Resilient Distributed H∞ Estimation via Dynamic
Rejection of Biasing Attacks ∗
V. Ugrinovskii†

arXiv:1802.09168v1 [] 26 Feb 2018

February 27, 2018

Abstract
We consider the distributed H∞ estimation problem with additional requirement of resilience to biasing attacks. An attack scenario is considered where an adversary misappropriates some of the observer
nodes and injects biasing signals into observer dynamics. Using a dynamic modelling of biasing attack
inputs, a novel distributed state estimation algorithm is proposed which involves feedback from a network of attack detection filters. We show that each observer in the network can be computed in real time
and in a decentralized fashion. When these controlled observers are interconnected to form a network,
they are shown to cooperatively produce an unbiased estimate the plant, despite some of the nodes are
compromised.

1 Introduction
Problems of resilient control and estimation came into prominence after situations were discovered where
an adversary was able to interfere with the control task by covertly injecting false information into the
measurement data [6, 1, 8, 16]. Networked control systems are particularly vulnerable to data injection
attacks, since operation of such systems depends on the integrity of communicated data.
The information shared between the nodes can also be utilized for monitoring integrity of the network.
We demonstrate in this paper that the information routinely collected and shared within a distributed observer network can be used to detect compromised observers and neutralize their biasing effect. For this,
we propose a novel distributed observer augmented with a network of attack detector filters; the latter filters
provide feedback to the node observers which neutralizes rogue biasing inputs, if such inputs are present.
The model of misappropriation attack considered here is the same as in [2]; it captures essential features
of the biasing attack described in [11]. It assumes that the adversary gains access to one or several nodes of
the observer network and injects biasing inputs directly into the state estimation algorithm; also, cf. [10].
Similarly to [2], our approach to detecting such biasing behaviour is by tracking changes in the behaviour
of the estimation errors caused by malicious biasing inputs. However different from [2], our method allows
each observer node to compute its attack detection filter in decentralized fashion, without communicating
with other nodes.
The decentralization of computations is accomplished in this paper using a decoupling technique which
we have developed previously for the distributed observer design in [15]. The technique involves an initial
‘setup’ step which requires the network to compute certain auxiliary parameters that are then distributed
among the nodes. Although this initial setup must be carried out centrally, it involves only the information
about the communication network, and does not require knowledge of the plant observed. The auxiliary
parameters computed at this setup step are then used for computing a collection of controlled node observers equipped with output feedback H∞ controllers; the latter can detect and cancel the attack using
the same sensory data that are available for estimation of the state of the observed plant. Also, our method
∗ This work was supported by the Australian Research Council and the University of New South Wales. The paper will appear in
the Proceedings of the 2018 American Control Conference.
† V. Ugrinovskii is with the School of Engineering and Information Technology, University of New South Wales at the Australian
Defence Force Academy, Canberra, ACT 2600, Australia. v.ougrinovski@adfa.edu.au

1

allows for the consideration of time-varying distributed filters. The robustness against uncertainties in the
sensors and the plant model is guaranteed as well.
The feedback-controlled nature of the proposed distributed observer distinguishes this paper from the
companion paper [14], where we used a similar decoupling technique. Here, the observers are computed
jointly with the attack detection filters. This requirement of co-design did not arise in the attack detection
problem considered in [2, 14], but arises in this paper since here the objective shifts from detecting and
signalling a biasing attack to ensuring the distributed observer network is resilient to this kind of attacks.
Notation: Rn denotes the real Euclidean n-dimensional vector space, with the norm kxk = (x′ x)1/2 ;
here the symbol ′ denotes the transpose of a matrix or a vector. The symbol I denotes the identity matrix.
For real symmetric n × n matrices X and Y , Y > X (respectively, Y ≥ X) means the matrix Y − X
is positive definite (respectively, positive semidefinite). The notation L2 [0, ∞) refers to the Lebesgue
space of Rn -valued vector-functions z(.), defined on the time interval [0, ∞), with the norm kzk2 ,
1/2
R∞
R∞
2
and the inner product 0 z1′ (t)z2 (t)dt.
0 kz(t)k dt

