Consensus formation in a switched Markovian dynamical system

Abstract

We address the problem of distributively obtaining average-consensus among a connected network of sensors that each respectively track, by linear stochastic approximation, the stationary distribution of an ergodic Markov chain with slowly switching regimes. A hyper-parameter modeled as a Markov process on a slow time-scale modulates the regime of each observed Markov chain, thus at any given time the hyper-parameter determines what stationary distribution will be estimated by each sensor. If the Markov chains share a common stationary distribution conditional on the regime, it is shown the sequence of sensor state-values weakly-converge to an average-consensus under the distributed linear consensus-filter for all network communication graphs. Conversely, if the Markov chains have unique stationary distributions in each regime, then the average-consensus can be achieved only when sensors communicate state-values at a frequency that is on the same time-scale as the frequency at which they observe the fast Markov chain. In this scenario, unlike a static consensus filter, the state-value communication graph need not be connected for an average-consensus to be reached, however this is true only when the communication graph of observation data satisfies a specific connectivity condition. Simulations illustrate our conclusions and observation model.

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Consensus Formation in a Switched Markovian Dynamical System
Kevin Topley, Vikram Krishnamurthy, George Yin
Abstract—We address the problem of distributively obtaining
average-consensus among a connected network of sensors that
each respectively track, by linear stochastic approximation,
the stationary distribution of an ergodic Markov chain with
slowly switching regimes. A hyper-parameter modeled as a
Markov process on a slow time-scale modulates the regime
of each observed Markov chain, thus at any given time the
hyper-parameter determines what stationary distribution will
be estimated by each sensor. If the Markov chains share a
common stationary distribution conditional on the regime, it is
shown the sequence of sensor state-values weakly-converge to
an average-consensus under the distributed linear consensus-
filter for all network communication graphs. Conversely, if the
Markov chains have unique stationary distributions in each
regime, then the average-consensus can be achieved only when
sensors communicate state-values at a frequency that is on the
same time-scale as the frequency at which they observe the fast
Markov chain. In this scenario, unlike a static consensus filter,
the state-value communication graph need not be connected for
an average-consensus to be reached, however this is true only
when the communication graph of observation data satisfies
a specific connectivity condition. Simulations illustrate our
conclusions and observation model.
1. INTRODUCTION
We consider consensus formation in an ad-hoc network
of coupled sensors where each sensor individually tracks
by an adaptive stochastic (SA) approximation the stationary
distribution of a set of Markov chains with time-varying
regime. By sharing information the sensors supplement each
others incomplete knowledge of the entire set of observed
parameters, our focus is on how distributive linear aver-
aging can ensure all sensors eventually reach a common
empirically-based estimate of the average of all observed
stationary distributions as they vary in time, we refer to the
desired estimate as an average-consensus.
Initially we will assume the sensor estimates are updated
according to a distributed linear consensus filter similar to
that proposed in [7], although in contrast to [7] we model
the observed parameter as a finite-state Markov chain with
piece-wise constant stationary distributions, also known as
slowly switching regimes. We show the model of [7] implies
the frequency of communication occurs on a slower time-
scale than that of sensor observation, and in this case the
average-consensus is generally unattainable. As an alterna-
tive we consider the same algorithm when the frequency
V. Krishnamurthy is with the Department of Electrical Engineer-
ing, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
(vikramk@ece.ubc.ca).
G. Yin is with the Department of Mathematics, Wayne State University,
Detroit, MI 48202 (gyin@math.wayne.edu).
of communication occurs on the same time-scale as sensor
observations, and in this case we derive connectivity condi-
tions on the network communication graph required to attain
average-consensus.
Analogous to [7], each sensor i = 1,...,n observes the
state Xi∈ RS×1of a respective S-state Markov chain Xi.
By a linear SA adaptive algorithm with fixed step-size 0 <
µ ≪ 1 each sensor forms their local estimate of the stationary
distribution πiassociated with Xi,
si
k+1= si
k−µ(Xi
k−si
k) , si
0= Xi
0, k = 0,1,.... (1.1)
The sensor estimates {s1,...,sn} will, as the number of
observations approaches infinity, then take the form of an
empirical distribution, thus the sensors maintain “type” data
[2].
