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Stability analysis of distributed Kalman filtering algorithm for stochastic regression model

Siyu Xie, Die Gan, Zhixin Liu

TL;DR

The work provides the stability analysis of the proposed distributed KF algorithm without independent and stationary signal assumptions, which implies that the theoretical results are able to be applied to stochastic feedback systems.

Abstract

In this paper, a distributed Kalman filtering (DKF) algorithm is proposed based on a diffusion strategy, which is used to track an unknown signal process in sensor networks cooperatively. Unlike the centralized algorithms, no fusion center is need here, which implies that the DKF algorithm is more robust and scalable. Moreover, the stability of the DKF algorithm is established under non-independent and non-stationary signal conditions. The cooperative information condition used in the paper shows that even if any sensor cannot track the unknown signal individually, the DKF algorithm can be utilized to fulfill the estimation task in a cooperative way. Finally, we illustrate the cooperative property of the DKF algorithm by using a simulation example.

Stability analysis of distributed Kalman filtering algorithm for stochastic regression model

TL;DR

The work provides the stability analysis of the proposed distributed KF algorithm without independent and stationary signal assumptions, which implies that the theoretical results are able to be applied to stochastic feedback systems.

Abstract

In this paper, a distributed Kalman filtering (DKF) algorithm is proposed based on a diffusion strategy, which is used to track an unknown signal process in sensor networks cooperatively. Unlike the centralized algorithms, no fusion center is need here, which implies that the DKF algorithm is more robust and scalable. Moreover, the stability of the DKF algorithm is established under non-independent and non-stationary signal conditions. The cooperative information condition used in the paper shows that even if any sensor cannot track the unknown signal individually, the DKF algorithm can be utilized to fulfill the estimation task in a cooperative way. Finally, we illustrate the cooperative property of the DKF algorithm by using a simulation example.

Paper Structure

This paper contains 18 sections, 14 theorems, 99 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Graph Theory
  4. Observation Model
  5. Distributed KF Algorithm
  6. Some Preliminaries
  7. Error Equation for the Distributed KF Algorithm
  8. Some definitions
  9. Assumptions
  10. The Main Results
  11. Proofs of the main results
  12. Proof of Theorem \ref{['thm1']}
  13. Proof of Lemma \ref{['lem1']}
  14. Proof of Lemma \ref{['lem2']}
  15. Simulation Results
  16. ...and 3 more sections

Key Result

Theorem 4.1

Let $\{\bm{A}_{k}\}$ be a sequence of $mn\times mn$ random matrices, and $\{\bm{Q}_{k}\}$ be a sequence of positive definite random matrices. Then for $\{\bm{P}_{k}\}$ and $\{\bar{\bm{P}}_{k}\}$ recursively defined by and we have for all $t>s$, Hence if $\{\bm{P}_{k}\}$ satisfies the following two conditions: then

Figures (1)

  • Figure 1: Tracking errors of the three sensors

Theorems & Definitions (24)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Definition 3.3
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • Lemma 4.1
  • Corollary 4.1
  • ...and 14 more