Dynamic Event Triggered Discrete Time Linear TimeVarying System with Privacy Preservation

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arXiv:2210.15875v1 [eess.SY] 28 Oct 2022

Dynamic Event−Triggered Discrete−Time Linear

Time−Varying System with Privacy−Preservation

Xuefeng Yang, Li Liu, Member, IEEE, Wenju Zhou, Member, IEEE, Jing Shi, Yinggang Zhang, Xin Hu and

Huiyu Zhou

Abstract—This paper focuses on discrete-time wireless sensor

networks with privacy-preservation. In practical applications,

information exchange between sensors is subject to attacks. For

the information leakage caused by the attack during the informa-

tion transmission process, privacy-preservation is introduced for

system states. To make communication resources more effectively

utilized, a dynamic event-triggered set-membership estimator

is designed. Moreover, the privacy of the system is analyzed

to ensure the security of the real data. As a result, the set-

membership estimator with differential privacy is analyzed using

recursive convex optimization. Then the steady-state performance

of the system is studied. Finally, one example is presented to

demonstrate the feasibility of the proposed distributed ﬁlter

containing privacy-preserving analysis.

Index Terms—Set-membership estimation, wireless sensor net-

works, privacy-preservation, event-triggered scheme.

I. INTRODUCTION

DISTRIBUTED computing is the sharing of information

among multiple pieces of software, which can run on a

single machine or connecting by multiple computers over a

network. Distributed computing applications are decomposed

into multiple small parts and distributed to multiple comput-

ers for processing, which allows for reducing running time

and sharing resource. Therefore, state estimation based on

distributed computing has become a popular research topic

[1]–[3].

Wireless sensor networks (WSNs) are mainly multi-hop

self-organized distributed sensing networks, which are formed

by large amount sensor nodes on the basis of wireless com-

munication technology. Due to the advantage of the unre-

stricted formation, the uncertain network structure and the

decentralized control among sensor nodes, WSNs are widely

used in military [4], industrial [5] and commercial ﬁelds [6].

However, performing efﬁcient distributed processing is an

extremely challenging topic, a great deal of studying on this

topic are conducted, such as Kalman ﬁlter [7], [8] and H∞

ﬁlter [9], [10]. Kalman ﬁlter is mainly applied to systems

with deterministic noise or models [11], [12]. Considering the

different applications in practice, to improve the performance

of the traditional Kalman ﬁlter, novel methods are proposed,

such as the unscented Kalman ﬁlter, the extended Kalman

ﬁlter and the cubature Kalman ﬁlter [13]–[15]. When the

noise or system model is uncertain, H∞ﬁlter is applied to

obtaining more accurate estimates and ensures the robustness

of the system [16]–[19]. However, in practical engineering

applications, due to the noise complexity and modeling im-

precision, the inaccurate values or even very terrible results

are caused during the estimation process. The set-membership

estimator (SME) is more suitable for solving such problems

under unknown but bounded (UBB) noise [20]. In recent

years, the ellipsoid algorithm in the SME algorithm has been

extensively studied [21]–[23]. The ellipsoid algorithm mainly

restricts the bounded noise to a set of ellipsoids. However, The

ellipsoid algorithm is ﬁrst transformed into a recursive convex

optimization problem, which is then solved using interior point

polynomials.

Information exchange between sensors is required in most

estimation algorithms, which means that sensors need to

broadcast information to their neighbors within a speciﬁc

sampling period. However, continuous or periodic information

exchange between sensors consumes a lot of communication

resources, which can even lead to network congestion or

packet loss [24], [25]. Therefore, designing a feasible algo-

rithm to decrease the frequency of data transmission between

sensors is important for the sustainable use of communication

resources. In practical research, event-triggered schemes (ETS)

are divided into static event-triggered schemes (SETS) [26]

and dynamic event-triggered schemes (DETS) [27]–[29]. The

threshold parameter of SETS is a ﬁxed scalar, while DETS

introduces an auxiliary parameter for the threshold. Auxiliary

dynamic variables and dynamic threshold parameters are two

typical algorithms for DETS. Since the DETS has higher

resource utilization than SETS, DETS is more widely used.

Meanwhile, the advent of information era has infringed

the network information security. During the information

exchange, information tampering and leakage, the transmis-

sion procedure is confronted with the main threats to net-

work communication [30]–[33]. Similarly, the openness of

interactive channels inevitably brings threats to information

security. Therefore, it is necessary to protect the privacy of

information. There are two main privacy-preserving meth-

ods: specially designed random noise and differential privacy

methods. However, differential privacy methods have been

widely studied due to the rigorous formulation derivation in

the application and the proven security [34], [35]. Differential

privacy methods protect the privacy of information using

random states. It is very difﬁcult for an attacker to deduce

the real information. To prevent the data from being tampered

and leaked during the exchange of sensor information, the ﬁlter

estimation that incorporates privacy-preserving still needs to

be further explored.

However, in reality, the existing models are unable to

address several challenges at the same time. Firstly, the com-

plexity of the noise leads to inaccurate models, despite simple

assumptions for noise. It becomes a challenge to build more

accurate models. Secondly, data transmission when sensors

exchange information consumes too many resources, but it is

difﬁculty to reduce the frequency of data transmission. Thirdly,

the open nature of the information exchange channel poses a

threat to data security. It is a challenge to ensure data security.

