Sensor selection schemes for consensus based distributed estimation over energy constrained wireless sensor networks

Neurocomputing 87 (2012) 132–137

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Sensor selection schemes for consensus based distributed estimation over energy constrained wireless sensor networks$ Wen Yang n, Hongbo Shi Key Laboratory of Advanced Control and Optimization for Chemical Processes (East China University of Science and Technology), Ministry of Education, Shanghai, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 November 2011 Received in revised form 3 February 2012 Accepted 4 February 2012 Communicated by Z. Wang Available online 24 February 2012

In the applications of wireless sensor networks (WSNs), sensor energy saving is essential to increase the life of sensor networks. In this paper, we consider the problem of performing consensus based estimation over energy constrained WSNs, in which energy is conserved by selecting only a subset of sensors to observe the state of the dynamical system at each time step. First, we derive a sufficient condition for the convergence of the state estimation covariance. Then we propose a sensor selection strategy to schedule sensors to measure the system state for next step with the goal of minimizing the state estimation error subject to sensor energy constraint. Finally, we provide some numerical examples to illustrate the performance and effectiveness of the proposed strategy. & 2012 Elsevier B.V. All rights reserved.

Keywords: Consensus problem Distributed estimation Sensor selection

1. Introduction Recent years have witnessed an increasing interest in the distributed estimation over wireless sensor networks (WSNs). A WSN is composed by a large number of geographically distributed sensors that can measure certain parameters of interest, process data and communicate with sensors within a certain range. With the development of integrated micro-sensor technology, the WSNs due to its fast mobility and self-organizing properties has been applied into many areas including environment monitoring and control, health care and battlefield surveillance. In particular, a lot of efforts have been directed toward trying to design distributed estimation algorithms under different network conditions. Most of the researches proposed distributed Kalman filtering algorithms, which combines consensus protocol into the standard Kalman filter. A typical example of such algorithm was proposed by Olfati-Saber [1], which used local aggregation of the sensor data but attempts to reach a consensus on estimates with other nodes in the network. Other work on optimal estimation, adaptive gain filtering could be found in [2,3]. A basic assumption in these researches is that sensor network is ideal, namely, no communication fault, no energy limitation and no time delay. However, these problems happen frequently in practice. Some consensus based estimation algorithms with network constraint were proposed in [4–10], such as, a distributed

$ n

This paper was not presented at any IFAC meeting. Corresponding author. Tel./fax: þ 86 21 64253376. E-mail addresses: [email protected] (W. Yang), [email protected] (H. Shi).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2012.02.011

parameter estimation over a WSN with bit rate constraint was considered in [4], an algorithm for discrete-time decentralized state estimation with intermittent observations and communication faults was proposed in [6], a new distributed H1 consensus filtering over a finite-horizon for networks with multiple missing measurements was studied in [7], the distributed H1 filtering problem is addressed for a class of polynomial nonlinear stochastic systems in sensor networks in [9]. In real applications, sensor network energy limitation is one of the major obstacles to the adoption of such technology, especially in complex dynamic environment, such as the deserts, the battlefield and the disaster area, where is hard to recharge the sensor battery. In past decades, some efficient strategies on reducing sensor network energy consumption and extending network lifetime were proposed in [8–14], such as, the MAC protocol [11], sensor scheduling strategies [15], and stochastic sensor selection algorithm [16]. A unified framework to solve a large class of optimization problems over energy constrained WSNs was also proposed by Mo et al. [17], who designed a sensor selection strategy to minimize the network estimation error. Most of these researches assumed that one root sensor existing in the network received all the measurements from the sensors who could observe the state of the dynamical system, and estimated the state with recursive Kalman filter. In contrast, consensus based estimation does not rely on one single sensor, but each sensor estimates the state of dynamical system, and exchanges the estimations at each time step. In the literature, distributed consensus estimation with energy limitation seems less studied. In this work, we investigate a distributed consensus based estimation algorithm proposed in [18] over an energy constrained

