Distributed H Filtering with Consensus Strategies in Sensor Networks: Considering Consensus Tracking Error

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Brief Paper

ACTA AUTOMATICA SINICA

Distributed H ∞ Filtering with Consensus Strategies in Sensor Networks: Considering Consensus Tracking Error WAN Yi-Ming1

DONG Wei1

YE Hao1

Abstract The existing distributed H∞ filtering with consensus strategies consists of two steps: the consensus step through locally communicating with neighboring sensor nodes and the local filtering step. In this paper, the influence of consensus tracking error on the local estimation error is analyzed, and a distributed H∞ filtering algorithm considering the consensus tracking error is proposed. When the number of consensus iterations per sampling period is limited, the proposed method can suppress the effect of consensus tracking error on local estimation error; when the number of consensus iterations per sampling period goes to infinity, i.e., the consensus tracking error converges to zero, the local filtering in the distributed algorithm reduces to the centralized H∞ filtering. Simulation shows the effectiveness of the proposed method. Key words Sensor networks, distributed H∞ filter, average consensus, consensus tracking error Citation Wan Yi-Ming, Dong Wei, Ye Hao. Distributed H∞ filtering with consensus strategies in sensor networks: considering consensus tracking error. Acta Automatica Sinica, 2012, 38(7): 1211−1217 DOI 10.3724/SP.J.1004.2012.01211

The problem of distributed estimation plays an important role in the applications of wireless sensor networks. It is almost impossible to collect all the sensor measurements for the central node due to communication constraints in a large-scale sensor network. In addition to that, it is sometimes computationally infeasible to process all the measurements on one central node. Without having access to all the nodes0 sensing models and measurements, it is appealing to solve the estimation problem in a distributed fashion, i.e., every sensor node performs local estimation based on information communications with its neighbors on a network[1−2] . There have been many studies on distributed estimation of dynamic systems focusing on Kalman filtering. Pioneer work on distributed Kalman filtering with consensus strategies[1] has received wide attention. The consensus based Kalman filter approximates the information form of centralized Kalman filter with two steps: in the first step, each sensor node fuses the measurements and covariance matrices of its neighbors by using consensus algorithm; with the fused measurement and covariance matrix, local Kalman filtering is performed on each sensor node in the second step. References [3−4] proved that distributed Kalman filtering with consensus strategies yields unbiased local estimations, but its local estimation error covariance is no less than that of the centralized Kalman filter. Consensus algorithms have been also applied to deManuscript received July 19, 2010; accepted April 12, 2011 Supported by National Basic Research Program of China (973 Program) (2010CB731800) and National Natural Science Foundation of China (60974059, 60736026, 61021063) Recommended by Associate Editor GENG Zhi-Yong 1. Department of Automation, Tsinghua National Laboratory for Information Science and Technology (TNList), Tsinghua University, Beijing 100084, P. R. China

Vol. 38, No. 7 July, 2012

centralized robust Kalman filtering[5] , distributed Kalman filtering over randomized communication network[6] , distributed estimation for overlapping subsystems[7] , and distributed Kriged Kalman filter for spatial estimation[8−9] . In addition to communicating local measurements and covariance matrices[1] , another family of distributed Kalman filtering methods further improve each node0 s local estimation by fusing the local estimations of its neighbors[10−13] . This paper discusses distributed H∞ filtering in which each node only locally communicates with its neighbors. Compared to distributed Kalman filtering, the work on distributed H∞ filtering is still limited. It is worth while to notice that the communications within neighborhood in the proposed distributed H∞ filtering problem implies a sparse but connected communication network which is different from the communication network in existing decentralized H∞ filtering[14−17] . One kind of decentralized filtering involves no communications, i.e., each node obtains its local estimation with only its own measurement[14−16] . The other kind of decentralized H∞ filtering requires each node to communicate with all the other nodes[17] , which implies all-to-all communication links and may not be scalable for large-scale sensor networks[1−2] . The most related work to this paper is [18]. It follows the approach in [1−2] and proposes the distributed H∞ filtering approach with consensus strategies, which requires only local communications. In the aforementioned work on distributed filtering[1−8, 10−12, 18] , the consensus problem is to fuse the information from neighboring nodes by tracking the average of changing signals. In the consensus algorithm, each iteration requires local communication with the neighboring nodes. By using existing consensus algorithms with sufficiently large number of iterations per sampling period, we can achieve zero or sufficiently small tracking error for only limited classes of changing signals[19−22] . In cases that the number of consensus iterations per sampling period is limited, the consensus tracking error should not be ignored. The existing distributed filtering methods using consensus strategies[1−2, 18] have not considered the influence of consensus tracking error on local estimation. In this paper, we propose a distributed H∞ filter, trying to suppress the effect of consensus tracking error on local estimation. The outline of this paper is as follows: The existing distributed H∞ filtering approach in [18] is reviewed in Section 1. The main result of the consensus based distributed H∞ filter considering consensus tracking error is given in Section 2. Simulation results are given in Section 3. Finally, concluding remarks are made in Section 4.