2 Biasing misappropriation attacks on distributed observers
A distributed observer problem consists in obtaining an estimate of the state of a time varying plant
ẋ = A(t)x + B(t)w,

x(0) = x0 ,

(1)

which is subject to an unknown modeling disturbance w. The estimate is to be obtained from a collection
of measurements
yi = Ci (t)x + Di (t)vi , i = 1, 2, . . . , N,
(2)
taken at N nodes of a sensor network, each perturbed by a measurement disturbance vi . In the distributed
estimation setting a state estimate must be obtained at each network node without sending the data to a
central data processing facility, and the nodes must obtain the same estimate of the plant. This is achieved
by interconnecting the observers into a network. This way, the nodes can use the information which they
receive from their neighbours to correct their plant state estimates until all nodes reach an agreement.
Let the state x and the disturbance w be vectors in Rn , Rm respectively, and each measurement yi be
a vector in Rpi . The disturbances w and vi ∈ Rmi will be assumed to be L2 integrable signals defined on
the interval [0, ∞). The initial state x0 is also assumed to be unknown. A typical distributed estimation
problem involves a network of filters connected over a graph with vertices in the set {1, . . . , N }, each
representing a node of the network:
x̂˙ i = A(t)x̂i + Li (t)(yi − Ci (t)x̂i )
X
+
Kij (t)(cij − Wij x̂i ), x̂i (0) = ξi .

(3)

j∈Ni

Each observer (3) produces an estimate x̂i (t) of the plant state x(t). For this, it uses its measurement
yi and the information received from the neighbours; the latter information is communicated over noisy
communication channels in the form of pij -dimensional signals
cij = Wij x̂j + Hij vij ,

j ∈ Ni .

(4)

Since the plant is time-varying, the observer gains Li , Kij in (3) are allowed to be time-varying.
The signals cij complement the local measurements yi at node i and assist it in obtaining a high fidelity
estimate of the plant. Each such signal contains information about the neighbour’s estimate x̂j of the plant
state x. That is, the observers (3) are coupled via the signals cij , forming a distributed observer network.
Such a coupling between the observer nodes is essential in situations where the plant is not detectable from
local measurements at some of the nodes, and these nodes require additional information which can only be
obtained from their neighbours. The matrix Wij determines the part of the vector x̂j which node j shares
with node i. Since this information is usually delivered over noisy communication links, a disturbance vij
is included in (4) which is also assumed to be an L2 integrable signal.
2

PSfrag replacements
−νi
+
fi

1
s

Gi (s)

−

fˆi

Figure 1: An auxiliary ‘input tracking’ representation of a biasing attack input introduced in [2].
The task of distributed estimation using the observer network (3) is to ensure that each estimate x̂i (t)
converges to x(t) as t → ∞ in some sense, with some robustness against disturbances in the plant model,
measurements and interconnection channels. A large body of literature is dedicated to the question as to
how the observers (3) can be constructed which achieve this objective; e.g. [7, 3, 5, 4]. However, the
dependency on information sharing leaves the distributed observers vulnerable to attacks seeking to disrupt
the estimation task. A scenario of such attacks usually considers an injection of false signals into sensor
measurements or communicated data [8]. Here we follow [2] and consider a different scenario where the
adversary substitutes one or several observers (3) with their biased versions
x̂˙ i = A(t)x̂i + Li (t)(yi (t) − Ci (t)x̂i )
X
+
Kij (t)(cij − Wij x̂i ) + Fi fi ,

x̂i (0) = ξi ,

(5)

j∈Ni

Here Fi ∈ Rn×nfi is a constant matrix and fi ∈ Rnfi is an unknown signal representing the attack input.
Following [2], we consider a class of attacks on the filter (5) consisting of biasing inputs fi (t) of the form
fi (t) = fi1 (t) + fi2 (t),

