Distributed filtering for time-varying systems over sensor networks with randomly switching topologies under the Round-Robin protocol

Accepted Manuscript

Distributed Filtering for Time-Varying Systems over Sensor Networks with Randomly Switching Topologies Under the Round-Robin Protocol Xianye Bu, Hongli Dong, Fei Han, Nan Hou, Gongfa Li PII: DOI: Reference:

S0925-2312(19)30110-9 https://doi.org/10.1016/j.neucom.2018.07.087 NEUCOM 20387

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Neurocomputing

Received date: Revised date: Accepted date:

14 May 2018 2 July 2018 27 July 2018

Please cite this article as: Xianye Bu, Hongli Dong, Fei Han, Nan Hou, Gongfa Li, Distributed Filtering for Time-Varying Systems over Sensor Networks with Randomly Switching Topologies Under the Round-Robin Protocol, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2018.07.087

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Distributed Filtering for Time-Varying Systems over Sensor Networks with Randomly Switching Topologies Under the Round-Robin Protocol

Abstract

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Xianye Bu, Hongli Dong∗ , Fei Han, Nan Hou and Gongfa Li

This paper focuses on the filtering problem for nonlinear systems over sensor networks whose topologies are

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changeable subject to Round-Robin (RR) protocol in the finite horizon case. The phenomenon of switching topologies for sensor networks is considered due to the weak connections between the node and its neighbors. The Round-Robin communication strategy is employed to save the limited bandwidth and reduce the network resource consumption. It is our target to design distributed filters so that the dynamics of augmented error system satisfies the given level of average H∞ performance. We get the sufficient conditions in order to guarantee the existence of the desired

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distributed filters whose parameter matrices would be attained by a recursive method in terms of a set of matrix inequalities. In the end, a numerical simulation is exploited to describe the validity of the raised filter design

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technique.

Index Terms

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Distributed filtering; Time-varying system; Round-Robin protocol; Nonlinearity; Switching topologies.

I. I NTRODUCTION

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In the past decades, due to the rapid development of sensor technology, the sensor network (SN) control systems have achieved astonishing development in industrial production, disaster prevention, environmental monitoring and

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This work was supported in part by the China Postdoctoral Science Foundation under Grant 2017M621242, the PetroChina Innovation Foundation under Grant 2018D-5007-0302, the Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Ministry of Education in Wuhan University of Science and Technology under Grant 2017C01, the Natural Science Foundation of Heilongjiang Province of China under Grant F2018005, the Northeast Petroleum University Innovation Foundation For Postgraduate YJSCX2017-026NEPU and the Alexander von Humboldt Foundation of Germany. X. Bu, H. Dong, F. Han and N. Hou are with the Institute of Complex Systems and Advanced Control, Northeast Petroleum University, Daqing 163318, China, and also with the Heilongjiang Provincial Key Laboratory of Networking and Intelligent Control, Northeast Petroleum University, Daqing 163318, China (Email: [email protected]). G. Li is with the Key Laboratory of Metallurgical Equipment and Control Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China. ∗

Corresponding author.

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other fields. As the key research area for SN systems, the filtering/estimation issue has aroused widespread interest of scientists and engineers in recent years. The distributed estimator/filter which obtains the information by the sensors dispersed in the target zone is designed to detect the dynamical behavior of the systems. Fruitful studies have been carried out for the distributed estimators/filters design so far, see e.g. [2], [3], [6], [8], [12], [25], [38]. In [2], the distributed filter has been considered for discrete systems over SNs suffering from partial measurements under the event-triggered communication method. In [6], the state estimator design method has been proposed for

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sensor networks in which the communication channels are redundant under the event-triggered protocol. In [8], the distributed fault estimator has been obtained for time-invariant systems subject to uncertainty over sensor networks. As is well known, the scattered sensor nodes access information not only from themselves but also from their neighbors according to the topology. However, the majority of studies on sensor networks are based on the fixed topologies, see e.g. [6], [23], [32], [35]. In reality, the connections between the node and its neighbors are not stable, because of the limited bandwidth, node fault, power exhaustion, sensor movement, such that the topologies of sensor

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networks would be flexible. For example, biologists have applied the sensor network techniques to track migratory trajectories of migratory birds, where the wearable sensors are installed on the bodies of migratory birds. It is no doubt that the topology structure of these series of wearable sensor networks is varying due to the sensor movement. However, allowing for the complexity of mathematical calculations, only a few studies have been reported on the phenomenon of switching topologies, see e.g. [7], [36], [38]. It’s remarkable that the phenomenon of switching

