Distributed Kalman Filter in a Network of Linear Dynamical Systems
Damián Marellia,b , Mohsen Zamanic , Minyue Fuc,a
a Department

arXiv:1711.07625v1 [] 21 Nov 2017

of Control Science and Engineering and State Key Laboratory of Industrial Control Technology, Zhejiang University, 388
Yuhangtang Road Hangzhou, Zhejiang Province, 310058, P. R. China.
b French-Argentinean International Center for Information and Systems Sciences, National Scientific and Technical Research Council, Ocampo
Esmeralda, Rosario 2000, Argentina.
c School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.

Abstract
This paper is concerned with the problem of distributed Kalman filtering in a network of interconnected subsystems
with distributed control protocols. We consider networks, which can be either homogeneous or heterogeneous, of
linear time-invariant subsystems, given in the state-space form. We propose a distributed Kalman filtering scheme for
this setup. The proposed method provides, at each node, an estimation of the state parameter, only based on locally
available measurements and those from the neighbor nodes. The special feature of this method is that it exploits the
particular structure of the considered network to obtain an estimate using only one prediction/update step at each time
step. We show that the estimate produced by the proposed method asymptotically approaches that of the centralized
Kalman filter, i.e., the optimal one with global knowledge of all network parameters, and we are able to bound the
convergence rate. Moreover, if the initial states of all subsystems are mutually uncorrelated, the estimates of these
two schemes are identical at each time step.
Keywords: Estimation, Kalman Filter, Distributed Systems.

1. Introduction
There has been an increasing activity in the study of
distributed estimation in a network environment. This
is due to its broad applications in many areas, including
formation control Subbotin and Smith [16], Lin et al.
[8], distributed sensor network Zhang et al. [23] and
cyber security Teixeira et al. [18], Zamani et al. [22].
This paper examines the problem of distributed estimation in a network of subsystems represented by a finite
dimensional state-space model. Our focus is on the scenario where each subsystem obtains some noisy measurements, and broadcasts them to its nearby subsystems, called neighbors. The neighbors exploit the received information, together with an estimate of their
internal states, to make a decision about their future
states. This sort of communication coupling arises in
different applications. For example, in control system
security problems Teixeira et al. [18], distributed state
estimation is required to calculate certain estimation er-

Email addresses: Damian.Marelli@newcastle.edu.au
(Damián Marelli), Mohsen.Zamani@newcastle.edu.au (Mohsen
Zamani), Minyue.Fu@newcastle.edu.au (Minyue Fu)
Preprint submitted to Systems and Control Letters

ror residues for attack detection. Similarly, for formation control Lin et al. [10], Zheng et al. [24], Lin et al.
[9], each subsystem integrates measurements from its
nearby subsystems, and states of each subsystem need
to be estimated for distributed control design purposes.
The main objective of this paper is to collectively estimate the states of all subsystems within such a network.
We will propose a novel distributed version of the celebrated Kalman filter.
The current paper, in broad sense, belongs to the large
body of literature regarding distributed estimation. One
can refer to Lopes and Ali [11], Kar et al. [6], Conejo
et al. [4], Gómez-Expósito et al. [5], Marelli and Fu
[12], Olfati-Saber [13], Ugrinovskii [19, 20], Zamani
and Ugrinovskii [21], Khan and Moura [7], Olfati-Saber
[14] and the survey paper Ribeiro et al. [15], as well as
references listed therein, for different variations of distributed estimation methods among a group of subsystems within a network. A consensus based Kalman filter
was proposed in Olfati-Saber [13]. The author of Ugrinovskii [19] utilized a linear matrix inequality to minimize a H∞ index associated with a consensus based estimator, which can be implemented locally. Some of the
results there were then extended to the case of switching
November 22, 2017