1

Distributed H ∞ filtering with consensus strategies

First, the centralized H∞ filter is reviewed. Then, it is reformulated so that the centralized algorithm is decomposed into the consensus step and the local H∞ filtering step. Suppose that the monitored process dynamics and the sensing model are as follows: xk + Gw wk x k+1 = Ax

(1)

xk + v k z k = Hx

(2) n

where k denotes the sampling time, x k ∈ R is the state variable, and z k ∈ Rm is the measurement. The process noise w k ∈ Rl and the measurement noise v k ∈ Rm are assumed to be bounded with unknown statistics. The initial

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ACTA AUTOMATICA SINICA

ˆ 0 . The state x0 of the system is unknown with estimation x object is to estimate xk s k = Lx

(3)

In the game theory approach to H∞ filtering, the object is to make the cost function less than a user-specific bound γ 2[23] : NP −1

sup J = ˆk s

kssk − sˆk k2F k=0 NP −1 ³ x0 − x ˆ 0 k2P −1 + w k k2Q−1 kw kx 0 k=0

+

kvv k k2R−1

2 ´ 0 Ns T

And the following condition must hold at each time k:

Consider a sensor network with Ns nodes, where only local communications with neighbors are used. The sensing model for each sensor is

where z i,k

z i,k = Hix k + v i,k , i = 1, 2, · · · , Ns P s ∈ Rmi , N i=1 mi = m, and £ T ¤T z k = z 1,k z T · · · zT 2,k Ns ,k £ ¤T vk = vT vT · · · vT 1,k 2,k Ns ,k ¤T £ T H = H1T H2T · · · HN s

yk =

(21) (22) (23)

(11) (12)

2

(9)

(10)

Ns 1 T −1 1 X T −1 Hi Ri Hi H R H= Ns Ns i=1

(13)

Ns 1 T −1 1 X T −1 H R zk = Hi Ri z i,k Ns Ns i=1

(14)

˜ = Ns Q P˜k = Ns Pk , Q

(20)

Remark 1. The global communication network is unknown to each node in distributed filtering. Given a different but fixed communication network, the parameters in each local filtering do not need to be redesigned, but are automatically determined by information from its neighbors[1−2, 18] . It is assumed in this paper that the communication network is fixed in the course of filtering. The problem of distributed filtering over changing topology of communication network is out of scope of this paper.

in correspondence with (2). If R in (4) is chosen as R = diag {R1 , R2 , · · · , RNs }, then define S=

µ ¶−1 1 −2 ¯ −1 Pi,k + Si,k − γ F Ns ¡ ¢ ˆ i,k = Aˆ x i,k + AMi,k y i,k − Si,kx

Mi,k =

GQGT Pk−1 − γ −2 F¯ + H T R−1 H > 0

Vol. 38

(15)

The centralized filtering algorithms (5) ∼ (7) can be equivalently reformulated as µ ¶−1 1 −2 ¯ Mk = P˜k−1 + S − γ F (16) Ns ˆ k+1 = Aˆ x x k + AMk (yy k − Sˆ xk ) (17) ˜ T P˜k+1 = AMk AT + GQG (18) And at each time k, the following condition should be satisfied: 1 −2 ¯ γ F >0 (19) P˜k−1 + S − Ns

