IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 1
Coupled Distributed Estimation and Control for
Mobile Sensor Networks
Reza OlfatiSaber and Parisa Jalalkamali
Abstract—In this paper,we introduce a theoretical framework
for coupled distributed estimation and motion control of mobile
sensor networks for collaborative target tracking.We use a Fisher
Information theoretic metric for quality of sensed data.The
mobile sensing agents seek to improve the information value of
their sensed data while maintaining a safedistance from other
neighboring agents (i.e.perform informationdriven ﬂocking).
We provide a formal stability analysis of continuous Kalman
Consensus ﬁltering (KCF) algorithm on a mobile sensor network
with a ﬂockingbased mobility control model.The discretetime
counterpart of this coupled estimation and control algorithm
is successfully applied to tracking of two types of targets with
stochastic linear and nonlinear dynamics.
Index Terms—mobile sensor networks,distributed Kalman
ﬁltering,ﬂocking,informationdriven control,collaborative target
tracking
I.INTRODUCTION
Collaborative tracking of multiple targets (or events) in an
environment arise in a variety of surveillance and security
applications and intelligent transportation.Most of the past
research on target tracking has been focused on the use of
centralized algorithms that run on static multisensor platforms
[1].Centralized Kalman ﬁltering plays a crucial role in such
target tracking algorithms.
Distributed estimation for static sensor networks has at
tracted many researchers in recent years [2],[3],[4],[5],
[6],[7].The existing distributed algorithms for target tracking
using mobile sensor networks are extremely limited to a few
instances [8],[9].In [10] the KCF algorithm of the ﬁrst author
is successfully applied to multitarget tracking using camera
networks.
In this paper,we present a systematic analysis framework
for mobile sensor networks with a ﬂockingbased mobility
control model that run a novel distributed Kalman ﬁltering
algorithm [11] for collaborative tracking of a single target.
The sensors in our framework have an information value
function I
i
= f(
i
) where
i
denotes the target range and
deﬁned as the distance between the agent and the predicted
position of target .In addition,f() is a decreasing function
of the target range.According to this model of quality of
sensed data,the information value of a sensor increases as
the sensor comes closer to the target.This notion of the
information value that was also used in [8] is the same as
R.OlfatiSaber is an Assistant Professor of Engineering at the Thayer
School of Engineering at Dartmouth College,Hanover,NH.Email:ol
fati@dartmouth.edu.This work was supported in part by the NSF CAREER
award of the author.
P.Jalalkamai is a PhD candidate at the Thayer School of Engineering at
Dartmouth College,Hanover,NH.Email:parisa.jalalkamali@dartmouth.edu
the trace of the Fisher Information Matrix (FIM) of sensed
data for target tracking applications [12],[13].
We propose a solution to the problem of collisionfree
tracking of a mobile target via mobile sensor networks using a
combination of the ﬂocking and KalmanConsensus Filtering
algorithms [2],[11] of the ﬁrst author.
The major challenge in analysis of the resulting coupled
estimation and control algorithm for mobile sensor networks
that we call informationdriven ﬂocking is that each sensing
agent
i
has its own dedicated agent called ^
i
(See [14]
for the deﬁnition of  and agent).The state of ^
i
is
the estimate of the state of target by agent i and the n
different estimates ^
i
of the target are distinct.In the ﬂocking
algorithms presented in [14],all n agents are the same.This
change results in a perturbed structural dynamics of the ﬂock
where the perturbation terms depend on the estimation errors.
Our main result is to establish that the coupled distributed
estimation and control algorithm for a mobile sensor network
has a combined cost (Lyapunov function) that is monotonically
decreasing in time and guarantees reaching a consensus on
estimates of the state of the target by all mobile sensors.We
also introduce a cascade nonlinear normal form and stability
analysis for structural dynamics of mobile sensor networks
performing informationdriven ﬂocking.
The outline of the paper is as follows.Some basic notations
and problem setup are discussed in Section II.Our main
theoretical results on distributed target tracking algorithms for
mobile sensor networks are provided in Section III.Our exper
imental results are presented in Section V.Finally,concluding
remarks are made in Section VI.