In summary, this paper studies a dynamic event-triggered

discrete-time linear time-varying system (DTLVS) containing

privacy-preservation. Considering the accuracy of the calcula-

tion and alleviating the complexity of the system, this paper

uses an optimal bounding ellipsoid algorithm. The primary

work is summarized as follows.

First, privacy-preserving noise is used to protect the initial

state information from being stolen and leaked, and then

analyze the privacy of the system. Secondly, the DETS is

investigated to decrease the consumption of communication

resources, and to achieve sustainable utilization of exchange

resources. Finally, the event-triggered SME is designed to

obtain accurate estimates even under the inﬂuence of uncertain

noise. Note that the conditions required to be satisﬁed by the

SME after introducing differential privacy are analyzed, so

that one-step prediction state is always contained within the

estimation ellipsoid.

The rest of the paper is summarized as the following parts.

Section II establishes the DTLVS model under WSN and

describes the design of an event-triggered SME. In section

III, the conditions of the SME are presented after differential

privacy. Based on this, the system stability is analyzed to verify

its performance. Section IV simulates the proposed model,

which can be efﬁciently applied to ship navigation. Section V

summarizes the entire work.

Notation: Rnrepresents the n-dimensional vector space, k.k

is the Euclidean norm, colN{.}denotes the column vector

consisting of Nblocks, diagN{.}signiﬁes the diagonal matrix

consisting of Nblocks, A1g(.)expresses the algorithm exe-

cution, and Ξindicates the execution domain of the algorithm.

II. MAIN WORK

In a distributed WSN, the network topology is used for the

model, which is deﬁned as follows.

Let the index set of nodes in the directed graph be denoted

as V={1,2,...,N}and an edge set of nodes be denoted

as ε∈V×V. The topological graph is a weighted directed

graph, A= [aij ]∈RN×Nexpresses the adjacency matrix,

where each element aij represents the weight of edge between

adjacent nodes. When data transfer is possible between two

neighboring nodes, aij >0, otherwise, aij = 0. The set of all

neighbors for the node iis denoted by Ni={j∈V: (i, j)∈

ε}. The directed graph consisting of Nnodes is denoted by

D= (V, ε, A).

Considering the noise from the external environment is

described as UBB, and the ellipsoidal ensemble form is

introduced as Z,{a:a=b+Ec, kck ≤ 1}, where the

center of the ellipsoid is represented by b∈RN. Meanwhile,

E∈Rn×mis a lower triangular matrix satisfying rank(E) =

m≤n. Assuming that each element in the diagonal of

matrix Eis greater than zero, the ellipsoidal representation can

Sensor i

Relay site

Shore site

data with

privacy noise

Hacker

Fig. 1. Transmission procedure of ship navigation system.

be rewritten as Z,a: (a−b)TP−1(a−b)≤1, where

P=EETaccording to the cholesky decomposition.

A. System Model

In the ﬁeld of ship navigation, some noise is inevitably

generated because of unknown environmental changes during

sailing. At the same time, the resistance generates changes so

that the speed of the sailing ship is affected. Consequently, the

data of the sailing speed and resistance, which are sensed or

transmitted by the sensor, are containing noise. The disturbed

data is processed by the sensor to obtain the original data.

And as the sensors send and receive signals in a distributed

manner, the open nature of their transmission channels may

allow attacks to be made in the process causing distortion of

the data. The security of the data is not guaranteed. Erroneous

data is transmitted and sensed by neighboring sensors resulting

in inaccurate data being obtained by the ﬁnal estimator. It

is necessary to ensure the security of its initial state. Conse-

quently, introducing privacy noise protects the data security

with mixing noise for the state value of the system state.

The aim is to prevent data from being altered in the event

of an attack. When the data is leaked, the data obtained by

the stealer is the data containing privacy noise, nevertheless,

the real original data cannot be obtained, i.e. the security of

the original data is protected. The data transmission procedure

of a ship navigation system is shown in Fig. 1.

The system model of the ship navigation is established as

follows.

ζk=xk+ηk

xk+1 =Ckζk+Fkwk,(1)

where xk∈Rnxdescribes the state value. Affected by the

wind and waves, the ship sways. The ship swaying changes

periodically due to the wave activities. Ckand Fkare deﬁned

as the time-varying periodic matrices, and ζk∈Rnζdescribes

the state after privacy-preserving, ηk∈Rnηdescribes the

random privacy noise obeying the Laplace distribution. When

the state information is leaked or stolen, the stealer obtains

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摘要：

arXiv:2210.15875v1[eess.SY]28Oct20221DynamicEvent−TriggeredDiscrete−TimeLinearTime−VaryingSystemwithPrivacy−PreservationXuefengYang,LiLiu,Member,IEEE,WenjuZhou,Member,IEEE,JingShi,YinggangZhang,XinHuandHuiyuZhouAbstract—Thispaperfocusesondiscrete-timewirelesssensornetworkswithprivacy-preservation.In...

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Dynamic Event Triggered Discrete Time Linear TimeVarying System with Privacy Preservation

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