Distributed communication of type data to a single fusion
center is considered in, for instance, [11]. Due to the pre-
sumption of a fusion base node, the algorithms for processing
communicated data proposed in works such as [11] are
in general quite different from the distributed algorithms
presented in [6], [12], [3]. The latter works assume no base
node, rather it is assumed individual sensors are connected
by a limited number and arbitrarily placed set of coupling
links. This is identical to the ad-hoc network design that we
consider. For consensus it is required that, in some sense, the
union of all coupling links implies a path between any two
sensors, but we emphasize there is no fusion center assumed,
as this might imply consensus trivially.
A. Consensus Algorithm
As an averaging algorithm we consider the one-hop dis-
tributed linear consensus-algorithm proposed in [7], that is
at every iteration k ∈ {0,1,2...,} each sensor i computes
the state-value si
weighted average of their previous state-value si
current observed value Xi
k, and the state and observed values
received from all sensors that are adjacent to sensor i. In
combination with (1.1) this algorithm has been expressed
in [7] as a discrete-time iteration with constant step-size
0 < µ ≪ 1,
sk+1= sk− µ(Lv+ Do)sk+ µWoXk
where we define the network communication weight ma-
trices, Do= diag(Wo1 1), Lv= Dv− Wv, and Dv=
diag(Wv1 1). The elements of the matrices {Wv,Wo} are
the linear weights attributed to each transmission of a state-
value (si
Each Wv
identity matrix such that wv
ijare non-zero only when a
k+1∈ RS×1by taking an element-wise
k, their
(1.2)
k) or observation values (Xi
ij= wv
k), respectively.
ijI, i.e., a scalar multiple of the S × S
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47th IEEE Conference on Decision and Control
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communication link exists between the respective sensors i
and j. The collection of all communication links comprises
an edge set E of the network communication graph G =
{n,E,Wv}. We initially assume this graph is connected,
undirected, and fixed, thus (1.3) possesses the following
constraints,
1) Wv
2) (i,j) ∈ E ⇔ (j,i) ∈ E,
3) ∀ i, ∃j such that (i,j) ∈ E
The same constraints apply to Wo, although we note the
diagonals of Woare assumed non-zero, whereas by (1.2)
the diagonal elements of Wvare irrelevant and are set equal
to zero for convenience.
ij= 0 ⇔ (i,j) / ∈ E,
B. Background
Although much research has explored the properties of the
“static” consensus algorithm
si
k+1= si
k+ µ
n
?
j=1
Wv
ij(sj
k− si
k),i = 1,...,n,
(1.3)
under time-varying or stochastic graphs (for instance see
[4], [10], [5]) significantly less research has considered
distributed averaging across nodes that track an external
parameter while simultaneously seeking consensus. An es-
sential example of such research is [7], which considers
the continuous-time limit of (1.3) when each node shares
and is also linearly updated by a continuous signal γt ∈
R of bounded rate, observed in i.i.d. Gaussian noise. All
nodes are then shown to remain within a bounded distance
ε of the observed parameter at all times. This framework
has subsequently been expanded in a number of respects,
such as with implementation of local Kalman-filtering, time-
varying communication graphs, as well as dynamic tracking
algorithms for mobile sensors, to name a few [9], [8].
The preliminary difference between [7] and the current
model is that we now replace the communally observed
continuous signal γ by a set of S-state Markov chains
{X1,...,Xn}, each of which is privately observed by a
corresponding sensor si, i = 1,...,n. Secondly, we con-
sider the Markov chains {Xi} are dependent on a parameter
θ, where the sequence {θk} evolves according to a Markov
chain whose transitions take place infrequently and with state
space represented by Mθ= {θ1,...,θm}.
As the essential results of this paper, we show that
provided the transition probability matrix of θk is “near”
identity, or specifically that θ evolves on a time-scale of
the same order O(µ) as that of the sensor consensus-
tracking algorithm (1.2), then under a suitably weighted
communication graph the continuous-time linear interpo-
lation of the state-values si
kwill converge weakly to a
stochastically switching consensus of the average observed
stationary distribution ¯ π(θ) =
the sequence of tracking errors between each sensor and
the average-consensus, when properly scaled, is shown to
converge weakly to the solution of a switching diffusion.
These results constitute an altogether distinct set of network
consensus dynamics than yet reported in the literature.
1
n
?n
i=1πi(θ). In addition,
The paper is organized as follows: in §2 we present the
main convergence theorems, §3 investigates some ramifica-
tions of these. Simulations are provided in Section §4 to
illustrate our results, and Section §5 provides a summary.
