Decentralized observer with a consensus filter for distributed discrete-time linear systems

Abstract

This paper presents a decentralized observer with a consensus filter for the state observation of a discrete-time linear distributed systems. In this setup, each agent in the distributed system has an observer with a model of the plant that utilizes the set of locally available measurements, which may not make the full plant state detectable. This lack of detectability is overcome by utilizing a consensus filter that blends the state estimate of each agent with its neighbors' estimates. We assume that the communication graph is connected for all times as well as the sensing graph. It is proven that the state estimates of the proposed observer asymptotically converge to the actual plant states under arbitrarily changing, but connected, communication and sensing topologies. As a byproduct of this research, we also obtained a result on the location of eigenvalues, the spectrum, of the Laplacian for a family of graphs with self-loops.

Full text

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Decentralized Observer with a Consensus Filter for Distributed
Discrete-Time Linear Systems
Behc ¸et Ac ¸ıkmes ¸e and Milan Mandi´ c
Abstract—This paper presents a decentralized observer with
a consensus filter for the state observation of a discrete-time
linear distributed systems. In this setup, each agent in the
distributed system has an observer with a model of the plant
that utilizes the set of locally available measurements, which
may not make the full plant state detectable. This lack of
detectability is overcome by utilizing a consensus filter that
blends the state estimate of each agent with its neighbors’
estimates. We assume that the communication graph is con-
nected for all times as well as the sensing graph. It is proven
that the state estimates of the proposed observer asymptotically
converge to the actual plant states under arbitrarily changing,
but connected, communication and sensing topologies. As a
byproduct of this research, we also obtained a result on the
location of eigenvalues, the spectrum, of the Laplacian for a
family of graphs with self-loops.
I. INTRODUCTION
Decentralized estimation [1] has long been an active area
of research with an increased recent interest in distributed
systems. Here we focus on a decentralized estimation prob-
lem for a distributed system with multiple agents, where
each agent estimates the state of the whole system. In
this problem setup, each agent represents a physical entity
such as a spacecraft or an aircraft in a formation. Some
of the earlier research in decentralized estimation focused
on combining the state estimates of a system with multiple
agents into a single central estimate [2], [3], [4], where
all the information is communicated to all agents in the
system back and forth. This is a communication intensive
approach and may not be appropriate for distributed systems
with a large number of agents. The main idea behind these
algorithms is to blend independently obtained state estimates
into a single better state estimate, which has been the main
idea behind the more recent algorithms as well. In the
covariance intersection method [5], [6], the state estimates
and their error covariance matrices are exchanged without
the exact knowledge of correlation between the estimates of
the different agents. The unknown correlation between the
exchanged state estimates is bounded by a bound on the
intersection of the error covariance matrices. This method
ensures that the unknown correlations are accounted for,
but it requires the computation of the error covariances and
their inverses, which can be computationally demanding. In
a recent approach to distributed system state estimation, Ref.
Guidance and Control Analysis Group, Jet Propulsion Laboratory,
Pasadena, CA. Email: behcet@jpl.nasa.gov
Mechanical and Aerospace Engineering Department, UCLA. Email:
mandicm@ucla.edu
c ?2011 California Institute of Technology. Government sponsorship
acknowledged.
[7] considers a fusion center that combines measurements or
state estimates from the agents into a single estimate by using
a Kalman filter with a particular structure. However we have
to treat each agent as a fusion center if this approach is to
be adapted, which considerably increases the complexity in
the information routing problem.
A large number of recent research study the consensus
problems in distributed systems in a graph theoretical frame-
work [8], [9], [10], [11], [12], [13] to tackle the difficulties
of the distributed estimation and control. The distributed
Kalman filters with embedded consensus filters are studied
in [14], [15]. Particularly [15] introduces a state estimator
for continuous time linear time systems with a consensus
filter that blends state estimates of neighboring agents, which
motivated the particular observer structure we adapted in this
paper. Our paper provides a stable observer with a consensus
filter for a discrete-time distributed system with time varying
communication topologies and measurement matrices. The
synthesis of stable observers for discrete-time distributed lin-
ear time systems is non-trivial. In [15] a quadratic Lyapunov
function of the estimation error is constructed by using the
time-varying covariance matrix that is computed via a Riccati
matrix differential equation. However this Lyapunov function
has the right properties to be a valid candidate under non-
trivial observability conditions on the linear system [16].
These conditions must be satisfied by the system matrices
that are time-varying, which is non-trivial to verify. This
happens in distributed systems due to constantly changing
sensing topology, i.e., the set of measurements available to
each agent changes as an unknown function of time.
This paper presents an observer with an embedded con-
sensus filter for a class of discrete-time linear systems. Each
agent utilizes its local measurements and its neighbors’ com-
municated state estimates to update its own state estimate.
This observer architecture makes the information routing
problem straight-forward. The local measurement vectors are
described linearly as a function of the plant state via time
varying matrices. The local measurements do not provide the
full state detectability that is required to have asymptotically
convergent local observers. The consensus filters of each
agent blends the state estimates with its neighbors’ estimates
to overcome this limitation by seeking consensus among
neighbors’ estimates. The consensus filters update their in-
ternal states more frequently then the local observers, that is,
there are multiple consensus updates in between the observer
state updates. This ensures that a sufficient level of consensus
is reached for the stability of the observers. The number of
consensus state updates between consecutive observer state
2011 American Control Conference
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June 29 - July 01, 2011
978-1-4577-0079-8/11/$26.00 ©2011 AACC4723