W. Yang, H. Shi / Neurocomputing 87 (2012) 132–137

network. To reduce the sensor energy consumption, we select only a subset of sensors to observe the state of the dynamical system at each time step. This method has similar manifestation as pinning control within synchronization area [19–21]. A related work in this literature was proposed by Yu et al. [22], who considered using a small fraction of sensors to measure the target information by the pinning control method. However, it is assumed that the pinned sensors were fixed over arbitrary time step, which leads to the energy of those pinned sensors run out soon. Different from [22], this work considers reselecting the sensors to observe for the next time, which keeps the balance of sensor energy consumption. We also derive a sufficient condition to guarantee the convergence of the estimation error covariance. To minimize the network estimation error with energy constraint, we pose a convex optimization problem which is addressed by reformulating the estimation algorithm that directly explores the dependence of the estimation error covariance on the selected sensors. The paper is organized as follows. In Section 2, we formalize consensus based distributed estimation problem in a energy constrained network. In Section 3, we analyze the convergence properties of the consensus based estimation algorithm with a subset of sensors selected to observe the state of the system. In Section 4, we solve a convex optimization problem which minimizes the estimation error subject to sensor energy constraint. Finally, in Section 5, we provide some simulations to verify the theoretical results and state out conclusions.

133

bi0 ¼ 0 8i. In Eq. (3), mik ¼ 1 indicates that sensor i is selected with x at time step k, and otherwise, mik ¼ 0. Moreover, E is the consensus gain and is in the range of ð0; 1=DÞ where D ¼ maxi ðdi Þ with di ¼ 9Ni 9 be the number of neighbor sensors of sensor i, and Kik is the measurement gain matrix which is given by K ik ¼ Sik C iT ðC i Sik C iT þRi Þ1 :

Sik is computed from Sik þ 1 ¼ ASik AT ASik C iT ðC i Sik C iT þ Ri Þ1 C i Sik AT þQ , with Si0 ¼ P0 8i. Since ðA,C i Þ is observable and ðA,Q 1=2 Þ is controllable, Sik converges to a unique positive semi-definite matrix Si . Hence, the steady-state gain matrix is given by K i ¼ Si C iT ðC i Si C iT þ Ri Þ1 : i Let K~ k ¼ A  Sik C iT ðC i Sik C iT þ Ri Þ1 and M k ¼ diagfm1k , . . . ,mlk g. Define the estimation error at sensor i as i eik ¼ x^ k þ 19k xk : 1 2 n Denoting e^ k ¼ ½e^ k , e^ k , . . . , e^ k T and vk ¼ ½v1k ,v2k , . . . ,vnk T as the concatenation of the vectors eik, vik, the estimation error equation for the entire system is i

ek þ 1 ¼ ðIn  AÞek EðL  AÞek ðMk  Im ÞDiagðK~ k C i Þek i

þ ðM k  Im ÞDiagðK~ k Þvk 1n wk ,

ð5Þ

~ k ek þ W ~ k, ek þ 1 ¼ G

ð6Þ

where  represents the matrix Kronecker product, and 2. Problem formulation

G~ k ¼ ðIn  AÞ  ðIn  Im E  L  Im ðMk  Im ÞDiagðK ik C i ÞÞ,

In this section, we model the wireless sensor network as an undirected graph G ¼ ðV,EÞ with the nodes V ¼ f1; 2 . . . ,ng being the n sensors, and the edges E  V  V representing the available communication links. An edge (i,j) means sensor i can receive data from sensor j. Define the neighbor sensors of sensor i by Ni ¼ fj : ði,jÞ A Eg. Thus sensor i can only receive local state estimates from the sensors in Ni at each time step. We use the wireless sensor network consisting of n sensors to estimate the state of the linear discrete-time system as following:

~ k ¼ ðM k  Im ÞDiagðK~ i Þ  vk 1n wk W k

xk þ 1 ¼ Axk þ wk ,

3. Stability analysis

ð1Þ

where xk A Rm is the state vector, wk A Rm is the process noise which is assumed to be zero-mean white Gaussian with covariance matrix Q Z 0. Assume ðA,Q 1=2 Þ is controllable. The initial state x0 is also zero-mean Gaussian with covariance P0 , and is independent of wk 8k. The measurement equation of the i-th sensor is given by yik ¼ C i xk þ vik ,

ð2Þ m

where vik A R is a zero-mean white Gaussian with covariance matrix Ri 4 0. Assume Ci is invertible. Assume vik is independent of x0 ,wk 8k,i and vik is independent of vis when ia j or k a s. Further assume that ðA,C i Þ is observable for 8i. In real applications, the sensor network energy is limited, and it is inconvenient or impossible to replenish the energy source frequently. Thus it is essential to conserve the sensors energy as much as possible. To reduce the energy consumption, we select 1 rl on sensors to observe the state of system (1) at each time step k, and introduce an index mik into the distributed estimation algorithm [18] for sensor i, X i i i j i b x k9k ¼ b x k9k1 Þ, x k9k1 E ðb x k9k1 b x k9k1 Þ þmik K ik ðyik C i b ð3Þ j A Ni i b x k þ 19k

i ¼ Ab x k9k

ð4Þ

and 0 i DiagðK~ k C i Þ ¼

B B @

1 K~ k C 1

...

0

^

&

^

0

...

n K~ k C n

1 C C: A

The following preliminary results [23] are needed before we state the main result. Lemma 1. If Ag0 and C in invertible, then C T ACg0. Lemma 2. Let Ag0 and B be a Hermitian matrix. Then AB is diagonalizable and all of its eigenvalues are real. Furthermore AB has the same number of positive, negative, and zero eigenvalues as B. Lemma 3. Let A,B A M n be Hermitian matrices. Suppose Ag0 and Bk0. Then AkB if and only if rðBA1 Þ r 1, and AgB if and only if rðBA1 Þ o1. Definition 1. The matrix norm 999  999s is defined on Mn by 999A999s ¼ 999S1 AS999 for all A A M n , where S A M n is nonsingular. We are now ready to analyze the convergence of the estimation error covariance of (3). Note that

Eðek þ 1 Þ ¼ G~ k  Eðek Þ,

ð7Þ T

T

~ Þ: ~ kW Eðek þ 1 eTk þ 1 Þ ¼ G~ k  Eðek eTk Þ  G~ k þ EðW k

ð8Þ

Without loss of generality, we assume the initial condition Eðe0 Þ ¼ 0, and then Eðek þ 1 Þ ¼ 0. Hence, the estimator (3) is unbiased. From Eq. (6), it is easy to see that the estimation error

134

W. Yang, H. Shi / Neurocomputing 87 (2012) 132–137

~ Þ o 1. In the later analysis, we try to covariance converges if rðG k ~ k Þ o1. Here, we firstly find conditions for guaranteeing rðG ~ Þ o1, and further analyze the upper bound for assume that rðG k ~ k is independence the estimation error covariance. Note that G over time, thus we have 0 1 k1 X k1 X i T jT T ~ k1i W ~ ~ A Eðek  e Þ ¼ E@ G~ W G k1

k

k1j

k1

i¼0j¼0

¼

k1 X

i

iT

G~ k1 Q~ G~ k1 ,

where 0

R1 B Q~ ¼ DiagðK~ Þ  @ ^ 0

... & ...

0 Q i T B^ C ~ Þ þ  Diagð K @ A ^ Q Rn 0

1

1

...

Q

&

^C A: Q

...