2. DYNAMIC CONSENSUS FORMATION
Let Xi
kbe an S-state Markov chain with a state space
{e1,...,eS}, where {ei} are orthogonal S × 1 standard
unit vectors. Each Xi
kis θ dependent for θ ∈ Mθ =
{θ1,...,θm}, that is the transition matrix of Xi
on θ is given by Ai(θ) = (ai
lj(θ)), where
kconditioned
ai
lj(θ) = P(Xi
k+1= ej|Xi
k= el,θk= θ).
(2.4)
In the following section we first assume the weights
{Wo,Wv} are fixed such that H = Lv+ Dohas only
eigenvalues with positive real parts, denoted H ≻re0, thus
providing bounded stability of the consensus algorithm (1.2)
in the limit µ approaches zero [1]. Under this condition we
describe how the state-values skof (1.2) will then weakly-
converge as µ vanishes.
A. Asymptotic Properties of Stochastic Approximation Algo-
rithms
To proceed, we pose the conditions below.
(A) The following conditions hold.
1) For each θ ∈ Mθ, the transition matrix Ai(θ) is
irreducible and aperiodic with stationary measure
πi(θ).
2) Parameterize the transition probability matrix of θ
as
Pε= I + εQ,
where ε is a small parameter satisfying 0 < ε ≪ 1
and Q is the generator of a continuous-time finite-
state Markov chain.
3) The process θkis slow in the sense ε = O(µ). For
simplicity, we take ε = µ henceforth.
(B) H ≻re0 where H = (Lv+ Do).
Although the proof is omitted for brevity (see [13]), we
shall show that it is feasible for the network to track changes
of a linear combination of {π1,...,πn}. In particular, as
the step-size goes to zero, each sensor state value converges
weakly to the solution of a stochastically switching ODE and
possesses a scaled tracking error with a switching diffusion
limit.
Theorem 2.1: Assume conditions in (A) - (B) hold and
ε = µ. Define the interpolated sequences of iterates
sµ
t= sk, θµ
t= θk for t ∈ [kµ,(k + 1)µ).
Then as µ → 0, (sµ(·),θµ(·)) converge weakly to (s(·),θ(·))
such that θ(·) is a continuous-time Markov chain with
generator Q and s(·) satisfies
dst
dt
where
π(θ) = [π1(θ),...,πn(θ)]′.
= −Hst+ Woπ(θt), t ≥ 0,
(2.5)
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We next study the associated tracking errors. Define
vk=sk− GE(π(θk))
õ
,
(2.6)
where G = H−1Wo. Define the interpolated sequence of
scaled tracking errors by
vµ
t= vk, t ∈ [kµ,(k + 1)µ).
(2.7)
Theorem 2.2: Under the same conditions as Theorem 2.1,
vµ(·) converges weakly to v(·), which is a solution of the
switching diffusion
dvt= −Hvtdt + WoΣ1/2(θt)dw,
where w(·) is a standard Brownian motion and for a fixed
θ, Σ(θ) is a covariance matrix.
Theorem 2.1 and Theorem 2.2 both present a convergence
result for small µ and large k such that µk remains bounded,
we refer to this time-scale as O(1). From the theorems it is
clear how the asymptotic sensor dynamics depend on both
the variation in θ and the choice of edge weights {Wo,Wv}
of the communication graph. In the following section we
focus on Theorem 2.1 regarding the sufficient or necessary
weights for all sensors to reach the average-consensus ¯ π(θ).
(2.8)
3. RAMIFICATIONS
It is clear by (2.5) that each sensor in the network
weakly-converges to an equilibrium, or steady-state, that is
modulated by θ and can be represented as the product Λπ(θ),
where
Λ = H−1Wo= (Lv+ Do)−1Wo.
If each observed Markov chain has an identical stationary
distribution π∗(θ) conditioned on θ, then ¯ π(θ) = π∗(θ) and
the above theorems imply weak-convergence to the average-
consensus ¯ π(θ) by all sensor state-values, that is, si
for i = 1,...,n. This is in fact true regardless of the network
communication graph G, provided (B) holds.
Corollary 3.1: If πi(θ) = π∗(θ) for all i and θ ∈ M,
then provided (A) - (B) and ǫ = µ, an average-consensus is
reached for all communication graphs G, including the null
graph G = {n,0,0}.
Proof. By the first assumption π(θ) = [π∗(θ),...,π∗(θ)]′
for some π∗(θ) ∈ RS×1, thus Λπ = π for all Λ1 1 = 1 1.