(6)

where the Laplace transform of fi1 (t), fi1 (s), is such that supω |ωfi1 (jω)|2 < ∞ and fi2 ∈ L2 [0, ∞).
Obviously, biasing inputs with rational Laplace transforms which have no more than one pole at the origin
and the rest of the poles located in the open left half-plane of the complex plane have this property. We will
use the notation F for the set of such inputs. It includes biasing attack inputs introduced in [11] consisting
of a steady-state component and an exponentially vanishing component generated by a low pass filter.
The following lemma characterizes the properties of biasing inputs of this class. Its proof is given in
the journal version of [2]. Let Gi (s) be a proper transfer function for which the system in Fig. 1 is stable,
and fˆi be an output of that system.
Lemma 1 (i) Consider a class of signals fi (t) that admit the decomposition (6). Then for all such
signals fi (t) it holds that
Z ∞
kfi − fˆi k2 dt < ∞.
(7)
0

(ii) If in addition, Gi is selected so that
1
lim k(I + Gi (s))−1 k = 0,
s→0
s

(8)

then limt→∞ kfi (t) − fˆi (t)k = 0 for all inputs fi ∈ F .
According to Lemma 1, biasing inputs that have the form (6) can be ‘tracked’ using a system shown in
Fig. 1. Of course, in reality it is not possible to track covert attack inputs. Nonetheless, the model in Fig. 1
allows us to associate the class of biasing attack inputs with the minimal realization of the strictly proper
transfer function 1s Gi (s), of the form
ǫ̇i = Ωi ǫi + Γi νi ,
fˆi = Υi ǫi ,

3

ǫi (0) = 0,

(9)

where νi = fˆi − fi is an L2 -integrable input, according to (7). Clearly, each signal νi corresponds to a
certain unknown biasing input fi ; it represents a mismatch error between the attack input fi and the output
fˆi of the system (9). In the sequel, this error will be regarded as an additional L2 -integrable disturbance
which will arise when we replace fi with fˆi in the derivation of our attack detection and resilient estimation
algorithms.
Apart from ensuring stability of the system in Fig. 1, according to Lemma 1, the proper transfer function
Gi (s) can be selected arbitrarily.

3 Problem Formulation
In this paper we are concerned with the design of resilient version of the distributed observer (3). Our
approach is to augment each node observer with additional dynamic feedback controllers to suppress the
attack inputs. To accomplish this task, we introduce the following controlled modification of the observers
(3), (5):
x̂˙ i = A(t)x̂i + Lri (t)(yi (t) − Ci (t)x̂i )
X
r
+
Kij
(t)(cij − Wij x̂i ) + Fi fi + ui ,

(10)

j∈Ni

x̂i (0) = ξi ,

r
The superscript r is to emphasize that the gains Lri (t), Kij
(t) are to be different from the gains Li (t),
Kij (t) of the original observer (5). Also, ui denotes the control input.
We propose the following observer-based feedback structure for generating the controls ui :

ui = −Fi ϕi ,

(11)

µ̇i = Ad (t)µi + Ld,i (t)(ζi − Wd,i µi )
X
+
Kd,ij (t)(ζij − Wd,ij (µj − µi )),

(12)

Here ϕi denotes an output of a filter

j∈Ni

ϕi = Cd,i (t)µi ,

µi (0) = µi,0 ;

where Ad (t), Ld,i (t), Kd,ij (t), Wd,i Wd,ij , Cd,i are matrix coefficients to be found. Each filter (12) is
governed by the innovation signals ζi , ζij :
ζi = yi − Ci (t)x̂i ,

(13)

ζij = cij − Wij x̂i .

(14)

The filter (12) must generate ϕi so that when node i is under attack, the signal ui counters the biasing input
fi . Also, at the nodes which are not attacked directly, ui must not interfere with the state observer. This
requires the output ϕi of the filter (12) to track the biasing signal fi , turning (12) into an attack detector.
The problem of resilient estimation under consideration is now formally stated as the problem of constructing a network of filters (12) which, when interconnected with the modified state observers (10) via
the feedback control (11), achieve the following properties
(i) In the absence of disturbances and when the system is not under attack, at every node i, kx(t) − x̂i k
and ϕi converge to 0 exponentially.
(ii) In the presence of uncertainties and/or attack,
Z +∞
kϕi − fi k2 dt < +∞ ∀i,
0
Z +∞
e′ P edt < +∞;
0

