Distributed Consensus Filtering in Sensor Networks - Semantic Scholar

1568

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 6, DECEMBER 2009

Distributed Consensus Filtering in Sensor Networks Wenwu Yu, Student Member, IEEE, Guanrong Chen, Fellow, IEEE, Zidong Wang, Senior Member, IEEE, and Wen Yang

Abstract—In this paper, a new filtering problem for sensor networks is investigated. A new type of distributed consensus filters is designed, where each sensor can communicate with the neighboring sensors, and filtering can be performed in a distributed way. In the pinning control approach, only a small fraction of sensors need to measure the target information, with which the whole network can be controlled. Furthermore, pinning observers are designed in the case that the sensor can only observe partial target information. Simulation results are given to verify the designed distributed consensus filters. Index Terms—Consensus, distributed consensus filter, pinning control, pinning observer, sensor network.

I. I NTRODUCTION

S

ENSOR networks have attracted increasing attention from many researchers in different fields due to their widescope applications in robotics, surveillance and environment monitoring, information collection, wireless networks, and so on. A sensor network consists of a large number of sensor nodes that are distributed over a spatial region. Each sensor performs some level of communication, intelligence for signal processing, and data fusion, which build up a sensing network. Due to the limited single-sensor energy, computational ability, and communication capability, a large number of sensor nodes are commonly used in a wide region in practical applications. In general, each sensor node is equipped with a microelectronic device with limited power source; hence, it might not be possible to transmit some messages over a large sensor network. To save energy, a natural way is to carry out data fusion to reduce the communication overhead. Therefore, distributed estimation and tracking is one of the most important problems in largescale sensor networks. Since sensor networks are usually of large scale, it is literally impossible to employ a centralized signal processor, so distributed algorithms and devices are more preferable [1], [2]. Manuscript received August 18, 2008; revised December 30, 2008. First published May 15, 2009; current version published November 18, 2009. This work was supported in part by the Hong Kong Research Grants Council under CERG Grant CityU 1117/08E, by the NSFC-HKRGC Joint Research Scheme under Grant N-CityU107/07, and by the Royal Society of the United Kingdom under an international joint project. This paper was recommended by Associate Editor S. X. Yang. W. Yu and G. Chen are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: wenwuyu@gmail. com; [email protected]). Z. Wang is with the Department of Information Systems and Computing, Brunel University, UB8 3PH Uxbridge, U.K. (e-mail: Zidong.Wang@brunel. ac.uk). W. Yang is with the School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2009.2021254

With regard to filtering, the now-classical Kalman filtering [3], H∞ filtering [4], and finite-horizon filtering [5] have been widely investigated. However, most of the existing works are based on centralized schemes which can collect information from all the sensor nodes. In the past decades, on the other hand, the large-scale system theory has intensively been studied, and a lot of interesting results have been obtained for the design of decentralized controllers [6]. Compared with centralized control approaches, decentralized control has its advantages such as low dimensions, fast implementation, and low cost, and generally, communication among subsystems is not needed. However, the decentralized control is not effective without involving information exchange among the subsystems. Recently, distributed estimation and control have been shown to be very effective for consensus [7]–[9]. In a distributed framework, each node can only communicate with its neighboring peers, and the objective of filtering or control can be achieved in a distributed way. Noticeably, the convergence analysis for distributed filtering is still lacking today, so this paper aims to provide some basic theoretical analysis of a new class of distributed consensus filters. From a network-theoretic point of view, a large-scale sensor network can be viewed as a complex network with each node representing a sensor and each edge performing information exchange between sensors. It would be interesting to see how synchronization of complex networks [10]–[19] can be used in the distributed consensus filtering design. In such a network, each node communicates with its neighboring nodes to exchange information, and eventually, all the states could achieve the expected synchronization, which achieves distributed consensus filtering. In 1998, Pecora and Carroll proposed a master stability function to study the synchronization of coupled complex systems [20]. Thereafter, stability and synchronization of small-world and scale-free networks have extensively been investigated by using this master stability function method. In [10] and [11], local synchronization was studied based on the transverse stability to the synchronization manifold, where synchronization was discussed with respect to small-world and scale-free network topologies. In [21], a distance from the collective states to the synchronization manifold was defined and then utilized to obtain conditions for global synchronization of coupled systems [12], [13], [22]. More recently, a general criterion has been given in [23] by using linear matrix inequality (LMI) techniques. In the case where the whole network cannot synchronize, some controllers may be designed and applied to force the network to be synchronized or stabilized [24]. However, it is practically impossible to add controllers to all nodes. To reduce the number of controlled nodes, some local feedback

1083-4419/$25.00 © 2009 IEEE

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

YU et al.: DISTRIBUTED CONSENSUS FILTERING IN SENSOR NETWORKS

1569

injections may be applied to a small fraction of the network nodes. Such an idea is known as pinning control [25]–[29]. In [28], both specific and random pinning schemes were studied and compared. In [26], several new pinning schemes were proposed. It was found that the nodes with low degrees should be pinned first, and the derived pinning condition with controllers given in a higher dimensional setting can be reduced to a lower dimensional condition without the pinning controllers involved. Practically, it is usually difficult to observe all the states of the target, so pinning observers may be designed in the case where the sensors can only measure partial states of the target. In this paper, based on the theory of synchronization and consensus in complex networks and systems, some distributed consensus filters are designed. From the pinning control approach, only a small fraction of sensors are used to measure the target information, thereby achieving the intended control. Furthermore, pinning observers are designed when the sensor can observe only partial states of the target. Several theoretical results and design methods are new to the existing research literature. The rest of this paper is organized as follows. In Section II, some preliminaries are given. Distributed consensus filter design problems in sensor networks with fully equipped controllers, pinning controllers, and pinning observers are discussed in Sections III–V, respectively. In Section V, simulation examples are illustrated. Finally, Section VI concludes the paper. II. P RELIMINARIES