We now show that all Λ satisfying (3.9) are row-stochastic,
hence the asymptotic equilibrium will be the consensus π∗.
Rearranging (3.9) yields the necessary condition
(3.9)
k→ ¯ π(θ)
Wo= (Lv+ Do)Λ.
(3.10)
If Λ1 1 = 1 1 then we have
Wo1 1 = Dv1 1 − Wv1 1 + Do1 1 ,
which is true by the definitions Dv= diag(Wv1 1) and Do=
diag(Wo1 1). If Λ1 1 ?= 1 1 then, as the inverse of a matrix is
unique, (3.9) could not hold.
If the observed Markov chains do not have identical
stationary distributions conditioned on θ, that is πi(θ) ?=
(3.11)
2
πj(θ) for all i ?= j and θ ∈ M, then average-consensus
is achieved only if Λ =
constraint than Λ1 1 = 1 1, and in fact it can only be achieved
in approximation if the edge set E is not complete, that is
there exists an ordered pair (i,j) such that (i,j) / ∈ E, we
refer to such a graph as “unsaturated”.
Corollary 3.2: Under the assumption each sensor will
asymptotically observe a unique stationary distribution, con-
sensus cannot be achieved by a linear consensus-filter if G
is unsaturated.
1
n1 11 1′. This is a much stronger
Proof. Since any consensus requires Λ have identical rows,
left-multiplication of Λ by (Lv+ Do) results in DoΛ by
definition of Lv. Since Dois diagonal, by (3.9) the matrix
Wowill have every element non-zero, thus implying a
saturated communication graph.
Contrary to the above result, by increasing the frequency
of communication, consensus can be achieved. We inspect
the consequences of this scenario in the following subsection.
2
A. Communication Step-Size
Although the observation update weights Womust be
scaled by the step-size 0 < µ ≪ 1 in order for weak-
convergence of sensor state-values, there is no apparent
reason the update weights Wvalso need to be scaled by
µ. To see this we note,
1) communication of state-values consists implicitly of
a communication of past observed values as scaled
by µ(1 − µ)l, where l denotes the lthmost recent
observation since last communication.
2) if (Lv+ Do) ≻re0 then (mLv+ Do) ≻re0 for all
m > 0 only under the conditions {Lv?re0,Do≻re
0}. We assume the latter conditions.
Our analysis until now has assumed Wvis scaled by µ, we
now explain how this can be seen to imply a distributed linear
consensus-filter for which the frequency of communication
is much smaller than the frequency of observation.
1) Slow Communication: In §2− A we assert that as the
step-size µ approaches zero if ǫ = µ the continuous-time
piece-wise constant interpolation of the sequence {sk},
st= sk
t ∈ [kµ,(k + 1)µ),
will convergeweakly to solutions of the switching ODE (2.5)
where Q is the generator of a continuous-time Markov chain,
which we emphasize is continuous on the time-scale O(1).
Taking µ to zero when ǫ = µ implies the transition matrix
Pǫis “near” identity on the time-scale O(µ). By (1.1) this
implies the observed parameter θ evolves on the same time-
scale as the adaptive SA algorithm, or equivalently the time-
scale of sensor observation.
We now inspect the time-scale O(1), which is the time-
scale of reference for (2.5). On this time-scale the matrix Pǫ
is not near identity, and the parameter θ is seen to moderately
vary. We once again assume the observation model (1.1), but
now we explicitly model sensor communication to occur on
the slow time-scale O(1) and will subsequently show that
this is equivalent to (1.2).
(3.12)
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First consider the n sensors are independently updated as
(1.1) by n Markov chains Xi. By (2.5) each sensor would
then weakly-converge to the solutions of
dst
dt
= −st+ π(θt), t ≥ 0,
(3.13)
where t is measured on the time-scale O(1). Suppose that
at every time t∗∈ T = {t : t ∈ N}, each sensor
i communicates its observation value Xi
current state-value si
t∗, through all out-going communication
links; the receiving sensors then compute a weighted linear
average of their own state-value and the elements of data
(si
t∗) they just received.
The continuous-time sensor dynamics are then expressed,
t∗, along with its
t∗,Xi
st=
?
e−At(s¯ t∗ − A−1π(θt)) + A−1π(θt) ,
(I − L − Do
where¯t∗= maxt∗{t∗< t}, Wo
on the diagonal, and A = Do−Do
each sensors individual observation update weight. We also
denote Lv= L for remainder of this section.