4

(15)

here P = P ′ ≥ 0 is an nN × nN matrix and e = [e′1 . . . e′N ]′ , where ei = x − x̂i denotes the
estimation error of the observer (10) at node i.
The first condition in (15) formalizes the requirement for the filters (12) to track the corresponding
attack inputs in the L2 sense. Therefore, by monitoring the behaviour of the outputs ϕi , it will be possible
to establish which node has been attacked. The second condition in (15) describes the desired resilience
property of the observers. The matrix P is considered to be given. The resilience of the modified observers
(10), (11), (12) requires (15) to hold for any collection of admissible biasing inputs fi described in Section 2. The problem in this paper is to determine the characteristics Ad (t), Ld,i (t), Kd,ij (t), Wd,i , Wd,ij ,
Cd,i (t) of the filter (12) which guarantee that the above conditions (i) and (ii) hold.

4 Design of resilient distributed observers
4.1 Analysis of error dynamics
To construct suitable filters (12) consider the dynamics of the estimation errors of the controlled observers
(10), (11). It is easy to see that these errors evolve according to
X
r
ėi = (A(t) − Lri (t)Ci (t) −
Kij
(t)Wij )ei
j∈Ni

+

X

r
(t)Wij ej
Kij

+ B(t)w − Lri (t)Di (t)vi

j∈Ni

−

X

Kij (t)r Hij vij − Fi fi + Fi ϕi ,

(16)

j∈Ni

ei (0) = x0 − ξi .
Noting that fi = Υi ǫi − νi , combine the dynamics of the systems (16) and (9) into an augmented system
with (e′i , ǫ′i )′ as a state vector:
X
r
Kij
(t)Wij )ei
ėi = (A(t) − Lri (t)Ci (t) −
j∈Ni

+

X

r
Kij
(t)Wij ej − Fi Υi ǫi + Fi ϕi

j∈Ni

+ B(t)w − Lri (t)Di (t)vi −

X

r
Kij
(t)Hij vij + Fi νi ,

j∈Ni

ǫ̇i = Ωi ǫi + Γi νi ,
fˆi = Υi ǫi ,
ei (0) = x0 − ξi ,

(17)

ǫi (0) = 0.

Observe that the system (17) at node i depends on the estimation errors at the neighboring nodes j ∈ Ni .
Therefore, we propose a distributed observer of the form (12) to estimate the state of the extended system

5

(17) and its outputs fˆi from the outputs (13), (14):
ê˙ i = (A(t) − Lri (t)Ci (t) −

X

r
Kij
(t)Wij )êi

j∈Ni

+

X

r
Kij
(t)Wij êj

+ L̄ri (ζi − Ci (t)êi )

j∈Ni

+

X

r
K̄ij
(t)(ζij − Wij (êi − êj )),

j∈Ni

ǫ̂˙i = Ωi ǫ̂i + Ľri (ζi − Ci (t)êi )
X
r
+
Ǩij
(t)(ζij − Wij (êi − êj )),

(18)

j∈Ni

êi (0) = 0,

ǫ̂i (0) = 0.

The outputs of this observer
ϕi = Υi ǫ̂i ,

(19)

will be constructed so that each output signal ϕi approximates the attack input fi at the corresponding node
i. This will allow it to be used for feedback to compensate the attack as well as for signalling the biasing
attack. Note that the innovation signals (13), (14) can be written as
ζi = Ci (t)ei + Di vi ,

(20)

ζij = −Wij (ej − ei ) + Hij vij ,

j ∈ Ni ,

(21)

and can be regarded as outputs of the interconnected large-scale uncertain system comprised of systems
(17).

4.2 The design algorithm
To present the procedure for constructing a resilient observer of the form (10), let us consider the error
dynamics of the observer (18). Define zi = ei − êi , δi = ǫi − ǫ̂i . Then it follows from (17), (18) that
X
r
żi = (A(t) − L̂ri (t)Ci (t) −
K̂ij
(t)Wij )zi − Fi Υi δi
j∈Ni

+

X

r
K̂ij
(t)(Wij zj

− Hij vij ) + Bw

j∈Ni

−L̂ri (t)Di (t)vi + Fi νi ,
δ̇i = Ωi δi − Ľri (t)Ci (t)zi −

X

r
Ǩij
(t)Wij zi + Γi νi

j∈Ni

+

X

r
Ǩij
(t)(Wij zj

− Hij vij )

(22)

j∈Ni

zi (0) = x0 − ξi ,

δi (0) = 0.

