Distributed robust estimation over randomly switching networks
using H∞ consensus ⋆
V. Ugrinovskii a

arXiv:1512.01294v1 [] 4 Dec 2015

a

School of Engineering and IT, University of NSW at the Australian Defence Force Academy, Canberra, ACT, 2600, Australia

Abstract
The paper considers a distributed robust estimation problem over a network with Markovian randomly varying topology. The objective
is to deal with network variations locally, by switching observer gains at affected nodes only. We propose sufficient conditions which
guarantee a suboptimal H∞ level of relative disagreement of estimates in such observer networks. When the status of the network is
known globally, these sufficient conditions enable the network gains to be computed by solving certain LMIs. When the nodes are to
rely on a locally available information about the network topology, additional rank constraints are used to condition the gains, given this
information. The results are complemented by necessary conditions which relate properties of the interconnection graph Laplacian to the
mean-square detectability of the plant through measurement and interconnection channels.
Key words: Large-scale systems, distributed robust estimation, worst-case transient consensus, vector Lyapunov functions.

1 Introduction
One of the motivations for using distributed multisensor networks is to make the network resilient to loss of communication. This has led to an extensive research into distributed
filtering over networks with time-varying, randomly switching topology. In particular, the Markovian approach to the
analysis and synthesis of estimator networks has received
a significant attention in relation to the problems involving
random data loss in channels with memory which are governed by a Markov switching rule [17,8].
In addition to capturing memory properties of physical communication channels, Markovian models allow for other random events in the network, such as sensor failures and recovery, to be considered in a systematic manner within the
Markov jump systems framework. However, the Markov
jump systems theory usually assumes the complete state of
the underlying Markov chain to be known to every controller or filter [7] . In the context of distributed estimation
and control, this requires each node of the network to know
the complete instantaneous state of the network to be able
to deploy suitable gains. To circumvent such an unrealistic
assumption, the literature focuses on networks whose communication state is governed by a random process decomposable into independent two-state Markov processes de⋆ This work was supported by the Australian Research Council.
Email address: v.ugrinovskii@gmail.com
(V. Ugrinovskii).

Preprint submitted to Automatica

scribing the status of individual links [6,17], even though
this typically leads to design conditions whose complexity grows exponentially [6]. Also, the assumption of independence between communication links may not always be
practical, e.g., when dealing with congestions.
The objective of this paper is to develop a distributed filtering technique which overcomes the need for broadcast of
global communication topology and does not require Markovian segmentation of the network. Our main contribution is
the methodology of robust distributed observer design which
enables the node observers to be implemented in a truly distributed fashion, by utilizing only locally available information about the system’s connectivity, and without assuming
the independence of communication links. This information
structure constraint is a key distinction of this work, compared with the existing results, e.g., [17,6]. In addition, the
proposed methodology allows to incorporate other random
events such as sensor failures and recoveries.
The paper focuses on the case where the plant to be observed, as well as sensing and communication models are
not known perfectly. To deal with uncertain perturbations in
the plant, sensors and communications, we employ the distributed H∞ filtering framework which has received a significant deal of attention in the recent literature [15,18,19].
The motivation for considering H∞ observers in this paper,
instead of Kalman filters [17], is to obtain observers that have
guaranteed robustness properties. It is well known that the
standard Kalman filter is sensitive to modelling errors [12],

8 January 2018

matrix pairs formed by parameters of the observers and interconnections.

and consensus Kalman filters may potentially suffer from
the same shortcomings. This explains our interest in robust
performance guarantees in the presence of uncertainty.

The paper is organized as follows. The problem formulation is given in Section 2. Section 3 studies an auxiliary distributed estimation problem without the information structure constraints. The results of this section are then used
in Section 4 where the main results of the paper are given.
Section 5 discusses requirements on the observer communication topology. Section 6 presents an illustrating example.

In contrast to [17,15], in this paper the node estimators are
sought to reach relative H∞ consensus about the estimate
of the reference plant. As an extension of the consensus
estimation methodology [9], our approach responds to the
challenge posed by the presence of uncertain perturbations
in the plant, measurements and interconnections. Typically,
a perfect consensus between sensors-agents is not possible
due to perturbations. To address this challenge, we employ
the approach based on optimization of the transient relative H∞ consensus performance metric, originally proposed
in [18]. We approach the robust consensus-based estimation
problem from the dissipativity viewpoint, using vector storage functions and vector supply rates [4]. This allows us to
establish both mean-square robust convergence and robust
convergence with probability 1 of the distributed filters under consideration and guarantee a prespecified level of H∞
mean-square disagreement between node estimates in the
presence of perturbations and random topology changes.

Notation Rn is the real Euclidean n-dimensional vector
space, with the norm kxk , (x′ x)1/2 ; ′ denotes the transpose of a √matrix or a vector. Also, for a given P = P ′ ,
kxkP = x′ P x. 1k , [1 . . . 1]′ ∈ Rk , and Ik is the
identity matrix in Rk ; we will omit the subscript k when
this causes no ambiguity. For X = X ′ , Y = Y ′ , we write
Y > X (Y ≥ X), when Y − X is positive definite (positive
semidefinite). ⊗ denotes the Kronecker product of matrices.
diag[P1 , . . . , PN ] is the block-diagonal matrix, whose diagonal blocks are P1 , . . . , PN . The symbol ⋆ in position (k, l)
of a block-partitioned matrix denotes the transpose of the
(l, k) block of the matrix. L2 [0, ∞) is the Lebesgue space
of Rk -valued vector-functions z(·), defined on [0, ∞), with
1/2
R∞
.
the norm kzk2 , 0 kz(t)k2 dt

The information structure constraint, where the filters must
rely on the local knowledge of the network topology, poses
the main challenge in the derivation of the above-mentioned
results. The standard framework of Markov jump systems is
not directly applicable to the problem of designing locally
constrained filters whose information about the network status is non-Markovian. To overcome this difficulty, we adopt
the approach recently proposed for decentralized control of
jump parameter systems [20]. It involves a two-step design
procedure. First, an auxiliary distributed estimation problem
is solved under simplifying assumption that the complete
Markovian network topology is instantaneously available at
each node. However, we seek a solution to this problem
using a network of non-fragile estimators subject to uncertainty [5]. Resilience of the auxiliary estimator to uncertain
perturbations is the key property to allow this auxiliary uncertain estimator network to be modified, at the second step,
into an estimator network which satisfies the information
structure constraint and retains robust performance of the
auxiliary design.

