OPTIMAL DECENTRALIZED ESTIMATION THROUGH SELF-SYNCHRONIZING
NETWORKS IN THE PRESENCE OF PROPAGATION DELAYS
Gesualdo Scutari, Sergio Barbarossa, and Loreto Pescosolido
Dpt. INFOCOM, Univ. of Rome "La Sapienza", Via Eudossiana 18, 00184 Rome, Italy.
e-mail: {aldo.scutari,sergio,loreto}@infocom.uniromal.it.
ABSTRACT
In this paper we focus on a sensor network scheme whose nodes are
locally coupled oscillators that evolve in time according to a differential equation, whose parameters depend on the local estimate. The
proposed system is capable, by self-synchronization, to reach the
network consensus that coincides with the globally optimum maximum likelihood estimate, even though each sensor is only locally
coupled with nearby nodes. Our main contribution is to study the
effect of propagation delay on both the synchronization capability
of the system and the final estimate. We provide delay-independent
conditions for the proposed system to synchronize, and we derive
closed-form expression of the synchronized state. Interestingly, the
effect of propagation delays is simply to introduce a bias on the final
estimate, that depends on the network topology and on the values of
the delays. The analysis of this bias, suggest us how to design the
coupling mechanism in order to alleviate it or even remove it.
1. INTRODUCTION
invariant, inhomogeneous delays. Our new contributions are the following. In the case of local linear coupling, we prove that, if the
network is connected, 1 differently from [6], the system of [3] always synchronizes, for any set of finite (eventually heterogeneous)
delays. We also derive closed form expression for the synchronized
state (i.e. final estimate) reached by the system, and suggests ideas
about how to design the coupling mechanism in order to avoid the
undesired bias on the final estimate. In the more general case of
nonlinear coupling, using the small perturbation analysis, we study
the impact of the propagation delay on the final estimate, provided
that the system reaches a constant synchronized state. Finally, we
corroborate our theoretical findings with simulation results.
2. NONLINEAR COUPLING MECHANISM WITHOUT
DELAYS
In this section, we briefly overview the mathematical model describing the state equation of the dynamic system given in [3].
The proposed sensor network is composed of N nodes, each
composed of four basic components: i) a transducer that senses the
physical parameter of interest (e.g., temperature, concentration of
contaminants, radiation, etc.); ii) a local detector or estimator that
processes the measurements taken by the node; iii) a dynamical system (termed oscillator, for simplicity) whose state evolves in time
according to a differential equation, whose parameters depend on
the local estimate, and it is coupled with the states of nearby sensors; iv) a radio interface that transmits the state of the oscillator
and receives the state transmitted by nearby nodes.
Denoting by wi the initial local decision (either the result of a
detection or estimation) taken by node i, the dynamical system (oscillator) present in node i evolves according to the following equation
One of the main problems in current sensor networks research is how
to convey the necessary amount of data from the network nodes to a
fusion center in the most efficient manner.
An innovative direction was taken by Hong and Scaglione [1]
and Lucarelli and Wang [2], who suggested the use of local mutually coupled oscillators as the basic mechanism to reach network
consensus by self-synchronization, without the need for sending the
data to a fusion center. The oscillator and coupling model proposed
in [1, 2] associates the local estimate to the time shift of a pulse oscillator. However, especially for large scale network, this may create
a problem, as the information bearing time shift may become indistinguishable from the propagation delay.
To overcome these limitations, a more general approach was
then proposed in [3], where it was showed how to reach the global
maximum likelihood estimates through local coupling, with a scheme
that is valid for the general case of vector parameter estimation, and
it is much more flexible than the ones suggested in [1, 2]. The stability analysis of the system in [3] was addressed in [4].
However, the self-synchronization of systems proposed in [1, 2]
and [4] is guaranteed only under the assumption that the coupling
among the oscillators occurs instantaneously, which falls short when
the propagation delay in the network is not negligible. Unfortunately, this happens in many practical applications where a largescale network is required. In all these cases, the systems in [1, 2, 3]
may become unstable and thus unusable in practice.