Here we used the notation
L̂ri (t) = Lri (t) + L̄ri (t),

r
r
r
K̂ij
(t) = Kij
(t) + K̄ij
(t).

(23)

Although the equations describing the evolution of zi and δi look identical to the equations describing
r
dynamics of the detector errors in [2, 14], the distinction lies in how L̂ri , K̂ij
are split to provide the
gains for the state observer and the attack detection filter at node i. Contrast to [2, 14], in this paper the
r
r
r
matrices Lri , Kij
are not considered to be given; they are determined jointly with L̂ri , K̂ij
, Ľri , Ǩij
using
the following procedure.

6

r
r
1. First, the coefficients L̂ri (t), K̂ij
(t), Ľri (t), Ǩij
(t) for each system (22) are derived, to stabilize the
uncertain interconnected system comprized of the systems (22) in an L2 sense. Then with these coefficients, one has Υi (ǫi − ǫ̂i ) ∈ L2 [0, ∞).
r
2. The coefficients Lri (t), Kij
(t) for the controlled distributed plant observer (10) are computed in parallel
with the previous step. Since with the parameters derived in the previous step, the signal fi − ϕi =
Υi (ǫi − ǫ̂i ) − νi is L2 integrable for every admissible attack input fi , this will be accomplished by
r
treating fi − ϕi as a disturbance perturbing the error dynamics (16). The coefficients Lri (t), Kij
(t) are
computed to attenuate these disturbances, along with w, vi , vij . Essentially, we redesign the original
unbiased distributed plant observer (3) to make it is robust against attack tracking errors which will arise
as a result of applying the attack cancelling control (11).
r
3. Finally, the remaining coefficients L̄ri (t), K̄ij
(t) of the attack detector (18) are obtained from (23), using
r
r
r
r
the values L̂i (t), K̂ij (t) and Li (t), Kij (t) obtained in the previous steps.

4.2.1 Stabilization of the detector error dynamics (22)
This step is identical to the corresponding step in [14]. Introduce the following notation:




A(t) −Fi Υi
B(t) Fi
Ai (t) =
, Bi =
,
0
Ωi
0 Γi


Ci (t) 0
#
"
 Wij1 0 
r
r
r
.
.
.
K̂
K̂
L̂


ijqi
ij1
i
r
,
Ci (t) =  ..
..  , Li =
r
r
. . . Ǩij
Ľri Ǩij
 .
.
qi
1
Wijqi 0


Di (t) 0 . . . 0
0 ... 0
 0 H . . . 0 Z 1/2 . . . 0 
ij1


ij1
.
Di (t) = 
.. . .
..
.. . .
.. 
 ..
. .
. . 
.
.

 .
1/2
0
0 . . . Hijqi 0 . . . Zijq

(24)

i

It is assumed that Ei (t) = Di (t)Di ′ (t) > 0 for all t.
Also, following [15], introduce a collection of positive definite (n + nfi ) × (n + nfi ) block-diagonal
matrices Ri , Xi , i = 1 . . . , N , partitioned as




Ri 0
Xi 0
Ri =
, Xi =
,
0 Ři
0 X̌i

with n × n matrices Ri , Xi and nfi × nfi matrices Ři , X̌i . Also, define the block matrix Φ = [Φij ]N
i,j=1 ,


i = j,
∆i ,
−1
′
Φij = −Wij Uij Wij , i =
6 j, j ∈ Ni ,


0
i=
6 j, j ∈
6 Ni ,
′
Uij = Hij Hij
+ Zij ,

∆i =

X

Wij′ Uij−1 Zij Uij−1 Wij ,

(25)

(26)

j∈Ni

Zij i = 1, . . . , N , j ∈ Ni are square pij × pij positive definite matrices. Also, let R = diag[R1 , . . . , RN ],
∆ = diag[∆1 , . . . , ∆N ].
Lemma 2 (cf. [14, 15]) Suppose there exists a constant γ > 0 and symmetric matrices Ri ≥ 0, Ři ≥ 0,
Zij > 0, j ∈ Ni , i = 1, . . . N , such that