~ s ~ k is independent of W The second equality holds because W when k a s. ~ k Þ o 1, 999E½ek  eT 999 o1, then Again since rðG k lim E½ek  eTk -e,

k-1

e ¼ e T 40,

with T

G~ k  e  G~ k e þ Q~ ¼ 0:

ð9Þ

~ Þ o1, where e satisfies (9). converges to e if rðG Thus, Eðek  k Define D ¼ diagfDi , . . . ,Di g with Di ¼ Si C iT ðC iT Si C i þ Ri Þ1 C i . According to Lemma 1, C iT ðC iT Si C i þ Ri Þ1  C i g0. We assume that limk-1 Sik ¼ Si g0 (refer to [25]). By Lemma 2, Di is diagonalizable and all of its eigenvalues are positive real. Let a nonsingular ~ matrix Sk A M n such that ðM k  Im ÞD ¼ S1 k L k Sk , and the eigenvalues of ðMk  Im ÞD be arranged in increasing order as L1k r    r Lnk . Note that L1k ¼    ¼ Llk ¼ 0 since we only select l sensors. Let Llkþ 1 be the second smallest eigenvalue in fLik g, i ¼ 1, . . . ,n. Next we are ready to derive a sufficient condition for rðG~ k Þ o1. eTk Þ

Theorem 1. Assume that the original network topology G is undirected and connected. If A is stable, or even A is unstable with lþ1 999A999 o 1=ð999Sk 999  999S1 Þ, then Eðek  eTk Þ converges to e. k 999Lk Proof. Since G is connected, L has only one zero eigenvalue. Thus, the eigenvalues of In EL must all be between 0 and 1 since E A ð0; 1=DÞ. Moreover, it is easy to see that 0!ðC iT Si C i þ Ri Þ1 !ðC iT Si C i Þ1 , and C iT ðC iT Si C i þRi Þ1 C i !C iT ðC iT Si C i Þ1 C i ¼ ðSi Þ1 (note that Ci is invertible). By Lemma 3, we have rðDi Þ o 1. Let Gk ¼ In  Im E  L  Im ðMk  Im ÞDiagðK ik C i Þ. According to Definition 1 999Gk 999s ¼ 999Sk Gk S1 k 999

¼ max 9x Sk ðIn E r max 9999Sk 999  xT x ¼ 1

x^ k9k ¼ x^ k9k1 EðL  Im Þx^ k9k1 þ ðM k  Im ÞK k ðyk C x^ k9k1 Þ, K k ¼ diagfK 1k ,

. . . ,K nk g

where and x 1 ¼ x^ 190 and Y 1 ¼ y1 , for k¼1,

1

n

C ¼ diagfC , . . . ,C g.

ð10Þ Denoting

^ 1x1, x^ 191 ¼ x 1 EðL  Im Þx 1 þ ðM 1  Im ÞK 1 ðy1 Cx 1 Þ ¼ G^ 1 Y 1 þ H x^ 291 ¼ ðIn  AÞx^ 191 , ^ 1 ¼ ðIn ELÞ  Im ðM 1  Im ÞK 1 C. Denotwhere G^ 1 ¼ ðM1  Im ÞK 1 , H ing Y k ¼ ½yT1 ,yT2 , . . . ,yTk T , we have ^ kx1, x^ k9k ¼ G^ k Y k þ H

ð11Þ

where G^ k ¼ ½ððIn ELÞ  Im ðM k  Im ÞK k CÞ  ðIn  AÞ  G^ k1 ,ðMk  Im ÞK k , ^ k ¼ ½ðIn ELÞ  Im ðM k  Im ÞK k C  ðIn  AÞH ^ k1 : H The above equation is obtained by induction. Suppose that (11) holds for k ¼ 1, . . . ,N. When k ¼ N þ 1, we have

þ ðM N þ 1  Im ÞK N þ 1 yN þ 1 :