II.PRELIMINARIES:NOTATIONS AND PROBLEM SETUP
Consider n mobile sensors
i
with the dynamics
(
_q
i
= p
i
_p
i
= u
i
(1)
where q
i
;p
i
;u
i
2 R
d
and the goal to track the state of a mobile
target with dynamics
_x = Ax +Bw;x 2 R
m
(2)
The sensing agents make the following partialstate noisy
measurements of the state of
z
i
= H
i
x +v
i
;i = 1;2;:::;n;z
i
2 R
l
(3)
where the matrices A,B,and H
i
are generally timevarying
and of appropriate dimensions and w and v
i
are zeromean
Gaussian noise.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 2
Let G = G(q) be the proximity graph (network) of the
mobile sensors.The set of vertices of G is V = f1;2;:::;ng.
Let r > 0 be the interaction range of every sensor.Then,the
set of edges of G is a timevarying set deﬁned as
E(q) = f(i;j) 2 E:kq
j
q
i
k < rg (4)
and the set of neighbors N
i
of sensor i on this proximity
network is given by
N
i
= fj 2 V:kq
j
q
i
k < rg:
The main problem of interest is to design distributed motion
control and estimation algorithms that achieve two objectives:
i) the group of sensing agents improve their collective in
formation value
P
i
I
i
and ii) avoid collisions during track
ing of target .We refer to this problem as “information
driven ﬂocking.” We propose a solution to this problem using
a combination of ﬂocking and KalmanConsensus Filtering
algorithms [11].
III.DISTRIBUTED TRACKING WITH MOBILE SENSORS
The KalmanConsensus ﬁltering algorithm (or Algorithm 1)
relies on reaching a consensus on estimates obtained by
local Kalman ﬁlters rather than distributed averagingbased
Kalman ﬁltering.Algorithm 1 is the discretetime analog of
the continuoustime KalmanConsensus ﬁlter described in the
following.
Theorem1.(KalmanConsensus Filter [2]) Consider a sensor
network with a continuoustime linear sensing model in (3).
Suppose each node applies the following distributed estimation
algorithm
_
^x
i
= A^x
i
+K
i
(z
i
H
i
^x
i
) +P
i
X
j2N
i
(^x
j
^x
i
)
K
i
= P
i
H
T
i
R
1
i
; > 0
_
P
i
= AP
i
+P
i
A
T
+BQB
T
K
i
R
i
K
T
i
(5)
with a KalmanConsensus estimator and initial conditions
P
i
(0) = P
0
and ^x
i
(0) = x(0).Then,the collective dynamics
of the estimation errors
i
= x ^x
i
(without noise) is
a stable linear system with a Lyapunov function V () =
P
n
i=1
T
i
P
1
i
i
.Moreover,
_
V 2
G
() 0 where
G
(^x) = ^x
T
^
L^x =
1
2
X
(i;j)2E
k^x
j
^x
i
k
2
and
^
L = L
I
m
is the mdimensional Laplacian of the
network (
denotes the Kronocker product).Moreover,all
estimators asymptotically reach a consensus,i.e.^x
i
= x;8i.
The following ﬂocking algorithm is a modiﬁed form of
Algorithm 2 in [14].
Algorithm 2:(ﬂocking with n distinct agents) Let ^x
i
=
col(^q
i;
;^p
i;
) be the estimate of the state of target by mobile
sensor i obtained via KalmanConsensus ﬁltering.Then,each
sensing agent
i
with dynamics in (1) applies the following
distributed control to interact with its neighboring sensors on
G(q):
u
i
=
X
j2N
i
(kq
j
q
i
k
)n
ij
+
X
j2N
i
a
ij
(q)(p
j
p
i
)+f
i
(6)
Algorithm 1 KalmanConsensus Filter [11] (one cycle)
Given P
i
,x
i
,and messages m
j
= fw
j
;W
j
;x
j
g;8j 2 J
i
=
N
i
[ fig,
1:Obtain measurement z
i
with covariance R
i
.
2:Compute information vector and matrix of node i
w
i
= H
T
i
R
1
i
z
i
W
i
= H
T
i
R
1
i
H
i
3:Broadcast message m
i
= (u
i
;U
i
;x
i
) to neighbors.
4:Receive messages from all neighbors.