2 Problem formulation

2.1 Networks with Markovian switching topology
Consider a directed weakly connected graph G = (V, E),
where V = {1, . . . , N } is the set of nodes, and E ⊆ V × V
is the set of edges. The edge (j, i) originating at node j and
ending at node i represents the event “j transmits information to i”. In accordance with a common convention, we
consider graphs without self-loops, i.e., (i, i) 6∈ E. However,
each node is assumed to have complete information about
its filter, measurements and the status of incoming communication links.
We consider two types of random events at each node.
Firstly, node neighborhoods change randomly as a result of
random link dropouts and recovery. Also, to account for
sensor adjustments in response to these changes, as well as
sensor failures/recoveries, we allow for random variations of
the sensing regime at each node. Letting x(t), yi (t) denote
an observed process and its measurement taken at node i at
time t, and using a standard linear relation between these
quantities

An important question in connection with our distributed
observer architecture is concerned with requirements on
the communication topology under which the consensus of
node observers is achievable. For networks of one- or twodimensional agents, and networks consisting of identical
agents, conditions for consensus are tightly related to properties of the graph Laplacian matrix [10,13,22]. In a more
general situation involving nonidentical node observers, the
role of the interconnection graph is often hidden behind the
design conditions, e.g., see [17,15]. Our second contribution
is to show that for the distributed estimation problem under
consideration to have a solution, the standard requirement
for the graph Laplacian to have a simple zero eigenvalue
must be complemented by detectability properties of certain

˜ ξ,
yi = C̃i x + D̃i ξ + D̄
i i

yi ∈ R r ,

(1)

such adjustments are associated with randomly varying co˜ . These random events are additional to
efficients C̃i , D̃i , D̄
i
link dropouts. This leads us to consider the combined evolution of each node’s neighbourhood and sensing regime.

2

˜ ) be its neighDefinition 1 For a node i, let Vi , (C̃i , D̃i , D̄
i
bourhood set and the measurement matrix triplet, respec˜ )}, is
tively, at a certain time t. The pair {Vi , (C̃i , D̃i , D̄
i
said to represent the local communication and sensing state
(or simply the local state) of node i at time t. Two states of
˜ 1 )}, {V2 , (C̃ 2 , D̃2 , D̄
˜ 2 )}
i at times t1 , t2 , {Vi1 , (C̃i1 , D̃i1 , D̄
i
i
i
i
i
˜ 1 ) 6= (C̃ 2 , D̃2 , D̄
˜ 2 ).
are distinct if Vi1 6= Vi2 , or (C̃i1 , D̃i1 , D̄
i
i
i
i

and only if j ∈ Vi i . Here and hereafter, the symbol Viki
describes the neighbourhood of node i when this node is in
Φ (k)
local state ki . In accordance with this notation, Vi i is the
neighbourhood of node i when the network is in global state
PN
PN
k. Also, pki = j=1 akij , qik = j=1 akji , and L k denote
the in- and out-degrees of node i and the Laplacian matrix
of the corresponding graph Gk , respectively.

From now on, we associate with every node i the ordered
collection of all its feasible distinct local states and denote
the corresponding index Ii , {1, . . . , Mi }. The time evolution of each local state will be represented by a random
mapping ηi : [0, ∞) → Ii .

We will use the notation (η, G , Φ) to refer to the switching
network described above. Since η(t) is stationary, then each
process ηi (t) is also stationary. However, in general the local
state processes ηi (t) are not Markov, and the components of
the multivariate process (η1 (t), . . . , ηM (t)) may statistically
depend on each other. Hence our network model allows for
dependencies between links within the network.

Φ (k)

The global configuration and sensing pattern of the network
at any time can be uniquely determined from its local states.
This leads us to define the global state of the network as
an N -tuple (k1 , . . . , kN ), where ki ∈ Ii . Consider the ordered collection of all feasible global states of the network
and let I = {1, . . . , M } denote its index set. In general,
not all combinations of local states correspond to feasible
global states. Owing to dependencies between network links
and/or sensing regimes, the number of feasible global states
may be substantially smaller than the cardinality of the set
I1 × . . . × IN of all combinations of local states. The oneto-one mapping between the set of feasible global states
{(k1 , . . . , kN )} and its index set I will be denoted Φ, i.e.,
(k1 , . . . , kN ) = Φ(m), where m is the index of the N tuple (k1 , . . . , kN ). Also, we write ki = Φi (m), whenever
(k1 , . . . , kN ) = Φ(m).

2.2 Distributed estimation with H∞ consensus
Consider a plant described by the equation
ẋ = Ax + B2 ξ(t),

x(0) = x0 .

(2)

Here x ∈ Rn is the state, ξ(t) ∈ Rl is a deterministic
disturbance. We assume that ξ(·) ∈ L2 [0, ∞), and that the
solution of (2) exists on any finite interval [0, T ], and is L2 integrable on [0, T ].
Also, consider an observer network {η, G , Φ} whose nodes
take measurements of the plant (2) as follows
η (t)

yi = C̃i i

Using the one-to-one mapping Φ, define the global process η(t) = Φ−1 (η1 (t), . . . , ηN (t)) to describe the evolution of the network global state. The local state processes
ηi (t) are related to it as ηi (t) = Φi (η(t)) ∀t ≥ 0. Throughout the paper, we assume that {η(t), t ≥ 0} is a stationary Markov random process [0, ∞) → I defined in a filtered probability space (Ω, F , {Ft }, P), where Ft denotes
a right-continuous filtration with respect to which {η(t), t ≥
0} is adapted 1 [1]. The σ-algebra F is the minimal σalgebra which contains all measurable sets from the filtration
{Ft , t ≥ 0}. The transition probability rate matrix of the
Markov chain {η(t), t ≥ 0} will be denoted Λ = [λkl ]M
k,l=1 ,
P
with λkl ≥ 0, k 6= l and λkk = −
λkl ≤ 0, ∀k ∈ I [1].

