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Distributed Observer Design over Directed Switching Topologies

Haotian Xu, Shuai Liu, Bohui Wang, Jingcheng Wang

TL;DR

A network transformation mapping method is developed whereby each local observer can classify itself into an independent subgraph based on independent judgment, and asymptotic omniscience is proven using a developed recursive proof method.

Abstract

The distributed observer design problem holds significant importance in cases in which the output information of a system is decentralized across different subsystems. Each subsystem has a local observer and access to one part of the measurement outputs and information exchanged through communication networks. This paper focuses on the design of distributed observer with jointly connected directed switching networks. The problem presents challenges due to passive switching modes and the open-loop unboundedness that results from local observability. To overcome these challenges, we develop a network transformation mapping method whereby each local observer can classify itself into an independent subgraph based on independent judgment. Next, an observable decomposition and reorganization method is developed for the digraph case to ensure that each subgraph possesses independent dynamic properties. Asymptotic omniscience is then proven using a developed recursive proof method. This paper includes many previous results as special cases, because most are only suitable for undirected switching topologies or fast-switching cases. An adaptive coupling gain design is proposed to simplify the calculation and verification of conditions that guarantee asymptotic omniscience. Finally, simulation results with the power system show the validity of the developed theory.

Distributed Observer Design over Directed Switching Topologies

TL;DR

A network transformation mapping method is developed whereby each local observer can classify itself into an independent subgraph based on independent judgment, and asymptotic omniscience is proven using a developed recursive proof method.

Abstract

The distributed observer design problem holds significant importance in cases in which the output information of a system is decentralized across different subsystems. Each subsystem has a local observer and access to one part of the measurement outputs and information exchanged through communication networks. This paper focuses on the design of distributed observer with jointly connected directed switching networks. The problem presents challenges due to passive switching modes and the open-loop unboundedness that results from local observability. To overcome these challenges, we develop a network transformation mapping method whereby each local observer can classify itself into an independent subgraph based on independent judgment. Next, an observable decomposition and reorganization method is developed for the digraph case to ensure that each subgraph possesses independent dynamic properties. Asymptotic omniscience is then proven using a developed recursive proof method. This paper includes many previous results as special cases, because most are only suitable for undirected switching topologies or fast-switching cases. An adaptive coupling gain design is proposed to simplify the calculation and verification of conditions that guarantee asymptotic omniscience. Finally, simulation results with the power system show the validity of the developed theory.
Paper Structure (11 sections, 5 theorems, 56 equations, 8 figures, 2 tables)

This paper contains 11 sections, 5 theorems, 56 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Design of distributed observer
  4. Dynamics of the local observer
  5. Network transformation mapping
  6. Asymptotic omniscience of distributed observer
  7. State decomposition and reorganization
  8. Stability analysis of the error dynamics of distributed observer
  9. Achievement of distributed observer with adaptive coupling gain
  10. Simulation with a power system
  11. Conclusions

Key Result

Lemma 1

Consider a linear time-variant system where $x\in\mathbb{R}^n$, $A(t),~M\in\mathbb{R}^{n\times n}$, and $\xi(t)\in\mathbb{R}^n$ satisfies $\lim_{t\to\infty}\xi(t)=0$, then $\lim_{t\to\infty}x(t)=0$ if the solution $\mathbbm{x}(t)$ of $\dot{x}(t)=A(t)x(t)$ satisfies $\mathbbm{x}(t)\leq a_1e^{-a_2(t-t_0)}$, where $a_1,a_2$ are positive co

Figures (8)

  • Figure 1: Illustration of the mapping on an adjacency matrix
  • Figure 2: Augmented communication networks with $4$ agents
  • Figure 3: Error dynamics of distributed observer with network transformation mapping
  • Figure 4: Error dynamics of distributed observer without network transformation mapping
  • Figure 5: Time-varying trajectory of coupling gain
  • ...and 3 more figures

Theorems & Definitions (12)

  • Remark 1
  • Lemma 1
  • Lemma 2: Hong2008Distributed
  • Lemma 3
  • Remark 2
  • Remark 3
  • Lemma 4
  • Remark 4
  • Remark 5
  • Theorem 1
  • ...and 2 more