Defining s0= X0and t∗
arbitrary t∗
−the expected sensor state-values,
if t / ∈ N
−d)st+ Wo
−dXt,
if t ∈ N
(3.14)
−dequals Wobut with zeros
−dis the diagonal matrix of
−= limt ր t∗, we then have for
E(st∗
−) =Wt∗−1e−At∗s0+?t∗−1
(I + e−A(Wo− I))π(θt∗
l=0Wle−Al
−)
(3.15)
where we define W = I − L + Doand note that, without
prior knowledge, the expected value of Xi
For large t∗
−it is clear that under §2−A (B) the coefficient
of X0vanishes whereas the coefficient of π converges to
tis πi(θt).
(I −We−A)−1(I +e−A(Wo
−d−I)) =
eA− I + Wo
eA− I + L + Do
−d
−d
(3.16)
which we find is equal to the equilibrium (3.9) under
the continuous model (1.2) when either A = ln(2) or
equivalently when the rate of continuous-time is scaled
∆tdisc= ln(2)∆tcont.
We note that in general if sensor i has an individual
observation update weight Wo
(1.2), then the corresponding update weights of the sensors
in the discrete model (3.14) is given by Aii= ln(Wo
If Aii ?= Ajj then the sensors (i,j) might operate on
time-scales that are uniquely scaled and thus would possess
asynchronous times of communication to yield (3.14).
Specifically, we find that under the discrete model there
exist an infinite subsequence of sensor state-values that, with
the correct time factors, will converge in expectation to the
sensor state-values under the continuous model. To illustrate
this fact we simulate a distributed ad hoc network of 10
sensors operating under both (1.2) and (3.14). For clarity in
Fig.1 we plot the linear sum of each sensors state-value, that
is we plot the the scalar 1 1′sirather than the S × 1 vector
sifor each sensor i. A jump in θ occurs mid-way in our
simulation, as signified by the vertical line.
iiin the continuous model
ii+1).
Fig. 1.
suming slow communication. As t increases from the most recent transition
time of θ, both models have identical equilibriums at the times t∗
Comparison of Discrete (3.14) and Continuous (1.2) Models as-
−.
2) Consensus Properties under Slow Communication: It
is evident from Fig.1 and proven by Corollary 3.1 that in
either the discrete or continuous model, consensus cannot be
reached when A ?= 0. However, when A = 0 no tracking
occurs, which implies that linearly tracking a set of distinct
parameters by the SA (1.1) is fundamentally opposed to
distributed linear consensus formation. Since consensus can
only be partially achieved in this setting, we refer to an
arbitrary graph has having an inherent “consensus ability”;
although we leave this term formally undefined for the
moment, it is intuitive that graphs with many strategically
placed communication links should have better consensus
ability than graphs with only a few randomly placed links,
and indeed it is only the saturated graph that can actually
attain consensus in this framework.
We also note that the discrete model requires unscaled
communication of observation data, thus the sensor iterates
are stochastic in the sense that at each time t∗
in expectation that (3.14) yields the same equilibrium as
(1.2). At any given time t∗
−the sensors under (3.14) will be
sharing realizations of {X1,...,Xn}, not their stationary
measures πi(θ). Thus the iterates st∗
model do not actually approach ¯ π(θ), rather their average
?m
1 (w.p.1) as m increases, where lθ denotes the most recent
transition time of θt.
In Fig.1 we parameterized the variability of X as relatively
small such that the stochastic property we are discussing
cannot be noticed. Fig.2 displays a more accurate illustration.
−it is only
−under the discrete
l=min(l∗>lθ)
1
msl∗
−will approach (3.16) with probability
Fig. 2.
assuming slow communication but when X has large variation. As m
increases the average?m
of (1.2) have identical equilibriums w.p.1.
Comparison of Discrete (3.14) and Continuous (1.2) Models
l=min(l∗>lθ)
1
msl∗
−of discrete iterates and st∗
−
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The continuous algorithm results in no visible stochastic
element, although both models possess the scaled diffusion
(2.8). The stochastic element of (3.14) is the result of
updating the sensors by the parameter X unscaled by µ. We
refer to this stochastic element as “stochastic variation” and
note that it is the essential difference between the two models
(1.2) (3.14), since otherwise their equivalence depends only
on scaling the rate at which the sensors of either algorithm
measure time.
To avoid the undesirable stochastic variation present in
(3.14) we can set Wo
used to update the sensor state-values and thus we need
not take their expectation. When Wo
(3.16) becomes
eA− I
eA− I + L.