η (t)

x + D̃i i

˜ ηi (t) ξ ,
ξ + D̄
i
i

y i ∈ R ri ,

(3)

where ξi (t) ∈ Rli represents the deterministic measurement
uncertainty at sensing node i, ξi (·) ∈ L2 [0, ∞). The coefficients of equation (3) take values in given sets of constant
matrices of compatible dimensions,
η (t)

(C̃i i

η (t)

, D̃i i

˜ k ), k ∈ I }.
˜ ηi (t) ) ∈ {(C̃ k , D̃k , D̄
, D̄
i
i
i
i
i

It will be assumed throughout the paper that Ẽik =
˜ k (D̄
˜ k )′ > 0 for all i and k ∈ I .
D̃ik (D̃ik )′ + D̄
i
i
i
The measurements yi are processed at node i according to
the following estimation algorithm (cf. [17,18,19]):

l6=k

Using the global state process η(t), the time evolution of the
communication graph can be represented by a random graphvalued process Gη(t) , whose value at every time instance is
a directed subgraph of G. It is assumed that for all t, Gη(t)
is weakly connected and has the same vertex set as G. When
η(t) = k ∈ I , Ak = [akij ]i,j=1,N will denote the adjacency
matrix of the digraph Gk = Gη(t) . Note that akij = 1 if

η (t)
η (t)
x̂˙ i = Ax̂i + L̃i i (yi (t) − C̃i i x̂i )
X
η (t)
+
K̃iji (vij − Hij x̂i ), x̂i (0) = 0,

(4)

η (t)

j∈Vi i

where vij is the signal received at node i from node j,
vij = Hij x̂j + Gij wij ,

1

In the sequel, we will consider the filtration generated by a
composite Markov process consisting of η and error dynamics of
the estimator introduced in the next section.

vij ∈ Rrij ,

(5)

wij ∈ Rsij describes the channel uncertainty affecting the
information transmission from node j to i. It is assumed

3

that wij belongs to the class of mean-square L2 -integrable
random disturbances, adapted to the filtration {Ft , t ≥ 0}.

converge asymptotically with probability 1:
Ex0 ,m0 kx̂i (t) − x(t)k2 ≤ ce−ǫt , (∃c, ǫ > 0),
Px0 ,m0 ( lim kx̂i (t) − x(t)k2 = 0) = 1.

It will be further assumed that Fij = Gij G′ij > 0 for all i
η (·)
η (·)
and j ∈ Viki , ki ∈ Ii . Also in (4), L̃i i , K̃iji are matrix-

t→∞

(ii) Given a constant γ > 0, the following mean-square
H∞ consensus performance is guaranteed
R∞
Ex0 ,m0 0 Ψη(t) (x̂(t))dt
sup
≤ γ 2 .(7)
x0 ,(ξ,ξi ,wij )6=0 µP (x0 , ξ, [ξi , wij ]i,j=1,...,N )

valued functions of the local state process ηi (t). These functions are the design parameters of the algorithm describing
innovation and interconnection gains of the observer (4).
(·)
(·)
Note that the coupling and observer gains K̃ij , L̃i are required to be functions of the local state (i.e., functions of
ηi ), rather than the global state. This ‘locality’ information
structure constraint is additional to the assumption about
the Markov nature of the communication graph; cf. [17]
where the complete communication graph was assumed to
be known at each node. The problem in this paper is to determine these functions to satisfy certain robust performance
criteria to be presented in Definition 2 below.

(iii) All estimators converge in the mean-square and with
probability 1:
Z ∞
Ex0 ,m0
kx(t) − x̂i (t)k2 dt < ∞,
(8)
0

Px0 ,m0 ( lim kx(t) − x̂i (t)k2 = 0) = 1.

(9)

t→∞

Remark 1 In equation (5), the matrices Hij ∈ Rri ×n and
Gij ∈ Rri ×sij do not depend on ηi (t). This is to reflect a
situation where node j always broadcasts its information to
node i, but node i randomly fails to receive this information,
or chooses not to accept it, e.g. due to random congestion.
It is possible to consider a more general situation where the
matrices Hij and Gij also depend on ηi (t). Technically, this
more general case is no different from the one pursued here.

Properties (8) and (9) refer to different types of asymptotic behaviour of the estimation errors. Condition (8)
states that x̂i (t) converges to x(t) in the mean-square L2
sense. From the Chebyshev
inequality, this also
R∞
 implies that
limR→∞ Px0 ,m0 0 kx(t) − x̂i (t)k2 dt > R = 0, that
is, almost all estimator trajectories converge in L2 sense.
Property (9) states that kx(t) − x̂i (t)k2 converges to zero
asymptotically for almost all realizations of the global state
process η(t). This is a stronger property; in general, it does
not follow from the a.s. L2 convergence. For that reason,
both convergence properties are considered in Definition 2.

Associated with the system (2) and the set of filters (4) is
the disagreement function (cf. [10])
Ψk (x̂) =

[x̂′1

N
1 X
N i=1

X

j∈VΦi (k)

kx̂j − x̂i k2 ,

k ∈ I,

3 An auxiliary global distributed estimation problem

(6)

3.1 Non-fragile distributed estimation

x̂′N ]′ .

x̂ ,
...
It represents the average (over the set of
all nodes) of the total disagreement between the estimate at
each node, and the estimates computed at the neighbouring
nodes, when the network is in state k. Following [18], we
adopt Ψk (x̂) to define the transient consensus performance
metric in the distributed estimation problem defined below.

In this section, we temporarily lift the locality information
structure constraint and assume the global communication
and sensing state process η(t) to be available at every node.
Φ (k)

Φ (k)

For every k ∈ I define Cik = C̃i i Dik = D̃i i , D̄ik =
˜ Φi (k) . Note that E k , Dk (Dk )′ + D̄k (D̄k )′ = Ẽ Φi (k) >
D̄
i
i
i
i
i
i
i
0. Then, the measurements taken at node i can be rewritten
in terms of the global state process η(t):

Let Px0 ,m0 , Ex0 ,m0 denote the conditional probability and
conditional expectation, given x(0)− x̂i (0) = x0 ∀i, η(0) =
m0 . Also, given a matrix P = P ′ > 0, let

η(t)

yi = Ci

µP (x0 , ξ, [ξi , wij ]i,j=1,...,N ) , kx0 k2P + kξk22

N 
M
X
1 X
η(·)
+
kξi k22 +
Ex0 ,m0 kaij wij k22 .
N i=1
j=1

η(t)

x + Di

η(t)

ξ + D̄i

ξi .