topology in Ugrinovskii [20]. The same problem was
solved using the minimum energy filtering approach
in Zamani and Ugrinovskii [21]. A common drawback
of the state estimation methods described above is that,
being based on consensus, they require, in theory, an infinite number of consensus iterations at each time step.
This results in computational and communication overload. To avoid this, in this paper we exploit the network
structure to achieve a distributed Kalman filter method
which requires only one prediction/update step at each
time step. Moreover, it is worthwhile noting that there is
a major difference between the above-mentioned works
and the problem formulation in the current paper. More
precisely, in the former, the aim of each subsystem is
to estimate the aggregated state which is common to all
subsystems. In contrast, in the problem studied here,
each subsystem is dedicated to the estimation of its own
internal state, which in general is different from those
of other subsystems. This allows the distributed estimation algorithm to be scalable to networked systems with
a large number of subsystems where requiring each subsystem to estimate the aggregated state is both computationally infeasible and practically unnecessary.
To show the effectiveness of the proposed algorithm,
we compare our method with the classical (centralized)
Kalman filter, which is known to be optimal (in the minimum error covariance sense). The classical method requires the simultaneous knowledge of parameters and
measurements from all subsystems within the network
to carry out the estimation. In contrast, our proposed
distributed estimation algorithm runs a local Kalman filter at each subsystem, which only requires the knowledge of local measurements and parameters, as well as
measurements from neighbor subsystems. Hence, it can
be implemented in a fully distributed fashion. We show
that the state estimate, and its associated estimation error covariance matrix, produced by the proposed distributed method asymptotically converge to those produced by the centralized Kalman filter. We provide
bounds for the convergence of both the estimate and the
estimation error covariance matrix. A by-product of our
result is that, if the initial states of all subsystems are
uncoupled (i.e., they are mutually uncorrelated), the estimates produced by our method are identical to that of
the centralized Kalman filter.
The rest of the paper is structured as follows. In Section 2, we describe the network setup and its associated centralized Kalman filter. In Section 4, we describe
the proposed distributed Kalman filter scheme. In Section 5, we demonstrate the asymptotic equivalence between the proposed distributed filter and the centralized
one, and provide bounds for the convergence of the esti-

mates and their associated estimation error covariances.
Simulation results that support our theoretical claims
are presented in Section 6. Finally, concluding remarks
are given in Section 7.
2. System Description
In this paper we study networks of I linear timeinvariant subsystems. Subsystem i is represented by the
following state-space model
(i)
xk+1

=

(i)
A(i) xk(i) + z(i)
k + wk ,

y(i)
k

=

C (i) xk(i)

+

v(i)
k .

The subsystems are interconnected as follows
X
z(i)
=
L(i, j) y(kj) ,
k

(1)
(2)

(3)

j∈Ni

where xk(i) ∈ Rni is the state, y(i)
∈ R pi the outk
(i)
put, wk is an i.i.d Gaussian disturbance process with
(0, Qi ), and v(i)
w(i)
k ∼ N
k is an i.i.d. Gaussian measurement noise process with v(i)
∼ N (0, Ri ). We further
 (i) ( j)>  k

>
suppose that E wk wk
= 0 and E v(i)
v(kj) = 0,
k


>
>
∀i , j and E xk(i) wk( j) = 0, E xk(i) v(kj) = 0 ∀i, j.
We also
n denote theo neighbor set of the subsystem i by
Ni = j : L(i, j) , 0 .
Remark 1. We note in (1)-(2) that the coupling between neighboring subsystems is solely caused through
the zk(i) term in (3). The main motivation for considering such coupling comes from distributed control
where (1) represents the model of an autonomous subsystem (or agent) with the zk(i) being the control input
and that (3) represents a distributed control protocol
which employs feedback only from neighboring measurements. This type of distributed control is not only
common for control of multi-agent systems (see, for example, [8, 10, 9, 24]), but also realistic for large networked systems in the sense that only neighbouring information is both easily accessible and most useful for
each subsystem. We emphasize that the distributed state
estimation problem arises for the networked system (1)(3) because of our allowance for measurement noises v(i)
k
in (2). This consideration is very important for applications because measurement noises are unavoidable in
practice. This also sharply distinguishes our distributed
control formulation from most distributed control algorithms in the literature where perfect state measurement
is often implicitly assumed.
 >
 > 
We define ξk> = ξk(1) , · · · , ξk(I)
and Ξk =
{ξ1 , · · · , ξk }, where (ξ, Ξ) stands for either (x, X), (y, Y),
2

(z, Z),n (w, W) or (v,
o V); moreover, we denote Υ =
diag Υ(1) , · · · , Υ(I) , where Υ stands for either A, B, C,
h
i
Q or R, and L = L(i, j) : i, j = 1, · · · , I .
Using the above notation, we let the initial state x1 of
all subsystems have the joint distribution x0 ∼ N (0, P).
We can also write the aggregated model of the whole
network as
xk+1
yk