¼1

xT x ¼ 1

In Section 3, we analyze the convergence properties of the state estimation error when only l sensors are selected to observe the state. How to select a subset of sensors minimizing the state estimation error with sensor energy constraint is an interesting but challenge problem. In this section, we try to solve this problem if each sensor has limited energy. 1 2 n 1 2 Denoting x^ k9k ¼ ½x^ k9k , x^ k9k , . . . , x^ k9k T , x^ k9k1 ¼ ½x^ k9k1 , x^ k9k1 , . . . , n x^ k9k1 T and yk ¼ ½y1k ,y2k , . . . ,ynk  as the concatenation of the vectors i i x^ k9k , x^ k9k1 and yik, the estimation algorithm for the entire system is

x^ N þ 19N þ 1 ¼ ½ðIn ELÞ  Im ðM N þ 1  Im ÞK N þ 1 Cx^ N þ 19N

~ ¼ max 9xT ðSk ðIn ELÞS1 k L k Þx9 T

4. Sensor selection strategy for energy constrained network

^ N x1 x^ N þ 19N ¼ ðIn  AÞG^ N Y N þ ðIn  AÞH

9xT Sk Gk S1 ¼ max JSk Gk S1 k xJ ¼ max k x9 JxJ ¼ 1 xT x ¼ 1 xT x

Remark 2. If A is stable, the estimation error covariance converges even if selecting one sensor to observe. However, when A is unstable, if we randomly select the sensors to observe at each time step, it is hard to guarantee the convergence of (3) by Theorem 1. Remark 3. It is worth to point out that we choose Kik as the standard Kalman filter, but the proposed algorithm is obviously not the optimal filter for distributed state estimation. Since the estimates are coupled, it is very hard to find an optimal solution minimizing the estimation error. In Section 4, we try to minimize the state estimation error when only part of sensors are selected to observe the state of the system over energy constrained network.

i¼0

i

Remark 1. Note that if we select all the sensors to observe the state of the system, then L1 40 8k. Thus, 999Gk 999s r 999S999  999S1 999L1 .

T ~ LÞS1 k xx L k x9

999S1 k 999

~ x9  ðx ðIn ELÞxÞx L k T

T

^ k x 1 xk Þ ¼ CovðG^ k Y k xk Þ, Pk9k ¼ Covðx^ k9k xk Þ ¼ CovðG^ k Y k þ H

¼ max 9999Sk 999  999S1 k 999 xT x ¼ 1

2

Hence, Eq. (11) holds for k ¼ N þ 1. Let xk ¼ 1n  xk . Define the estimation error covariance

2

Sni¼ 1 li ðIn ELÞ9ðU T xÞi 9 Snj¼ 1 Ljk 9xj 9 9 r 999Sk 999  999S1 k 999: The last inequality holds because 1Eln ðLÞ r li ðIn ELÞ r1 and 0 r Ljk r1 8i,j A 1, . . . ,n. When A is stable, there always exists a ~ k Þ o1. When consensus gain E such that the spectral radius rðG ~ Þ o 1 holds if 999A999 o 1=ð999S 999  999S1 999 A is unstable, rðG k k k Llkþ 1 Þ. &

ð12Þ

^ k x 1 is a deterministic vector. the last equality uses the fact that H Let Lc be a complete graph. Define G~ k ¼ ½ððIn ELc Þ  Im K k CÞ  ðIn  AÞ  G~ k1 ,K k , ~ k ¼ ½ðIn ELc Þ  Im K k C  ðIn  AÞH ~ k1 , H with G~ 1 ¼ K 1 ,

~ 1 ¼ ðIn ELc Þ  Im K 1 C: H

W. Yang, H. Shi / Neurocomputing 87 (2012) 132–137

Furthermore, we have P k9k ¼ Covðx^ k9k xk Þ ¼ CovðG^ k Y k xk Þ, ~ k x 1 xk  ¼ Cov½ðG^ k G~ k ÞY k þ G~ k Y k þ H T ¼ ðG^ k G~ k Þ CovðY k ÞðG^ k G~ k Þ þ P~ k9k

ð13Þ

where zk ¼ ½CovðY k Þ1=2 because CovðY k Þ is positive semi-definite. We assume that each sensor has initial energy Ei, and consumes energy ei on each observation. Therefore each sensor can do at most Ei =ei observations till its battery runs out. We describe energy constraints for sensor i as following: T X

mik r Ei =ei ,

i ¼ 1, . . . ,n:

ð14Þ

k¼1

^ ¼ ½mi  has columns for each i with k ¼ 1, . . . ,T Define a matrix M k and rows for each k with i ¼ 1, . . . ,n. Denote b ¼ ½E1 =e1 , . . . ,En =en T as the concatenation of Ei =ei . We restate the constraint (14) as ^ M1r b,

ð15Þ

^ where 1 ¼ ½1, . . . ,1 . Moreover, according to the definition of G, we have T

mik ZJJG^ k,ðk1Þnm þ im J1 J0 ,

ð16Þ

where G^ k,ðk1Þnm þ im is the (ðk1Þnmþ im)th column of G^ k , and J  J0 is the L0 norm of a scalar. If the scalar is 0, then its L0 norm is zero, and it is 1 otherwise. Here, we pose a minimization problem as following: ðP0 Þ :

min

G^ k ,mik

T X

JG^ k zk G~ k zk J2F

ð17Þ

k¼1

^ rb s:t: M1 i m Z JJG^

k,ðk1Þnm þ im J1 J0 ,

k

i ¼ 1, . . . ,n:

Remark 5. With the assistance of simulations, we find that the value of mik ðlÞ almost approximates the binary after 15 iterations. As in Algorithm 1, the P0 problem has been transformed to a quadratic programming problem P1 with reweighted L1 iterations. There are some existed methods to solve the P1 problem, such as, interior point method, gradient projection method. For instance, if we use the interior point method, then the time complexity of Algorithm 1 is Oðn5 m5 T 5 Þ. 5. Simulation results First, we illustrate the results derived in Sections 3 and 4 by numerical simulations, and then we compare the performance of consensus based estimation algorithm with random and suboptimal sensor selections strategy (get from Algorithm 1). We give two examples. In the first example, the conditions in Theorem 1 are verified. Example 1. Consider a wireless sensor network with n ¼50 sensors. The discrete-time system and sensing model configurations are as given as follows: ! !   0 9ni 2di 0 10 0 Q¼ , Ci ¼ , Ri ¼ , 0 9ni 0 2di 0 10 where di , ni A ð0; 1 for 8i. We choose a network topology G with its second eigenvalue l2 ðLÞ ¼ 0:2786 and the maximal degree D ¼ 22. Here, we select l¼25 sensors to observe the state of the system (1). Furthermore, Ll þ 1 ¼ 0:2773 and 999S9992 ¼ 1:0374. According to Theorem 1, if 999A9992 o 1:3837, then the estimation error converges 1 0:008 1 0:95 as Fig. 1 shows ((a) A ¼ ð0:009 1 Þ and (b) A ¼ ð0:9 1 Þ). Example 2. In this example, we investigate the performance of random and suboptimal sensor selection strategy. At each time

ð18Þ

However, it is an combinatorial problem since it contains a L0 norm, which is hard to be solved when the sensors number is large. In the past decade, some researchers provided several methods to replace L0 norm by L1 norm. Here, we further relax the P0 optimization problem into a convex problem based on a reweighted L1 minimization method [24]. The reweighted L1 algorithm is as following:

250 200 150 ||eik||

¼ ðG^ k zk G~ k zk ÞðG^ k zk G~ k zk ÞT þ P~ k9k ,

100

Algorithm 1. Sensor selection strategy with reweighted L1 minimization. 1: 2: 3:

50

Initialization: l ¼ 0, wki ð0Þ ¼ 1 for i ¼ 1, . . . ,n, k ¼ 1, . . . ,T; while new data exists do Solve the weighted L1 minimization problem: PT min JG^ z G~ z J2 ðP1 Þ : G^ k ,mik

s:t:

k¼1

k k

135

0 0

50

100 150 200 250 300 350 400 k

k k F

3500

^ rb M1

mik Zwki ðlÞJJG^ k,ðk1Þnm þ im J1 J0 ;

3000

Updatethe weights: wki ðl þ1Þ ¼ mk ðlÞ1 þ E , E 4 0; i

5: 6:

l ¼ l þ 1; end while

||eik||

2500

4:

2000 1500 1000 500

Remark 4. As we can see, the problem of minimizing the estimation error subject to sensor energy constraint has been converted into a convex optimization problem P0. By solving this problem, we get a group of optimized mik for sensor selection strategy.

