditions, including for DMHE-1 and a scaled gradient projection criterion for DMHE-2, ensuring asymptotic stability of the estimation error under suitable tuning of , , and . A case study on a reactor-separator benchmark demonstrates improved estimation accuracy and scalability over existing partition-based approaches, validating the practical relevance of the proposed method.

Abstract

In this paper, partition-based distributed state estimation of general linear systems is considered. A distributed moving horizon state estimation scheme is developed via decomposing the entire system model into subsystem models and partitioning the global objective function of centralized moving horizon estimation (MHE) into local objective functions. The subsystem estimators of the distributed scheme that are required to be executed iteratively within each sampling period are designed based on MHE. Two distributed MHE algorithms are proposed to handle the unconstrained case and the case when hard constraints on states and disturbances, respectively. Sufficient conditions on the convergence of the estimates and the stability of the estimation error dynamics for the entire system are derived for both cases. A benchmark reactor-separator process example is introduced to illustrate the proposed distributed state estimation approach.
Paper Structure (20 sections, 8 theorems, 70 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 20 sections, 8 theorems, 70 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. System description
  5. Centralized moving horizon estimation
  6. Two DMHE designs
  7. Construction of partition-based objective functions
  8. Design of DMHE-2
  9. Design of DMHE-1
  10. Implementation of DMHE algorithm
  11. Stability analysis
  12. Convergence of the unconstrained iterative DMHE (DMHE-1)
  13. Stability of the unconstrained DMHE (DMHE-1)
  14. Convergence of the constrained iterative DMHE (DMHE-2)
  15. Stability of the constrained DMHE (DMHE-2)
  16. ...and 5 more sections

Key Result

Theorem 1

In the $p$th iteration step of sampling instant $k$, given $\{y\}_{k-N}^{k}$, $\{u\}_{k-N}^{k-1}$, $\bar{x}_{k-N}^{i}$, and $\hat{x}_{k-N|k}^{l,[p-1]}$, $l\in\mathbb{N}\setminus \{i\}$, the $i$th estimator of DMHE-1 computes the optimal estimates of the subsystem state and disturbances as below:

Figures (6)

  • Figure 1: A schematic diagram of the partition-based distributed moving horizon estimation scheme.
  • Figure 2: A schematic of the reactor-separator process.
  • Figure 3: Trajectories of the actual system states and state estimates for the three vessels.
  • Figure 4: The trajectories of the average RMSEs for the proposed DMHE-1 and the DMHE algorithm in schneider2015convergence calculated based on 500 simulation runs with the standard deviation of the system disturbances being $\sigma_w=0.01$.
  • Figure 5: The mean values of the average RMSEs for the proposed DMHE-1 and the DMHE algorithm in schneider2015convergence with different magnitudes of the system disturbances.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Definition 1
  • Lemma 3
  • Theorem 4
  • ...and 4 more