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Collaborative Kalman Filtration
Bayesian Perspective
Kamil Dedecius
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic
Pod Vod
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arenskou v
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e
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z
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ı 4, 182 08 Prague, Czech Republic
Keywords:
Bayesian analysis, Estimation Theory, Distributed estimation, Kalman filter.
Abstract:
The contribution studies the problem of collaborative Kalman filtering over distributed networks with or with-
out a fusion center from the theoretically consistent Bayesian perspective. After presenting the Bayesian
derivation of the basic Kalman filter, we develop a versatile method allowing exchange of observations among
the network nodes and their local incorporation. A probabilistic nodes selection technique based on prior
knowledge of nodes performance is proposed to reduce the communication requirements.
1 INTRODUCTION
The theory of distributed parameter estimation has at-
tained tremendous attention in the last decade, par-
ticularly due to still cheaper and increasingly pow-
erful (wireless) sensor networks. According to the
network topology, three main types of networks and
hence algorithms can be distinguished. First, the net-
works with a fusion center, responsible for informa-
tion processing. In these networks, the nodes do not
necessarily evaluate any modelling/estimation. Sec-
ond, the networks with a Hamiltonian cycle, similar
to the token ring computer networks. There exists
only one path in these networks; the information cir-
culates in the network and the nodes incorporate own
information (observations) into it. Third, the diffu-
sion networks, avoiding both the fusion center and
the Hamiltonian cycle. These networks with a higher
degree of connectedness employ cooperation among
nodes within subsets called neighborhoods. The other
two topology types can be viewed as highly degraded
diffusion networks. Unlike them, the (non-degraded)
diffusion networks have the highest robustness due
to the avoidance of single points of failure (SPOFs).
Therefore, we focus on filtering in diffusion networks,
while keeping in mind that the centralized and Hamil-
tonian types can be solved with the proposed results
as well.
Distributed Kalman filtering we focus on is
closely related to the distributed recursive least
squares, first proposed for the diffusion networks in
(Cattivelli et al., 2008) in the classical paradigm and
in (Dedecius and Se
ˇ
ck
´
arov
´
a, 2013) from the Bayesian
point of view. For a totally connected (hence decen-
tralized) network, the Kalman filter was proposed by
(Speyer, 1978; Ribeiro et al., 2006). However, the
requirement of total connectedness is relatively pro-
hibitive. Three types of the consensus Kalman fil-
ters, avoiding this requirement, were proposed, e.g.,
in the seminal paper (Olfati-Saber, 2007). These so-
lutions rely on the so-called microfilter architecture.
The consensus algorithms typically impose the need
of intermediate averaging iterations among nodes, de-
manding additional in-network communication. The
diffusion Kalman filter, (Cattivelli and Sayed, 2010),
avoids them.
The main problem associated with distributed
estimation is the communication burden. Several
strategies for its alleviation were proposed, however,
mostly for centralized networks. For example, (Gupta
et al., 2006) considers the case where only one node
can take measurements at a time and proposes a
stochastic scheme for its selection. Another, also cen-
tralized scheme, was proposed in (Mo et al., 2006),
with the goal of minimizing an objective function re-
lated to the Kalman filter error covariance matrix. The
most recent distributed solution (Yang et al., 2014)
considers minimization of the mean square estimation
error.
The purpose of this paper is twofold: first, we re-
view the formal derivation of the Kalman filter as-
similating measurements obtained from the network
(or its part), given in a detail in (Dedecius, 2014).
This derivation follows the basic Bayesian approach
468
Dedecius K..
Collaborative Kalman Filtration - Bayesian Perspective.
DOI: 10.5220/0005018104680474
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 468-474
ISBN: 978-989-758-039-0
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)