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updates is analytically determined ahead of time. The main
contribution of this paper is the proof of the exponential
stability of the observer error dynamics under time-varying
communication and sensing topologies. We provide a method
to compute quadratic Lyapunov functions that prove the
exponential stability of the observer error dynamics. We also
present a useful graph theoretic result, which is a byproduct
of this research, on the smallest eigenvalue of the Laplacians
for a class of undirected graphs with self-loops.
Notation
The following is a partial list of notation used in this paper:
Q = QT> (≥)0 implies Q is a symmetric positive (semi-
)definite matrix; G = (V,E) represents a finite graph with
a set of vertices V and edges E with (i,j) ∈ E denoting
that there is an edge between the vertices i and j; L(G) is
the Laplacian matrix for the graph G; a(G) is the algebraic
connectivity of the graph G, which is the second smallest
eigenvalue of L(G); Rnis the n dimensional real vector
space; ?v? is the 2-norm of the vector v; I is the identity
matrix of appropriate dimension and Imis the identity matrix
in Rm×m; 1m is a column vector of ones in Rm; ei is a
vector with its ith entry +1 and the rest of entries zero;
σ(A) is the set of all eigenvalues of the matrix A and
σ+(A) is the set of all of its positive eigenvalues; max(σ(P))
and min(σ(P)) are maximum and minimum eigenvalues
of symmetric matrix P; “⊗” is the Kronecker product;
(v1,v2,...,vm) represents a vector obtained by augmenting
vectors v1,...,vmsuch that:
(v1,v2,...,vm) ≡?vT
E is the vertex-edge adjacency matrix, A adjacency ma-
trix, and D is the diagonal matrix of node in-degrees for
G, then the following gives a relationship to compute the
Laplacian matrix
1
vT
2
...vT
m
?T.
L(G) = ETE = D − A.
The Laplacian matrix is a symmetric matrix with non-
negative diagonal and non-positive off-diagonal entries.
The following relationships are well known in the litera-
ture [17] and [18] for a connected undirected graph G with
N vertices and without any self-loops or multiple edges
a(G)≥
≤
2(1 − cos(π/N))
2d(G)
(1)
max(σ(L(G)))
(2)
where d(G) is the maximum in-degree of G. Indeed the
inequality (2) is valid for any undirected graph without
self-loops or multiple edges whether they are connected
or not. Next we characterize the location of the Laplacian
eigenvalues for a connected undirected graph G with self-
loops. Having a self-loop does not change whether a graph is
connected or not, that is, a graph with self-loops is connected
if and only if the same graph with the self-loops removed
is connected. Furthermore we define the Laplacian of an
undirected graph with at least one self-loop as
L(G) = L(Go) +
?
(i,i)∈E
eieT
i
(3)
where Gois the largest subgraph of G with the self-loops
removed, and
L(Go) =
?
(i,j)∈E,i?=j
(ei− ej)(ei− ej)T.
(4)
II. SYSTEM DESCRIPTION
We consider the problem of decentralized state observation
for the following discrete-time linear system representing a
group of N collaborative agents:
xk+1
yi,k
= Axk
Ci,kxk,
(5)
=i = 1,...,N
(6)
where xk ∈ Rnis the state vector at time instance k and
yi,k ∈ Rmi,kis the measurement vector of the ith agent
at time instance k. In this scenario, each agent has its own
measurements determined by the measurement matrix Ci,k
and it has direct communication links with a subset of other
agents, which will be referred as the “neighbors”. The set
of communication links in between the agents determine
the communication topology and an associated graph, Gc,k,
where each agent is represented by a vertex of Gc,k, and each
communication link is represented by an edge of Gc,k. We
assume that the graph Gc,kis a undirected connected graph
[19] without self-loops or multiple edges for all times, which
implies that [17] a(Gc,k) > 0 for all k = 0,1,...
We consider a “core” set of m measurements zk,