(1)

where s(t) ∈ Rn is the state of the target, v(t) ∈ Rn is an external noise intensity function, and ν(t) is a 1-D Brownian motion with expectation E{dν(t)} = 0 and variance D{dν(t)} = 1. The model is defined on a complete probability space (Ω, F, P) with a natural filtration {Ft }t≥0 generated by {ν(s) : 0 ≤ s ≤ t}, where Ω is associated with the canonical space generated by ν(t), and F is the associated σ-algebra generated by {ν(t)} with probability measure P. Consider a sensor network of size N and assume that if sensor i can measure the signal s(t), then yi (t) = s(t) + σi (t)dωi /dt

x˙ i (t) = f (xi (t), t) + c

N 

aij (xj (t) − xi (t)) + ui (t),

j=1,j=i

i = 1, 2, . . . , N

(2)

where yi (t) ∈ Rn is the measurement of sensor i on target s(t), σi (t) is an external noise intensity function of agent i, and ωi (t) is an independent 1-D Brownian motion with expectation E{dωi } = 0 and variance D{dωi } = 1, i = 1, 2, . . . , N . Note that there are a large number of sensors that can measure the target s(t) from observations yi (t), i = 1, 2, . . . , N . However, it is still a challenging problem as how to carry out the data fusion if there is not a centralized processor capable of collecting all the measurements from the sensors. The objective here, therefore, is to design a distributed filter to track the state s(t) of the target.

(3)

where xi (t) is the estimation of the target s(t) in sensor node i, c is the coupling strength, and ui (t) is the designed controller that is dependent on the measurement yi . If sensor i is in the sensing range of sensor j, then there is a connection between sensor i and sensor j, i.e., aij = aji > 0 (i = j); otherwise,  aij = aji = 0. Let aii = − N j=1,j=i aij for i = 1, 2, . . . , N , and N (i) = {j|aij > 0} denote the set of neighbors of sensor i. Then, (3) can be written as x˙ i (t) = f (xi (t), t) + c

N 

aij xj (t) + ui (t),

j=1

i = 1, 2, . . . , N.

(4)

From (4), it is easy to see that the sensor i can only receive estimated signals from its neighbors in N (i). This paper only considers the situation where the sensor network coupling matrix A = (aij )N ×N is irreducible. Assumption 1: For all x, y ∈ Rn , there exists a constant θ such that (x−y)T (f (x, t)−f (y, t)) ≤ θ(x − y)T (x − y),

Let the target be described by the following model: ds(t) = f (s(t), t) dt + v(t)dν

Suppose that each sensor can only communicate with the neighboring sensors. Taking the measurement yi as the input, the following filter is designed:

∀t ∈ R. (5)

Assumption 2: Both v(t) ∈ Rn and σi (t) ∈ Rn belong to L∞ [0, ∞), i.e., v(t) and σi (t) are bounded vector functions satisfying v T (t)v(t) ≤ α,

∀t ∈ R

(6)

≤ βi ,

∀t ∈ R

(7)

σiT (t)σi (t)

where α and βi are positive constants, i = 1, 2, . . . , N . Assumption 3: For all x ∈ Rn , there exists a constant γ > 0 such that f (x, t) ≤ γ,

∀t ∈ R.

(8)

Definition 1: The designed controllers ui , i = 1, 2, . . . , N , are said to be distributed bounded consensus controllers if there exist constants φ > 0, ηi > 0, and μ > 0 such that N  N   1 xi (t) − s(t)2 ≤ φγ + ηi βi +μα. (9) lim E t→∞ N i=1 i=1 If φ = 0, then they are called distributed consensus controllers. Definition 2: The designed filters (3) or (4) are said to be distributed bounded consensus filters (distributed consensus filters) if the controllers in (3) or (4) are distributed bounded consensus controllers (distributed consensus controllers). In thefollowing, the convergence of the bound for 2 (1/N )E( N i=1 xi (t) − s(t) ) will be analyzed.

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

1570

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 6, DECEMBER 2009

Lemma 1 [27]: If A = (aij )N ×N is irreducible, aij =  aji ≥ 0, for i = j, and N j=1 aij = 0, for all i = 1, 2, . . . , N , then all eigenvalues of the matrix ⎞ ⎛ a11 − ε a12 . . . a1N a22 . . . a2N ⎟ ⎜ a21 ⎜ .. .. .. ⎟ .. ⎝ . . . . ⎠ aN 2

aN 1

...