Recently, [5] and [6] provide sufficient conditions for the synchronization of linearly coupled oscillators, under the idealistic assumption of homogeneous delays and identical oscillators. Differently from [5, 6], in this paper, we study the synchronization capability of the system proposed in [4], in the presence of finite, time-
where 7i(t) is the state function of the i-th sensor, that is initialized as a random number 9io, i.e. i (0) = 9io; f( ) is, typically, a
monotonically increasing (possibly) nonlinear odd function of its argument that takes into account the mutual coupling between the sensors. Without loss of generality, f (x) is normalized so that df (O) ldx
= 1. A different value of df(O)Idx can always be included in K,
where K is a control loop gain; ci is a positive coefficient that quantifies the attitude of the i-th sensor to adapt its values as a function of
the signals received from the other nodes: The higher is ci, the less
is the attitude of the i-th node to change its original decision wi. The
running decision, or estimate, of each sensor is encoded in its pulsation tj (t). The coefficients aij take into account the local coupling
between oscillators. We assume that two oscillators are coupled (i.e.,
This work has been partially funded by ARL, Contract N62558-05-P0458.
posed by several hops, between each pair of nodes.
KN
wit i +-
j =1
ai f (Vj(t) -Vi (0))
i
=
1, ...,IN, (1)
iA network is said to be connected if there exists a path, possible com-
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ai ? 0O), only if their distance is smaller than the coverage radius of
each sensor2.
When each sensor measures more, let us say L, physical parameters, the coupling mechanism (1) generalizes according to the
following expression3
where ( is the common unknown parameter to be estimated and wi,
i = 1, ... , N are a set of i.i.d. Gaussian random variables with zero
mean and variances (72 , ... , o72). Setting, in (1), ci = b 2/o2 and
Wi = Xi/bi, the network synchronized state in (4) becomes:
i (t)
= w + KQi 1jaij f (ijW (t) -'i(t)),
for i = 1, ... , N, where Vi(t) is the L-size vector state of the ith node, that is initialized as a random vector Vio; wi is the L-size
vector, function ofthe L measurements taken by node i; Qi is an L x
L non-singular matrix that depends on the observation model; the
symbol f (x) has to be intended as the vector whose k-th component
is f(Xk). In Section 2.2, we show how to choose the vectors wi and
the matrices Qi to guarantee the convergence of (2) to the global
optimal maximum likelihood (ML) estimate.
2.1. Self-Synchronization of Coupled Oscillators
Differently from [1, 2], where the network synchronization was intended to be the situation where all oscillators reach the same state,
we define the network synchronization as follows:
Definition 1 The overallpopulation ofoscillators (1) and (2) is said
to synchronize if there exists a vector V (t), called the synchronized
state of the system, such that
lim
I1%i(t) -*(t)
0,
Vi=
1,2,
...
, N,
(3)
where * 1 denotes some vector norm. This state is said globally
asymptotically stable if the system synchronizes, for any set of initial
conditions.
From Definition 1, it follows that, if there exists a synchronized
state that is globally asymptotically stable, then it must necessarily be unique. Interestingly, the synchronized state, if it exists, can
be computed in closed form, without explicitly solving the system
of differential equations (1) and (2). Specifically, for system (1) we
obtain a synchronized state V (t) given by [3, 4]
N
ZZ
i=l
Ci
whereas, for system (2), we have
1
9*(t)
A w
= (± Q)
i=l
(
QI:Qwi
(5)
i=l
2.2. Reaching global ML estimate through self-synchronization
In the model of sensor network we propose, the self-synchronization
process represents the basic mechanism to reach globally optimal
estimates through local transmissions, without passing through afusion center when the network observes a common phenomenon. In
particular, let us consider the scalar observation
Xi= bi+wi,
2The
(6)
coverage radius is assumed to be the same for all sensors, even
though this could be changed to accommodate for different network topological models, like small worlds or scale-free networks.