?
such that all locally available “actual” measurements for each
agent can be formed as a linear combination of the core
measurements as follows
zk=

z1,k
...
zm,k

 =




C1
...
Cm
??


?

C
xk
where zi,k∈ Rp∀i. (7)
yi,k= (Ei,k⊗ Ip)zk,
where yi,k ∈ Rmi,k, Ei,k ∈ Rqi,k×m, i = 1,...,N, are
“vertex-edge adjacency” matrices, hence mi,k= qi,kp, with
p being the size of the local, core measurement vector, zi,k.
A vertex-edge adjacency matrix describes an edge between
two vertices or a single vertex (for a self-loop) in a graph
on each of its rows, whose entries corresponding to these
vertices are +1 and −1 (it does not matter which entry is +
or −) and the rest of the entries are zeros. Note that if the
edge described by a row is a self-loop then there is only one
non-zero entry with +1.
The assumption that all actual measurements can be ex-
pressed in terms of the core measurements adds more struc-
ture to the problem at hand without losing much generality,
and its use will become apparent in later sections.
Next we collect the set of all distinct local measurements
into a global measurement vector ykas follows
i = 1,...,N.
(8)
yk= (Ek⊗ Ip)zk= (Ek⊗ Ip)Cxk,
(9)
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where the vertex-edge adjacency matrix Ekcontains all the
distinct rows of all Ei,k, i = 1,...,N, that is, Ekis a vertex-
edge adjacency matrix of a graph without multiple edges.
Therefore, ykis not necessarily an augmentation of all local
measurements in general, that is, yk ?= (y1,k,...,yN,k) in
general. Moreover a local measurement vector yi,k for any
agent can simply be obtained by picking the right entries of
the vector yk. Consequently, a row of Ek can correspond
to a measurement that belongs to multiple agents, that is, a
measurement can be available to multiple agents. For each
agent we will define a vector hi,k∈ Zqi,kthat contains the
positive integer numbers representing how many agents each
measurement is available to. This implies that
ET
kEk=
N
?
i=1
ET
i,k(diag(hi,k))−1Ei,k.
(10)
In summary, the sensing graph Gs,k is constructed with
its vertices as the core set of measurements z1,k,...,zm,k
and its edges represent the actual measurements at time
instance k. Since a core measurement can also be an actual
measurement, e.g., yi,k= zj,k, a sensing graph can have self
loops, and in the case when all measurements are the core
ones, the sensing graph can be completely disconnected in
the usual sense. We introduce a concept of pseudo-connected
graphs to capture useful properties of the sensing graphs that
will be encountered (see Figure 8 for an example of a pseudo-
connected graph).
Definition 1: An undirected graph G(V,E) without mul-
tiple edges is defined to be pseudo-connected if every vertex
is connected to itself and/or to another vertex and if every
connected subgraph of G has at least one vertex with a self-
loop.
Given the above definition, the following conditions are
assumed to hold for the system defined by equations (5)
and (6):
A1) Gc,kis a connected graph without self-loops or mul-
tiple edges ∀k.
A2) Gs,kis pseudo-connected without multiple edges ∀k.
A3) The pair (C,A) is detectable.
A4) Each agent knows hi,kat any given time instance k.
Assuming a pseudo-connected sensing graph implies that
one or more of the core measurements are among the actual
measurements at any given time. The assumption of having
a connected communication graphs can be relaxed to, for
example, having jointly connected communication graphs
[20]. Such relaxations can lead to some generalizations
of the forthcoming results, which is beyond the scope of
this paper. The detectability of (C,A) pair ensures that an
exponentially stable observer exists by utilizing only the core
measurements. The last assumption of each agent having the
information of the vectors hi,ksimply means that each agent
knows how many other agents have access to the information
that it has. The hi,kvectors can be routed to each agent in