By Assumption 2, the weak infinitesimal operator L of the stochastic process gives LV (t) =

N 





eTi (t)

f (xi (t), t) − f (s(t), t)

i=1 N 

+c

aN N

 1 T c2 ki2 T + v (t)v(t) + σ (t)σi (t) 2 2 i ≤θ

N 

eTi (t)ei (t) + c

i=1

1 c + Nα + 2 2

III. D ISTRIBUTED C ONSENSUS F ILTER D ESIGN FOR S ENSOR N ETWORKS W ITH F ULLY E QUIPPED C ONTROLLERS

i = 1, 2, . . . , N

(10)

where ki > 0 is the feedback control gain [26]. Let K = diag(k1 , k2 , . . . , kN ), B = (bij )N ×N = A − K, and λmax (B) denote the largest eigenvalue of matrix B. Subtracting (1) from (4) with controllers (10) yields the following error dynamical network: ⎡ ⎤ N  aij ej (t)−cki ei (t)⎦ dt dei (t) =⎣f (xi (t), t)−f (s(t), t)+ c

+

(12)

1 eTi (t)ei (t) + N α 2 i=1

N c2  2 k βi . 2 i=1 i

LV (t) ≤ (θ+cλmax (B))

(16)

N 

c2 1 eTi (t)ei (t)+ N α+ 2

i=1

2

N 

ki2 βi

i=1

= N (θ+cλmax (B))   2  N 2 α+ cN N 1 T i=1 ki βi . e (t)ei (t)− × N i=1 i −2 (θ+cλmax (B))

(11)

where ei (t) = xi (t) − s(t), i = 1, 2, . . . , N . Theorem 1: Suppose that Assumptions 1 and 2 hold. The designed controllers (10) are distributed consensus controllers if

ki2 βi

In view of Lemma 1, λmax (B) < 0. From (12), it follows that

j=1

− v(t)dν +cki σi (t)dωi

bij eTi (t)ej (t)

i=1 N 

≤ (θ + cλmax (B))

For simplicity, consider linear state-feedback controllers

N  N  i=1 j=1

N 2 

θ + cλmax (B) < 0.

aij ej (t) − cki ei (t)

j=1

are negative for any positive constant ε. Lemma 2 [21]: Assume that an undirected network is irreducible. Then, matrix A has an eigenvalue of zero with an algebraic multiplicity of one, and all the other eigenvalues are negative: 0 = λ1 (A) > λ2 (A) ≥ · · · ≥ λN (A).

ui (t) = −cki (xi (t) − yi (t)) ,



(17)

From the Itô formula, it follows that EV (t) − EV (0) t LV (s)ds

=E 0

The estimated bound is given by N  2  2  α + cN N 1 2 i=1 ki βi . xi (t) − s(t) ≤ lim E t→∞ N −2 (θ + cλmax (B)) i=1 (13) Proof: Consider candidate:

the

V (t) =

following

1 2

N 

Lyapunov

eTi (t)ei (t).

N 

≤ N (θ + cλmax (B)) 0

function

N 1  T E e (t)ei (t) N i=1 i

 2  2 α + cN N i=1 ki βi ds. − −2 (θ + cλmax (B))

 2 2 (13), if (1/N )E( N i=1 ei (t) ) > ((α + (c /N ) 2 i=1 ki βi )/ − 2(θ + cλmax (B))), then EV (t)−EV (0) < 0. This completes the proof. By letting ki = k > 0, i = 1, 2, . . . , N , in (10), one has the following corollary. Corollary 1: Suppose that Assumptions 1 and 2 hold. The designed controllers (10) are distributed consensus controllers if Under N

(14)

i=1

From the Itô formula [2], [30], one obtains the following stochastic differential: dV (t) = LV (t)dt +

t 

eTi (t) [−v(t)dν + cki σi (t)dωi ] .

i=1

(15)

θ − ck < 0.

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

(18)

YU et al.: DISTRIBUTED CONSENSUS FILTERING IN SENSOR NETWORKS

1571

The estimated bound is given by N  2  2  α + cN N 1 2 i=1 k βi . xi (t) − s(t) ≤ lim E t→∞ N 2(ck − θ) i=1 (19) Proof: By Lemma 2 and Theorem 1, the proof can easily be completed.  2 Denote g(k) = ((α+(c2 /N ) N − θ)). Then, i=1 k βi )/2(ck  N

3 2 one has g (k) = (((c /N ) i=1 βi k − 2(c2 /N ) N i=1 βi θk − 2

). Let g (k) = 0. Then, it follows that k¯1,2 = cα)/2(ck − θ)   N ((θ ± θ2 + (α/(1/N ) i=1 βi ))/c). Since k¯1 > k > θ/c > k¯2 , if k¯1 > k > 0, then g (k) < 0; if k > k¯1 , then g (k) > 0. Therefore, there is a minimum value of g(k) at k = k¯1 . If N = 1, i.e., only one sensor is used to track the target, one has the following corollary. Without loss of generality, let the first node be this sensor. Corollary 2: Suppose that Assumptions 1 and 2 hold. The designed controllers (10) are distributed consensus controllers if θ − ck1 < 0.

From Lemma 1, it follows that λmax (B) < 0. Note that N 

⎛  ⎞2  N 2  c2 2 −cλ /2 γ + γ (B) α+ k β max i=1 i i N ⎜ ⎟ ⎟ . (22) ≤⎜ ⎝ ⎠ −cλmax (B)

Proof: Choose (14) as the Lyapunov function candidate. By Assumption 3, the weak infinitesimal operator L of the stochastic process yields LV (t) ≤

N 

eTi (t) [f (xi (t), t) − f (s(t), t)]

i=1

+ cλmax (B)

≤ 2γ

N 

N 

N c2  2 1 eTi (t)ei (t) + N α + k βi 2 2 i=1 i i=1

ei (t) + cλmax (B)

i=1

N 

ei (t)2

i=1 N 2 

c 1 + Nα + 2 2

i=1

ki2 βi .