3We assume that the coupling coefficients aij are the same for all estimated parameters. This assumption is justified by the fact that aij depends
on the coverage radius of each transmitter and not on the measurements.
SML =-
(2)
J1
N
N
i=l
bixzi
2
Ei=l
72
VIL
N
(7)
b2'
which coincides with the global maximum likelihood estimate.
Similarly, let us consider the linear vector case, where we observe vectors of the form
(8)
Xi = Aif + wi,
where xi is the M x 1 observation vector, ( is the unknown common
parameter vector, Ai is the mixing matrix of sensor i, and wi is the
observation noise vector, modeled as a circularly symmetric Gaussian vector with zero mean and covariance matrix Ci. We assume
that the noise vectors affecting different sensors are statistically independent of each other, and that each Ai is full column rank, which
implies M > L. In this case, setting, in (2), Qi = A HC- 1Ai and
wi = (AiYC-1Ai) 1'Af'C1 x, the equilibrium (5) becomes
WL
=(E,
1A) i
i=l
ini
(9)
,
i=l
that coincides with the globally optimal ML estimate.
It is important to emphasize that, following the proposed approach, if the network converges, each node goes to the optimal estimate sending no data (i.e. all matrices Ai and Ci) to any sink node,
but simply exchanging the state vectors Vi (t) with nearby nodes.
The critical issue is now to show under what conditions the synchronized state, as defined above, exists and it is globally asymptotically stable. We address this issue in the following Section.
2.3. Globally Asymptotic Stability of the Synchronized State
The study of conditions for the existence and globally asymptotic
stability of the synchronized states (4) and (5) was carried out in [4].
The main result of [4] can be resumed in the following4
Theorem 1 Given the system (1), assume that the following conditions are satisfied: al) The graph associated to the network is connected; a2) The nonlinearfunction f() : R i-- R is a continuously
differentiable, odd, increasingfunction in R;5 a3) The nonzero coefficients aij are positive.
Then, there exist two unique critical values ofK, denoted by KL
and KU, with 0 < KL < KU, such that the synchronized state existsfor all K > KU, and it does notfor all K < KL. Furthermore,
if it exists, the synchronized state is globally asymptotically stable.
Upper and lower bounds of KL and KU are
KL >
ID,Awllo
fmaxdmax
and
KU <
2
ID,Aw 112
fmax A2(LA )
(10)
where Aw A- W*w N, with w defined in (4); fmax-im1,±
f (x); dmax and A2 (LA) are the maximum degree and the algebraic
connectivity of the graph, respectively.
4For the vector system (2) we have a similar result [4, Theorem 2].
5For the lack of space, we consider only asymptotically convex or concave
functions f( ), i.e. functions that can not change their concavity infinitely
often. Observe that this constraint does not represent a strong restriction in
the choice of the function f (.). However, the general case is studied in [4].
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Remark 1. Conditions in (10) state an important property of the
whole system: If one wants the network to reach a global consensus
(common estimate), it is sufficient to take K greater than the upper
bound in (10); conversely, if one does not want the network to reach
a global consensus, we need to take K smaller than the lower bound
in (10). This was used, for example, in [4] to get spatial smoothing
of the observed phenomenon. This is indeed a unique possibility
offered by nonlinear systems. Our nonlinear model contains, in fact,
as a particular case, linear dynamic systems, corresponding to the
choice f(x) = x. However, in the case of unbounded coupling
function, as in the linear case, the lower and upper bounds in (10)
coincide, so that there exists a unique critical value of K, given by
KL = KU = 0. Thus, a linear system always converges to the
equilibrium, for any positive values of K.
Remark 2. From (10), it is evident that the synchronization properties depend on the graph topology through the second-smallest
eigenvalue of the graph representing the network. This means that,
for a given K, different topologies give rise to different behaviors
[4].