5:Fuse information matrices and vectors
y
i
=
X
j2J
i
w
j
;S
i
=
X
j2J
i
W
j
:
6:Compute the KalmanConsensus state estimate
M
i
=
P
1
i
+S
i
1
;
^x
i
= x
i
+M
i
(y
i
S
i
x
i
) +F
i
G
i
X
j2N
i
(x
j
x
i
);
= =(1 +kF
i
G
i
k);kXk = tr(X
T
X)
1
2
F
i
= I M
i
S
i
;
G
i
= AM
i
A
T
+BQB
T
+P
i
S
i
P
i
7:Update the state of the information ﬁlter (x
+
is the
updated x)
P
+
i
= AM
i
A
T
+BQB
T
x
+
i
= A^x
i
where f
i
is a linear feedback for tracking particle ^
i
with
state ^x
i
:
f
i
= c
1
(q
i
^q
i;
) c
2
(p
i
^p
i;
);c
1
;c
2
> 0 (7)
where n
ij
= (q
j
q
i
)=
p
1 +kq
j
q
i
k
2
is a subnormal
vector connecting agent i to agent j.Please,refer to [14] for
the deﬁnitions of
,the norm k k
,and smooth adjacency
elements a
ij
(q).
Remark 1.According to the ﬂocking framework in [14],there
exists a smooth potential function in explicit form
U
(q) =
X
j6=i
(kq
j
q
i
k
) +
2
X
kq
i
q
c
k
2
(8)
with q
c
= 1=n
P
n
i=1
q
i
such that u
i
can be stated as a
distributed gradientbased control:
u
i
= r
q
i
U
(q) +
X
j2N
i
a
ij
(q)(p
j
p
i
) +f
i
:(9)
r
q
i
denotes the partial derivative with respect to q
i
.
Note that the state estimates generated by Algorithm 1 is
directly used in equation (7) of Algorithm 2 for distributed
mobilitycontrol of the sensors.We refer to the combined
Algorithms 1 and 2 as the cascade distributed estimation and
control algorithm for collisionfree distributed tracking of a
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 3
mobile target .The analysis of the this discretetime coupled
estimation and control algorithm is tremendously challenging
and is one of our future research objectives.
In this paper,we seek to provide the stability analysis of the
continuoustime version of this coupled distributed estimation
and control algorithm.
IV.STABILITY ANALYSIS:COUPLED DISTRIBUTED
ESTIMATION AND CONTROL ALGORITHMS
The formulation of our main analytical result as well as
the following assumptions are inspired by our experimental
observations and consistent collective behavior of a group of
mobile sensors tracking two types of mobile targets:1) a
linear target and 2) a maneuverable nonlinear target called
particleinthebox.Both models of the motion of targets will
be discussed in detail in Section V.The notions of ﬂocks,
structural stability,and cohesion of ﬂocks are used in the
following proposition and deﬁned in [14].
A ﬂock is a connected network of dynamic agents.Flocking
is the collective behavior of a network of dynamic agents
with the objective to selfassemble and maintain a connected
network in a collisionfree manner.A ﬂock is called cohesive
if all the agents can be contained in a ball of ﬁnite radius.
Assumption 1.Assume there exists a ﬁnite time T
1
> 0 such
that the proximity graph G(q(t)) becomes connected for all
t T
1
.
The following deﬁnition clariﬁes that the Laplacian and
algebraic connectivity of the networks used in ﬂocking and
KCF algorithms are not the same.
Deﬁnition 1.(Laplacian and
2
of the proximity networks
in ﬂocking vs.KCF) Let a
ij
(q) be the smooth adjacency
elements of the proximity network of mobile agents with
conﬁguration q = col(q
1
;:::;q
n
).We represent the adjacency
matrix of ﬂocking with A
f
(q) = [a
ij
(q)] and its Laplacian and
algebraic connectivity with L
f
and
f
2
=
2
(L
f
),respectively.
The adjacency matrix A
e
= [a
e
ij
(q)] of networked ﬁlters in
KCF has 01 elements,i.e.a
e
ij
= 1 if a
ij
(q) > 0 and
a
e
ij
= 0,otherwise.Similarly,we denote the Laplacian and
algebraic connectivity of the networked ﬁlters with L
e
(q) and
e
2
=
2
(L
e
),respectively.