=

Axk + LCxk + wk + BLvk

=

Ãxk + ek ,

(4)

= Cxk + vk ,

(5)

4. Distributed Kalman Filter
Consider the i-th subsystem (1)-(2). Notice that,
since the measurements y(kj) , j ∈ Ni , are known by the
i-th subsystem, they can be treated as external inputs.
This observation leads us to the following intuitive approach for a distributed Kalman filter scheme.
Let, for all i = 1, · · · , I and k, l ∈ N,


(i)
, E xk(i) |y(mj) ; j ∈ Ni ∪ {i}, m = 1, · · · , l ,
x̂k|l
h
i h (i)
i 
(i)
(i)
(i) >
Σ(i)
,
E
x
−
x̂
x
−
x̂
,
k
k
k|l
k|l
k|l

with
Ã = A + LC

ek = wk + Lvk .

and

It then follows that
"
#
ek h >
ek
cov
vk

v>k

i!

"
=

Q̃
S̃ >

S̃
R

(6)

and Σ(i)
= P(i) . Then, the prediction and update steps
0|0
for the proposed distributed Kalman filter are:

#
,

(14)

(7)

1. Prediction:
(i)
x̂k+1|k

where Q̃ = Q + LRL> and S̃ = LR.

=

(i)
+
A(i) x̂k|k

X

Li,(i,j j) yk( j) ,

(15)

j∈Ni

Σ(i)
k+1|k

3. Centralized Kalman Filter



(i)
(i)
(i)
x̂k|k
= x̂k|k−1
+ Kk(i) yk(i) − C (i) x̂k|k−1
,

 (i)
(i)
(i) (i)
Σk|k = I − Kk C Σk|k−1 ,

(8)

Since the distributed Kalman filter approach given in
Section 4 is motivated by intuition, the question naturally arises as to which extent it is optimal. In this
section
this question.
 we address

h (i)>To this end, we
i de?
?
?>
fine x̂k|l , Σk|l , where x̂k|l = x̂k|l : i = 1, · · · , N and
 (i)

Σ?k|l = diag Σk|l
: i = 1, · · · , N , to be the outcomes of

distributed filter and x̂k|l , Σk|l to be those of centralized
one. In Section 5.1, we show that the estimation error
covariance of the distributed filter Σ?k|k converges to that
of the centralized one Σk|k , and provide a bound for this
convergence. In Section 5.2, we do the same for the
?
convergence of x̂k|k
to x̂k|k .

(10)

= AΣk|k A + Q.
1. Update:
>

5.1. Convergence of Σ?k|k to Σk|k

(11)

In this section, we show that the covariance matrices
Σk|k and Σ?k|k exponentially converge to each other, and

(12)

with

−1
Kk = Σk|k−1C CΣk|k−1C > + R .

(18)

5. Optimality analysis




>
Σk+1|k = M − S̃ R−1C Σk|k M − S̃ R−1C

>

(17)



(i)>
(i)>
(i) (i)
(i) −1
C
C
Kk(i) = Σ(i)
C
Σ
+
R
. (19)
k|k−1
k|k−1

and


x̂k|k = x̂k|k−1 + Kk yk − C x̂k|k−1 ,
Σk|k = (I − Kk C) Σk|k−1 ,

(16)

with

and Σ0|0 = P. Our aim in this subsection is to compute
x̂k|k in a standard centralized way. Notice that equation (7) implies that, in the aggregated system formed
by (1)-(2), the process noise ek and the measurement
noise vk are mutually correlated. Taking this into account, it follows from [1, S 5.5] that the prediction and
update steps of the (centralized ) Kalman filter are given
by:
1. Prediction:


x̂k+1|k = Ã − S̃ R−1C x̂k|k + S̃ R−1 yk
(9)
= A x̂k|k + Lyk ,

+ Q̃ − S̃ R−1 S̃

>

(i) (i)
A(i) Σk|k
A + Q(i) .

2. Update:

Consider the standard (centralized) Kalman filter. For
all k, l ∈ N, let
x̂k|l , E (xk |Yl ) ,


 
Σk|l , E xk − x̂k|l xk − x̂k|l > ,

=

introduce a bound on Σk|k − Σ?k|k . To this end, we require the following definition from [3, Def 1.4].