(10)

The auxiliary problem in this section is concerned with estimation of the state of the uncertain plant (2), (10) using a
network of estimators subject to uncertainty, as follows

Definition 2 The distributed estimation problem under consideration is to determine switching observer gains L̃ki and
k
interconnection coupling gains K̃ij
, k ∈ Ii , for the filters
(4) which ensure that the following conditions are satisfied:

η(t)
x̂˙ i = Ax̂i + Li (yi (t) − Ci (η(t))x̂i )
X
η(t)
+
Kij (vij − Hij x̂i )
j∈VΦi (η(t))

+

(i) In the absence of the uncertainty, all node estimators
converge exponentially in the mean-square sense and

X

j∈VΦi (η(t))

4

(1)

(2)

(ωij + ωij ) + ωi ,

x̂i (0) = 0.

(11)

(·)

(·)

(iii) All estimators converge in the mean-square and with
probability 1; i.e., conditions (8), (9) hold.

Here, Li , Kij are matrix-valued functions of the state
(1)
ωij ,

(2)
ωij ,

of the global Markov chain η to be found, and
and ωi are estimator perturbations. It is assumed that these
perturbations are random processes adapted to the filtration {Ft , t ≥ 0} and such that the multivariate process
(x̂1 , . . . , x̂N , η) is Markov with respect to that filtration.
Also in this section, it will be assumed that these uncertainties satisfy the following norm-bound conditions:
η(t)

kωi (t)k2 ≤ α2i Ci

η(t)

ei (t) + Di

η(t)

ξ(t) + D̄i

(1)

ξi (t)

A solution to this auxiliary problem is given in Theorem 1
below. The conditions of the theorem involve the following
k
linear matrix inequalities in the variables τik > 0, θij
> 0,
k
k
k ′
ϑij > 0, Xi = (Xi ) > 0, i = 1, . . . , N , k ∈ I , j ∈
Φ (k)

Vi i

2

k 2
γ 2 I − τik α2i Eik > 0, γ 2 I − θij
βij Fij > 0,


Qki
⋆
⋆
⋆
⋆



Nik
−γ 2 I ⋆
⋆
⋆ 





Sik
0 −γ 2 I ⋆
⋆  < 0,


1
k
0
0 −Ti ⋆ 

 1+2Mi ⊗ Xi
0
0
0 −Zi
Ξ′i

,

2

2
kHij ei (t) + Gij wij k ,
kωij (t)k2 ≤ βij
(2)

2
kHij ej (t)k
kωij (t)k2 ≤ βij

2

a.s. ∀t ≥ 0,

(12)

where αi , βij are given constants, and ei = x − x̂i is the
estimation error of the auxiliary estimator at node i, which
evolves according to the equations
η(t)

η(t)

)ei
η(t)
Kij (Hij (ej

ėi = (A − Li
X
+

Ci

Φi (η(t))

η(t)

η(t)

Di

j∈VΦi (η(t))

η(t)

)ξ − Li

(1)

(14)

(15)

where
− ei ) − Gij wij )


Nik , I − (Dik )′ (Eik )−1 Dik B2′ Xik ,

j∈Vi

+ (B2 − Li

X
−

:

(2)

Sik , −(D̄ik )′ (Eik )−1 Dik B2′ Xik ,


k
k
k
k
,
.
.
.
,
ϑ
,
.
.
.
,
θ
,
ϑ
Tki , diag τik , θi,j
i,j k ,
i,j k
i,j1
1

η(t)

D̄i

ξ
i

(wij + wij ) + wi  ,

p

ei (0) = x0 . (13)

Qki

It will be shown in Section 4 that when the locality information structure constraints are imposed, this will result in an
uncertainty due to the mismatch between filter error dynamics in the network subject to these constraints, and the errors
which would arise in the same network if its communication state was known globally. It will be shown in the proof
of Theorem 2 that this uncertainty satisfies conditions (12).
The resilience of the constraint-free auxiliary network (11)
to this uncertainty will be used in the next section to show
that the network (4), constructed from the auxiliary solution,
maintains the same convergence and robust H∞ consensus
performance properties when the information structure constraints are enforced.

p

i

i

Xik (A

,
+ δi I − B2 (Dik )′ (Eik )−1 Cik )
+ (A + δi I − B2 (Dik )′ (Eik )−1 Cik )′ Xik +
M
X
X
2
′
λkl Xil
ϑkji βji
Hji
Hji +
+
Φ (k)
l=1
j:i∈Vj j

− γ 2 (Cik )′ (Eik )−1 Cik − γ 2

X

(pki + qik )I

′
Hij
Fij−1 Hij ,

Φi (k)

j∈Vi


−1
−1
2 ′
′
F
−
I
.
.
.
γ
H
F
H
−
I
H
,
Ξi = γ 2 Hij
ij
ij
1
ijpk ijpk
ij1
1
pk
i
i
i


2δjpk
2δ
j
Zi = diag  k 1 Xjk1 , . . . , k i Xjk k  .
qj1 + 1
qj k + 1 pi


p

i

Theorem 1 Suppose the network (η, G , Φ) and the constants γ > 0, αi , βij and δi > 0 are such that the couk
pled LMIs (14) and (15) in the variables τik > 0, θij
> 0,

Definition 3 The auxiliary distributed consensus estimation
k
, k ∈ I , for
problem is to determine sets of gains Lki , Kij
the filters (11) to ensure the following:

Φ (k)

ϑkij > 0, Xik = (Xik )′ > 0, j ∈ Vi i , i = 1, . . . , N ,
k ∈ I , are feasible. Then the network of observers (11) with

(i) When (ξ, ξi , wij ) ≡ 0, the interconnected system consisting of subsystems (13) must be exponentially stable in the mean-square sense and asymptotically stable
(1)
with probability 1 for all estimator perturbations ωij ,

k
′
= γ 2 (Xik )−1 Hij
Fij−1 ,
Kij


Lki = γ 2 (Xik )−1 (Cik )′ + B2 (Dik )′ (Eik )−1

(2)
ωij ,

and ωi for which the correspondingly modified
constraints (12) hold.
(ii) In the presence of exogenous disturbances ξ, ξi , wij ,
the mean-square consensus performance condition in
(7) is satisfied for all admissible estimator perturba(2)
(1)
tions ωij , ωij , and ωi subject to (12).

(16)

solves the auxiliary estimation problem in Definition 3. The
matrix P in condition (7) corresponding to this solution is
PN
P = γ 21N i=1 Xim0 , where m0 = η(0).
The proof of Theorem 1 is given in the Appendix.

5

When the global Markov state of the network is available
at every node, the solution to the distributed H∞ consensus
estimation problem can be obtained from Theorem 1 by
(2)
(1)
letting ωi = ωij = ωij = 0 and taking αi = βij = 0.