Distributed H ∞ filtering considering consensus tracking error

As shown in Section 1, there are two steps in each iteration of the distributed H∞ filtering algorithm[18] : the consensus step[18−19, 24] and the local H∞ filtering step in (20) ∼ (23). Recent studies on the convergence of consensus algorithms[19−21] show that zero or sufficiently small consensus tracking error can be achieved with sufficiently large number of consensus iterations per sampling period. However, there are cases where the consensus tracking error may not be sufficiently small, e.g., the number of consensus iterations per sampling period per node is limited as required in battery-powered wireless sensor networks[12] . The presence of the consensus tracking error, i.e., Si,k − S and y i,k − y k , can degrade the performance of the local estimation, but is not considered in the existing distributed filtering with consensus strategies[1−2, 18] . In the following, the presence of the consensus tracking error in local estimation error dynamics is analyzed, and then the distributed H∞ filter considering consensus tracking error is given. Finally, it is further shown that by incorporating the structure of the consensus tracking error, the proposed distributed H∞ filter in this paper reduces to the centralized H∞ filter if the number of consensus iterations per sampling period goes to infinity.

No. 7 2.1

WAN Yi-Ming et al.: Distributed H∞ Filtering with Consensus Strategies in · · · Consensus tracking error in local error dynamics

The local filter for each node can be described by ¡ ¢ ˆ i,k+1 = Aˆ ˆ i,k x x i,k + Ki,k y i,k − Si,kx sˆi,k = Lˆ x i,k

(24)

ˆ i,k . From (13) where i = 1, 2, · · · , Ns . Define e i,k = x k − x and (14), there is ˆ i,k = y i,k − Si,kx ¢ ¡ 1 T −1 xk + ˆ i,k + y i,k − y k = Sx H R v k − Si,kx Ns ¢ ¡ 1 T −1 Si,ke i,k + H R v k + (S − Si,k ) x k + y i,k − y k = Ns 1 T −1 Si,ke i,k + H R v k + q i,k Ns (25) ¢ ¡ where q i,k = (S − Si,k ) x k + y i,k − y k contains the consensus tracking error S − Si,k and y i,k − y k due to limited number of consensus iterations per sampling period. By defining s˜i,k = sˆi,k − s k , from the monitored dynamics (1), the sensing model (9) and the local filter (24), the error dynamics for each node is ˜ i,k w ˜ i,k ei,k+1 = A˜i,k ei,k + G s˜i,k = Lke i,k where

(26)

A˜i,k = A − Ki,k Si,k · ˜ i,k = G

G



˜ i,k = w

£

1 Ki,k H T R−1 Ns wT k

vT k

qT i,k

(27) ¸ −Ki,k ¤T

(28) (29)

As shown in (26) and (29), the local estimation error s˜i,k ˜ i,k . is influenced by consensus tracking error contained in w 2.2

Distributed H ∞ filtering considering consensus tracking error

The object of local filtering on each node is to suppress the influence of initial state estimation error, process noise, measurement noise and consensus tracking error on the local estimation error as shown in (30)[23, 25] : NP −1

k=0 NP −1 ³

sup J = ˆi,k s

k˜ s i,k k2F

keei,0 k2P −1 + 0

k=0

˜ i,k k2W −1 kw

2 ´ 0 to the difference Riccati equation:

˜ ˜T P¯i,k+1 = A˜i,k P¯i,k A˜T i,k + Gi,k W Gi,k + ³ ´−1 ¯ P¯i,k L ¯T ¯ P¯i,k A˜T ¯ T I − γ −2 L L γ −2 A˜i,k P¯i,k L i,k (31) ¯ P¯i,k L ¯ T > 0 where L ¯ = F 21 L. such that I − γ −2 L Proof. The error dynamics (26) can be rewritten as ˜ 0i,kw ˜ 0i,k e i,k+1 = A˜i,ke i,k + G ¯ ei,k s˜0i,k = Le

(32)

where 1 ˜ 0i,k = G ˜ i,k W 12 , w ¯ = F 12 L, G ˜ 0i,k = W − 2 w ˜ i,k L

(33)