(13)
3

Proof. Let Σ̄? = limk→∞ Σ?k|k and
n
o
σ̃ = max kPk , P? , Σ̄ , Σ̄? ,
(25)


−1
−1
. (26)
ω̃ = max P−1 , P? , Σ̄−1 , Σ̄?

Definition 2. For n × n matrices P, Q > 0, the Riemannian distance is defined by
v
t n
X

log2 σk PQ−1 ,
δ (P, Q) =
k=1

We can then appeal to the fact that the Riccati equation
is monotonic Bitmead et al. [2], to conclude that, for all
k ∈ N,
n
o
Σk|k ≤ max kPk , Σ̄ ≤ σ̃,
(27)
n
o
(28)
Σ?k|k ≤ max P? , Σ̄? ≤ σ̃,
n
o
−1
−1
Γk|k ≤ max P , Σ̄
≤ ω̃,
(29)


−1
−1
≤ ω̃.
(30)
Γ?k|k ≤ max P? , Σ̄?

where σ1 (X) ≥ · · · ≥ σn (X) denote the singular values
of matrix X.
Several properties of the above definition, which we
use to derive our results, are given in the following
proposition.
Proposition 3. [17, Proposition 6] For n × n matrices
P, Q > 0, the following holds true:
1. δ(P,
 P) = 0.
2. δ P−1 , Q−1 = δ (Q, P) = δ (P, Q) .
3. For any m × m matrix W > 0 and m × n matrix B,
we have
α
δ(P, Q),
δ(W + BPB> , W + BQB> ) ≤
α+β

Recall that
Σk+1|k = AΣk|k A> + Q.
Also, from [1, p. 139], we have
Γk|k = Γk|k−1 + U.
Clearly, similar relations hold for Σ?k|l and Γ?k|l . Then, it
follows from Proposition 3-3 that,




δ Σk+1|k , Σ?k+1|k =δ AΣk|k A> + Q, AΣ?k|k A> + Q


≤υ̃1 δ Σk|k , Σ?k|k ,
(31)




δ Γk|k , Γ?k|k =δ Γk|k−1 + U, Γ?k|k−1 + U


≤υ̃2 δ Γk|k−1 , Γ?k|k−1 ,
(32)

where α = max{kBPB> k, kBQB> k} and β =
σmin (W).


4. If P > Q, then kP − Qk ≤ eδ(P,Q) − 1 kQk .
The main result of this section is given in Theorem 5.
Its proof requires the following technical result.
?
?
Lemma 4. Let Γk|l = Σ−1
k|l and Γk|l = Σk|l . Then
−1

Σk|k , Σ?k|k

≤

σ,

Γk|k , Γ?k|k

≤

ω,

with
υ̃1 =

and


δ Σk|k , Σ?k|k ≤


δ Γk|k , Γ?k|k ≤



υk δ P, P? ,


υk δ P, P? ,

(21)

ω

=

o
n
max kPk , P? , Σ̄ ,


−1
max P−1 , P? , Σ̄−1 ,

υ2

=
=

υ1 υ1 ,

υ2 =
ω

ω + U −1

−1

,

σ kAk2
σ kAk +
2

−1
Q−1

(23)

,

−1
Q−1

and

υ̃2 =

ω̃
ω̃ + U −1

−1

.

with υ̃ = υ̃1 υ̃2 . Finally, the above implies that Σ̄? = Σ̄.
Hence, the parameters σ̃ and ω̃ given in (25)-(26) are
equivalent to σ and ω in (22)-(23), respectively, and the
result follows.

(22)

with P? denoting the diagonal matrix formed by the
block diagonal entries of the matrix P,
υ

σ̃ kAk +
2

It then follows from (31)-(32) and Proposition 3-2, that




δ Σk|k , Σ?k|k ≤ υ̃k δ P, P? ,




δ Γk|k , Γ?k|k ≤ υ̃k δ P, P? .

(20)

where
σ =

σ̃ kAk2

We now introduce the main result of the section, stating a bound on Σk|k − Σ?k|k .
Theorem 5. Let Σ̃k|l = Σk|l − Σ?k|l and Γ̃k|l = Γk|l − Γ?k|l .
Then

(24)

U = C > R−1C,

Σ̃k|k ≤ κσυk
where

and Σ̄ = limk→∞ Σk|k .
4

and

Γ̃k|k ≤ κωυk ,

?
κ = eδ(P,P ) − 1.