real-time, or the distributed system at hand may have the
working assumption that each measurement is known by a
fixed number of agents at any given time.
III. DECENTRALIZED OBSERVER WITH CONSENSUS
FILTER
We propose the following local observers with a consensus
filter that process both the locally available measurements
and the neighbors’ state estimates:
Local Observers with Consensus Filter
ˆ xi,k+1
=Asi,k+ Li,k(Ci,ksi,k− yi,k)
ξi,l−
j∈Si,k
ˆ xi,k,l = 1...r
ξi,r,i = 1...N.
(11)
ξi,l+1
=
?
δ(ξi,l− ξj,l)
(12)
ξi,0
si,k
=
=
where r is the number of iterations, consensus state updates,
per single time step, δ > 0 is a design parameter, Si,kis the
index set of neighbors for the agent i. The gain matrices
Li,kare computed by using the matrix L, which is defined
as the core observer gain matrix corresponding to the core
measurement zk, as follows
Li,k= L?ET
i,kdiag(hi,k)−1⊗ Ip
?.
(13)
The choices for the scalars r and δ will be explained later
in the paper. Here we assume that the consensus filter
can be iterated as many times as the integer r dictates
during a single time step. Hence r can be seen as a design
parameter that determines how fast the consensus dynamics
need to be for the stability of this observation algorithm.
Clearly the number r can be too large to be handled by the
available communication hardware. Hence one of our design
objectives is to determine the least conservative upper bound
on the number of consensus updates r.
IV. SYNTHESIS FOR A STABLE OBSERVER
In this section, we present a constructive proof of the
exponential stability of the proposed decentralized observa-
tion algorithm. As a by-product of this proof, we obtain
synthesis procedures to compute the observer gain matrix
L and parameter r.
Let ξl be the overall (stacked-up) vector of ξi,k ∈ RnN
where N is the number of spacecraft and n is the number
of states per agent.
ξl= (ξ1,l,ξ2,l,...,ξN,l)
Similarly, ˆ xkand Xk(both in RnN) can be expressed as
ˆ xk
Xk
:=(ˆ x1,k, ˆ x2,k,..., ˆ xN,k)
1N⊗ xk= (xk,xk,...,xk).:=
From equation (12):
ξl+1
=
=
(InN− δLc⊗,k)ξl,
(InN− δLc⊗,k)rˆ xk
l = 1,...,r
⇒ξr
where Lc⊗,k = Lc,k ⊗ In, and Lc,k is the Laplacian
matrix of the communication graph Gc,k at time k. Let
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the observation error be defined as ei,k := ˆ xi,k− xk and
ek= (e1,k,...,eN,k), then using the equation (11), we have
ei,k+1
=ˆ xi,k+1− xk+1
Asi,k+ Li,k(Ci,ksi,k− yi,k) − Axk.=
Since (Lc,k⊗ In)(1N⊗ xk) = Lc,k1N⊗ xk= 0 and
(InN− δLc⊗,k)r= I + (...)Lc⊗,k+ (...)L2
c⊗,k+ ...,
we have
(InN− δLc⊗,k)rXk
=(InN− δLc⊗,k)r(1N⊗ xk)
(1N⊗ xk).=
Then, the overall error dynamics, ekcan be expressed as:
ek+1
=Ac(InN− δLc⊗,k)rek
Ac[(IN− δLc,k)r⊗ In]ek
=
(14)
where
Ac = diag{A + Li,kCi,k; i = 1,...,N}
= diag?A + L(ET
where i=1,...,N
(15)
i,kdiag(hi,k)−1Ei,k⊗ Ip)C;?
Theorem 1: Suppose that the sensing graph is pseudo-
connected and communication graph is connected for all
k = 1,2,..., and there exist some L, P = PT> 0, and
λ ∈ [0,1) such that the following inequality holds for the
Laplacian, Ls, of any pseudo-connected sensing graph Gs,
λP − Aa(Ls)TPAa(Ls) ≥ 0
Aa(Ls) := A +1
where(16)
NL(Ls⊗ Ip)C
Let δ ∈ (0,1/(N − 1)). Then there exists a large enough
positive integer r ≥ 1 such that the error dynamics of the
observer given by the equation (14) are globally exponen-
tially stable (GES), hence the observer given by equations
(11) and (12) is GES and, for i = 1,...,N,
?ˆ xi,k− xk? ≤ ci˜λk
for some ci> 0 and˜λi∈ (0,1).
Proof: We consider the following Lyapunov function
for the error dynamics
i?ˆ xi,k− xk?,∀k = 0,1,...
(17)
V (ek) = eT
k(Iα⊗ P)ek where Iα=
?10
0 αIN−1
?
(18)
and α>0. The first step is to find an appropriate transfor-
mation that will split the error vector ekinto agreement and
disagreement subspaces. These subspaces bring a geometri-
cal insight to the observer synthesis, and help clarifying the
roles of the measurement and communication feedback terms
in the observer. Consider the following transformation