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topologies, investigated in [7], are governed by a Markovian jump chain. Different from [7], a Bernoulli-distributed white sequence is employed to describe the switching topologies in this paper. As a result, the phenomenon of

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switching topologies is a new complex research focus for SNs. It is noteworthy that SNs are composed of abundant sensor nodes scattered around the target system. Thus,

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the amount of measurements from the sensors is voluminous, considering the limited bandwidth in the networked system, such huge pressure on the data transmission may cause packet dropout, nonlinearity, channel fading, and then result in negative effects on the filtering ( or estimation) performance. In order to tackle this challenge, we utilize a

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novel equal communication rule for measurements, called Round-Robin protocol, to arrange when the information could be delivered from the sensor nodes to the filter/estimator. As one of the most popular communication rules, the

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Round-Robin mechanism has found a wide utilization in computing, communication, adjusting the load of server and so on. It has been proved to be quite effective in engineering fields that substantial research achievements have been presented on the Round-Robin protocol in recent literature, for instance [19], [24], [35], [42]. The state estimator has been achieved in [42] for dynamical networks subject to by employing the Round-Robin communication strategy. The researchers have devoted their energy on analysis the state estimation for artificial neural networks in [24] with Round-Robin communication strategy. Compared with [24], [42], the Round-Robin protocol has not been well investigated for sensor networks with switching topologies. In view of this reason, we would do some research on Round-Robin protocol for sensor networks. On another research frontier, the emergence of nonlinearity would not be prevented in practical engineering

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systems. Thus, a lot of attention has been paid to the negative influences from the nonlinearity on the target systems. For instance, in [4], the security control problem has been concerned for nonlinear systems suffering from spoofing attacks, and the prescribed probatilistic security is guaranteed by designing a dynamic controller of output feedback. In [9], the problem of fault estimation has been considered for nonlinear systems in the finite horizon case. The filter design method have been addressed in [34] for discrete systems suffering from time-delay with measurement dropout as well as stochastic disturbances. However, the researchers get some preliminary results on On the basis of the above facts, we are motivated to do some research.

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the problem of distributed filtering for nonlinear systems over SNs where the topologies are changing abundantly. Encouraged by the above discussions, in this paper, a set of finite-horizon distributed filters is designed for a class of nonlinear systems over SNs. Due to the physical characteristics of sensor network, data transmission would be affected by the limited bandwidth. Besides, the framework of sensor network may change because of the sensor fault, sensor movement as well as energy depletion. For these reasons, the Round-Robin method is exploited

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to relieve the pressure of data transmission, and the changeable topologies governed by a random sequence are introduced to describe the phenomenon of switching topologies. The parameters are derived by using a specific Matlab software package. The major innovations proposed in this paper are indicated as follows: 1) a set of distributed filters is obtained for nonlinear systems over sensor networks whose topologies are switching; 2) the Round-Robin communication strategy is applied to adjust the communication load for sensor networks; and 3) a

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recursive algorithm is achieved for the finite-horizon distributed filter design over sensor networks. We would organize this paper in those aspects. The distributed time-varying filter model is established for

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nonlinear systems in Section II. In Section III, the main results are presented by using recursive linear matrix inequalities (RLMIs). In Section IV, the usefulness of the designed filters would be demonstrated by a simulation

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example. At the end of this paper, we make a conclusion in Section V. Notation: In this paper, the notation is mostly standard except special circumstances that are stated. Rn stands for

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the n-dimension Euclidean space while all the n × m matrices is descried by Rn×m . l2 [0, N ) stands for the space √ obeying square summable case and the norm of matrix B is denoted by ∥B∥ = trace(B T B). For real symmetric matrices U ≥ V , U − V represents positive semi-definite matrix. The transpose of L is defined by LT . The unit

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matrix is defined by I and 0 is the null matrix. For event “⋆”, E{⋆|⋆} represents the conditional expectation. The

notation ⊗ describes the Kronecker product. ∥x∥ denotes the Euclid norm when the vector is x. “∗” is utilized

as an ellipsis for symmetry-induced terms in symmetric block matrices. 1n = [1, 1, . . . , 1]T ∈ Rn . For event “ · ”,

Prob{·} shows its probability, and Prob{·|·} stands for the conditional probability. Matrices are supposed that they would have relevant dimensions if we don’t mentioned specially.