Then, the object function (30) can be rewritten as NP −1 ° k=0

sup J = ˆk s

keei,0 k2P −1 0

+

° °s˜0i,k °2 NP −1 ³° k=0

< γ2 ° ´

(34)

2 °w ˜ 0i,k °

By applying Theorem 8.1 in [25] to the error dynamics (32), it is obvious that (31) is the sufficient and necessary condition for the existence of an H∞ filter (24) to achieve (34). ¤ Theorem 1. For a given γ 2 > 0, there exists an H∞ filter (24) satisfying (30) if and only if there exists a solution T ¯ P¯i,k L ¯ T > 0, to the > 0, satisfying I − γ −2 L P¯i,k = P¯i,k difference Riccati equation P¯i,k+1 = AP¯i,k AT + GQGT − ³ ´−1 T T ¯ i,k ¯+H ¯ i,k P¯i,k H ¯ i,k ¯ i,k P¯i,k AT AP¯i,k H R H (35) where · ¯ i,k = H

Si,k L





¸

1 ¯ ¯ =  W + Ns S R 0

,

0 −γ 2 F −1

 (36)

If the above condition is satisfied, the filter (24) can be given with the filter gain of

where

T −1 Ki,k = APi,k Si,k Ui,k

(37)

³ ´−1 −1 Pi,k = P¯i,k − γ −2 LT F L

(38)

2

where γ is a prescribed scalar, P0 and W are symmetric, positive definite matrices chosen according to the specific © ª ¯ is chosen, where Q application. Here, W = diag Q, R, W ¯ is the weighting and R are the same as those in (4), and W matrix for consensus tracking error q i,k . Remark 2. Equation (30) requires local estimation of each node to satisfy the same performance criterion. Some related works on distributed H∞ consensus filtering[26−28] adopted a different performance criterion which considers the ratio of the averaged local estimation error to the averaged initial state estimation error and the averaged noises. Lemma 1. For a given γ 2 > 0, there exists an H∞ filter (24) satisfying (30) if and only if there exists a solution P¯i,k

T Ui,k = Si,k Pi,k Si,k +

1 ¯ S+W Ns

(39)

Proof. Defining −1 −1 −1 ¯TL ¯ = P¯i,k − γ −2 LT F L Pi,k = P¯i,k − γ −2 L

(40)

With matrix inversion lemma, (31) can be reformulated as ˜ ˜T P¯i,k+1 = A˜i,k Pi,k A˜T i,k + Gi,k W Gi,k Substituting (27) and (28) into (41), we have

(41)

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ACTA AUTOMATICA SINICA

P¯i,k+1 = (A − Ki,k Si,k ) Pi,k (A − Ki,k Si,k )T + 1 T T ¯ Ki,k GQGT + 2 Ki,k H T R−1 HKi,k + Ki,k W = Ns T −1 APi,k AT + GQGT − APi,k Si,k Ui,k Si,k Pi,k AT + ³ ´ ³ ´T T −1 T −1 Ki,k − APi,k Si,k Ui,k Ui,k Ki,k − APi,k Si,k Ui,k

time-invariant system, since there is limk→∞ Si,k = S, the condition that the solution of (50) converges to the solution satisfying the following algebraic Riccati equation ´−1 ³ ¯ +H ¯ P¯i H ¯T ¯ P¯i AT ¯T R P¯i = AP¯i AT + GQGT − AP¯i H H · ¯ = H

(42) where 1 ¯ = H T R−1 H + W Ns2 1 T ¯ Si,k Pi,k Si,k + S+W Ns

2.3 (43)

Sufficiency. with matrix inversion lemma, (35) can be reformulated as ³ ´−1 −1 T ¯ −1 ¯ ¯ i,k P¯i,k+1 = A P¯i,k +H R Hi,k AT + GQGT (44) Substituting (36), (40), and (39) into (44), we have P¯i,k+1 = APi,k AT + GQGT − ¶−1 µ T T ¯ + 1 S + Si,k Pi,k Si,k Si,k Pi,k AT = APi,k Si,k W Ns T −1 Si,k Pi,k AT + GQGT APi,k AT − APi,k Si,k Ui,k