7

(i) the following linear matrix inequalities are satisfied
R + γ 2 (Φ + Φ′ − ∆) > 0,

Ři > Υ′i Υi ;

(27)

1
Ri )Yi + Bi B′i ,
γ2

(28)

(ii) each differential Riccati equation
Ẏi = Ai Yi + Yi A′i
−Yi (C′i E−1
i Ci −
Yi (0) = X−1
i ,
has a positive definite symmetric bounded solution Yi (t) on the interval [0, ∞), i.e., for all t ≥ 0,
α1 I < Yi (t) = Yi′ (t) < α2 I, for some α1,2 > 0.
r
r
Then the network of systems (22) with the coefficients L̂ri , K̂ij
, Ľri , Ǩij
, obtained by partitioning the
matrices

Lri (t) = Yi (t)Ci (t)′ E−1
i (t).

(29)

according to (24), guarantees that the noise- and attack-free network is exponentially stable, and in the
presence of disturbances or an attack it holds that
N Z ∞
N 
X
X
2
2
ˆ
kfi − ϕi k dt ≤ γ
(x0 − ξi )′ Xi−1 (x0 − ξi )
0

i=1

+

Z

i=1

∞

2

2

2

kwk + kvi k + kνi k +

0

X

j∈Nij



kvij k dt .
2

(30)

The proof of Lemma 2is omitted
for brevity, it uses a completion of squares argument to establish that

PN
zi
−1
′ ′
V = i=1 [zi δi ]Yi (t)
is a Lyapunov function for the large-scale system comprised of systems
δi
(22). Also, it trivially follows from (30) that each signal
ηij = −Wij zj ,

j ∈ Ni ,

(31)

is L2 -integrable; each such signal ηij connects the system (22) at node i with the analogous system at
node j, j ∈ Ni . It then follows from Lemma 2, condition (ii), that each detector ensures the following
decentralized disturbance attenuation performance:
Z ∞
(kzi k2Ri + kδi k2Ři )dt
0

2
≤ γ (x0 − ξi )′ Xi−1 (x0 − ξi )

Z ∞
X

2
2
2
2
kwk + kvi k +
+
(32)
(kηij kZ −1 + kvij k dt .
0

j∈Nij

ij

This explains the role of the matrices Zij as weights on the contribution of the interconnection signals ηij
in the individual performance of each detector component (18).
r
r
According to Lemma 2, each node computes the matrix Lri and its components L̂ri , K̂ij
(t), Ľri , Ǩij
(t)
independently from other nodes. For this, the respective Riccati differential equation (28) must be solved
on-line; this allows the matrix Lri to be computed and partitioned according to (24) in real time. Unlike [2],
the nodes do not need to communicate to solve these Riccati equations. To setup these equations, the matrices Ri must be determined first from the LMIs (27). Even though this step must be performed centrally,
it does not require the knowledge of the parameters of the system observed; only the matrices Wij and Hij
are required which characterize the communication network. Compared with [2], this reduces substantially
the amount of information that the node must agree upon in advance. As long as the matrices Wij , Hij
and Zij and the disturbance attenuation parameter γ 2 do not change, the same matrices Ri and Xi can
be utilized even when the plant changes substantially. In the case of such an event, each node must only
update its Riccati equation (28); it can do so without communicating with its neighbours.
8

4.2.2 Design of the resilient distributed plant observer (10)
Now that we able to guarantee that fi − ϕi = (fˆi − ϕi ) − νi ∈ L2 , the large-scale system comprised
of the error dynamics (16) of the observer (10), (11) can be stabilized in an H∞ sense, while attenuating
r
this disturbance. The coefficients Lri , Kij
which accomplish this task can be computed in parallel with the
coefficients of the attack detector, using the same approach based on the results of [15]. To present this
step of our algorithm, introduce the notation
B1,i =
L1,i =



h

h
i′
C1,i (t) = Ci′ (t) Wij′ 1 . . . Wij′ qi ,
i
r
. . . Kij
.
qi


B(t) Fi ,
r
Lri Kij
1

(33)