II. P ROBLEM F ORMULATION In this paper, we suppose that networked system contains n sensor nodes. Besides, all the sensors are distributed in space in the light of 2 switching topologies denoted by directed graphs G(r) = (V(r) , E(r) , A(r) ) (r = 1, 2), in

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which the number of nodes V(r) is defined by {1, 2, . . . , n}, the number of edges is described as E(r) ⊆ V(r) × V(r) , (r)

(r)

and A(r) = [aij ] (r = 1, 2) denotes the weighted adjacency matrices with nonnegative elements aij , and r is the number of weighted adjacency matrices. (i, j) stands for an edge of G(r) . Define that all the elements in (r)

the adjacency matrices are positive, i.e., aij > 0 (r = 1, 2), which shows that for topology r (r = 1, 2), the (r)

information could be transmitted from sensor j to sensor i. Also, suppose that aii = 1 for all i ∈ V(r) and (r)

Ni

= {j ∈ V(r) : (i, j) ∈ E(r) } which contains the sensor i itself.

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r = 1, 2, and therefore (i, i) would be defined as a new edge for E(r) . The number of neighbors is described by

In this section, we define the time horizon as [0, N − 1] := {0, 1, 2, · · · , N − 1}. Then, the target system on

instant u ∈ [0, N − 1] is considered as follows:   x(u + 1) = A(u)x(u) + h(x(u)) + B(u)w(u) 

z(u) = L(u)x(u)

(1)

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where the state vector is indicated by x(u) ∈ Rnx ; w(u) ∈ Rnw which is owned by l2 [0, N − 1] stands for the

noise signal and the nonlinear function is defined by h(x(u)).

The nonlinear function h : Rnx → Rnx is a continuous function which satisfies sector boundary conditions shown

as:

[h(x) − h(y) − M1 (x − y)]T [h(x) − h(y) − M2 (x − y)] ≤ 0

(2)

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where M1 and M2 are matrices that have proper dimensions.

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The measurement output from ith node is expressed:

yi (u) = Ci (u)x(u) + Di (u)v(u)

(3)

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the measurement from sensor i is presented by yi (u) ∈ Rny , v(u) ∈ l2 [0, N − 1] denotes the external disturbance. Besides, Ci (u), B(u), Di (u), A(u) and L(u)) are given matrices with homologous dimensions.

Remark 1: Due to the limited bandwidth of network, data collisions occur inevitably in the process of data

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transmission. In order to resolve this problem, it is a commonly method that only one sensor has access to communication channel at the data delivery instant. The Round-Robin method would be suitably used to determine

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which sensor deliver the information via SNs at a certain instant in such a situation. Considering the characteristic that the Round-Robin method is a static protocol employed by sensor networks, every sensor has equal opportunity to transmit the information. Suppose that the ith sensor deliver the data at the first transmission instant i. When all the sensors have been given the access to the network one after one in a loop, which means that a round is completed. Then, the transmission sequence in the (γ + 1)th (γ = 0, 1, 2, . . .) round can be described as y1 (γn + 1), y2 (γn + 2), y3 (γn + 3), . . . , yn (γn + n)

(4)

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which describes that the output yi (u), from the node i, is delivered at u only when (5)

mod(u − i, n) = 0 holds.

[ ]T The y˜(u) is defined as y˜1T (u) y˜2T (u) · · · y˜nT (u) ∈ Rny n . According to the Round-Robin protocol, the

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measurement y˜i (u) (i = 1, 2, . . . , n) is updated as follows:   yi (u), if mod(u − i, n) = 0 and u > 0 y˜i (u) =  y˜i (u − 1), otherwise.

(6)

In order to simplify the expression, we assume that y˜i (q) = 0 for q = −(n − 1), −(n − 2), . . . , −1, 0. The update

matrix is defined as:

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Γi = diag{ϑ(i − 1)I, ϑ(i − 2)I, . . . , ϑ(i − n)I}

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where ϑ(·) ∈ {0, 1} denotes the Kronecker delta function. In view of the communication process presented by (6), the y˜(u) can be described as

y˜(u) = Γθ(u) y(u) + (I − Γθ(u) )˜ y (u − 1)

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Let

x ˜(u) = 1n ⊗ x(u),

[ ]T wv (u) = w ¯ = 1n ⊗ w(u), v¯(u) = 1n ⊗ v(u), ¯ T (u) v¯T (u) , w(u) [ ]T z¯(u) = 1n ⊗ z(u), H(¯ x(u)) = hT (˜ x(u)) = 1n ⊗ h(x(u)). x(u)), 0 , h(˜

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[ ]T x ¯(u) = x ˜T (u) y˜T (u − 1) ,

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where θ(u) = mod(u − 1, n) + 1 stands for the selected node which can deliver the data to the filter.