(45) Then, (42) is obvious from (45) and (37). Necessity. it can be obtained following similar procedure in the proof of Theorem 8.3 in [25]. ¤ Note that S in (36) and (39) is not directly available to each node. So, in the implementation of the distributed algorithm, the consensus estimation Si,k is used to replace S in (36) and (39). Then, the distributed H∞ filtering algorithm is as the following: ˆ i (0) = x ˆ 0 , P¯i,0 = P¯0 ; Step 1. Initialization, x Step 2. Perform consensus algorithm in each node to compute Si,k and y k according to [18−19, 24]; Step 3. Local H∞ filtering ´−1 ³ −1 − γ −2 LT F L (46) Pi,k = P¯i,k Ui,k =

1 ¯ + Si,k + W Ns

(48) (49)

P¯i,k+1 = AP¯i,k AT + GQGT − ³ ´−1 T T ¯ i,k ¯ i,k + H ¯ i,k P¯i,k H ¯ i,k ¯ i,k P¯i,k AT AP¯i,k H R H (50) where · ¯ i,k = H

Si,k L

¸ ,

¯ i,k = R

· ¯ W+

1 S Ns i,k

0

0 2 −1 −γ F

¸ (51)

I −γ

−2 ¯

LP¯i,k L > 0 ¯T





¸ ,

1 ¯ ¯ =  W + Ns S R 0

Incorporating the tracking error

0



−γ 2 F −1

(52)

Remark 3. As mentioned before, the consensus estimation Si,k is used to replace S in (36) and (39) to result in (51) and (47), respectively. For the distributed filtering of a

structure

of

consensus

As shown in (25), the consensus tracking error ¡ ¢ q i,k = (S − Si,k ) x k + y i,k − y k

(53)

is determined by the communication network, the number of consensus iterations per sampling period, and so on[19−21, 24] . If the communication network is timeinvariant, the structure of consensus tracking error can be incorporated in the distributed H∞ filter design. The consensus step in the proposed distributed H∞ filter is actually two consensus problems for calculating (13) and (14). For computing (14), the input to the consensus algorithm is iT h u1 = uT uT · · · uT 1,1 1,2 1,Ns (54) u 1,i = HiT Ri−1z i,k Then, y k in (14) can be represented as y k = N1s (1 ⊗ In ) (1 ⊗ In )T u 1 h iT 1 = 1 1 ··· 1 ∈ RNs where ⊗ means the Kronecker product. With Γ (p) ∈ R(nNs )×(nNs ) representing the consensus algorithm with p consensus iterations per sampling period, the consensus estimation y i,k can be denoted by Γ (p) u 1 . Then, the consensus tracking error for y k is[19, 24] y i,k − y k = Γ (p) u 1 −

1 (1 ⊗ In ) (1 ⊗ In )T Ns

(55)

u1 u 1 = E(p)u

(47)

T −1 Ki,k = APi,k Si,k Ui,k ¡ ¢ ˆ i,k+1 = Aˆ ˆ i,k x x i,k + Ki,k y i,k − Si,kx

S L

is left for future research.

T Ui,k = Si,k Pi,k Si,k +

T Si,k Pi,k Si,k

Vol. 38

where Γ (p) − N1s (1 ⊗ In ) (1 ⊗ In )T = E(p) ∈ RnNs ×nNs . When p goes to infinity, there is[19, 24] lim E(p) = 0

(56)

p→∞

Denote E=

£

E1T

E2T

···

T EN s

¤T

(57)

where Ei ∈ Rn×(nNs ) and p in E(p) is omitted here as well as in the following. From (55) and (57), in the consensus problem for (14), y i,k − y k = Ei u1 . Similarly, considering applying the same consensus algorithm Γ (p) for computing (13), there is u 2,i = HiT Ri−1 Hi ,

Si,k − S = Eiu 2

So, (53) can be rewritten as uqk q k = Eu

(58) (59)