Assumption 2.Assume there exist constant thresholds
1
;
2
2 (0;1) such that the algebraic connectivity functions
f
2
(t) =
2
(L
f
(q(t))) and
e
2
(t) =
2
(L
e
(q(t))) along
the trajectory of mobile agents cross the levels
1
and
2
,
respectively,at time T
2
= T
2
(
1
;
2
) > T
1
and remain above
those threshold values thereafter,i.e.
f
2
(t) ;
e
2
(t) for
all t T
2
.
Assumption 3.The parameters c
1
;c
2
> 0 in the tracking
feedback f
i
of the ﬂocking algorithm satisfy c
1
< c
2
< 1
and c
2
> 1
1
where
1
is deﬁned in Assumption 2.
Here is our main theoretical result:
Proposition 1.Consider a network of n mobile sensing agents
with dynamics (1),the sensing model in (3),and the proximity
graph G(q) with the set of edges (4).Suppose that the agents
apply the KalmanConsensus ﬁlter in (5) to obtain n estimates
^x
i
of the state of a mobile target with dynamics (2).These
state estimates of the target determine the states of n agents
^
i
.Suppose that every sensing agent i tracks its associated 
agent ^
i
by applying the ﬂocking algorithm in (6).Let
e
and
c
be the collective dynamics of the n networked estimators
and mobilitycontrolled agents,respectively,and denote their
cascade with .Then,the following statements hold:
(i) can be separated into three subsystems that consist of
the structural and translational dynamics of the group of
mobile sensors in cascade with the error dynamics of the
KalmanConsensus ﬁlter.
(ii) Given Assumption 1,the agents form a cohesive ﬂock in
ﬁnite time.
(iii) Suppose that Assumptions 1 through 3 hold.Then,the
solutions of the structural dynamics of the ﬂock of mobile
sensors are asymptotically stable.
(iv) Given the assumptions in part (iii),all estimators asymp
totically reach a consensus on the state estimates of the
target ^x
1
= = ^x
n
(for the error dynamics of KCF
with zero noise).
The proof of proposition 1 is relatively lengthy;therefore,
we present the proof in separate parts.
A.Proof of Part (i):
Let us ﬁrst determine the error dynamics of the Kalman
Consensus ﬁlter in (5).The estimation error of sensor i is
deﬁned as
i
= x ^x
i
,thus error dynamics of (5) (without
noise) is in the form:
_
i
= F
i
i
+P
i
X
j2N
i
(
j
i
)
with F
i
= AK
i
H
i
.Deﬁning block diagonal matrices F =
diag[F
i
] and P = diag[P
i
] and = colf
i
g,one can rewrite
the last equation as
_ = F P
^
L
e
= F
e
(10)
where F
e
= F P
^
L
e
.According to Theorem 1,the error
dynamics _ = F
e
is stable and has a quadratic Lyapunov
function V () =
T
P
1
=
P
i
T
i
P
1
i
i
.
The ﬂocking dynamics of the agents can be written as
8
>
>
>
<
>
>
>
:
_q
i
= p
i
_p
i
= r
q
i
U
(q) +
X
j2N
i
a
ij
(q)(p
j
p
i
)
c
1
(q
i
^q
i;
) c
2
(p
i
^p
i;
)
(11)
or
8
>
>
>
<
>
>
>
:
_q
i
= p
i
_p
i
= r
q
i
U
(q) +
X
j2N
i
a
ij
(p
j
p
i
)
c
1
(q
i
q
+q
^q
i;
) c
2
(p
i
p
+p
^p
i;
)
After deﬁning the block matrix C = [c
1
I
m
c
2
I
m
],one can
express the last equation in a form with an input
i
:
8
<
:
_q
i
= p
i
_p
i
= r
q
i
U
(q) +
X
j2N
i
a
ij
(q)(p
j
p
i
) +f
i
C
i
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 4
with a linear tracking feedback
f
i
= c
1
(q
i
q
) c
2
(p
i
p
):
This enables us to express the dynamics of as the cascade
of its estimation and control subsystems
e
and
c
:
c
:
(
_q
i
= p
i
_p
i
= rU
(q) D(q)p +f
^
C
e
:_ = F
e
(12)
where D(q) = c
2
I +
^
L
f
(q) is a positive deﬁnite damping
matrix,f = colff
i
g,and
^
C = C
I
n
is a constant matrix.