3. DELAYED LINEAR COUPLING
We study now the effect of the propagation delays on the existence
and the stability of the synchronized state (as defined in Definition
1) of systems (1) and (2), in the case of linear coupling.
3.1. Scalar Observations
Using the same notation as in (1), in the presence of propagation
delays, the dynamical system of the i-th node evolves according to
the following (delayed) differential equation
7gi(t)
=
wi
KN
+-
:aij (Vj (t -Tij)
-
Vi(1)
g =1
for i =1, .. , N, where Tij denotes the propagation delay of the
signal going from node j to node i. We assume that all the delays
are time-invariant and bounded above, with Tmax = maxij Tij
Because of the delays, the solution of (11) is uniquely defined at
time t > to6, provided that the value of each Vi(t) for the whole
interval [-Tmax, 0] is specified. We assume that initial conditions
of (11) are given by a set of continuous bounded functions Xi(t)
[-Tmax, 0] F-÷ R, i.e. 79i(t) = Xi(t), Vt C [-Tmax, 01.
Observe that the presence of the propagation delays makes system (11) quite different from the one given in (1) with f (x) = x, so
that the results obtained in Theorem 1 for (1) can not be applied. For
example, it is straightforward to see that expression (4) is no longer
solution of (1 1). Moreover, even if a time-independent synchronized
state exists, there is no guarantee for its globally asymptotic stability.
Interestingly, we can still derive closed form expression of the
synchronized state of (11) and its stability properties, without solving the system of differential equations, as given in the following.
Theorem 2 Given system (11), assume that thefollowing conditions
are satisfied: al) The graph associated to the network is connected;
a2) The nonzero coefficient aij are positive, and K > 0.
Then, system (1]) synchronizes for any set of initial conditions
and propagation delays. The synchronized state is given by
N
Liw
W
6We assume to
(12)
1
i=
=
c+K i=1
0 without loss of generality.
=1
Proof. See Appendix. U
Remark 3. Interestingly, the presence of propagation delays does
not affect the synchronization capability of system (11), that synchronizes for any set {Tij } and initial conditions, provided that the
network is connected, as the corresponding system without delays
(Theorem 1). Moreover, the synchronized state is still time-independent, irrespective of the value of the delays, as in (4). However,
comparing (4) with (12) we infer that the effect of the delays is to
introduce a bias in the final estimate. Such a bias depends on the
network topology (by aij) and on the values of the delays.
Remark 4. The analysis of (12) suggests also a way to remove the
bias, as given next. Let us consider the case where the system operates in a pipeline mode, so that the time axis is divided into periods
and the updating ofthe state variables performed in each period takes
into account only the state variables observed in the previous period.
In this way, the system in each period is able to distinguish between
delays and anticipations occurring in the previous period. In mathematical terms, if the time arrows is correctly taken into account,
we may assumeTij =-Tji. Under this assumption, it is easy to
check that the summation E E aijTij in (12) is zero. Hence,
the system synchronizes to the right value.
3.2. Vector Observations
When each sensor measures L physical parameters, the coupling
mechanism (11) generalizes according to the following differential
equation
N
vj(t)
=
aij f ('§j (t- Ti)-'i(t)),
wi + KQi 1
(13)
J=1
for i = 1, ... , N, where 0j(t), wi, and Qi are defined as in (2).
The initial conditions are given by some bounded continuous vector
functions i (t): [-Tmax,U ] ) RL
The existence and the stability properties of the synchronized
state of (13) are given in the following [7].
Theorem 3 Given system (13), assume that conditions of Theorem
2 are satisfied. Then, system (13) synchronizes for any set of initial
conditions and propagation delays. The synchronized state is
N
ZZT
=
1: Qi
i=l
+ IL
N N±±
T)
E E
aijTij)
KI
i=l
j=1
-1
N
I:Qiwi)
where x denotes the Kronecker product.