¯ A(u)

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where

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Then the time-varying system is formulated as follows:    x ¯(u + 1) = A(u)¯ x(u) + H(¯ x(u)) + B(u)wv (u)    y˜(u) = C1 (u)¯ x(u) + D1 (u)wv (u)      ¯ x(u) z¯(u) = L(u)¯ 0

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

¯ ¯ ¯ 1 (u) = diag{D1 (u), D2 (u), . . . , Dn (u)},  , B(u) = diag{B(u), Γθ(u) D(u)}, D ¯ Γθ(u) C(u) I − Γθ(u) [ ] [ ] [ ] ¯ ˜ ˜ ¯ ¯ L(u) = L(u), , L(u) = I ⊗ L(u), C (u) = , D = 0 Γθ(u) C(u) In − Γθ(u) 0 Γθ(u) D(u) , n 1 1

A(u) = 

¯ A(u) = In ⊗ A(u),

¯ B(u) = In ⊗ B(u),

¯ C(u) = diag{C1 (u), C2 (u), . . . , Cn (u)}.

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Then, the distributed filter is designed for sensor node i as follows:  ∑ (1) (1)  ˆi (u + 1) = A(u)ˆ xi (u) + π(u)aij Kij (u)(˜ yj (u) − ϑ(θ(u) − j)(Cj (u)ˆ xj (u)) − (1 − ϑ(θ(u) − j)) x    j∈ N  i   ∑   (2) (2)   × yˆj (u − 1)) + (1 − π(u))aij Kij (u)(˜ yj (u) − ϑ(θ(u) − j)(Cj (u)ˆ xj (u))     j∈ N i       − (1 − ϑ(θ(u) − j))ˆ yj (u − 1)) + h(ˆ xi (u))    ∑ (1) (1) yˆi (u) = ϑ(θ(u) − i)(Ci (u)ˆ xi (u)) + (1 − ϑ(θ(u) − i))ˆ yi (u − 1)) + π(u)aij Hij (u)(˜ yj (u)    j=i    ∑  (2) (2)   − ϑ(θ(u) − j)(Cj (u)ˆ xj (u)) − (1 − ϑ(θ(u) − j))ˆ yj (u − 1)) + (1 − π(u))aij Hij (u)     j=i       × (˜ yj (u) − ϑ(θ(u) − j)(Cj (u)ˆ xj (u)) − (1 − ϑ(θ(u) − j))ˆ yj (u − 1))       zˆi (u) = L(u)ˆ xi (u)

(11)

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where the state estimate from node i is denoted by x ˆi (u) ∈ Rnx , the estimation of measurement is showed by (r)

yˆi (u) ∈ Rny , the estimation of z(u) is described by zˆi (u) and h(ˆ xi (u)) is the estimation of h(x(u)). Here, Kij (u) (r)

and Hij (u) (r = 1, 2) are the filter parameters which need to be designed.

In order to present the phenomenon of switching topologies, the stochastic variable π(u) ∈ R, whose values are

0 or 1, is introduced that:

Prob{π(u) = 0} = 1 − π ¯

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Prob{π(u) = 1} = π ¯,

where π ¯ (0 ≤ π ¯ ≤ 1) is a known constant which can be obtained by statistical experiments.

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Remark 2: The filter (11) reflects the spatial nature of the distributed estimation problem where the information

from both the sensor node i and its neighbors is used for the updating. It can be observed from (11) that the topology of SN may switch in a probabilistic way by means of their intensity. It is worth mentioning that, with

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help from the stochastic variable π(u), it is easily capable of describing the randomly switching topologies in an Denote

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adequate way.

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[ ]T x ˇ(u) = x ˆT1 (u) x ˆT2 (u) · · · x ˆTn (u) yˆ1T (u − 1) yˆ2T (u − 1) · · · yˆnT (u − 1) , [ ]T zˆ(u) = zˆ1T (u) zˆ2T (u) · · · zˆnT (u) , ¯ (r) (u) = [K ¯ (r) (u)]n×n K ij

¯ (r) (u) = with K ij

{

¯ (r) (u) = [H ¯ (r) (u)]n×n H ij

¯ (r) (u) = H ij

{

with

(r)

(r)

(r)

aij Kij (u), i = 1, 2, . . . , n; j ∈ Ni ; r = 1, 2 i = 1, 2, . . . , n; j ∈ /

0, (r)

(r) Ni ;

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r = 1, 2

(r)

aij Hij (u), i = 1, 2, . . . , n; j = i; r = 1, 2 0,

i = 1, 2, . . . , n; j ̸= i; r = 1, 2.