Lemma 3 (see [15]) Suppose there exists a constant γ̄ > 0 and symmetric matrices R̄i ≥ 0, X̄i > 0,
Zij > 0, j ∈ Ni , i = 1, . . . N , such that
(i) the following linear matrix inequality is satisfied
R̄ + γ̄ 2 (Φ + Φ′ − ∆) > P,

(34)

where R̄ = diag[R̄1 , . . . , R̄N ], and Φ, ∆ are the matrices defined in (25), (26) which are the same as
in Lemma 21 ;
(ii) each differential Riccati equation
Ẏi = AYi + Yi A′
−Yi (C′1,i E−1
i C1,i −

1
R̄i )Yi + B1,i B′1,i ,
γ̄ 2

(35)

Yi (0) = X̄i−1 ,
has a positive definite symmetric bounded solution Yi (t) on the interval [0, ∞), i.e., for all t ≥ 0,
ᾱ1 I < Yi (t) = Yi′ (t) < ᾱ2 I, for some ᾱ1,2 > 0.
r
Then the network of systems (16), with the coefficients Lri , Kij
, obtained by partitioning the matrices

Lr1,i (t) = Yi (t)C1,i (t)′ E−1
i (t).

(36)

according to (33), guarantees that the noise- and attack-free network of error dynamics (16) is exponentially stable, and in the presence of disturbances or an attack it holds that
∞

Z

e′ P edt ≤ γ̄ 2

0

+

N 
X
(x0 − ξi )′ Xi−1 (x0 − ξi )
i=1

Z

∞

kwk2 + kvi k2 + kϕi − fi k2 +

0

X

j∈Nij



kvij k2 dt .

(37)

The proof of the lemma is analogous to the proof of the corresponding result in [15].
4.2.3 The main result
The main result of this paper follows from the properties of the observer errors (16) and the properties of
the errors of the attack detection filters (18).
1 Performance tuning of the algorithm may require one to choose different matrices Z in this step of the algorithm. In this case,
ij
the matrices Φ, ∆, and Ei will also need to be updated, and will not be the same as in Lemma 2. However, this does not have any
effect on the statement of Lemma 3.

9

r
Theorem 1 Suppose the conditions of Lemmas 2 and 3 hold. Let the coefficients Ľri , Ǩij
of the detectors
r
r
(18) be obtained as described in Lemma 2, and let the coefficients L̄i , K̄ij be obtained using the matrices
r
r
L̂ri , K̂ij
, Lri , Kij
from Lemmas 2 and 3, as

L̄ri = L̂ri − Lri ,

r
r
r
K̄ij
= K̂ij
− Kij
.

(38)

Then, the network of state observers (10), augmented with the network of attack detectors (18) produces
state estimates x̂i which have the following convergence properties.
(i) In the absence of disturbances and biasing attacks, kx − x̂i k → 0 exponentially as t → ∞.
(ii) When the plant and/or the network is subject to L2 -integrable disturbances and/or admissible biasing
attacks, the estimates x̂i converge to x in the L2 sense, and the resilient performance of this observer
network is characterized by the condition
Z ∞
e′ P edt
0

≤ γ̄ 2

N 
X
(x0 − ξi )′ (X̄i−1 + 2γ 2 Xi−1 )(x0 − ξi )
i=1

+(1 + 2γ 2 )

Z

∞

kwk2 + kvi k2 +
0

+2γ̄ 2 (1 + γ 2 )

X

j∈Nij
N Z ∞
X
i=1

kνi k2 dt.


kvij k2 dt


(39)

0

Also, the outputs ϕi of the distributed attack detector network (18) track the attack inputs in the L2
sense.

5 Conclusion
We have proposed a novel class of distributed observers for robust estimation of a linear plant, which are
resilient to biasing misappropriation attacks. The observers involve feedback from an additional network of
attack detection filters, which can also signal the attack. The design method is based on the methodology
of distributed and decentralized H∞ filtering which is combined with a decoupling technique to obtain
observers which attenuate benign disturbances, while sensing and compensating biasing inputs.

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