1
Distributed Kalman filtering based on consensus
strategies
Ruggero Carli, Alessandro Chiuso, Luca Schenato, Sandro Zampieri
Abstract
In this paper, we consider the problem of estimating the state of a dynamical system from distributed
noisy measurements. Each agent constructs a local estimate based on its own measurements and estimates
from its neighbors. Estimation is performed via a two stage strategy, the first being a Kalman-like
measurement update which does not require communication, and the second being an estimate fusion
using a consensus matrix. In particular we study the interaction between the consensus matrix, the
number of messages exchanged per sampling time, and the Kalman gain. We prove that optimizing
the consensus matrix for fastest convergence and using the centralized optimal gain is not necessarily
the optimal strategy if the number of exchanged messages per sampling time is small. Moreover, we
showed that although the joint optimization of the consensus matrix and the Kalman gain is in general
a non-convex problem, it is possible to compute them under some important scenarios. We also provide
some numerical examples to clarify some of the analytical results and compare them with alternative
estimation strategies.
I. INTRODUCTION
The recent technological advances in wireless communication and the decreasing in cost and
size of electronic devices, are promoting the appearance of large inexpensive interconnected
systems, each with computational and sensing capabilities. These complex systems of agents
can be used for monitoring very large scale areas with fine resolution. However, collecting
measurements from distributed wireless sensors nodes at a single location for on-line data
R. Carli, L. Schenato and S. Zampieri are with the Department of Information Engineering, Universit
`
a di Padova, Via Gradenigo
6/a, 35131 Padova, Italy. A. Chiuso is with the Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit
`
a di Padova,
Stradella S. Nicola, 3 - 36100 Vicenza, Italy, {carlirug|chiuso|schenato|zampi}@dei.unipd.it
This work has been supported in part by the national project New techniques and applications of identification and adaptive
control funded by MIUR, and by European Union project SENSNET founded by Marie Curie IRG grant n.014815
May 17, 2007 DRAFT
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processing may not be feasible due to several reasons among which long packet delay (e.g.
due to multi-hop transmission) and/or limited bandwidth of the wireless network, due e.g. to
energy consumption requirements.
This problem is apparent in wireless ad-hoc sensor networks where information needs to be
multi-hopped from one node to another using closer neighbors. Therefore there is a growing need
for in-network data processing tools and algorithms that provide high performance in terms of
on-line estimation while (i) reducing the communication load among all sensor nodes, (ii) being
very robust to sensor node failures or replacements and packet losses, and (iii) being suitable
for distributed control applications.
The literature is very rich of contributions addressing several aspects of distributed estimation
and, for obvious reasons, we shall mention only a few. Most works focus on static estimation
problems: [41] derives conditions under which one can reconstruct the global sufficient statistic
from local sufficient statistics; [20] investigates how much information two sensors (say S
1
and S
2
) have to transmit regarding their measurements (say y
1
and y
2
) in order for a fusion
center to be able to evaluate certain functions of the measured data y
1
and y
2
; this latter
paper introduces the concept of communication complexity, also shedding some new light on
well-known data fusion formulas; [14], [31], [32], [40], [39] address quantization issues and
optimal estimation using quantized data; [3] studies the problem of distributed estimation from
relative measurements, with applications to localization and time-synchronization; [29] aims at
reconstructing, in a decentralized manner, a field (say the temperature in a certain area) from
local measurements (taken for instance from temperature sensors deployed in the environment).
Concerning estimation of dynamic processes we should mention the so called Sign of Innovation
(SOI) Kalman filter [33], [17]; in this approach each sensor broadcasts to all sensors in the
network the new information acquired by sensing the environment, so that each sensor solves
a “centralized” (i.e. with all information available) estimation problem; however, in order to
limit the bandwidth requirement, only the sign of the innovation process is transmitted. It
is also worth recalling the paper [19], in which the authors study a decentralized problem
of joint estimation and control; the overall system is decomposed into “local” subsystems of
fixed structure. They restrict to (suboptimal) linear estimation and control schemes in which
the estimators and controllers gains are found by solving (off-line) a constrained parameter
optimization problem, aiming at optimizing an asymptotic (infinite horizon) cost.