The extent to which this reduces the sensor ability to form
consensus is considered negligible, and not discussed here.
3) Fast Communication: We now inspect the sensor
equilibrium as the frequency of communication in (3.14)
increases. For some m ∈ N we take t∗∈ T = {t : mt ∈ N}
and consider again (3.14). In this case the equilibrium (3.16)
becomes
eA/m− I + Wo
eA/m− I + L + Do
If the communication weights Woare not scaled, then
an expectation must be taken to attain equivalence between
the discrete and continuous models. As m increases the
stochastic element described above will affect the sensors
at more frequent time points and thus perpetuate a random
dynamic behavior of the sensor iterates, pictured in Fig.3
below.
−d= 0, then an unscaled X is no longer
−d= 0 the equilibrium
(3.17)
−d
−d
.
(3.18)
Fig. 3.
communication weights of observation data. The consensus ability is not
significantly improved as compared to the slow communication model
pictured in Fig.3−A.1−3−A.2. However, the stochastic variation present
in Fig.3 − A.2) now perturbs the sensor iterates with greater frequency as
m increases.
A connected network under fast communication with unscaled
The sensor equilibrium remains given by nearly the same
expression as it is under the slow communication models,
that is as m approaches infinity (3.18) becomes
lim
m→∞
A/m + Wo
A/m + L + Do
−d
−d
.
(3.19)
Although the above limit can be seen to aid in consensus for-
mation, we will not explore this here. We also assert without
proof that Cor.3.2 applies to (3.19) just as it did to (3.16),
thus consensus can again be attained only in approximation,
similar to the slow communication algorithms (1.2) (3.14).
Due to both the unwanted stochastic variation and the
network inability to form consensus, we now scale the
communication weights Woby a factor of order O(m−1).
In this case, as m approaches infinity the sensor equilibrium
(3.18) becomes
lim
m→∞(mL + Do)−1Wo.
(3.20)
In contrast to Cor.3.2, the following lemma shows (3.20)
implies that weak-convergenceto the average-consensus may
be attained by sensors operating under (3.14) when t∗∈
T = {t : mt ∈ N} as the frequency of communication m
approaches infinity.
Lemma 3.3: For any connected network graph with Lapla-
cian L, there exist left and right eigenvectors ωr and ωℓ
satisfying
Lωr= 0 , ω′
ℓL = 0 , ω′
ℓωr= 1,
(3.21)
such that
lim
m→∞(mL + Do)−1Wo= (ωℓDoωr)−1ωrω′
Proof. For a connected network the matrix L is guaranteed by
its definition to have eigenvectors {ωr,ωℓ} satisfying (3.21),
furthermore ωr= c1 1 for any c ∈ R .
Denoting the eigendecomposition of L(Do)−1as UJU−1
we note the argument in (3.22) may be rearrangedas follows,
ℓWo.
(3.22)
(mL + Do)−1Wo= (Do)−1U(I + mJ)U−1.
Since L(Do)−1will have the eigenvalue pair {Doωr,ωℓ}
satisfying the same conditions as (3.21) in regard to L,
the rearranged eigendecomposition U−1L(Do)−1= JU−1
implies that if the ithrow of U−1equals ω′
only when j = i. This in turn implies that only the ith
diagonal of the matrix (I +mJ) will remain bounded in the
limit m → ∞, and furthermore, since this element is unity,
the inverse (I + mJ)−1will have this precise term as its
only non-zero element.
By the same reasoning, rearranging the eigendecomposi-
tion as L(Do)−1U = UJ implies the ithcolumn of U is
Doωr. Since U−1U = I we have ω′
(3.22) is obtained by multiplying Woon both sides of (3.23)
and taking the limit.
2
It is clear then under fast communication the sensors
may achieve consensus if and only if the weight matrices
{Wv,Wo} are such that
(ωℓDo1 1)−11 1ω′
Due to this, we note that when communicating on the fast
time-scale O(µ), sensor communication of observation data
is not required to attain the average-consensus, provided L
is connected. To see this we may take the Laplacian Lcof
an arbitrarily connected graph and set Wo
ωℓsatisfies (3.21).
We also note that Lcneed not be balanced, which by [6] is
in contradiction to the average-consensus requirements of the
(3.23)
ℓthen Jjj = 0
ℓDoωr = 1 and thus
ℓWo=1
n1 11 1′.
(3.24)
ii=1
n(ωℓ)iwhere
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008WeC12.5
3551