0
√6
...
0
T =

?

1
√N
1
√2
−1
√2
1
√6
1
√6
−2
...
1
√
√
√
N(N−1)
1
N(N−1)
1
N(N−1)
1
1
√N
...
...
...
√
√
N(N−1)
−(N−1)
N(N−1)
000


?

??
Tc
⊗In. (19)
It can easily be shown that columns of T and Tchas 2-norm
equal to one and they are orthogonal to each other, hence T
and Tc are orthogonal matrices such that TTT = TTT=
InN and TcTT
c
= TcTT
c
= IN. Note that, for any graph
without self loops or multiple edges G, such as the graph of
a communication topology, we have
?
for some vector v ≥ 0 and matrix V = VT, which are related
by V 1 = v, and we can express matrix Tcas follows
?
with appropriately defined vector w and matrix U. Now we
can show that
?
where Lp(G)∈R(n−1)×(n−1)is a symmetric matrix given by
Lp(G) = (1Tv)wwT− wvTU − UTvwT+ UTV U. (22)
This can be shown as follows:
L(G) =
1Tv
−v
−vT
V
?
(20)
Tc=
1/√N
1/√N
wT
U
?
TT
cL(G)Tc=
0
0
0T
Lp(G)
?
(21)
TT
cL(G)Tc =
?
1
w UT
1T
?



1Tv−vT1
?
?
−vTU +
?
??
??
?
0
1TvwT−vTU
−v+V 1
?
0
−vwT+V U



=



01TvwT− 1TvwT
?
???
0T
vT
????
1TV U
???
0T
0
Lp(G)

.

Note that Tc (hence T) is a universal transformation that
does not depend on the graph at hand, and it generates
Lp(G) (that is a function of the graph), which is symmetric.
Furthermore, since Tcis used as a similarity transformation,
for any connected graph G without self-loops or multiple
edges, σ(L(G)) = σ(TT
σ(Lp(G))=σ(L(G))\{0}⊂[2(1−cos(π/N)),2d(G)].
cL(G)Tc). This implies that
(23)
Define transformed error as ˜ ek? TTek. Then the equation
(14) can be written as:
˜ ek+1
=
=
TTAc[(IN− δLc,k)r⊗ In]T˜ ek
TTAcT
? ?? ?
:=˜Ac
TT[(IN− δLc,k)r⊗ In]T
? ???
:= Λr
k⊗ In
˜ ek (24)
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Next we derive an expression for Λk. Noting that T = Tc⊗
In, we have
TT[(IN− δLc,k)r⊗ In]T =[TT
Here we have,
c(IN− δLc,k)rTc] ⊗ In.
TT
TT
IN+ c1˜Lc,k+ c2˜L2
where c1 and c2 etc., are some constants, and the newly
defined˜Lc,kis
c(IN− δLc,k)rTc
cINTc+ c1T−1
=
c Lc,kTc+ c2T−1
c,k+ ... = (IN− δ˜Lc,k)r,
c L2
c,kTc+ ...
=
˜Lc,k? TT
cLc,kTc=
?
0
0
0T
Lp,k
?
which is obtained by noting that the first column of T is in
the null space of Lk. Consequently
??
?
This transformation allows us to project the overall estima-
tion error vector ˜ ek into its components in the agreement
subspace, ǫk, and the disagreement subspace, ηk:
(IN− δ˜Lc,k)r
=
1
0
0T
IN−1
?
− δ
?
0
0
0T
Lp,k
??r
=
1
0
0T
IN−1− δLp,k
?r
.
˜ ek=
?
ǫk
ηk
?
⇒
?ǫk+1
ηk+1
?
=
?Aa,k FkΛr
?
p,k
Gk Ad,kΛr
??
p,k
?
?
Ae
?ǫk
ηk
?
,
(25)
where Λp,k=(IN−1−δLp,k)⊗Inand
Aa,k= A +1
NL
?N
i=1
?
ET
i,kEi,k⊗ Ip
?
C.
Gk
=


?