(r) (r) ¯ (r) ¯ (r) (u) can be showed as It is easy to see that, because aij = 0 when j ∈ / Ni . K (u) and H

¯ (r) (u) ∈ Tnx ×ny , K

¯ (r) (u) ∈ Tny ×ny H

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{ } (r) where Tp×q = T¯ = [Tij ] ∈ Rnp×nq | Tij ∈ Rp×q , Tij = 0 if j ∈ / Ni .

Letting e(u) = x ¯(u) − x ˇ(u) and z˜(u) = z¯(u) − zˆ(u), the following augmented system is obtained for the sensor

network:





(15)

¯ z˜(u) = L(u)e(u)

 A11 (u, π(u)) A12 (u, π(u)) , A(u, π(u)) =  A21 (u, π(u)) A22 (u, π(u))

¯ A12 (u, π(u)) = −K(u, π(u))(I − Γθ(u) ),



C(u, π(u)) = 

¯ B(u)

¯ ¯ −K(u, π(u))Γθ(u) D(u)

0



, ¯ ¯ ¯ Γθ(u) D(u) − H(u, π(u))Γθ(u) D(u)

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where

  e(u + 1) = A(u, π(u))e(u) + C(u, π(u))wv (u) + H(e(u))

¯ ¯ ¯ A21 (u, π(u)) = Γθ(u) C(u) − H(u, π(u))Γθ(u) C(u),

¯ A22 (u, π(u)) = (I − Γθ(u) ) − H(u, π(u))(I − Γθ(u) ),

¯ ¯ ¯ A11 (u, π(u)) = A(u) − K(u, π(u))Γθ(u) C(u),

¯ ¯ (1) (u) + (1 − π(u))H ¯ (2) (u). H(u, π(u)) = π(u)H

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¯ ¯ (1) (u) + (1 − π(u))K ¯ (2) (u), H(e(u)) = H(¯ x(u)) − H(ˆ x(u)), K(u, π(u)) = π(u)K [ ]T [ ]T h(ˆ x(u)) = hT (ˆ x(u)) = hT (ˆ x1 (u)) hT (ˆ x2 (u)) · · · hT (ˆ xn (u)) , H(ˆ x(u)) 0 ,

(16)

The following definition is introduced in order to process further.

Definition 1: We suppose that a disturbance attenuation level γ > 0 and a matrix R(u) > 0 are given, the H∞

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performance index satisfies the following inequality, namely:

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J := E{∥˜ z (u)∥2[0,N −1] } − γ 2 {∥wv (u)∥2[0,N −1] + eT (0)R(u)e(0)} < 0

where e(0) is defined as x ¯(0) − x ˇ(0).

(r)

(17)

(r)

In this paper, we are committed to obtain the filter gains Kij (u) and Hij (u) such that the error dynamics

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matches the preset performance constraint (17).

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III. M AIN R ESULTS

In this section, we need to investigate the filtering problem for SNs whose topologies are described by G(r) =

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(V(r) , E(r) , A(r) ) (r = 1, 2). We would establish the main results by using the following lemmas. T Lemma 1: (Schur Complement) [1] Given constant matrices S1 , S2 , S3 where S1 = ST 1 and 0 < S2 = S2 , then −1 S1 + ST 3 S2 S3 < 0 if and only if

 

S1

ST 3

S3 −S2



 < 0 or

 

− S2 S3

ST 3

S1



 < 0.

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The following theorem gives a sufficient condition under which the augmented system (15) satisfies the performance constraint 17. Theorem 1: In view of the time-varying system (1) and SNs whose topologies are changeable under Round-Robin (r)

(r)

protocol. Assume the parameters of designed filter Kij (u) and Hij (u)(r = 1, 2) are given, the filtering dynamics

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system (15) satisfies the given performance constraint (17) if there exists a scalar ϵ > 0 and a class of positive ˜ (u)}0≤u≤N +1 with the initial condition P(0) ≤ γ 2 R(0) satisfying definite matrices {P   Θ11 (u, π(u)) ∗ ∗     Θ(u, π(u)) = Θ21 (u, π(u)) Θ22 (u, π(u))  < 0, for 0 ≤ u ≤ N − 1 ∗   Θ31 (u, π(u)) Θ32 (u, π(u)) Θ33 (u, π(u))