System (12) is the cascade normal form of estimation and
control subsystems of a mobile sensor network in which its
sensing agents apply the ﬂocking algorithm for mobility con
trol and the KalmanConsensus ﬁlter for distributed tracking.
According to [14],since f
i
is a linear feedback,the ﬂocking
dynamics
c
can be further decomposed as the cascade of
structural and translational dynamics of particles.The position
and velocity of the center of mass (CM) of the particles is
given by
q
c
=
1
n
X
i
q
i
;p
c
=
1
n
X
i
p
i
:
Consider a moving frame centered at q
c
.Then,the position
and velocity of agent i can be written as x
i
= q
i
q
c
and
v
i
= p
i
p
c
.We refer to the dynamics of the motion of the
group of agents in the moving frame coordinates as structural
dynamics.The structural and translational dynamics of
c
can
be written as
s
:
(
_x = v
_v = rU
(x) D(x)v +
1
n
with 1
n
2 R
n
representing the column vector of ones and
t
:
(
_q
c
= p
c
_p
c
= c
1
(q
c
q
) c
2
(p
c
p
) +
where the perturbation terms = colf
i
g and
depend on the
target estimation errors by the sensors and are deﬁned as
i
= c
1
(q
^q
i;
) c
2
(p
^p
i;
) = C
i
=
1
n
X
i
i
= C; =
1
n
X
i
i
=
1
n
(1
T
n
)
The normal form of can be written as follows
s
:
(
_x = v
_v = rU
(x) D(x)v
^
C +C(1
T
n
)
1
n
t
:
(
_q
c
= p
c
_p
c
= c
1
(q
c
q
) c
2
(p
c
p
) C(1
T
n
)
e
:_ = F
e
B.Proof of parts (ii) to (iv)
The solutions of the structural dynamics in cascade with
e
is called cohesive for all t 0 if the position of all
agents remains in a ball of radius R
0
for t 0.Note that
this cascade nonlinear system is globally Lipschitz and all of
its solutions are bounded for arbitrary initial conditions.The
global Lipschitz property is a byproduct of the design of the
smooth potential function U
(q) which has a globally bounded
gradient.This implies that over the interval [0;T] the solutions
of the cascade system and therefore the position of all agent
remain bounded.For all t T,the proximity graph G(q(t))
is connected and thus has a ﬁnite diameter d(t) (n 1) at
any time t.Deﬁne the diameter of the ﬂock as
d
max
(t) = max
j6=i
kq
j
(t) q
i
(t)k;t T
Then,d
max
= d(t)r (n 1)r and by setting R
0
= (n
1)r=2 the position of the agents remain cohesive for all t T
inside a ball of radius R
0
.
To establish stability of the ﬂock,we need to construct an
energytype Lyapunov function'for the cascade of
s
and
e
.Let H
(x;v) = U
(x) +
1
2
kvk
2
be the Hamiltonian of
the unperturbed structural dynamics
s
and V () =
T
P
1
be the Lyapunov function of
e
.We propose the following
Lyapunov function for the cascade nonlinear system (
s
;
e
):
'(x;v;) = H
(x;v) +
k
2
V () (13)
Before computing _',let us state a simple inequality.For an
n m matrix M and two vectors x 2 R
n
and y 2 R
m
,the
following inequality holds:
jx
T
Myj
1
2
(kxk
2
+kMyk
2
)
1
2
(kxk
2
+
2
max
(M)kyk
2
)
In the special case of M = C = [c
1
I
m
c
2
I
m
],we have
jx
T
Cyj
1
2
(kxk
2
+c
2
3
kyk
2
):
where c
3
= max(c
1
;c
2
).By direct differentiation,we obtain
_'=
_
H
+
k
2
_
V ():
From Theorem 1 and Assumptions 1 and 2,for all t T
2
,
one gets
_
V () 2(
T
^
L
e
) 2
e
2
kk
2
where
e
2
= min
tT
2
2
(L
e
(q(t)) always exists based on
Assumption 2.