4. DELAYED NONLINEAR COUPLING
In the presence of propagation delays, the nonlinear system (1) can
be written as7
K N
i(t) wi +
Zaij fWj(t- Ti) - (t)) , (14)
,}j=1
= 1, ... , N, where the initial conditions are given as in (1 1).
In [5] a special case of (14), with Tij = T, Vi, j, was studied. Specifically, in [5], the authors provide the stability analysis
of the linearization of system (14) around the synchronized state
9i (t)= Qt, where Q is determined implicitly by the algebraic equation obtained introducing 79i(t)= Qt in (14). However, since the
equilibrium points of the linearized system are not hyperbolic, the
for i
7Because of the space limitation, in this section we focus only on scalar
observations. The vector case is studied in [7].
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local stability of the solution 9i (t)= Qt (as proved in [5]) does not
provide any useful information on the (even local) stability property of t9i (t)= Qt for the original nonlinear system. Moreover, also
assuming that the solution 9i (t) = Qt exists for (14) and that the system asymptotically converges to this solution, such a synchronized
state, in principle, is not useful in practice, since a simple relationship between Q and the global estimate (4) is not known.
In this section, we focus on the latter issue. Specifically, we
simply try to answer the following question: If system (14) synchronizes, with a constant final synchronized state, what is this state ? In
the following we report a small perturbation analysis, valid for small
delays, that gives an approximate answer to the previous question.
Simulation results are then shown to corroborate our findings.
We assume that the delays Tij are sufficiently small to approximate the function 7 (t- Ti) in (14), around the point 7 (t) with
its first order Taylor's series expansion
Vj (t- Tij )
-_
V
(t)
Tij
0jt
(1 5)
Then, we take the full Taylor series expansion ofthe function f (Wg (t)
-ti(t) -Ti (t)) around the point W9 (t) -9i(t), thus obtaining
(t
wi
~j
+E aij f (I) (t)
i
J=1
(t))
5. NUMERICAL RESULTS
As an example of synchronization in the presence of delays, in Fig.
1 we report the behavior of 9i (t) as a function of time, for a network
composed of N = 16 oscillators: The red curve refers to the case
with no delays; the blue curve refers to the delayed case, where the
maximum delay is equal to 100 times the sampling step used to implement the continuous dynamic system in discrete time; the green
line reports the value given from (18). We can see that, in spite of
the approximations, (18) predicts quite well the numerical result. As
N
oo
-
KN
Ci
that, from the bias point of view, a linear system with f (x) = x is
not an optimal choice since it weights equally well all contributions
in the summation in the denominator of (18). Conversely, a nonlinear
function, as, e.g. the sigmoid function f(x) = tanh(x) provides a
lower bias for any given set of 79jO, as it assigns a smaller weight,
with respect to the linear case, to all values of the argument different
from zero.
Remark 6. It is straightforward to see that, as for the linear system,
also for system (14), the bias in (18) can be removed if we may
assumeTij =-Tji . Hence, if the system synchronizes to a constant
state, it synchronizes to the right value.
f (k)
ZZai
al
jk!1 (Wg (t)
kik=i j}=1
Xk! .(t). (16)
i (t))
with f (k)(X) = d kf(x)IdXk. Multiplying each equation in (16)
by ci and taking the summation over i, if the system synchronizes
(according to Definition 1), then the synchronized state W(t), which
in general does not coincide with X in (4), must be solution of
N
~(t)Z1:ci
i=i
I:ciwi
Ciii
i=l
oo _
0k(t)
K Z,
(17)
Fig. 1. Behavior of 9j (t) as a function of time, for a network of 16 nodes.