L1C1−L2C2
√2N
L1C1+L2C2−2L3C3
√6N
...
?N−1
N2(N−1)
i=1LiCi−(N−1)LNCN
√



,
Fk
=
L1C1−L2C2
√2N
...
?N−1
i=1LiCi−(N−1)LNCN
√
N2(N−1)
?
.
Λp,k is a symmetric matrix with 2(1 − cos(π/N))I ≤
Λp,k≤ 2d(Gc,k)I. Hence a choice of δ ∈(0,1/(N −1))
renders the eigenvalues of the Laplacian of any connected
communication graph inside the unit circle, i.e., σ(Λp,k)<1.
With the transformed state we can express the Lyapunov
function in equation (18)˜V (˜ ek) as follows
˜V (˜ e) = V (Tα˜ ek)
k(I−1
= ˜ eT
= ˜ eT
2
α TT
cI−1
cI−1
2
α
α IαI−1
⊗ In)(Iα⊗ P)(I−1
2
α TcI−1
α
2
α TcI−1
2
α
⊗ In)˜ ek
k(I−1
k(I−1
2
α TT
22
⊗ P)˜ ek= ˜ eT
α ⊗ P)˜ ek.
where I−1
(λ,1), we have
α
is a positive definite matrix. Consider some γ ∈
γ˜V (˜ ek) −˜V (˜ ek+1) =
˜ eT
?
?γP
?Aa,kFkΛr
= S1− S2
where S1and S2are symmetric matrices defined by
?
?
α−1GT
Λr
p,k(α−1AT
=
k(λ(I−1
α ⊗ P) − AT
e(I−1
??
?
0
α ⊗ P)Ae)
?
:= S
˜ ek.
(26)
Then we can express the matrix S as
S =
0
0 γ(α−1IN−1⊗ P)
?T?P
−
p,k
Gk Ad,kΛr
p,k
0 α−1IN−1⊗ P
??Aa,k FkΛr
p,k
Gk Ad,kΛr
p,k
?
S1=
γP − AT
a,kPAa,k− α−1GT
0
kˆPGk
0
α−1γˆP
?
S2=
0
(α−1GT
kˆPAd,kΛr
p,k+ Aa,kPFΛr
p,k)T
?
kˆPAd,kΛr
a,kˆPAa,k+FTPF)Λr
p,k+ Aa,kPFΛr
p,k
p,k
.
withˆP =IN−1⊗P. By using the inequality (16), for any k
γP − AT
?
⇒ γP − AT
Hence, α > 0 can be chosen large enough such that S1=
ST
1>0.
Next observe that Λp,k is a symmetric matrix with all
its eigenvalues in (−1,1). Hence limr→∞Λr
all nonzero blocks of S2 contain Λp,k as a multiplier,
limr→∞S2= 0. This implies that the spectral radius of the
matrix S2can be made arbitrarily small by choosing r large
enough for a given α. Consequently, since S1 = ST
by a choice of large enough α, we can guarantee that
S = S1−S2> 0, and then choosing r large enough, which
then implies that
a,kPAa,k= λP − AT
a,kPAa,k
???
≥ 0
+(γ−λ)P
a,kPAa,k≥ (γ−λ)P > 0.
p,k= 0. Since
1> 0
γ˜V (˜ ek) −˜V (˜ ek+1) = ˜ eT
for some˜λ ∈ (0,1). Since˜V is positive definite quadratic
function of ˜ ek, this implies the exponential stability of the
error dynamics. The equation (17) is a direct consequence
of the exponential stability of the error dynamics, which
concludes the proof.
kS˜ ek≥˜λ˜V (˜ ek)>0 for ˜ ek?=0,
Next we will focus on how to design observer gain matrix
L and a valid lower bound on the number of consensus
iterations r so that the results of Theorem 1 apply.
A. Computation of the Observer Gain Matrix L
This section presents the result for the gain matrix L that
quadratically stabilize the agreement subspace of the error
dynamics. Note that the equation (25) implies that the error
in the agreement subspace evolves, when ηk= 0, as follows,
ǫk+1= Aa,kǫk,
where
Aa,k= A(Ls,k).
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