(23)

N  N 

ei (t) ej (t)

i=1 j=1

 1  ei (t)2 + ej (t)2 2 i=1 j=1 N

=N

N 

N

ei (t)2 .

i=1

Then, it follows that LV (t) ≤ cλmax (B)

N 

 N  √ 2 ei (t) + 2γ N  ei (t)2

i=1

i=1

c2  2 1 + Nα + k βi 2 2 i=1 i  N 1  2γ = cN λmax (B) ei (t)2 + N i=1 cλmax (B)   N 1  ei (t)2 × N i=1  2  2 α + cN N i=1 ki βi . − −2cλmax (B) N

(21)

N   1 2 lim E xi (t)−s(t) t→∞ N i=1

=

≤

The estimated bound is given by

Corollary 3: Suppose that Assumptions 2 and 3 hold. The designed controllers (10) are distributed bounded consensus controllers, and the estimated bound is given by

ei (t)

i=1

(20)

 α + c2 k 2 β  1 1 . lim E x1 (t) − s(t)2 ≤ t→∞ 2(ck1 − θ)

2

(24)

  2 (1/N ) N and g(z) = z 2 + (2γ/ i=1 ei (t)  2 cλmax (B))z − ((α + (c2 /N ) N i=1 ki βi )/ − 2cλmax (B)). It is easy to see that g(z) = 0 has two solutions    2  2 −γ ± γ 2 − cλmax (B) α + cN N i=1 ki βi /2 z1,2 = cλmax (B) (25) Let

z=

where z1 < 0, and z2 > 0. If z(t) ≥ z2 , then g(z) ≥ 0, and EV (t) − EV (0) < 0. The proof is completed. Remark 1: One may wonder that if only one sensor can achieve the filtering performance as shown by Corollary 2, then why are so many (N ) sensors used in Theorem 1 and Corollary 1? This can be explained as follows. First, in a sensor network consisting of a large number of sensor nodes in a wide spatial region, it is impossible to have only one centralized processor that can collect the measurements from all the sensors, particularly in a remote area. Usually, each sensor may only be able to use local information and communicate with neighbors so that the estimation can be achieved in a distributed way. Second, in a sensor network, using a large number of sensors to observe the target can usually obtain more accurate estimation than using a single sensor node. Third, in (2), every state of the target can be observed, which is not always the case in applications. The next few sections will deal with the case where a sensor can only measure some states of

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

1572

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 6, DECEMBER 2009

the target. In that case, using only one sensor cannot accurately estimate the target.   Remark 2: From k¯1 = (1/c)(θ+ θ2 +(α/(1/N ) N βi )), i=1

it follows that the minimal bound increases as α or θ increases, while it decreases when c or (1/N ) N i=1 βi increases. Therefore, it is unreasonable to use a very large control gain if the noise intensity βi is relatively high. IV. D ISTRIBUTED C ONSENSUS F ILTER D ESIGN FOR S ENSOR N ETWORKS W ITH P INNING C ONTROLLERS In many cases, it is literally impossible to add controllers to all sensors. To reduce the number of controlled sensors, some local feedback injections may be applied to a small fraction of sensors. This approach is known as pinning control [26]. Here, the pinning strategy is applied to a small fraction δ (0 < δ < 1) of the sensors in network (3). Without loss of generality, let the first l = δN  nodes be controlled, where · is the integer part of a real number. Thus, the designed pinning controllers can be described by ui (t) = − cki (xi (t) − yi (t)) , ui (t) = 0,

(26)

where ki > 0 is the feedback control gain. Let K = diag(k1 , . . . , kl , 0, . . . , 0), B = A − K = (¯bij )N ×N , and       N −l

l

λmax (B) denote the largest eigenvalue of matrix B. Subtracting (1) from (4) with controllers (26) yields the following error dynamical network: ⎡ dei (t) = ⎣f (xi (t), t) − f (s(t), t) + c

N 

⎤ ¯bij ej (t)⎦ dt

j=1

− v(t)dν + cki σi (t)dωi , ⎡

i = 1, 2, . . . , l ⎤ N  ¯bij ej (t)⎦ dt dei (t) = ⎣f (xi (t), t) − f (s(t), t) + c j=1

− v(t)dν,

i = l + 1, l + 2, . . . , N

(27)

where ei (t) = xi (t) − s(t), i = 1, 2, . . . , N . Theorem 2: Suppose that Assumptions 1 and 2 hold. The designed controllers (26) are distributed consensus controllers if θ + cλmax (B) < 0.



⎛ ⎜γ + ≤⎜ ⎝

γ 2 −cλ

 max (B)

2 α+ cN

l

2 i=1 ki βi

−cλmax (B)



⎞2 /2 ⎟ ⎟ . (30) ⎠

To further simplify the result, the following lemma is needed. Lemma 4 (Schur Complement [31], [32]): The following LMI: # " Q(x) S(x) >0 S(x)T R(x) where Q(x) = Q(x)T and R(x) = R(x)T , is equivalent to one of the following conditions: (i)Q(x) > 0 R(x) − S(x)T Q(x)−1 S(x) > 0

i = 1, 2, . . . , l,

i = l + 1, l + 2, . . . , N

Corollary 4: Suppose that Assumptions 2 and 3 hold. The designed controllers (10) are distributed bounded consensus controllers, and the estimated bound is given by N   1 2 xi (t) − s(t) lim E t→∞ N i=1

(28)

The estimated bound is given by N  2   α + cN li=1 ki2 βi 1 2 !. lim E ≤ xi (t) − s(t) t→∞ N −2 θ + cλmax (B) i=1 (29) Proof: From Lemma 1, one knows that B is negative definite. Thus, using Theorem 1 completes the proof.