N
N
Z Z aij f (k) (I) (t)
k
Vi(t))
i=lj1=l
k=1
where we have only exploited the oddness of f (x) and the symmetry
of aij. From (17), retaining only the term with k = 1, and assuming
that a constant synchronized state f(t) = t exists, we obtain the
approximate expression
N
ci + K
ENi I EI
f
aij (Gjo
-Vio) Tij
(18)
101
where {t79o} and t are also related by the following system of nonlinear equations (assumed to be feasible)
f(jo
-Eaij
J1
Vio
. Tij
=
.
wi,
i
=
an example of statistical behavior, in Fig. 2 we report the variance in
the final estimate, as a function of the SNR present at each node. The
maximum delay is 80 times the sampling step. The two curves refer
to a linear coupling function (dashed line) and to a hyperbolic tangent function (solid line). We can see that the variance decreases as
1/SNR. In this case, the differences between the linear and nonlinear case are very small because the final values of the state variables
are very close to each other.
N.
Remark 5. As for the linear coupling system in (11), (18) shows
that, also for the nonlinear system (14), the delays introduce a bias
on the final estimate, always smaller (in absolute sense) than the true
value (since f (x) > 0, Vx
JR). Observe that expression (18)
coincides with (12), in the case of linear coupling, i.e. if f(x) =
x. In the case of nonlinear function f (), even though we cannot
have a closed form expression for t and {79j0}, (18) still provides an
interesting result: since f (x) < 1, Vx 7? 0 and f (0) = 1, we infer
10 '
10
10o-
5
10
SNR (dB)
15
20
Fig. 2. Estimation variance as a function of the number of sensors.
6. CONCLUSION
In this paper, we proposed a sensor network scheme capable to reach
global optimal estimate, through local exchange of information among
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the nodes. This is particularly useful when the network observes one
common phenomenon, as it relaxes the need for sending data to a
fusion center. The proposed system was shown to be robust against
propagation delays, since it always synchronizes, irrespective of the
value of the delays, provided that the network is connected. This
feature makes the proposed system appealing in all the applications
that need a large scale network, where the propagation delays are not
negligible.
APPENDIX
A complete proof of Theorem 1 is given in [7]. Because of the space
limitation, here we provide only the stretch of the proof.
We first show that, under assumptions al-a2, there exists a unique
constant synchronized state for system (11), denoted by -, and that
such a state must necessarily satisfy (12). Then, invoking standard
results of stability theory for functional differential equations [8], we
prove that such a Z is also globally asymptotically stable (according
to Definition 1).
Let us assume that conditions al-a2 are satisfied and consider
the following change of variables 4i'(t) A ti(t) -t, with i
1, ... , N, so that the system (11) can be equivalently rewritten as
4(t) Awi
=
KN
+-
aij
j =1
('(t-Ti) -'i(t)),
(19)
Wi_-
I
K N
+-ci I:1I
ii
i=l
ii=l
aij Tij
['1l, .,4N], DC
diag(c,.. , CN), Aw
[A. ,... , AW N]T, with A wi given in (20), and LA is the weighted
Laplacian of the graph with diagonal weighted matrix DA. Because
of al-a2, we have rank(LA) = n-1, and LAlN = 0, with
1N denoting the N-length vector of all ones. Hence, system (21)
is feasible if and only if 1T D,A w = OT,which provides a unique
expression for X, as given in (12). Denoting by T* the solution of
system (21) given by
A
,7=1
(s) ds.
(24)
-i
Observe that V(Tb) is continuous on the closure C of C. We denote
with V(+b) the upper right-hand derivative of V(+b) along the solutions of system (23), defined as [8, Section 5.2]
I
(25)
h [V(Xh (10)) V(10)1 .
where Xh (sb) denotes the solution x(t) of (23) on the interval [hTmax, h] with initial function b at time to = 0. Using (25) with
V(+b) given in (24), we obtain
VOO)
=
"M SUPh,O+
V(+)=-K E Eaij, 2(o) + KaZZai
E E X(0) j (- Tij )
i=l =1
i=
j=1
aij/'(-ij )
+2KZ ZaiX(0)- 2K ZZ
i=l =1
-2KZ Zai2
(i2(o)
i=l =1
i=l
Any constant synchronized state of (1 1) must necessarily be an equilibrium of (19), and thus a solution, for any fixed -, of the following
system of linear equations
LA4
-D,A ,
(21)
K
where
(0) +Ilj2
2K Zaij X
)=E2
IKEl
where S is a free parameter to be determined, and
AWi
+(t), mapping the interval [-vmax 0] into RN, and consider the
following functional V: C i-- R, defined as
NZ
N
N
)
ai
j=1
-
20i (°)ojX(-Tigj) + 0A2 (-Tij) )
(Xi(°)- S -Tij))2
<
.