"

rank

k
Kij
,

l:Φi (l)=ki

λ̄l

.

=

P

P

l
λ̄l Kij

l:Φi (l)=ki

λ̄l

,

L̃ki i

P

l:Φi (l)=ki

= P

(18)

λ̄l Lli

l:Φi (l)=ki

λ̄l

′
(∆K,k
ij )

I

#

> 0,

(20)

l:l6=k,
Φi (l)=Φi (k)

P

P

λ̄l

 ′ −1

Fij
γ 2 λ̄l Yik − Yil Hij

l:l6=k,
Φi (l)=Φi (k)

P

l:Φi (l)=Φi (k)

λ̄l

.

"

Yik I
I Xik

#

≤ n,

(21)

(22)

Next, using (19) and (22), construct the estimator network
(4). The resulting distributed estimatior network solves the
distributed robust estimation problem in Definition 2.
Proof The result follows from Theorem 1 in a manner
similar to the proof of Theorem 4 in [20].

Lki ,

l:Φi (l)=ki

∆K,k
ij

′
k
Fij−1 ,
Kij
= γ 2 Yik Hij


Lki = γ 2 Yik (Cik )′ + B2 (Dik )′ (Eik )−1 .

Now let
k ∈ I , be the coefficients of the auxiliary distributed estimator obtained in Theorem 1. Using
Lemma 1, for each i = 1, . . . , N and ki ∈ Ii we define
ki
K̃ij

2
βij
I

Theorem 2 Given a Markovian switching network (η, G , Φ)
and a collection of constants γ, αi , βij and δi > 0,
i = 1, . . . , N , associated with each node, suppose there
exist matrices Xik = (Xik )′ > 0, Yik = (Yik )′ > 0, and posΦ (k)
k
itive scalars τik , θij
, ϑkij , i = 1, . . . , N , k ∈ I , j ∈ Vi i
which satisfy the matrix inequalities (14), (15), (20), and
the rank constraint (21). Using the solution matrices Yik ,
define the auxiliary gains

Lemma 1 Suppose the Markov process η(t) is irreducible
and has a unique invariant distribution λ̄. Given a matrixvalued function K (·) : I → {K 1 . . . , K M } ⊂ Rn×s , for
every node i and for all ki ∈ Ii we have:
P

> 0,

"

Also, consider the rank constraints

In this section, the solution to the auxiliary distributed estimation problem developed in Section 3 will be used to obtain a distributed estimator whose nodes utilize only locally
available information. This will be achieved by taking the
asymptotic conditional expectation of the auxiliary gains,
given a local state. Our method is based on the following
technical result of [20].

t→∞

I

∆K,k
ij ,

(2)

λ̄l K l

′
(∆L,k
i )

#

l:Φi (l)=Φi (k)

4 The main result

l:Φi (l)=ki

∆L,k
i

∆L,k
,
i

Then the network of observers (11) with ωi = ωij = ωij =
k
0 and Kij
, Lki defined in (16) solves the estimation problem
in Definition 3. The matrix P in condition (7) corresponding
PN
to this solution is P = N1γ 2 i=1 Xim0 , where m0 = η(0).

P

α2i I

where αi , βij are the same constants as those employed in
the LMIs (14), (15), and

P 2  k k ′ k −1
γ λ̄l Yi (Ci ) (Ei ) − Yil (Cil )′ (Eil )−1

Φ (k)
j∈Vi i



lim E K η(t) | ηi (t) = ki =

η (t)

To formulate the main result of this paper, consider the colk
, ϑkij , Xik and
lection of the LMIs in the variables τik , θij
Yik , consisting of the LMIs (14), (15), and the following
additional LMIs,

Q̄ki , Xik (A + δi I − B2 (Dik )′ (Eik )−1 Cik )
+(A + δi I − B2 (Dik )′ (Eik )−1 Cik )′ Xik + (pki + qik )I
M
X
X
′
Hij
Fij−1 Hij .
+
λkl Xil − γ 2 (Cik )′ (Eik )−1 Cik − γ 2
(1)

η (t)

η(t)

η(t)

Kij , Li , the evolution of K̃iji , L̃i i is governed by
the local communication and sensing state process ηi .

Corollary 1 Suppose the network (η, G , Φ) and the constants γ > 0 and δi > 0 are such that the following couΦ (k)
pled LMIs in the variables Xik = (Xik )′ > 0, j ∈ Vi i ,
i = 1, . . . , N , k ∈ I , are feasible:


Q̄ki
⋆
⋆
⋆


 N k −γ 2 I ⋆
⋆ 

 i
(17)
 < 0,
 k
2

S
0
−γ
I
⋆

 i
Ξ′i
0
0 −Zi

l=1

η (t)

η (t)

are then
From Lemma 1, the processes K̃iji , L̃i i
the asymptotic minimum variance approximations of the
η(t)
η(t)
corresponding processes Kij , Li . However, unlike

3.2 Special case: Broadcast of the global state

k
First we observe that the observer gains Kij
, Lki constructed
in this theorem, also satisfy the conditions of Theorem 1,
since (Xik )−1 = Yik in view of (21). This allows us to claim
that the network of auxiliary estimators (11), (22) solves the
auxiliary robust estimation problem in Definition 3.

. (19)

6

,

ki
does not depend on j. Hence we will also assume
case K̃ij
ki
K̃ij = K̃iki . Define Ā = IN ⊗ A, Ak , A + 21 λkk I,
Āk , Ā + 21 λkk InN , H̄ k = L k ⊗ H.

Next, consider the observer gains defined using (19) and
Φ (k)
(22). Note that for all i = 1, . . . , N , k ∈ I , and j ∈ Vi i ,
k
Kij

Lki

−

Φ (k)
K̃ij i

−

Φ (k)
L̃i i

=
=

P

l:l6=k,Φi (l)=Φi (k)

P

P

l:Φi (l)=Φi (k)

l:l6=k,Φi (l)=Φi (k)

P


 k
l
− Kij
λ̄l Kij
λ̄l



λ̄l Lki − Lli

l:Φi (l)=Φi (k)

λ̄l

,

Let Cik , Ō k , OH denote the undetectable subspace of
(Cik , Ak ) and the unobservable subspaces of (H̄ k , Āk ) and
(H, A), respectively. The following theorem shows that for
the problem in Definition 2 to have a solution, every combination of undetectable states of the pairs (Cik , Ak ) must
necessarily form an observable vector of (H̄ k , Āk ). The
proofs of this and subsequent results are removed for the
sake of brevity.