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This is very similar, in spirit, to the approach followed in this paper; we will focus on
distributed estimation of dynamical systems for which sensor nodes are not physically co-located
and can communicate with each other according to some underlying communication network. For
example, suppose that we want to estimate the temperature in a building that changes according
to a random walk, i.e T (t + 1) = T (t) + w(t), where w(t) is a zero mean white noise process
with covariance q, and we have N sensors that can measure temperature corrupted by some
noise, i.e. y
i
(t) = T (t) + n
i
(t), where n
i
(t) are zero mean white noise processes with same
covariance r. If all measurements were instantaneously available to a single location, it is well
known from the centralized Kalman filter that the optimal steady state estimator would have the
following structure:
ˆ
T (t + 1) = (1 − `
0
)
ˆ
T (t) + `
0
mean(y(t))
where mean(y(t)) :=
1
N
P
N
i=1
y
i
(t), and 0 < `
0
< 1 is the optimal Kalman gain that depends
on the process noise covariance q and the equivalent measurement noise variance r /N. This
expression already shows two important features of the optimal estimator. The first feature
is that the optimal state estimate
ˆ
T (t + 1) is a weighted average between the previous state
estimate
ˆ
T (t) and the average of the sensor measurements, thus implying that averaging reduces
uncertainty. The second is that the optimal gain needs to be tuned to optimally balance process
noise and the equivalent noise of the averaged measurements. In a distributed setting, it is not
possible to assume that all measurements are instantaneously available at a specific location,
since communication needs to be consistent with the underlying communication graph G, and
each sensor node has its own temperature estimate
ˆ
T
i
(t). However, if it was possible to provide
an algorithm that computes the mean of set of numbers only through local communication, then
the optimal estimate could be computed at each sensor node as follows:
ˆ
T
i
(t + 1) = (1 − `
0
)mean(
ˆ
T (t)) + `
0
mean(y(t))
= mean
¡
(1 − `
0
)
ˆ
T (t) + `
0
y(t)
¢
Algorithms able to compute the average of a set of numbers in a distributed way are known
as average consensus algorithms. They consist in iterations like z
+
= Qz, where z is the
vector whose entries are the quantities to be averaged and Q is a doubly stochastic matrix,
i.e. a matrix with properties Q
ij
≥ 0,
P
j
Q
ij
= 1 and
P
i
Q
ij
= 1. The consensus problem
has been widely studied in terms of convergence of Markov Chains [12] [38] [28], and it has
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been recently proposed as an effective approach to solve many control problems ranging from
flocking [16][21] to robot rendezvous [18][8]. Many interesting results have appeared recently
[30][15][25][10] just to name a few. However, a detailed discussion about the average consensus
problem is beyond the scope of this paper and we address the interested reader to the survey
paper [22] and references therein. Under some weak connectivity properties [6], these matrices
guarantee that lim
m→∞
[Q
m
z]
i
= mean(z), i.e. all elements of vector Q
m
z converge to their
initial mean mean(z). Therefore, provided it is possible to communicate sufficiently fast within
two subsequent sensor measurements, i.e. m À 1, then intuitively we can assume that the
following distributed estimation strategy yields the optimal global state estimate:
z = (1 − `
0
)
ˆ
T
i
(t) + `
0
y
i
(t) measur. & predict. stage
ˆ
T
i
(t + 1) = [Q
m
z]
i
consensus stage
Olfati-Saber [27] and Spanos et al. [35] were the first to propose this two-stage strategy based
on computing first the mean of the sensor measurements via consensus algorithms, and then
to update and predict the local estimates using the centralized Kalman optimal gains. This
approach can be extended to multivariable systems where the process evolves according to
T (t + 1) = AT (t) + w(t) and the state is only partially observable, i.e. y
i
(t) = C
i
T (t) + v
i
(t), as
shown in the static scenario by Xiao et al. [44] (A = I, w(t) = 0) and in the dynamic scenario
in [36][24]. In this context, i.e. m À 1, it natural to optimize Q for fastest convergence rate of
Q
m
, which corresponds to the second largest singular value of Q, for which there are already
very efficient optimization tools available [42] [43]. The assumption m À 1 is reasonable in
applications for which communication is inexpensive as compared to sensing. This is the case,
for example, in rendezvous control or coordination of mobile sensors where moving and sensing
the position is energetically more expensive than transmitting it to their neighbors. However,
there are many other important applications in which the number m of messages exchanged per
sampling time per node needs to be small, as required in static battery-powered wireless sensor
networks. Therefore the assumption that [Q
m
z]
i
≈ mean(z) is not valid. In this context, for
example, it is not clear whether maximizing the rate of convergence of Q is the best strategy.
Moreover, also the optimal gain ` becomes a function of the matrix Q and the number of
exchanged messages m, which is unlikely to coincide with the optimal centralized Kalman gain
proposed in all the aforementioned papers [27][35][36][24][44].
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Recently, Alriksson at al. [1] and Speranzon et al. [37], considered the case m = 1, i.e. sensors
are allowed to communicate once between sampling instants. In particular, in [1], the authors
consider a general MIMO scenario where the matrix Q = Q(t) (= W in their terminology) is
selected at each time step in order to minimize the estimation error covariance of each sensor
for the next time step, with the only constraint to leave the estimate unbiased corresponding to
P
j
Q
ij
= 1. Also the gain ` = `
i
(t) (= K in their terminology) is different for each sensor
and chosen at each time-step using the standard Kalman prediction and correction procedures in
order to minimize each sensor estimation error. Simulations show that this recursive algorithm
converges and provide good performance, thus providing a methodology to jointly optimize Q
and `. However the authors do not provide any proof of convergence nor any global optimality
guarantee. In fact, this distributed optimization approach greedily minimizes the error covariance
of each sensor at each time step, which might not be globally optimal. Differently, in [37] the
authors do not separate the algorithm between a consensus stage and an update and correct stage,
but they consider a single update equation
ˆ
T (t + 1) = K
ˆ
T (t) + Hy(t)
where
ˆ
T = [
ˆ
T
∗
1
. . .
ˆ
T
∗
N
]
∗
and y = [y
∗
1
. . . y
∗
n
]
∗1
, with the additional unbiasedness constraint
P
j
(K
ij
+ H
ij
) = 1, i.e. row-sum equal to unity. Using our terminology we note that we
would have K = (1 − `)Q and H = `Q, which satisfy the constraint. Then they propose to
compute the design matrices K, H by formulating an optimization problem where at each time
step the sum of all sensor node covariance errors is minimized. Similarly to [1], this approach
seems to converge and to provide good performance, but once again without any proof of global
optimality and insight about the connectivity properties of the underlying graph.
The consensus-based approach to distributed estimation is not the only approach. In fact,
recently Skizas et al. [34] proposed an iterative algorithm based on local estimates and on the
quality of the local estimates through special nodes called bridges, and they proved converge to
the centralized optimal estimator in the ideal scenario and to maintain good performance even
under quantization and non-gaussian noise measurements.
In this paper, we want to study the interaction between the consensus matrix Q, the number
of messages per sampling time m, and the gain `. With respect with the aforementioned works,
1
The symbol “*” denotes the conjugate transpose.
May 17, 2007 DRAFT