Now,let us compute
_
H
(x;v;).We have
_
H
= v
T
^
L
f
(x)v c
2
kvk
2
X
i
(v
T
i
C
i
+v
T
i
):
Note that jv
T
i
C
i
j
1
2
(kv
i
k
2
+c
2
3
k
i
k
2
) thus
X
i
jv
T
i
C
i
j
1
2
(kvk
2
+c
2
3
kk
2
):
In addition,v
T
i
=
1
n
P
j
v
T
i
C
j
.Hence
jv
T
i
j
1
2
X
j
(kv
i
k
2
+c
2
3
k
j
k
2
) =
n
2
kv
i
k
2
+
1
2
c
2
3
kk
2
and
X
i
jv
T
i
j
1
2
(kvk
2
+c
2
3
kk
2
):
Based on the above upper bounds,we get
_
H
v
T
^
L
f
(x)v c
2
kvk
2
+kvk
2
+c
2
3
kk
2
:
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 5
Given the fact that
v
T
^
L
f
(x)v
2
(L(x))kvk
2
and setting
f
2
= min
tT
2
2
(L
f
(x(t))),one concludes
_' (1 c
2
f
2
)kvk
2
+(c
2
3
k
e
2
)kk
2
< 0;8(v;) 6= 0
if the following two conditions hold:
(
f
2
> 1 c
2
e
2
> c
2
3
=k
(14)
Given the deﬁnition of
f
2
and
e
2
and Assumption 2,we
have
f
2
=
1
and
e
2
=
2
.By choosing k 1=
2
and c
2
>
1
1
(as in Assumption 3) both conditions will be satisﬁed.
Thus
_'(x;v;) < 0;8(v;) 6= 0
Based on LaSalle’s invariance principle,for any set of initial
conditions,the solutions of the cascade system (
s
;
e
)
asymptotically converge to the largest invariant set in
E = f(x;v;):rU
(x) = 0;v = 0; = 0g = E
s
f0g
where E
s
is the equilibria of the unperturbed structural dy
namics.From the equilibria in E
s
,only the local minima of
U
(x) are asymptotically stable.
The proof of part (iv) is a byproduct of the above stability
analysis:the estimation errors
i
asymptotically vanish for all
sensors and therefore all state estimates become the same.
Remark 2.If in addition to Assumptions 1 through 3,Conjec
tures 1 and 2 in [14] hold,then almost every solution of the
structural dynamics of the ﬂock asymptotically converges to
a quasi lattice.In all of our experimental results,we have
observed ﬁnitetime selfassembly of quasi lattices.
V.EXPERIMENTAL RESULTS
In this section,we apply our coupled distributed estimation
and control algorithm—namely,KCF plus ﬂocking—to two
types of targets:1) a target with a linear model which is
a particle moving in R
2
and 2) a maneuvering target with
nonlinear dynamics.The later target remains in a rectangular
region (box) for all time t 0.
A.Linear Target
Consider a particle in R
2
with a linear dynamics
x(k +1) = Ax(k) +Bw(k)
with
A =
I
2
I
2
0 I
2
;B =
(
2
=2)I
2
I
2
:
where = 0:01 is the discretization stepsize.The sensor
makes noisy measurements of the position of the target,i.e.
z
i
(k) = H
i
(k)x(k) +v
i
(k);H
i
= [I
2
0]:
The noise statistics for zeromean Gaussian signals w(k) and
v
i
(k) are
E[w(k)w(l)
T
] = Q
k
kl
;E[v
i
(k)v
j
(l)
T
] = R
i
(k)
kl
ij
:
where
kl
= 1 if k = l and
kl
= 0,otherwise.According to
the model of information value in [8],the measurement error
covariance matrix of sensor i is R
i
=
2
f(
i
)
I
2
where f(
i
) is
the information value function
I
i
= f(
i
) = 2I
0
(a +b +(a b)
i
l
p
1 +(
i
l)
2
)
1
(15)
where
i
= kH
i
x
i
q
i
k,I
0
= 0:1,and a > b > 0.In our
experiment,we use a mobile sensor network with n = 20
agents.The parameters of R
i
are a = 8b,b = 1,and l = 10d.