(ii)R(x) > 0

Q(x) − S(x)R(x)−1 S(x)T > 0.

Rewrite B as

" B=

$ A1 − K AT2

A2 $ B

#

$ = diag(k1 , . . . , kl ), A1 and A2 are matrices with where K $ is obtained by removing the appropriate dimensions, and B 1, 2, . . . , l row–column pairs of matrix A. Corollary 5: Suppose that Assumptions 1 and 2 hold. If the control gain ki is sufficiently large, then the condition (28) is equivalent to $ < 0. θ + cλmax (B)

(31)

The estimated bound is given by N  2   α + cN li=1 βi 1 2  . xi (t) − s(t) ≤ lim E t→∞ N $ −2 θ + cλmax (B) i=1 (32) $ −1 AT )Il $ > λmax (A1 − A2 B For simplicity, one may select K 2 as in [26]. $ = λmax (B), one Proof: To derive the result λmax (B) $ − λIN −l < 0 is equivalent to B − only needs to prove that B λIN < 0 for any positive λ > 0. $ − λIN −l < 0. It is easy to see that if B − λIN < 0, then B $ − λIN −l < 0, then B − Therefore, it suffices to prove that if B $ −1 AT . Then, by Lemma 4, $ > A1 − A2 B λIN < 0. Choose K 2 one indeed has B − λIN < 0. The proof is completed. Remark 3: In this section, only a small fraction of sensors are used to measure the target, which is more practical in real applications. Note that the sensors with measurements can communicate with the neighboring sensors, some of which cannot observe the target. The whole process may be considered

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

YU et al.: DISTRIBUTED CONSENSUS FILTERING IN SENSOR NETWORKS

1573

as a leader–follower behavior, where the target is the true leader, the measured sensors are virtual leaders, and the other sensors are the followers. Remark 4: Although −θ − cλmax (B) < −θ − cλmax (B) in Theorems 1 and 2, the sum of the bound (from 1 to l) in Theorem 2 (14) is lower than that in Theorem 1 (from 1 to N ). Therefore, the bound in Theorem 2 is not necessarily lower than that in Theorem 1. In fact, a higher dimensional condition (29) can be equivalent to a lower dimensional condition (31). V. D ISTRIBUTED C ONSENSUS F ILTER D ESIGN FOR S ENSOR N ETWORKS W ITH P INNING O BSERVERS In many cases, it is impractical to assume that a sensor can observe all the states of the target in (2). Therefore, assume that the first l = δN  sensors can measure the signals by y$i (t) = Hi s(t) + σi (t)dωi /dt,

i = 1, 2, . . . , l

ui (t) = 0,

i = 1, 2, . . . , l

i = l + 1, l + 2, . . . , N

(34)

where Di ∈ Rn×m is the feedback control gain matrix. Subtracting (1) from (4) with observers (33) and controllers (34) yields the following error dynamics:  N  aij ej (t) dei (t) = f (xi (t), t) − f (s(t), t) + c

− cDi Hi ei dt − v(t)dν + cDi Hi σi (t)dωi ,

dei (t) = ⎣f (xi (t), t) − f (s(t), t) + c

N 

θ + cλmax (A − Ξ) < 0

⎤

(35)

−εj s2j

for any s = (s1 , s2 , . . . , sn ) ∈ R . Definition 4: The state j (1 ≤ j ≤ n) of the target is said to be observable by the sensor network if it is observable by a sensor k, where 1 ≤ k ≤ l. Definition 5: The target is said to be observable by the sensor network if each of its state is observable by the sensor network. T

(36)

(37)

where λmax (A−Ξ) = maxj (λmax (A−Ξj )), and Ξj = diag(0, . . . , εj1 j , . . . , εjp j , . . . , 0) ∈ RN ×N , i.e., the ji th diagonal   j1

jp

element is εji j , and ji ∈ Mj , j = 1, 2, . . . , n, i = 1, . . . , p. The estimated bound is given by N   1 2 lim E xi (t) − s(t) t→∞ N i=1 ≤

N 

α+

! λmax HiT DiT Di Hi βi . −2 (θ + cλmax (A − Ξ))

c2 N

$l

i=1

(38)

 {eTi (t)

f (xi (t), t) − f (s(t), t)  1 T +c aij ej (t) + v (t)v(t) 2 j=1 N 

& l %  T + ei (t) − cDi Hi ei

where ei (t) = xi (t) − s(t), i = 1, 2, . . . , N . Definition 3: The state j (1 ≤ j ≤ n) of the target is said to be observable by sensor i (1 ≤ k ≤ l) if there is a gain matrix Di and a positive constant εj such that −s Di Hi s ≤

εkj e2kj

where the state j is observable by sensors k (1 ≤ k ≤ $ l), εkj are positive constants, and Mj are the sets of all sensors that can observe the target state j, j = 1, . . . , n. Theorem 3: Suppose that Assumptions 1, 2, and 4 hold. The designed controllers (34) are distributed consensus controllers if

aij ej (t)⎦ dt

i = l + 1, l + 2, . . . , N

T

n   j=1 k∈Mj

i=1

j=1

− v(t)dν,

eTi (t)Di Hi ei ≤ −

i=1

LV (t) =

i = 1, 2, . . . , l

⎡

$l 

Proof: Consider the Lyapunov function candidate (14). The weak infinitesimal operator L of the stochastic process gives

j=1



−

(33)

where y$i (t) ∈ Rm is the measurement of sensor i by observing the target s(t), and Hi ∈ Rm×n , i = 1, 2, . . . , l. The designed distributed consensus filter controller is described by ui (t) = − cDi (Hi xi (t) − y$i (t)) ,