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where

˜ (u + 1)A(u, π(u)) + ϵM ¯, Θ21 (u, π(u)) =P

˜ (u + 1) − γ 2 I − ϵI, Θ22 (u, π(u)) = P

˜ (u + 1)A(u, π(u)), Θ31 (u, π(u)) =CT (u, π(u))P

˜ (u + 1)C(u, π(u)), Θ33 (u, π(u)) = CT (u, π(u))P

MT M2 + MT2 M1 ¯ = M2 + M1 , = 1 , M 2 2  0 ¯ 1 = In ⊗ M1 , M ¯ 2 = In ⊗ M2 , , M 0

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˜ (u + 1), M ˜ Θ32 (u, π(u)) =CT (u, π(u))P    ¯1 0 ¯ M M  , M2 =  2 M1 =  0 0 0

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˜ (u + 1)A(u, π(u)) − P(u) + L ¯ T (u)L(u) ¯ ˜, Θ11 (u, π(u)) =AT (u, π(u))P − ϵM

˜ (u + 1) =¯ P π P(u + 1|π(u) = 1) + (1 − π ¯ )P(u + 1|π(u) = 0).

Here, P(u + 1|π(u)) denotes certain positive definite matrix at the instant u + 1 given the stochastic variable π(u). Proof: Define

˜ (u + 1)e(u + 1) − eT (u)P(u)e(u). J(u) = eT (u + 1)P

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(20)

= E

{(

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Through the calculation of the expectation for function (22), we can achieve that { } T T ˜ E{J(u)} = E e (u + 1)P(u + 1)e(u + 1) − e (u)P(u)e(u) )T

A(u, π(u))e(u) + C(u, π(u))wv (u) + H(e(u))

˜ (u + 1) A(u, π(u))e(u) P

(21)

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} ) T +C(u, π(u))wv (u) + H(e(u)) − e (u)P(u)e(u) .

(

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Then, we introduce a calculation method that adding z˜T (u)˜ z (u) − γ 2 w ¯vT (u)w ¯v (u) − z˜T (u)˜ z (u) + γ 2 w ¯vT (u)w ¯v (u)

AC

to E{J(u)}, we have

E{J(u)} ≤ E{η T (u)Θ(u, π(u))η(u) − z˜T (u)˜ z (u) + γ 2 w ¯ T (u)w(u)} ¯

(22)

]T [ η(u) = eT (u) H T (e(u)) wvT (u) .

(23)

where

We can have the following inequation by summing up the equation (22) from 0 to N − 1, } { N −1 ∑ ˜ (N )e(N )} + eT (0)P(0)e(0) − γ 2 eT (0)R(0)e(0). η T (u)Θ(u, π(u))η(u) − E{eT (N )P J
(24)

u=0

˜ (N ) > 0 and the initial condition P(0) ≤ γ 2 R(0), we have It’s remarkable that the positive definite matrix P

J < 0 when (19) holds.

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As discussed in Theorem 1, the performance analysis has been conducted for the presented system (15). Then, we are motivated to achieve the filters gains. In the next theorem, we would deal with this problem. Theorem 2: Assume that a scalar µ > 0 and a matrix R(0) = RT (0) > 0 are given. For the target plant (1) over ˇ ˇ sensor networks, the proposed time-varying filtering problem is solved if there exist matrices {Ω(u)} and {Υ(u)} (r)

(r)

(0 ≤ u ≤ N + 1), a scalar ϵ > 0 and a set of gains Kij (u) ∈ Tnx ×ny and Hij (u) ∈ Tny ×ny (r = 1, 2) satisfying diag{Ω(0), Υ(0)} 6γ 2 R(0),

and the following RLMIs:



Π(u, π(u)) = 

∗

Π11 (u, π(u))

Π21 (u, π(u)) Π22 (u, π(u))

with the parameters updated by

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the condition showed in the following:



 < 0,

(25)

(26)

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ˇ −1 (u + 1) = π Ω ¯ Ω(u + 1|π(u) = 1) + (1 − π ¯ )Ω(u + 1|π(u) = 0),