No. 7

WAN Yi-Ming et al.: Distributed H∞ Filtering with Consensus Strategies in · · ·

where uqk = u1 − u2 xk with u1 and u2 defined in (54) and £ ¤T (58), respectively, and q k = q T . qT · · · qT 1,k 2,k Ns ,k Therefore, with (59), for the error dynamic (26), modifications should be made to (28) and (29) as ¸ · ˜ i,k = G − 1 Ki,k H T R−1 −Ki,k Ei (60) G Ns ˜ i,k = w

£

wT k

vT k

uqk )T (u

¤T

(61)

Accordingly, the distributed H∞ filtering algorithm follows the three steps in Subsection 2.2 except that modifications in (47) and (51) are T Ui,k = Si,k Pi,k Si,k +

· ¯ i,k = H

Si,k L

¸

1 ¯ EiT Si,k + Ei W Ns

 ¯ EiT + 1 Si,k Ei W ¯ , Ri,k =  Ns 0

 0

GQGT which is exactly the same as (7). From (46), (48), and (62), symmetry S > 0 due to positive definiteness of symmetry R, and it is obtained that ¶−1 µ 1 S = Ki,k = APi,k S SPi,k S + Ns ¡ −1 ¢−1 Ns A Ns S + Pi,k = ³ ´−1 −1 Ns A H T RH + P¯i,k − γ −2 LT F L = ³ ´−1 Ns AP¯i,k I − γ −2 LT F LP¯i,k + H T RH P¯i,k Then, it can be easily seen that (49) is the same as (6). ¤ From the object function (30) and Theorem 1, it can be concluded that when the number of consensus iterations p is limited, the distributed H∞ filter with modifications (62) and (63) can suppress the effect of consensus tracking error on local estimation. And Theorem 2 here further shows that when the number of consensus iterations p goes to infinity, the consensus error converges to zeros, and the local filtering reduces to the centralized H∞ filter shown in (5) ∼ (7).

Simulation results Consider the following dynamic system: xk + Gw w k , x 0 = (2, −1)T x k+1 = Ax · ¸ · −0.2 −0.5 1 A= , G= 1.5 1 0

£ ¤ xk , where L = 1 3 . The object is to estimate sk = Lx The sensor network is shown in Fig. 1. The sensing model for each sensor node is z i,k = Hix k + v i,k , i = 1, 2, · · · , 20 £ ¤ where Hi = Hx1 = 1 0 for some nodes and Hi = £ ¤ Hx2 = 0 1 for others. The process noise w k is zero mean white noise with covariance 2I2×2 (I is the identity matrix throughout this paper), and the measurement noise for each node v i,k is zero mean white noise with variance 0.1. It should be noted that the statistics of the noises are assumed to be unknown in H∞ filtering. Suppose that the sensing models of each sensor and its neighbors have both output matrices Hx1 and Hx2 .

(62)

 −γ 2 F −1 (63) Theorem 2. The distributed H∞ filter with modifications (62) and (63) reduces to the centralized H∞ (5) ∼ (7) if the number of consensus iterations per sampling period goes to infinity. Proof. According to (56) ∼ (58), when the number of consensus iterations per sampling period goes to infinity, Si,k → S and Ei → 0. By substituting (63) into (50), it is easily obtained that (50) becomes ³ ´−1 P¯i,k+1 = AP¯i,k I − γ −2 F¯ P¯i,k + H T R−1 H P¯i,k AT +

3

1215

0 1

¸

Fig. 1

A sensor network with 20 nodes and 45 communication links

To implement distributed filtering, the consensus algo± P T −1 R H 20 and yk rithm is used to estimate S = 20 H i i i i=1 ± P T −1 = 20 H R z 20. S is time invariant, the estimation i,k i i i=1 Si,k of each node will converge to S when the time k goes to infinity. But the consensus tracking error of the time varying signal y k would not be sufficiently small if the number of consensus iterations per sampling period is limited, as shown in Fig. 2.