Assumption 4: Suppose that the target is observable and, without loss of generality, suppose that the target is observable by the first $ l ($ l ≤ l) sensors. Then, there exist matrices Di , i = 1, 2, . . . , $ l, such that

i=1

+ ≤θ

N 

'( c2 T σi (t)(Di Hi )T (Di Hi )σi (t) 2

eTi (t)ei (t) + c

i=1

n

−c

+

N  N 

aij eTi (t)ej (t)

i=1 j=1

l 

1 eTi (t)Di Hi ei + N α 2 i=1

l ! c2  λmax HiT DiT Di Hi βi . 2 i=1

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

(39)

1574

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 6, DECEMBER 2009

Then, by Assumption 4, one has N  N 

aij eTi (t)ej (t) −

i=1 j=1

≤

l 

eTi (t)Di Hi ei

i=1

N  N 

aij eTi (t)ej (t) −

i=1 j=1

−

l 

n  

εkj e2kj

j=1 k∈Mj

eTi (t)Di Hi ei

i=$ l+1

=

n 

e$Tj (t)A$ ej (t) −

j=1

−

n 

e$Tj (t)Ξj e$j (t)

j=1

l 

eTi (t)Di Hi ei

(40) Fig. 1. Orbits of the measured data yi in sensors and filtering data xj , i = 1, 2, . . . , 20 and j = 1, 2, . . . , 100.

i=$ l+1

where e$j = (e1j , e2j , . . . , eN j )T , j = 1, 2, . . . , n. For simplicity, choose Di = 0, for i = $ l + 1, . . . , l. Under (37), it follows that LV (t) ≤ (θ + cλmax (A − Ξ))

+

N 

1 eTi (t)ei (t) + N α 2 i=1

$l ! c2  λmax HiT DiT Di Hi βi 2 i=1

= N (θ + cλmax (A − Ξ)) ⎡ $l 1  T ×⎣ e (t)ei (t) N i=1 i −

α+

c2 N

$l

T T i=1 λmax Hi Di Di Hi

−2 (θ + cλmax (A − Ξ))

VI. S IMULATION E XAMPLES !

⎤ βi

(38),

if

 2 2 (1/N )E( N i=1 ei (t) ) > (α + (c /N )

λmax (HiT DiT Di Hi )βi )/ − 2(θ + cλmax (A − Ξ))), then EV (t) − EV (0) < 0. This completes the proof. Corollary 6: Suppose that Assumptions 2–4 hold. Then, the designed controllers (34) are distributed bounded consensus controllers, and the estimated bound is given by i=1

N   2 )  1 $ γ + γ2 − α 2 xi (t) − s(t) ≤ lim E t→∞ N −cλmax (A − Ξ) i=1 $ where α $ = cλmax (A−Ξ)(α+(c2/N ) li=1 λmax (HiT DiT Di Hi ) βi )/2, λmax (A − Ξ) = maxj (λmax (A − Ξj )), and Ξj = diag(0, . . . , εj1 j , . . . , εjp j , . . . , 0) ∈ RN ×N , i.e., the ji th   j1

jp

In this section, some simulation examples are provided to verify the designed distributed (bounded) consensus filters.

⎦. (41)

Under $l

Remark 5: In (2), each sensor can observe the full state of the target. However, this is not always the case in practice. In this section, only partial state information about the target is assumed to be observed by the sensors in (34), and the target is observable by the sensor network (Definition 5), which is very reasonable for real sensor networks. Remark 6: If the coupling matrix A is not irreducible, which means that the network is not connected, then the whole network can be composed of many connected components. The filtering design in this paper can be extended to the disconnected network by studying its connected components.

diagonal element is εji j , and ji ∈ Mj , j = 1, 2, . . . , n, i = 1, . . . , p.

A. Distributed Bounded Consensus Filters in Sensor Networks With Pinning Controllers Consider an ER random network [33] and suppose there are N = 100 nodes with p = 0.1. This random network has about pN (N − 1)/2 ≈ 500 connections. If there is a connection between node i and j, then aij = aji = 1 (i = j), i, j = 1, 2, . . . , 100. The target and the measurement models are described by (1) and (2), respectively, where f (s(t)) = cos(t) with initial condition s(0) = 0, v(t) = 0.5, and σi (t) = 0.4. The designed distributed consensus filter is (3) with controllers (26), where c = 1, ki = 5, and l = 20. Here, a simulation-based analysis on the controlled random network is performed by using the random pinning scheme. In the random pinning scheme, one randomly selects δN  nodes with a small fraction δ = 0.2 to pin. The orbits of the measured data yi in sensors and the filtering data xj are shown in Fig. 1 for i = 1, 2, . . . , 20 and j = 1, 2, . . . , 100. By Corollary 4, the designed pinning controllers are distributed bounded consensus controllers. The orbits of the errors ei are illustrated in Fig. 2.

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

YU et al.: DISTRIBUTED CONSENSUS FILTERING IN SENSOR NETWORKS

Fig. 2.

Orbits of the errors ei , i = 1, 2, . . . , 100.

Fig. 3.

Chaotic orbits of Chua’s circuit.