,

Then it follows from (20) that
Φ (k)

kL̃i i

− Lki k2 ≤ α2i ,

Φ (k)

kK̃ij i

k 2
2
− Kij
k ≤ βij
.

(23)

Theorem 3 Suppose there exist output injection gains L̃ki i ,
ki
= K̃iki , j ∈ Viki , ki ∈ Ii , i = 1, . . . , N , such that the
K̃ij
first condition in Definition 2(i) holds. Then,

Therefore the particular perturbations in the estimators (11),
η (t)

ωi = (L̃i i
(1)
ωij
(2)
ωij

η(t)

− Li

η(t)

)(Ci

η(t)

ei (t) + Di

η(t)
η (t)
= (K̃iji − Kij )(Hij ei +
η(t)
η (t)
= −(K̃iji − Kij )Hij ej ,

η(t)

ξ + D̄i

ξi ),

Gij wij ),

Ō k ∩

(24)

satisfy (12). This means that the estimator (4) in which the
η (t)
η (t)
above particular set of gains K̃iji , L̃i i is employed, represents one instance of the auxiliary estimator (11), corresponding to the particular perturbation (24), which is an admissible perturbation, due to (12). Therefore, since the matrices Kik , Lki , m ∈ I solve the auxiliary H∞ consensus
estimation problem in Definition 3, then the distributed estimator (4) with the local gains selected above, solves the
robust consensus estimation problem in Definition 2.
✷

N
Y

i=1

Cik = {0} ∀k ∈ I .

(25)

We now present a necessary and sufficient condition for (25)
to hold. The sufficient condition is explicitly expressed in
terms of the network Laplacian matrices L k .
Theorem 4 (a) If (25) holds, then for every k ∈ I :
TN
(i) i=1 Cik = {0}, and
(ii) OH ∩ Cik = {0} for all i = 1, . . . , N ;
(b) Conversely, suppose the geometric multiplicity of the
zero eigenvalue of Lk is equal to 1. If the above properties (i) and (ii) hold for every k, then (25) is satisfied.

Remark 2 Due to the rank constraints (21), the solution
set to the matrix inequalities in Theorem 2 is non-convex.
In general, it is difficult to solve such problems. Fortunately,
several numerical algorithms have been proposed for this
purpose [3,11].

One can further specialize the sufficient conditions in Theorem 4, e.g., for the cases of a balanced graph or a graph
containing a spanning tree. Also, note that the information
structure constraint is not used in the proofs. Therefore, the
results in this section apply to more general distributed estimation problems, such as the auxiliary problem considered
in Section 3.

5 Requirements on the communication graph and interconnections
In this section, we briefly discuss necessary requirements
on the network topology. Recall that condition (i) of Definition 2 requires that in the absence of perturbations, estimation error dynamics must be asymptotically stabilizable via
output injection in the mean-square. This problem belongs
to the class of stochastic mean-square detectability problems for linear jump parameter systems [2]. Unfortunately,
even without the locality information structure constraint,
there is no easy direct algebraic test to verify this property.
Some conclusions can however be drawn to provide an insight into the connection between the graph Laplacian and
the existence of stabilizing output injection gains.

6 Example
h −3.2 10 0 i
1
−1 1 ,
Consider a plant of the form (2), with A =
0 −14.87 0
h −0.1246 i
B2 = −0.4461 . The nominal part of the plant describes
0.3350
one of the regimes of the so-called Chua electronic circuit.
The Chua circuit is an example of a system which switches
between three regimes of operation in a chaotic fashion. For
the sake of simplicity, here we consider only one regime.
The plant is observed by a 5-node switching observer network which operates intermittently in two regimes. Its graph
topologies are shown in Figure 1. The evolution of the network is modelled as a two-state Markov
chain with the tran
0.1
sition probability rate matrix Λ = −0.1
0.1 −0.1 . Note that the

To highlight such a connection, in this section we will make
the simplifying assumption Hij = H, Gij = Gi , rij =
Φ (k)
r̄i for all j ∈ Vi i . From (22), it follows that in this
7

g replacements 1

k=1

2

k=2

1

2

2

1.8

η(t)

1.6

3

3

1.4

1.2

5

4

5

4

1

0

1

2

3

Fig. 1. Switching graph topology for the example.

4
Time t

5

6

7

8

(a)

Table 1
Coefficients Cik for the example, C∗1 = 10−3 × [3.1923 −
4.6597 1], C∗2 = [−0.8986 0.1312 − 1.9703].

0.8
Filter error at Node 1
Filter error at Node 2
Filter error at Node 5

0.7
0.6
0.5

i=2

i=3

i=4

i=5

k=1

C∗1

C∗1

C∗2

C∗1

C∗2

k=2

C∗1

C∗2

C∗2

C∗1

C∗2

0.4

Filter error

i=1

0.3
0.2
0.1
0
−0.1

graph corresponding to state k = 2 was used in [18,21] to
demonstrate synchronization of Chua systems. Indeed, the
filters share the same matrix A as the plant, and can be interpreted as ‘slave’ Chua systems operating in the same regime
as the master. Accordingly, the convergence of the filters in
our example can be interpreted as the observer-based synchronization between the slaves and the master; see [18] for
further details. However different from [18], in this example
the graph topology is time-varying, as explained below.

−0.2

0

1

2

3

4
Time t

5

6

7

8

(b)
Fig. 2. One path of η(t) (Fig. (a)) and estimations errors for the
first coordinate at nodes 1, 2 and 5 (Fig. (b)).

The design of the observer network was carried out using Matlab and the LMI solver LMIrank based on [11]. To
obtain a set of non-switching gains for node 1, the normbounded uncertainty constraints of the form (23) were defined for the communication link (3, 1) at node 1, where we
2
set α213 = 102 , β13
= 4 × 102 . These constants as well as
δi = 0.365 were chosen by trial and error, to ensure that the
corresponding rank constrained LMIs in Theorem 2 were
feasible. The feasibility was achieved with γ 2 = 0.7407.
This allowed us to compute the nonswitching gains K̃13 and
L̃1 for node 1 using (22).