0 1

2

3

4

5

6

7

8

9

10

k Fig. 1. The estimation error of each sensor Jeik J ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /eik ,eik S with E ¼ 0:3576.

136

W. Yang, H. Shi / Neurocomputing 87 (2012) 132–137

step k, we select fixed number sensors. Denote p as the proportion of the selected sensors to all the sensors. Define Ek ¼ Sli ¼ 1 ei as the network energy consumption, where sensor i is selected at time step k. The system parameters are given as follows: A¼



1

0:008

0:009

1



,

Ri ¼

!

20ni

0

0

20ni

,

Q and Ci are the same as in the first example. As Fig. 2 shows, the estimation error decreases with the proportion p increasing, while the network energy increases with p increasing. Moreover, the estimation error gap increases before p ¼40% and then decreases after p¼70%. The heuristic reason for these phenomena lies in the fact that when p is small, only a very small fraction sensors are selected, and few selected sensors even in suboptimal strategy have small influence on the estimation error. However, with the increasing of p, more optimal sensors are selected, the estimation error gap increases between both strategies. When p 4 70%, most of the sensors are selected in both strategies, thus the gap decreases.

50

The authors would like to thank the associate editor and the anonymous reviewers for their helpful remarks that improved the presentation of the paper. This work was supported by the NSF of PR China under Grant No. 61074079, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grants No. 20100074120010, and Shanghai Municipal Natural Science Foundation under No. 11ZR1409700.

40 Ek

18

Mean squared error

In this paper, we consider the problem of consensus based distributed estimation with energy constrained network. We present an analysis of consensus based estimation algorithm when only l sensors are selected to observe the state of the dynamic system at each time step. We find that the network estimation error converges even if one sensor is selected when the system A is stable. We also transform sensor selection problem into an optimization problem to minimize the estimation error subject to energy constraint, which is further relaxed into a convex problem based on reweighted L1 approximation. With the assistance of the simulations, we verify the theoretical results in Section 3 and compare the estimation performance between the random sensor selection strategy with the proposed sensor selection strategy. The simulations and heuristic analysis indicate that the proposed strategy leads to the best performance when 40–70% sensors are selected. These results may aid the designer of the sensor network in the choice of the number and disposition of the sensor. For the future work, we will study the general case without the assumption of invertible C, and will extend the result to the directed network.

Acknowledgements

20

16

30 20 10

14

0 200

12

220

240

260

280

300

k p=10% p=30% p=50% p=70% p=90%

10 8

References

6 10%

20%

30%

40%

50%

60%

70%

80%

90% 100%

p 15 random strategy suboptimal strategy

14 13 Mean squared error

6. Conclusions

12 11 10 9 8 7 6 10%

20%

30%

40%

50%

60%

70%

80%

90% 100%

p P Fig. 2. (a) The mean squared error ð1=nÞð ni¼ 1 ðJeik J2 ÞÞ1=2 at the steady-state varies with the proportion p with random selection strategy. (b) The performance comparison of random selection strategy and suboptimal selection strategy.

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137 Wen Yang received the M.Sc. degree in control theory and control engineering from Central South University, Hunan, China, in 2005 and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2009. She was a Visiting Student with the University of California, Los Angeles, from 2007 to 2008. She is currently with the School of Information Science and Engineering, East China University of Science and Technology (ECUST), Shanghai. Her research interests include coordinated and cooperative control, consensus problems, multiagent systems, and complex networks.

Hongbo Shi received the M.Sc. degree and the Ph.D. degree in control theory and control engineering from East China University of Science and Technology, China in 1989 and 2000, respectively. He was the 2003 Shu Guang Scholar of Shanghai. His research interest covers modeling of industrial process and advanced control technology, theory and methods of integrated automation systems, condition monitoring and fault diagnosis of industrial process.