B. Distributed Consensus Filters in Sensor Networks With Pinning Observers Here, a simulation-based analysis on the controlled scale-free network is performed by using the high-degree pinning scheme. In the simulated scale-free network [34], N = 100, and m0 = m = 3, which contains about 3000 connections. In the highdegree pinning scheme, one first pins the node with the highest degree and then continue to choose and pin the other nodes in monotonically decreasing order of node degrees. The target model is described by Chua’s circuit [35] ⎧ ⎨ ds1 = [η (−s1 + s2 − l(s1 ))] dt + v1 (t)dν (42) ds2 = [s1 − s2 + s3 ]dt + v2 (t)dν ⎩ ds3 = [−βs2 ]dt + v3 (t)dν where l(x1 ) = bx1 + 0.5(a − b)(|x1 + 1| − |x1 − 1|) and vi (t) = 0.5, i = 1, 2, 3. System (42) is chaotic without noise when η = 10, β = 18, a = −4/3, and b = −3/4, as shown in Fig. 3. In view of Assumption 1, by computation, one obtains θ = 5.1623. The measurement model is described by (33), where Hi is chosen as (1, 0, 0), (0, 1, 0), or (0, 0, 1), with σi (t) = 0.5. The designed distributed consensus filter is (3) with controllers (34), where c = 15, l = 20, and the corresponding Di is chosen as

1575

Fig. 4. Orbits of estimation xi , i = 1, 2, . . . , 100.

Fig. 5. Orbits of the errors ei , i = 1, 2, . . . , 100.

(50, 0, 0)T , (0, 50, 0)T , or (0, 0, 50)T . Assumption 4 is satisfied for εkj = 50, k = 1, . . . , l, and j = 1, 2, 3. The orbits of the estimation xi are shown in Fig. 4 for i = 1, 2, . . . , 100. Since θ = 5.1623 < −cλmax (A − Ξ) = 6.8565, by Theorem 4, the designed pinning controllers are distributed consensus controllers. The orbits of the errors ei are illustrated in Fig. 5. VII. C ONCLUSION In this paper, some distributed consensus filters in sensor networks have been designed, analyzed, and simulated. Three scenarios have been considered. Under the condition that each sensor can measure the target, the distributed controllers could be added to all sensor nodes. By using the pinning control approach and assuming that only a small fraction of sensors can measure the target, the network could still be controlled. Moreover, pinning observers have been designed under the condition that a sensor can only observe partial states of the target. All these designs have been analyzed and validated by simulations. Distributed consensus filters are every effective, with many advantages for its easier implementation, low cost, fast speed,

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

1576

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 6, DECEMBER 2009

etc. A general framework for distributed consensus filter design has been presented in this paper, which is a promising approach, therefore deserving further investigation in the near future. R EFERENCES [1] W. Yu, G. Chen, J. Cao, J. Lü, and U. Parlitz, “Parameter identification of dynamical systems from time series,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 75, no. 6, p. 067 201, Jun. 2007. [2] W. Yu and J. Cao, “Synchronization control of stochastic delayed neural networks,” Physica A, vol. 373, pp. 252–260, Jan. 2007. [3] F. Yang, Z. Wang, and Y. S. Hung, “Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises,” IEEE Trans. Autom. Control, vol. 47, no. 7, pp. 1179–1183, Jul. 2002. [4] R. Yang, L. Xie, and C. Zhang, “H2 and mixed H2 /H∞ control of two-dimensional systems in Roesser model,” Automatica, vol. 42, no. 9, pp. 1507–1514, Sep. 2006. [5] Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust finite-horizon filtering for stochastic systems with missing measurements,” IEEE Signal Process. Lett., vol. 12, no. 6, pp. 437–440, Jun. 2005. [6] Z. Duan, J. Wang, G. Chen, and L. Huang, “Stability analysis and decentralized control of a class of complex dynamical networks,” Automatica, vol. 44, no. 4, pp. 1028–1035, Apr. 2008. [7] R. Olfati-Saber, “Distributed Kalman filter with embedded consensus filters,” in 44th IEEE Conf. Decis. Control/Eur. Control Conf., Dec. 2005, pp. 8179–8184. [8] R. Olfati-Saber and J. S. Shamma, “Consensus filters for sensor networks and distributed sensor fusion,” in 44th IEEE Conf. Decis. Control/Eur. Control Conf., Dec. 2005, pp. 6698–6703. [9] Y. Hong, G. Chen, and L. Bushnell, “Distributed observers design for leader-following control of multi-agent networks,” Automatica, vol. 44, no. 3, pp. 846–850, Mar. 2008. [10] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: Robustness and fragility,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 54–62, Jan. 2002. [11] X. Wang and G. Chen, “Synchronization in small-world dynamical networks,” Int. J. Bifurc. Chaos, vol. 12, no. 1, pp. 187–192, 2002. [12] W. Lu and T. Chen, “Synchronization of coupled connected neural networks with delays,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 12, pp. 2491–2503, Dec. 2004. [13] G. Chen, J. Zhou, and Z. Liu, “Global synchronization of coupled delayed neural networks and applications to chaotic CNN models,” Int. J. Bifurc. Chaos, vol. 14, no. 7, pp. 2229–2240, 2004. [14] J. Zhou, J. Lu, and J. Lü, “Adaptive synchronization of an uncertain complex dynamical network,” IEEE Trans. Autom. Control, vol. 51, no. 4, pp. 652–656, Apr. 2006. [15] J. Lü and G. Chen, “A time-varying complex dynamical network models and its controlled synchronization criteria,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 841–846, Jun. 2005. [16] J. Cao, W. Yu, and Y. Qu, “A new complex network model and convergence dynamics for reputation computation in virtual organizations,” Phys. Lett. A, vol. 356, no. 6, pp. 414–425, Aug. 2006. [17] J. Liang, Z. Wang, Y. Liu, and X. Liu, “Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 4, pp. 1073–1083, Aug. 2008. [18] Y. Liu, Z. Wang, J. Liang, and X. Liu, “Synchronization and state estimation for discrete-time complex networks with distributed delays,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 5, pp. 1314–1325, Oct. 2008. [19] W. Yu, J. Cao, G. Chen, J. Lü, J. Han, and W. Wei, “Local synchronization of a complex network model,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 1, pp. 230–241, Feb. 2009. [20] L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2109–2112, Mar. 1998. [21] C. Wu and L. O. Chua, “Synchronization in an array of linearly coupled dynamical systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 8, pp. 430–447, Aug. 1995. [22] J. Cao, G. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with hybrid coupling,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 2, pp. 488–498, Apr. 2008. [23] W. Yu, J. Cao, and J. Lü, “Global synchronization of linearly hybrid coupled networks with time-varying delay,” SIAM J. Appl. Dyn. Syst., vol. 7, no. 1, pp. 108–133, 2008.