From Figure 1, nodes 3, 4, and 5 have varying neighbourhoods. Also, in this example we suppose that node 2 changes
its sensor parameters when the network switches between
two configurations. As a result, in this example, each local state process, except for that of node 1, has two states
and always takes the same value as the global state process.
On the other hand, node 1 always maintains the same local
state, and its local process is constant. Therefore, we seek
to obtain nonswitching observer gains for node 1 only. According to this description, in this example, I = I2 =
I3 = I4 = I5 = {1, 2}, I1 = {1}, and the mapping Φ is
as follows: Φ(1) = (1, 1, 1, 1, 1), Φ(2) = (1, 2, 2, 2, 2). Numerical values of the matrices Cik , k = 1, 2, for this example are given in Table 1; they are assumed to take one of the
two values C∗1 , C∗2 , shown in the table. These values were
chosen so that the pairs (C1k , A+ 21 λkk I), (C4k , A+ 12 λkk I),
k = 1, 2, corresponding to nodes 1 and 4, had undetectable
modes, while node 2 was allowed to switch between detectable and undetectable coefficient pairs. Therefore, for
estimation these nodes were to rely on communication with
their neighbours. Also, we let Dik = 0, D̄ik = 0.025 for all
nodes and all k, and Hij = I3×3 , Gij = 0.5 × I3×3 .

To validate the design, the system and the designed filters
were simulated numerically, with a random initial condition x0 . All uncertain perturbations were chosen to be of
the form sin(aπt + ϕ)e−bt , with different coefficients a, ϕ
and b for each perturbation. Also we let wij (t) = wji (t),
assuming an undirected nature of the channels in this example. The graphs of one realization of the global state process
η(t), and the corresponding estimation errors at nodes 1 (the
nonswitching filter), 2 (the filter with the switching sensing
regime) and 5 (the filter with the varying neighbourhood)
are shown in Figures 2(a) and 2(b), respectively. The graph
in Figure 2(b) confirms the ability of the proposed node estimators to successfully mitigate the changes in the graph
topology and sensing regimes, as well as uncertain perturbations in the plant, measurements and interconnections.

Note that both instances of the network have spanning trees
with roots at nodes 3 and 5. These nodes have detectable
matrix pairs (C3k , A + 12 λkk I), (C5k , A + 12 λkk I), k = 1, 2,
respectively. Also, (H, A) is observable. It follows from
these properties that the conditions in part (b) of Theorem 4
are satisfied. Hence, the necessary condition for global detectability, stated in Theorem 3 holds.

7 Conclusions
The paper has presented sufficient conditions for the synthesis of robust distributed consensus estimators connected over

8

[V1 (e1 , k) . . . VN (eN , k)]′ , with quadratic components
PN
Vi (ei , k) = e′i Xik ei . Also, define V (e, k) = i=1 Vi (ei , k).
i
Since L is a linear operator and ∂V
∂ej = 0 for j 6= i, then
Pm
[LV ](e, k) = i=1 [Li Vi ](e, k), where

a Markovian network. The proposed estimator provides a
guaranteed suboptimal H∞ disagreement of estimates, while
using only locally available information about the communication and sensing state of the network. Our conditions
allow a robust filter network to be constructed by solving an
LMI feasibility problem. The LMIs are partitioned in a way
which opens a possibility for solving them in a decentralized manner. When the network’s global state is available at
every node, this feasibility problem is convex, and the corresponding LMIs are solvable, e.g., using the decentralized
gradient descent algorithm in [18]. However, the elimination of the network state broadcast has led to the introduction of rank constraints additional to the LMI conditions.
Therefore, new numerical algorithms need to be developed
to exploit the proposed partition of the LMIs and rank constraints. This problem is left for future research. Other possible directions for future research may be concerned with
an integration of our approach with other distributed H∞
filtering techniques, such as for example, techniques involving randomly sampled measurements [16].


T 
XM
∂Vi
λkl Vi (ei , l) +
[Li Vi ](e, k) ,
(A − Lki Cik )ei
l=1
∂ei
X
k
+
k Kij (Hij (ej − ei ) − Gij wij )
j∈Vi i

X
(2)
(1)
+(B̂2 − Lki D̂ik )ξˆi − ωi −
ki (ωij + ωij ) .
j∈Vi

k
For arbitrary τik , θij
, ϑkij > 0, consider the expression

XN h
τik (α2i kCik ei + D̂ik ξ̂i k2 − kωi k2 )
[LV ](e, k) +
i=1
X
(1) 2
2
k
2
+
k θ (β kHij ei + Gij wij k − kωij k )
j∈Vi i ij ij
i XN
X
(2) 2
2
k
2
Ri (e, k),
+
=
ki ϑij (βij kHij ej k − kωij k )

Acknowledgement

i=1

j∈Vi

Discussions with C. Langbort are gratefully acknowledged.

where we let

8 Appendix: Proof of Theorem 1.



Ri (e, k) , [Li Vi ](e, k) + τik α2i kCik ei + D̂ik ξˆi k2 − kωi k2


X
(1)
2
k
βij
kHij ei + Gij wij k2 − kωij k2
θij
+

The following continuous-time counterpart of the RobbinsSiegmund convergence theorem [14] will be used in the
proof of Theorem 1. Its proof is similar to [14].

k

j∈Vi i

+e′i

Lemma 2 Consider nonnegative random processes v(t),
φ(t) and ψ(t) adapted to a filtration {F¯t , t ≥ 0}, with the
following properties:



X

k

j: i∈Vj j


X
(2)
2
′
ϑkji βji
Hji
Hji ei −
ϑkij kωij k2 . (27)
k

j∈Vi i

By completing the squares, one can establish that
(a) v(t) is right-continuous on [0, ∞);
(b) ψ(t) is locally Lebesgue-integrable on [0, ∞) with probability 1, i.e., almost all realizations of ψ(t) have the
Rt
property s ψ(θ)dθ < ∞ for all t ≥ s ≥ 0;
R∞
(c) E 0 φ(s)ds < ∞;
(d) The following inequality holds a.s. for all t ≥ s ≥ 0

Ri (e, k) ≤ e′i Ui ei + 2e′i Xik

k
Kij
Hij ej

k
j∈Vi i

+γ 2 (kξk2 + kξi k2 ) + γ 2


Z t


Z t
.
E v(t) +
ψ(θ)dθ F¯s ≤ v(s) + E
φ(θ)dθ F¯s (26)
s

X

X

k
j∈Vi i

kwij k2 ,

(28)

where

s



Ui = Xik A − B̂2 (D̂ik )′ (Eik )−1 Cik

Then the limitRlimt→∞ v(t) exists and is finite with proba∞
bility 1. Also, 0 ψ(s)ds < ∞ a.s..