Fig. 2 Consensus tracking error of one component of y 20,k in case of 10 consensus iterations per sampling period

In the following, the distributed H∞ filter (20) ∼ (23) in [18] and the proposed filter (46) ∼ (52) in this paper are respectively called distributed Algorithms A and B for short. Both distributed algorithms use the same consensus algorithm with the same number of consensus iterations per ˆ0 = sampling period, staring from the same initial state x (0, 0)T , and Pi,0 = 200I2×2 for Algorithms A and B. And the predefined weighting matrices in (4) and (30) are chosen

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ACTA AUTOMATICA SINICA

as Q = I2×2 , Ri = 0.1, F = I2×2 , W = diag {Q, R, I} And γ 2 = 20 is chosen to make sure that the conditions (23) and (52) are satisfied. The simulation results are shown in Fig. 3 and Table 1 for different number of consensus iterations p. The root mean square (RMS) estimation error for node i, given n local estimations {ˆ s i,k }, is as v u n u1 X RM Si = t (ssk − sˆi,k )2 n

Vol. 38

From Figs. 3 (a) and 3 (b), it can be seen that the performance degradation of both distributed Algorithms A and B compared to the centralized algorithm is due to the consensus tracking error resulted from limited number of consensus iterations. It is shown in Figs. 3 (a) and 3 (c) that when the number of consensus iterations is large enough, i.e., the consensus tracking error is sufficiently small, both distributed algorithm perform nearly as good as the centralized algorithm. With consideration to suppress the effect of consensus tracking error on local estimation error, distributed Algorithm B performs better than Algorithm A, as shown in Table 1.

k=1

Table 1

RMS estimation errors for individual nodes with different numbers of consensus iterations p Algorithm A

Node

(a) Centralized algorithm

(b) Distributed Algorithms A and B with the number of consensus iterations p = 10

4

Algorithm B

p = 10

p = 100

p = 10

p = 100

1

10.8788

4.8479

9.9117

4.7337

2

6.7560

4.7783

5.8270

4.6645

3

10.6474

4.7188

9.7199

4.6004

4

7.1639

4.7148

6.1118

4.5963

5

8.6878

4.6071

8.0404

4.4642

6

8.9509

4.5824

7.5693

4.4120

7

10.6847

4.5884

9.7480

4.4220

8

9.1329

4.5820

7.7338

4.4117

9

8.9186

4.8910

8.2422

4.7761

10

4.8784

4.8378

4.5184

4.7248

11

7.4756

4.7222

7.0203

4.6099

12

7.0001

4.9016

6.0449

4.7864

13

8.9997

4.8947

8.3046

4.7798

14

6.8855

4.8673

5.9705

4.7538

15

8.9009

4.8947

8.2255

4.7798

16

8.5632

4.7230

7.2678

4.6130

17

10.2537

4.4779

9.3619

4.3389

18

9.3689

4.5159

7.9249

4.2719

19

10.5979

4.5062

9.6680

4.2699

20

11.6491

4.6592

9.7938

4.3328

Conclusions

The existing work on distributed H∞ filtering with consensus strategies is reviewed, and it is analyzed that there is performance degradation of each node0 s local estimation due to consensus tracking error. After that, the distributed H∞ filtering method considering consensus tracking error is proposed. When the number of consensus iterations per sampling period is limited, the proposed method can suppress the effect of consensus tracking error on local estimation; when the number of consensus iterations per sampling period goes to infinity, i.e., the consensus tracking error converges to zero, the local filtering in the distributed algorithm reduces to the centralized H∞ filtering. Comparison in the simulation results shows the proposed method is effective of suppressing the effect of consensus tracking error on local estimation. (c) Distributed Algorithms A and B with the number of consensus iterations p = 100

Fig. 3

Comparison of centralized algorithm, distributed Algorithms A and B

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WAN Yi-Ming Ph. D. candidate in the Department of Automation, Tsinghua University. His research interest covers networked control systems, model-based fault detection and diagnosis. E-mail: [email protected] DONG Wei Research associate in the Department of Automation, Tsinghua National Laboratory for Information Science and Technology, Tsinghua University. His research interest covers fault diagnosis, modeling and simulation of complex system. E-mail: [email protected] YE Hao Professor in the Department of Automation, Tsinghua University. His research interest covers model-based fault diagnosis and its application to networked control systems. Corresponding author of this paper. E-mail: [email protected]