Hence, V(g) is a valid Lyapunov function for the system (23) [8,
Definition 3.1]. According to [8, Theorem 3.2], any solution of the
system of equations in (23), with initial conditions in C,9 converges
as t -, +o0 to the largest invariant set with respect to (23), contained
in
S-{ C C: V(C)= O}
{
o
=
C:
Xi(O)
=
Oj(-Tij), Vi,j}
where the last equality follows from assumption al. The largest invariant set with respect to (23), contained in S, is the set of all initial
values solutions 'L(t) of (23) satisfying the conditions 4'i(t)
4'j(t- Ti), Vi,j and Vt, that, using (23), become 4i'(t) = 0, Vi,
i.e. 'i (t) = c, Vi, where c is any constant. This proves the globally
asymptotic stability of the synchronized state of system (11).
7. REFERENCES
where we used (21), with replaced by '* given in (22).
We show now that system (23) converges to an equilibrium,
[ (t),....
given any initial bounded continuous functions +b(t)
[-Tmax,0] -g R N, with Tmax = maxij Ti; which
I ON (t)]T
proves the globally asymptotic stability of Z for system (19), with
w given in (12). To this end, let C
C([-Tmax,0], Rn) denote
the Banach space of continuous bounded (in some norm)functions
[1] Y.-W. Hong, and A. Scaglione, "A Scalable Synchronization Protocol for Large
Scale Sensor Networks and Its Applications", IEEE Journal on Select Areas in
Comm., Vol. 23, No. 5, pp. 1085-1099, May 2005.
[2] D. Lucarelli, and I.-J. Wang, "Decentralized Synchronization Protocols with Nearest Neighbor Communication", Proc. of SenSys 2004, Baltimore MND, Nov. 2004.
[3] S. Barbarossa, "Self-organizing sensor networks with infonnation propagation
based on mutual coupling of dynamic systems", Proc. of IWWAN 2005, London,
UK, 2005.
[4] S. Barbarossa, and G. Scutari, "Decentralized maximum likelihood estimation for
sensor networks composed of self-synchronizing locally coupled oscillators", submitted to IEEE Trans. on signal Proc., December 2005.
[5] M.G. Earl, and S.H. Strogatz, "Synchronization in oscillator networks with delayed coupling: A stability criterion", Physical Rev. E, Vol. 67, pp. 1-4, 2003.
[6] R. Olfati-Saber, and R.M. Murray, "Consensus Problems in Networks of Agents
with Switching Topology and Time-Delays," IEEE Trans. on Automatic Control,
vol. 49, pp. 1520-1533, Sep., 2004.
[7] G. Scutari, S. Barbarossa, and L. Pescosolido, "Optimal decentralized estimation
through self-synchronizing networks in the presence of propagation delays", in
preparation.
[8] J. K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations,
Springer-Verlag, 1991.
8With a slight abuse of notation, we use the the same variables * to
denote also the system (19), after the shift around J*.
90bserve that, since the functions q$ E C are bounded, the set Ul, as
defined in [8, Theorem 3.1] coincides with C.
4*
=
L
KD
w
(22)
where LA denotes the generalized inverse of the weighted Laplacian LA, for the sake of convenience, we translate the origin of the
dynamic system (19) around T* and write8, for i = 1,
N,
,
qi (t)
K
=
N
-EI
ci
aij ( Pj (t-rTijg )-'i (t) ) '
(23)
I
A
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