ˇ −1 (u + 1) = π Υ ¯ Υ(u + 1|π(u) = 1) + (1 − π ¯ )Υ(u + 1|π(u) = 0),

where

Π11 11 (u, π(u)) = −

¯ TM ¯2 + M ¯ TM ¯ 1) ϵ(M 1 2 − Ω(u), 2

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  Π111 (u, π(u)) ∗ , Π11 (u, π(u)) =  2 0 Π11 (u, π(u))   Γ11 (u, π(u)) 0 ∗  11    1 Π11 (u, π(u)) =  0 −Υ(u) ∗ ,   ¯ 1 +M ¯ 2) ϵ(M 0 −ϵI 2

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Π22 (u, π(u)) = diag{−I, ΩΥ (u + 1)},

  2I −γ ∗ , Π211 (u, π(u)) =  2 0 −γ I 

Π21 (u, π(u)) = 

¯ L(u)

0

0

A(u, π(u)) I C(u, π(u))

ˇ + 1), −Υ(u ˇ + 1)}. ΩΥ (u + 1) = diag{−Ω(u

(27) 

,

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Proof: Assume that the positive definite matrix P(u) could be decomposed as follows: P(u) = diag{Ω(u), Υ(u)},

¯ + 1) = π Υ(u ¯ Υ(u + 1|π(u) = 1) + (1 − π ¯ )Υ(u + 1|π(u) = 0),

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¯ + 1) = π Ω(u ¯ Ω(u + 1|π(u) = 1) + (1 − π ¯ )Ω(u + 1|π(u) = 0),

˜ −1 (u + 1) = diag{Ω(u ˇ + 1), Υ(u ˇ + 1)}, (28) P

˜ (u + 1) = diag{Ω(u ¯ + 1), Υ(u ¯ + 1)}. P

Then, considering the aforementioned Lemma, it is easy to achieve (26) by complex matrix calculation. Therefore, the proof of this theorem is now completed. By using the Theorem 2, we can easily carry out the Distributed Filter Design (DF D) algorithm in the following table. Remark 3: Through the Theorem 2, the distributed filter gains have been obtained by calculating a set of inequalities (26). It is remarkable that the main innovations of this paper are presented in those aspects: 1) a

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Algorithm DF D Step 1.

Select suitable values for the performance index γ , the matrix R(0) > 0, the states x(0), x ˆi (0) and y˜i (−1) when time instant u = 0. Chose values for the matrices Ω(0) and Υ(0) under the condition

Step 2.

(25) as well as set the time instant u = 0. ˇ ˇ Calculate the matrices Ω(u) > 0, Υ(u) > 0. By solving the RLMIs (26), The gains matrices of (r)

(r)

Kij (u) and Hij (u) would be obtained at time instant u . Step 3.

Set u = u + 1 and acquire Ω(u) and Υ(u) by the formula (28). If u < N , execute the Step 2. or

Step 4.

Stop.

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execute Step 4.

original distributed filter model has been established for sensor networks whose topologies change in a random way; 2) the Round-Robin protocol has been applied in sensor networks in order to save precious communication resources; and 3) the DFD algorithm which is built in a recursive form is suitable for online applications. Apparently,

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the result derived in this paper is more valuable for practical application.

IV. A N I LLUSTRATIVE E XAMPLE

We would reveal the effectiveness of the addressed finite horizon distributed filter design scheme for SNs which contain switching topologies under the Round-Robin protocol.

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We employ a directed graph G(r) = (V(r) , E(r) , A(r) ) to describe network framework. With the set of nodes V(1) = {1, 2, 3} and V(2) = {1, 2, 3}, set of edges E(1) = {(1, 1), (1, 3), (2, 1), (2, 2), (3, 2), (3, 3)} and E(2) =

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{(1, 1), (1, 2), (2, 2), (2, 3), (3, 1), (3, 3)}. The respective adjacency matrices are shown as follows:     1 0 1 1 1 0         (1) (2) A = 1 1 0 , A = 0 1 1 .     0 1 1 1 0 1

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The parameters of considered time-varying nonlinear discrete system (1) are given by   [ ]T [ ] 0.02sin(2u) −0.03  , B(u) = 0.5 0.5 , L(u) = 0.5 0.3 A(u) =  0.03 −0.01

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and select the nonlinear function h(x(u)) as

[ ]T h(x(u)) = sin(x1 (u)) sin(x2 (u)) ,

where xl (u) (l = 1, 2) represents the lth element of the system (1). In order to satisfy the nonlinear function h(x(u)) presented in (2), the parameters are selected as follows:     −10 12 −11 10  , M2 =  . M1 =  15 16 13 11

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Let n = 3 and the parameters of measurements be defined as: [ ] [ ] C1 (u) = 0.02 sin(2u) 0.03 sin(2u) , C2 (u) = 0.03 sin(2u) 0.02 sin(2u) , D1 (u) = 0.8,

D2 (u) = 0.9,

D3 (u) = 0.7,

π ¯ = 0.8.