Lemma 7. Let

Proof. Using (22)-(23), together with (20)-(21), Proposition 3-4 and Lemma 11, we obtain
 k

?
Σ̃k|k ≤ eυ δ(P,P ) − 1 Σk|k ≤ κυk Σk|k
Γ̃k|k



>
∆k = E x̃k|k−1 x̃k|k−1
.

(34)

∆k ≤ Hk ∆k−1 Hk> + λυk I,

(35)

Then

≤

κσυk ,

≤

κυk Γk|k ≤ κωυk .

where I is the identity matrix, υ is defined in (24), and
!
q
λ , sup ζ + 2 ζ kHk k2 k∆k−1 k < ∞,
(36)

?
5.2. Convergence of x̂k|k
to x̂k|k
In this subsection, we study the convergence of state
?
estimate x̂k|k
, obtained through the distributed method,
and that of the centralized one x̂k|k . Moreover, we derive
?
a bound on the error x̂k|k - x̂k|k
. We start by introducing a
number of lemmas which are instrumental for establishing our main results.

k∈N

with

?
Lemma 6. Let x̃k|l = x̂k|l − x̂k|l
. Then

ζ

=

α

=

p
(α + β) + 2 αβ,


κ2 ω2 σ2 kAk2 σ kAk2 + kQk ,

β

=

κ ω σ kAk .
2

2

3

(37)

2

Hk

=


A I − Σk|k U ,

Proof. We split the argument in steps:
Step 1) From Lemmas 6 and 12






E ξk ξk>
≤ E ak a>k + E bk b>k
r




E ak a>k
E bk b>k .
+ 2

ξk

=

ak + bk ,

=

?
AΣk|k Γ̃k|k x̂k|k−1
,

Now, using Lemma 4,

ak
bk

=

?
.
AΣ̃k|k Γ?k|k x̂k|k

x̃k+1|k = Hk x̃k|k−1 + ξk .

(33)

where



E ak a>k ≤ kAk2 Σk|k

2

Γ̃k|k

2



?>
?
x̂k|k−1
E x̂k|k−1

≤ κ2 ω2 σ2 kAk2 Σ?k|k−1 υ2k ≤ αυ2k ,

?
?
and γ̃k|l = γk|l −
= Γ?k|l x̂k|l
Proof. Let γk|l = Γk|l x̂k|l , γk|l
?
γk|l . We can easily obtain

and

x̃k+1|k = A x̃k|k .



E bk b>k

Also, from [1, p. 140], we obtain
Then

γ̃k|k = γ̃k|k−1 .
Then it is easy to check that

2

2



? ?>
x̂k|k
E x̂k|k

≤

kAk2 Γ?k|k

≤

κ2 ω2 σ2 kAk2 Σ?k|k υ2k ≤ βυ2k .

Σ̃k|k



E ξk ξk> ≤ ζυ2k .

Step 2) From (33) and Lemma 12, we have

?
?
= Σk|k γk|k − Σ?k|k γk|k
x̃k|k = x̂k|k − x̂k|k



∆k ≤ Hk ∆k−1 Hk > + E ξk ξk>
r

?
= Σk|k γ̃k|k + Σ̃k|k γk|k
,

+2

and



kHk ∆k−1 Hk k E ξk> ξk I ≤ Fk (∆k−1 ),

with

?
γ̃k|k−1 = γk|k−1 − γk|k−1
?
?
= Γk|k−1 x̂k|k−1 −Γ?k|k−1 x̂k|k−1
= Γk|k−1 x̃k|k−1 +Γ̃k|k−1 x̂k|k−1
.

Fk (X) =

We then have

Hk XHk>

!
q
2
+ ζυ + 2 ζ kHk k kXk Iυk .
k

Clearly, if A > B then Fk (A) > Fk (B). Also, there
¯ < ∆,
¯ for all k ≥ k̄.
clearly exists k̄ and ∆¯ such that Fk (∆)
Hence, limk→∞ ∆(k) < ∞, and the result follows.

?
x̃k+1|k = AΣk|k γ̃k|k−1 + AΣ̃k|k γk|k
= AΣk|k Γk|k−1 x̃k|k−1 + ξk

= AΣk|k Γk|k − U x̃k|k−1 + ξk = Hk x̃k|k−1 + ξk .