M

′
X
λkl Xil
+ A − B̂2 (D̂ik )′ (Eik )−1 Cik Xik +
l=1

1
X
X 1
1  k k
2
′
+ k +
X
X
+
+
ϑkji βji
Hji
Hji
i
i
k
k
τi
θij
θij
k
k

Proof of Theorem 1 We will use the notation ki = Φi (k),
kj = Φj (k), where k ∈ I , ki ∈ Ii , kj ∈ Ij . Also,
D̂ik = [ Dik D̄ik ], B̂2 = [ B2 0 ], ξˆi = [ ξ ′ ξi′ ]′ .

j∈Vi i

j: i∈Vj j

1 k
X B̂2 I − (D̂ik )′ (Eik )−1 D̂ik B̂2′ Xik
γ2 i
X
′
Hij
Fij−1 Hij .
−γ 2 (Cik )′ (Eik )−1 Cik − γ 2

+

Let L denote the infinitesimal generator of the interconnected system consisting of subsystems (13) [1]. Consider the vector Lyapunov candidate for this system,





k

j∈Vi i

9

Here E [·|e(s), η(s)] is the expectation conditioned on the σfield generated by (e(t), η(t)), t ≤ s. We now observe that
the processes v(t) , V (x(t), η(t)),

We now observe that it follows from the LMI (15) that for
any nonzero collection of vectors ei , ej ∈ Rn
X

e′i Ui ei + 2e′i Xik

k

j∈Vi i

−2e′i

k
Kij
Hij ej + (pki + qik )kei k2

X

ej <

N
X

φ(t) ,
k
πij
(e′j Xjk ej ),

i=1

(29)



−2δi , j = i,
2δj
k
k
πij = qk +1 , j ∈ Vi i ,
j


0,
otherwise.

(30)

Together with (28), the latter inequality leads to
Ri (e, k) +
X

+γ 2

k
j∈Vi i

+

qik )kei k2

kwij k2 +

−

X

2e′i

X

k
j∈Vi i

ej < γ kξˆi k
2

k
πij
Vj (ej , k).

(31)

(32)
[1] A. Bain and D. Crisan.
Springer, NY, 2009.

k
j∈Vi i

[3] L. El Ghaoui, F. Oustry, and M. Ait Rami. A cone complementarity
linearization algorithm for static output-feedback and related
problems. IEEE Tran. Automat. Contr., 42:1171-1176, 1997.
[4] W. M. Haddad, V. Chellaboina, and S. G. Nersesov. Vector
dissipativity theory and stability of feedback interconnections for
large-scale non-linear dynamical systems. Int. J. Contr., 77:907-919,
2004.

+

j=1

[5] W. M. Haddad and J. R. Corrado. Robust resilient dynamic controllers
for systems with parametric uncertainty and controller gain variations.
Int. J. Contr., 73:1405-1423, 2000.
[6] C. Langbort, V. Gupta, and R. Murray. Distributed control over
failing channels. Networked Embedded Sensing and Control, pp.
325–342. Springer, 2006.

i=1

η(t)
aij kwij (t)k2

!

#

dt e(s), η(s) .

Fundamentals of Stochastic Filtering.

[2] E. F. Costa and J. B. R. do Val. On the observability and detectability
of continuous-time Markov jump linear systems. SIAM J. Contr.
Optim., 41:1295-1314, 2002.

h
i
E V (e(t), η(t)) e(s), η(s) − V (e(s), η(s))
Z t

+E
(ǫV (e(t), η(t)) + N Ψη(t) (e(t))dt e(s), η(s)
s
"Z
N
t X
2
≤γ E
kξi (t)k2 + kξ(t)k2
s

(e(t)) + ǫV (e(t), η(t))

References

The statement of Theorem 1 now follows from (32). This
can be shown using the same argument as that used to derive
the statement of Theorem 1 in [18] from a similar dissipation
inequality. Indeed, let ξ, ξi ∈ L2 [0, ∞), i = 1, . . . , N . Since
equation (13) defines (e(t), η(t)) to be a Markov process,
we obtain from (32) using the Dynkin formula that

N
X

j=1

aij kwij (t)k2  ,

Also, by taking the expectation conditioned on ei (0) = x0 ,
η(0) = m0 on both sides of (33) and then letting t → ∞,
PN
we obtain condition (7), in which P = N1γ 2 i=1 Xim0 .
Condition (8) follows from (33) in a similar manner.
Taking the expectation conditioned on ei (0) = x0 ,
η(0) = m0 on both sides of (33), then dropping the
Rt
nonnegative term 0 N Ψη(t) dt and letting t → ∞, we
R∞
establish that Ex0 ,m0 0 V (e(t), η(t))dt < ∞. Hence
R
∞
✷
Ex0 ,m0 0 ke(t)k2 dt < ∞.

It is easy to verify using (30) that all components of the
vector 1′N Πk are negative and do not exceed −ǫ, where
i
ǫ = mini,k q2δ
k +1 . Hence, it follows from (27), (31) that
i
the following dissipation inequality holds for all ei , ξ, ξi ,
(2)
(1)
wij , and for all uncertainty signals ωi (t), ωij (t), ωij (t)
satisfying the constraints (12)
N Ψk (e) + [LV ](e, k) ≤ −ǫV (e, k)


N
X
X

2
2
kwij k2  .
+γ 2
kξi k + kξk +

η(t)

η(t)



In the case where ξi = 0, wij = 0, ξ = 0, the above observation immediately yields the statement of the theorem about
internal stability of the system (13), (16) with probability
1. The claim of internal exponential mean-square stability
follows directly from (32), since Ψk ≥ 0 by definition.

2

j∈Vik ∪{i}

i=1

kξi (t)k2 + kξ(t)k2 +

N
X

satisfy the conditions
of Lemma 2. This leads to the conR∞
clusion that 0 (N Ψη(t) (e(t))) + εV (e(t), η(t)))dt < ∞
a.s., and also limt→∞ V (e(t), η(t)) < ∞ a.s.. Due
to the condition Xi > 0 Rfor all i, we conclude that
∞
limt→∞ kei (t)k2 exists and 0 kei (t)k2 dt < ∞ a.s.. This
implies limt→∞ ei (t) = 0 with probability 1 for all i and
arbitrary disturbances ξ, ξi , wij ∈ L2 [0, ∞); i.e., (9) holds.

k
where πij
are elements of the N × N matrix Πk , defined as

(pki



ψ(t) , N Ψ

j=1

k

j∈Vi i

N
X

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