[ ] C3 (u) = 0.02 0.01 ,

The H∞ performance index γ is chosen as 0.7 in this example. The parameters of filters is acquired by employing the designed DF D algorithm to solve the RLMIs presented ]T [ ]T [ ]T [ ]T [ ˆ1 (0) = 0 0 , x ˆ2 (0) = 0 0.5 , x ˆ3 (0) = 0 0 and by (26). Set the original states x(0) = 0.5 0.5 , x cos(2u+1) 3u+1 ,

and the disturbance v(u) is taken as

sin(10u+1) 3u+1 .

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yi (0) = 0 (i = 1, 2, 3). The noise w(u) is selected as

We would present the simulation results in Fig. 1 and Fig. 2. Fig. 1 shows the estimation errors of z(u) and Fig. 2 presents the estimation of z(u). Obviously, it is easy to confirm that the designed filters in this paper is useful for SNs. CONCLUSION

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V.

In this paper, the time-varying distributed filter has been obtained for a set of nonlinear systems over sensor networks subject to switching topologies. The Round-Robin protocol, in which all the sensors would be given identical probability to use the communication channels based on a fixed cycle, has been employed to allocate the limited bandwidth reasonably. Through stochastic analysis technique, sufficient conditions have been established

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such that the dynamics of augmented error system satisfies a given performance constraint. Besides, a numerical simulation has been conducted to show the usefulness of the acquired distributed filters. In the foreseeable future, the approach proposed in this paper would be extended to complex networks, gene regulatory networks, neural

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Xianye Bu received the B.Sc. degree in mathematics in 2013 from Northeast Petroleum University, Daqing, China, and is currently studying for a Ph.D. degree in Control Science and Engineering from Northeast Petroleum University. His research interests include control theory and control engineering, intelligent control theory and application, and networked control systems.

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Hongli Dong received the Ph.D. degree in Control Science and Engineering in 2012 from Harbin Institute of Technology, Harbin, China. From July 2009 to January 2010, she was a Research Assistant in the Department of Applied Mathematics, the City University of Hong Kong. From October 2010 to January 2011, she was a Research Assistant in the Department of Mechanical Engineering, the University of Hong Kong. From January 2011 to January 2012, she was a Visiting Scholar in the Department of Information Systems and Computing, Brunel University, London, U.K. From November 2012 to October 2014, she was an Alexander von Humboldt research fellow at the University of Duisburg-Essen, Duisburg, Germany. She is currently a professor with the Institute of Complex Systems and Advanced Control, Northeast Petroleum University, Daqing, China. She is also a director with the Heilongjiang Provincial Key Laboratory of Networking and Intelligent Control, Daqing 163318, China.

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Dr.~Dong's current research interests include robust control and networked control systems. She is a very active reviewer for many international journals.

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Fei Han received the B.Sc.~degree in mathematics and applied mathematics from China University Ming and Technology, Xuzhou, China, in 2003, M.Sc. ~degree in applied mathematics from Henan Normal University, Xinxiang, China, in 2009, and Ph.D. ~degree in system analysis and integration from University of Shanghai for Science and Technology, Shanghai, China, in 2017. From March 2018 to June 2018, he was a Senior Research Assistant in the Department of Electronic Engineering the City University of Hong Kong.

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He is currently an associate professor at Northeast Petroleum University, Daqing, China. His current research interests include distributed control and filtering.

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Nan Hou received the M.Eng. degree in Automation in 2015 from Northeast Petroleum University, Daqing, China. From March 2018 to June 2018, she was a Research Associate in the Department of Mathematics, the City University of Hong Kong. She is currently studying for the Ph.D. degree in Petroleum and Natural Gas Engineering at Northeast Petroleum University, Daqing, China. Her research interests include networked control systems, oil gas information and control engineering.

Gongfa Li was born in Hubei province, P. R. China, in 1979. He received B.S., M.S. and Ph.D. degrees in mechanical design and theory from Wuhan University of Science and Technology, China, in 2001, 2004 and 2008, respectively. He is an Associate Professor in the College of Machinery and Automation of Wuhan University of Science and Technology. His current research interests include intelligent control, computer aided engineering and optimization design.