The following result states a family of upper bounds
on the norm of the covariance matrix of x̃k|l .
5

Let dk = kDk k. Then

Theorem 8. Consider ∆k as defined in (34). Let Hk =
Vk Jk Vk −1 and H̄ = V̄ J¯V̄ −1 be the Jordan decompositions of Hk and H̄, respectively. Then for every  > 1,
there exists k ∈ N such that

dk ≤ λ

A =

λψ φ
,
ψ − υ

B =

υl Πl,k Π>l,k ≤ λφ

hk = φ ψk

d(z) = h(z)u(z) =
=

φ

=

m

=

H̄ = lim Hk ,
k→∞
2(k −1)


 m 
2
−1 2 
   
,
 V̄
V̄
ρ H̄
n
o
max 1, kH1 k , · · · , Hk −1 .

and

uk = λυk .

Taking z-transform we get

and
ψ

υl ψk−l = (hk ∗ uk ) ,

with

λυφ
.
υ − ψ

 
ρ H̄ ,

k
X
l=1

l=1

k∆k k ≤ A ψk + B υk ,
where

k
X

(38)

λφ


1 − ψ z−1 1 − υz−1
A
B
=
+
.
1 − ψ z−1 1 − υz−1

Hence,
dk = A ψk + B υk ,
and the result follows from the definition of dk and (39).

Proof. We split the argument in steps:
Step 1) Let
Theorem 8 states that the covariance of the difference
?
between x̂k|k−1
and x̂k|k−1 is bounded by two exponential terms. The term B υk is due to the convergence
of the Kalman gain Kk? to Kk , while the term A ψk is
due to the convergence of the states given by the system dynamics. In order to use this result to show the
?
asymptotic convergence of x̂k|k−1
to x̂k|k−1 , we need that
υ < 1 and ψ < 1, for some  > 0. While it is clear
from (24) that the former is true, guaranteeing the latter
is not that straightforward. The following proposition
addresses this issue.

Dk = Hk Dk−1 Hk > + λIυk .
with D1 = 0, From (35), and since ∆1 = D1 = 0, it
follows that
∆ k ≤ Dk .
(39)
Step 2) Let
Πl,k

=

Hk−1 Hk−2 · · · Hl

=

Vk−1 Jk−1 Vk−1 −1 · · · Vl Jl Vl −1 .

From (38), there exists k ∈ N such that, for all k ≥ k ,
√
 V̄ ,
kVk k ≤
√
−1
Vk
 V̄ −1 ,
≤
√
,
Vk+1 −1 Vk ≤
√  
ρ H̄ .
kJk k ≤

Proposition 9. If theh pair [A,
i C] is completely detectable and the pair A, Q1/2 is completely stabiliz 
able, then ρ H̄ < 1, where ρ(H̄) denotes the spectral
radius of matrix H̄.


Proof. Let Kk? = diag Kk(i) : i = 1, · · · , N . From Theorem 5,
lim Kk = lim Kk? , K̄.

Then, for all k ≥ l,
Πl,k

≤ mk  −l kVk−1 k kJk−1 k Vk −1 Vk−1 ×

k→∞

· · · × Vk +1 −1 Vk Jk Vk
  k−k
√
≤
 V̄ V̄ −1 mk  −l ρ H̄


 m k −l   k−l
√
−1 

   
=
 V̄ V̄
ρ H̄
ρ H̄
k−l
p
≤
φ ψ 2 .

Dk+1 = λ

x̂k+1|k

=

?
x̂k+1|k

=

A (I − Kk C) x̂k|k−1 + (AKk + L) yk ,




?
A I − Kk?C x̂k|k−1
+ AKk? + L yk .

Hence, if we had that Kk = Kk? = K̄, for all k ∈ N, then


x̃k+1|k = A I − K̄C x̃k|k−1 .

Step 3) We have
k
X

k→∞

Now,

However, under the same assumption, according to
Lemma 6, x̃k+1|k = H̄ x̃k|k−1 . Hence,


H̄ = A I − K̄C .

υl Πl,k Π>l,k .

l=1

6

0.07

S
5

S

4

S

0.06

1
0.05

S

3

S

x
~kjk!1

0.04

2

0.03

0.02

0.01

Figure 1: Communication graph of the five subsystems S i , i =
1, 2, . . . , 5.