[24] W. Yu, J. Cao, and G. Chen, “Robust adaptive control of unknown modified Cohen–Grossberg neural networks with delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 6, pp. 502–506, Jun. 2007. [25] J. Xiang and G. Chen, “On the V-stability of complex dynamical networks,” Automatica, vol. 43, no. 6, pp. 1049–1057, Jun. 2007. [26] W. Yu, G. Chen, and J. Lü, “On pinning synchronization of complex dynamical networks,” Automatica, vol. 45, no. 2, pp. 429–435, 2009. [27] T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 6, pp. 1317–1326, Jun. 2007. [28] X. Wang and G. Chen, “Pinning control of scale-free dynamical networks,” Physica A, vol. 310, no. 3/4, pp. 521–531, Jul. 2002. [29] J. Zhou, X. Wu, W. Yu, M. Small, and J. Lu, “Synchronizing delayed neural networks by pinning control,” Chaos, vol. 18, no. 4, p. 043 111, Dec. 2008. [30] Z. Schuss, Theory and Applications of Stochastic Differential Equations. New York: Wiley, 1980. [31] W. Yu, J. Cao, and J. Wang, “An LMI approach to global asymptotic stability of the delayed Cohen–Grossberg neural network via nonsmooth analysis,” Neural Netw., vol. 20, no. 7, pp. 810–818, Sep. 2007. [32] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [33] P. Erdös and A. Rényi, “On random graphs,” Publ. Math., vol. 6, pp. 290– 297, 1959. [34] A. L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, Oct. 1999. [35] L. O. Chua, “The genesis of Chua’s circuit,” Arch. Elektr. Ubertrag., vol. 46, no. 3, pp. 250–257, 1992.

Wenwu Yu (S’07) received the B.Sc. degree in information and computing science and the M.Sc. degree in applied mathematics from Southeast University, Nanjing, China, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. He held several visiting positions in China, Germany, Italy, The Netherlands, and the U.S. He is the author or a coauthor of about 30 referred international journal papers. He is a Reviewer of several journals. His research interests include multiagent systems, nonlinear dynamics and control, complex networks and systems, neural networks, cryptography, and communications. Mr. Yu is the recipient of the Best Master Degree Theses Award from Jiangsu Province, China, in 2008.

Guanrong Chen (M’89–SM’92–F’97) received the M.Sc. degree in computer science from Sun Yat-sen University, Guangzhou, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in 1987. He is currently a Chair Professor and the Founding Director of the Centre for Chaos and Complex Networks, Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. He served and is serving as Chief Editor, Deputy Chief Editor, Advisory Editor, Features Editor, and Associate Editor for eight international journals, including the International Journal of Bifurcation and Chaos, the IEEE TRANSACTIONS ON CIRCUITS AND S YSTEMS , and the IEEE T RANSACTIONS ON A UTOMATIC C ONTROL . Dr. Chen is the recipient of the 1998 Harden–Simons Prize for the Outstanding Journal Paper Award from the American Society of Engineering Education, the 2001 M. Barry Carlton Best Transactions Paper Award from the IEEE Aerospace and Electronic Systems Society, the 2002 Best Paper Award from the Institute of Information Theory and Automation, Academy of Science of the Czech Republic, the 2005 IEEE Guillemin-Cauer Best Transaction Paper Award from the Circuits and Systems Society, and the 2008 State Natural Science Award of China. He is an Honorary Professor at different ranks in more than 20 universities worldwide.

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.

YU et al.: DISTRIBUTED CONSENSUS FILTERING IN SENSOR NETWORKS

Zidong Wang (SM’03) was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics from Suzhou University, Suzhou, China, in 1986 and the M.Sc. degree in applied mathematics and the Ph.D. degree in electrical and computer engineering from Nanjing University of Science and Technology, Nanjing, China, in 1990 and 1994, respectively. He is currently a Professor of dynamical systems and computing with the Department of Information Systems and Computing, Brunel University, Uxbridge, U.K. He is the author of more than 100 papers in refereed international journals. His research interests include dynamical systems, signal processing, bioinformatics, control theory, and applications. He is currently serving as an Associate Editor for 11 international journals. Dr. Wang is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON NEURAL NETWORKS, IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C, and IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

1577

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, Shanghai. Her research interests include coordinated and cooperative control, consensus problems, multiagent systems, and complex networks.

Authorized licensed use limited to: CityU. Downloaded on September 22, 2009 at 22:12 from IEEE Xplore. Restrictions apply.