The interaction range of the agents in the ﬂock is r = 1:2d
and their desired interagent distance is d = 7.For the KCF
algorithm,P
0
= 100I
4
,x
0
N(0;
2
I
4
) with = 60,and
Q = 100I
2
.
Fig.1 shows the MSE of tracking error over 10 randomruns,
the average information value,and the algebraic connectivity
plots during tracking.From Fig.1 (c),one can readily verify
that Assumptions 1 through 2 hold.
0
10
20
30
40
50
0.2
0.4
0.6
0.8
1
1.2
MSE
time(sec)
average error value
ave
0
20
40
60
80
0.099
0.0992
0.0994
0.0996
0.0998
0.1
0.1002
0.1004
average info value
time(sec)
(a) (b)
0
50
100
150
200
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time(sec)
algebraic connectivity
2
f
2
e
ǫ
2
ǫ
1
T
1
T
2
0
50
100
150
200
250
300
0
50
100
150
200
250
300
x
est.x
fuse
(c) (d)
Fig.1.Experimental results for the linear target:(a) MSE for distributed
target tracking,(b) average information value,(c)
2
plots for ﬂocking
(smooth blue curve) and KalmanConsensus ﬁltering (piecewise constant red
curve),and (d) a target trajectory and fused estimates of 20 sensors.
B.Maneuvering Nonlinear Target:ParticleintheBox
We also consider a maneuvering target with the following
nonlinear dynamics:
x(k +1) = A(x(k))x(k) +Bw(k) (16)
where x(k) = (q
1
(k);p
1
(k);q
2
(k);p
2
(k))
T
denotes the state
of the target at time k.The target moves inside and outside
of a square ﬁeld [l;l]
2
.Matrix A(x) is deﬁned as
A(x) = M(x)
F
1
+(I
2
M(x))
F
2
F
1
=
1
0 1
;F
2
=
1
c
1
1 c
2
;
M(x) =
(x
1
) 0
0 (x
3
)
:
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 6
0
10
20
30
40
50
0
0.5
1
1.5
2
MSE
time(sec)
average error value
ave
0
10
20
30
40
50
0.0994
0.0996
0.0998
0.1
0.1002
average info value
time(sec)
(a) (b)
0
50
100
150
200
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
algebraic connectivity
2
f
2
e
T
1
T
2
ǫ
2
ǫ
1
50
0
50
20
10
0
10
20
30
40
50
x
est.x
fuse
(c) (d)
Fig.2.Experimental results for the nonlinear target:(a) MSE for distributed
target tracking,(b) average information value,(c)
2
plots for ﬂocking
(smooth blue curve) and KalmanConsensus ﬁltering (piecewise constant red
curve),and (d) a target trajectory and fused estimates of 30 sensors.
where F
1
and F
2
determine the dynamics of the target inside
and outside of the region,respectively,and (z) is a switching
function taking 01 values deﬁned by
(z) =
(a +z) +(a z)
2
(z) =
1;z 0;
1;z < 0
In addition,matrix B is given by
B = I
2
G;G =
2
0
=2
0
:
where = 0:03 is the stepsize,
0
= 2,a = 45,l = 50,c
1
=
7:5 and c
2
= 10 are the parameters of a PD controller,and
the elements of w(k) are normal zeromean Gaussian noise
with Q = 100I
2
.The initial condition of the target is x
0
N(0;
2
I
4
with = 2 and P
0
= 100I
2
.The parameters of the
information value function in (15) are I +0 = 0:1,a = 10b,
b = 1,l = 10d and d = 7.We consider a mobile sensor
network with n = 30 nodes with a linear sensing model and
H
i
=
1 0 0 0
0 0 1 0
:
Fig.2 illustrates the tracking estimation error,average
information value,and the algebraic connectivity plots for the
nonlinear target with snapshots shown in Fig.3.Similarly,
Assumptions 1 and 2 hold based on Fig.2 (c).
VI.CONCLUSIONS
We introduced a theoretical framework for coupled dis
tributed estimation and ﬂockingbased control of mobile sensor
networks for collaborative target tracking.The mobile sensing
agents seek to improve the information value of their sensed
data while avoiding interagent collisions.We demonstrated
that the coupled dynamics of the combined distributed estima
tion and control algorithm has a separable cascade nonlinear
normal form.Then,we provided the stability analysis of the
structural dynamics of a ﬂock with n dedicated agents
in cascade with the error dynamics of the continuoustime
KCF.Based on our experimental results,the discretetime
counterpart of the informationdriven ﬂocking algorithm is
effectively applicable to tracking both a linear and a nonlinear
maneuverable target.