0

-0.01

-0.02

0

1

2

3

4

5

6

7

k

i.e., H̄ equals the matrix that determines the asymptotic
dynamics of the centralized Kalman filter’s estimation
error. Then, in view of the model (4)-(5), the result follows from [1, S 4.4].

Figure 2: Difference between the estimated states obtained via both
filtering schemes.

6

5.3. The case when the initial covariance is block diagonal

#10 -17

4

It turns out that, when the initial covariance matrix
has a block diagonal structure both estimation methods
are completely identical. This is summarized in the following corollary.

x
~kjk!1

2

0

-2

10.inConsider
TexPointCorollary
fonts used
EMF. the network of subsystems (1)(2).
If
the
matrix
is block
thenbox.:
the disRead the TexPoint manualPbefore
youdiagonal,
delete this
AA

-4

-6

tributed Kalman filter scheme (15)-(19) produces, for
each i, the same estimate as the centralized Kalman filter (9)-(13).

0

1

2

3

4

5

6

7

k

Figure 3: Difference between the estimated states obtained via both
filtering schemes, when P is block-diagonal.

Proof. Recall that matrices A, Q, C and R are all block
diagonal. It then follows from (10) that, if Σk|k is block
diagonal, so is Σk+1|k . One can easily check from (12)
and (13) that the same holds for Kk and Σk|k if Σk|k−1
is block diagonal. Since Σ1|0 = P is block diagonal, it
follows that the matrices Σk|k−1 and Σk|k remain block
diagonal for all k. Now, it is straightforward to verify
that (9)-(13) become equivalent to (15)-(19), when Σk|k
and Σk|k−1 are block diagonal. Hence, the distributed
and centralized Kalman filters produce the same estimate and the result follows.

centralized one. To this end, we set the initial conditions
?
= 0.
for both schemes to be the same, i.e., x̂0|0 = x̂0|0

6. Simulations

In the first simulation, the initial covariance matrix is
chosen by randomly generating a positive-definite matrix using P = LL> + 0 I5 , where 0 = 0.1 and the entries L ∈ R5×5 are drawn from the uniform distribution
U(0, 1). Fig. 2 shows the time evolution of the entries
?
of the difference x̃k|k−1 = x̂k|k−1
− x̂k|k−1 between the es?
timation outcome x̂k|k−1 of the distributed Kalman filter
and that x̂k|k−1 of the centralized one. We see that how
this difference asymptotically converges to zero.

In this section, we present numerical results
to compare the performance of the proposed distributed scheme (15)-(19) and the optimal centralized
scheme (9)-(13). We assume that each subsystem is a
single integrator with a pole at 0.2. The communication
graph is undirected, as in Fig. 1, and the nonzero values
of L(i, j) are set to 0.3. Furthermore, vk ∼ N (0, 0.1I5 )
and wk ∼ N (0, 0.1I5 ).
We now compare the state estimation produced by the
proposed distributed scheme with that produced by the

In the second simulation we evaluate the same difference in the case where the initial covariance matrix is
block-diagonal. To this end, we choose P = 1 I5 , where
1 is a random scaler drawn from the uniform distribution U(0, 1). The time evolution entries of the difference x̃k|k−1 are shown in Fig. 3. We see that these differences are very negligible, only due to numerical errors. This confirms our claim that the estimates of both
Kalman filter schemes are the same when the matrix P
is block-diagonal.
7

7. Conclusion

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We studied the distributed Kalman filter problem in
a network of linear time-invariant subsystems. We proposed a distributed Kalman filter scheme, which uses
only local measurements, and we studied the extent to
which this scheme approximates the centralized (i.e.,
optimal) Kalman filter. It turned out that the covariance
matrix associated with the initial value of the state vector plays an important role. We showed that if this matrix is block diagonal, the proposed distributed scheme
is optimal. Moreover, if that condition is dropped, the
estimation error covariances, and the associated estimates, obtained through these two approaches approximate each other exponentially fast. We also established
proper bounds on error between estimates and its covariance matrix.
Appendix A. Some lemmas
Lemma 11. [17, Lemma 25] For every x ∈ R and 0 ≤
y ≤ 1, e xy − 1 ≤ (e x − 1) y.
"
#
A B>
Lemma 12. [17, Lemma 26] If
≥ 0, then
B C
√
kBk ≤ kAk kCk.
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