REFERENCES
[1] Y.BarShalom and X.R.Li,MultitargetMultisensor Tracking:Princi
ples and Techniques,YBS Publishing,Storrs,CT,1995.
[2] R.OlfatiSaber,“Distributed Kalman Filtering for Sensor Networks,”
Proc.of the 46th IEEE Conference on Decision and Control,Dec.2007.
[3] A.Speranzon,C.Fischione,B.Johansson,and K.H.Johansson,
“Adaptive distributed estimation over wireless sensor networks with
packet losses,” Proc.of the 46th IEEE Conf.on Decision and Control,
pp.5472–5477,Dec.2007.
[4] R.Carli,A.Chiuso,L.Schenato,and S.Zampieri,“Distributed Kalman
ﬁltering based on consensus strategies,” IEEE Journal on Selected Areas
in Communications,vol.26,no.4,pp.622–633,May 2008.
[5] R.OlfatiSaber and N.F.Sandell,“Distributed tracking in sensor
networks with limited sensing range,” Proc.of the 2008 American
Control Conference,pp.3157–3162,June 2008.
[6] U.A.Khan and J.M.F.Moura,“Distributing the Kalman ﬁlter for
largescale systems,” IEEE Trans.on Signal Processing,vol.56,no.
10,pp.4919–4935,Oct.2008.
[7] A.Speranzon,C.Fischione,K.H.Johansson,and A.Sangiovanni
Vincentelli,“A distributed minimum variance estimator for sensor
networks,” IEEE Journal on Selected Areas in Communications,vol.
26,no.4,pp.609–621,2008.
[8] R.OlfatiSaber,“Distributed tracking for mobile sensor networks with
informationdriven mobility,” Proc.of the 2007 American Control
Conference,pp.4606–4612,July 2007.
[9] P.Barooah,W.M.Russell,and J.Hespanha,“Approximate distributed
Kalman ﬁltering for cooperative multiagent localization,” Int.Conf.on
Distributed Computing in Sensor Networks (DCOSS ’10),June 2010.
[10] C.Soto,B.Song,and A.K.RoyChowdhury,“Distributed multitarget
tracking in a selfconﬁguring camera network,” 2009 IEEE Conference
on Computer Vision and Pattern Recognition (CVPR ’09),pp.1486–
1493,June 2009.
[11] R.OlfatiSaber,“KalmanConsensus ﬁlter:optimality,stability,and
performance,” Joint 48th Conference on Decision and Control and 28th
Chinese Control Conference,pp.7036–7042,Dec 2009.
[12] B.Grocholsky,A.Makarenko,and H.DurrantWhyte,“Information
Theoretic coordinated control of multiple sensor platforms,” Proceedings
of the 2003 IEEE Int.Conf.on Robotics and Automation,pp.1521–1525,
Sep.2003.
[13] S.Martinez and F.Bullo,“Optimal sensor placement and motion
coordination for target tracking,” Automatica,vol.42,no.4,pp.661–
668,April 2006.
[14] R.OlfatiSaber,“Flocking for MultiAgent Dynamic Systems:Algo
rithms and Theory,” IEEE Trans.on Automatic Control,vol.51,no.3,
pp.401–420,Mar.2006.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.57,NO.9,SEPTEMBER 2012 7
10
0
10
20
30
40
50
0
10
20
30
40
time= 3
10
20
30
40
50
60
50
40
30
20
time= 46.7
(a) (b)
20
10
0
10
20
30
0
10
20
30
40
time= 8
40
30
20
10
0
10
0
10
20
30
time= 17
(c) (d)
60
50
40
30
20
10
10
0
10
20
time= 22
10
20
30
40
50
60
50
40
30
20
time= 46.7
(e) (f)
Fig.3.Snapshots of a mobile sensor network tracking a maneuvering target with a ﬂockingbased motion control algorithm.The target is marked by a red
circle and the estimates are marked by green dots.
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