Distributed Estimation from Relative Measurements in Sensor Networks

Transcription

1 Distributed Estimation from Relative Measurements in Sensor Networks #Prabir Barooah and João P. Hesanha Abstract We consider the roblem of estimating vectorvalued variables from noisy relative measurements. The measurement model can be exressed in terms of a grah, whose nodes corresond to the variables being estimated and the edges to noisy measurements of the difference between the two variables. We take the value of one articular variable as a reference and consider the otimal estimator for the differences between the remaining variables and the reference. This tye of measurement model aears in several sensor network roblems, such as sensor localization and time synchronization. Two algorithms are roosed to comute the otimal estimate in a distributed, iterative manner. The first algorithm imlements the Jacobi method to iteratively comute the otimal estimate, assuming all communication is erfect. The second algorithm is robust to temorary communication failures, and converges to the otimal estimate when certain mild conditions on the failure rate are satisfied. It also emloys an initialization scheme to imrove accuracy in site of the slow convergence of the Jacobi method. I. INTRODUCTION We consider an estimation roblem that is relevant to a large number of sensor networks alications, such as localization and time synchronization. Consider the estimation of n vector-valued variables x 1, x 2,..., x n R k based on several noisy relative measurements ζ ij. The measurement indices (i, j) take values in some set E of airs of values from V := {1, 2,..., n}. The term relative comes from the measurement model considered: ζ uv = x u x v + ɛ uv, (u, v) E, (1) where the ɛ uv are uncorrelated zero-mean noise vectors with known covariance matrices P uv = E[ɛ uv ɛ T uv ]. Just with relative measurements, determining the x u s is only ossible u to an additive constant. To avoid this ambiguity, we assume that a articular variable (say x 1 ) is used as the reference and therefore x 1 = 0. The measurement equations (1) can be exressed in terms a directed grah G = (V, E) with V = n vertices (nodes) and E = m edges, with an edge (u, v) if the measurement ζ uv is available. The vector x u is called the u-th node variable. Our objective is to construct the otimal estimate ˆx u of x u for every node u V\{1}. The Otimal estimate refers to the estimate roduced by the classical Unbiased Minimum Variance Estimator (UMVE), which achieves the minimum variance among all linear unbiased estimators. When alied to the location estimation roblem, a node variable x u could be the osition of node u in 2-d or 3-d sace w.r.t. the reference node, and when alied to time synchronization, it could be time shift of u s local clock w.r.t. the clock of node 1. For a through discussion on how these roblems can be modeled with (1), see [1], [2] and references therein. To comute the otimal estimate directly, one seems to need all the measurements and the toology of the grah (see beginning of section III). Thus, if one node in the network has to comute it, all this information has to be transmitted to that node. For networks with a large number of measurements, doing so will be rohibitively exensive in terms of energy consumtion, bandwidth and communication time. Moreover, such a centralized comutation will be less robust to dynamic changes in toology resulting from link and node failures over time. In this aer we roose an iterative algorithm to comute the estimate of the node variables in a distributed manner. By distributed we mean that at every ste, each node comutes its own estimate and the data required for the comutation erformed at a node is obtained through communication with its one-ho neighbors. We show that the estimate roduced by the algorithm asymtotically aroaches the otimal estimate even in the resence of faulty communication links, as long as some mild conditions on the duration of faults are satisfied. We first roose an algorithm that imlements the Jacobi iterative method to comute the otimal estimate, assuming that all communication is erfect (no failure), for which we can establish erformance bounds. This algorithm was then imroved to handle dynamic changes in the communication toology brought about by temorary link failures. It also emloys an initialization scheme to achieve greater accuracy. Accuracy of an estimate is measured by the norm of the difference between it and the otimal one. A similar algorithm was alluded to in [1] where the roblem of time synchronization from measurements of time differences was considered; but the algorithm was not investigated. The algorithm roosed in this aer is more general, since it works for vector valued variables such as ositions and not just scalar valued ones such as clock times. Moreover, our algorithm is roven to work even in the resence of faulty communication. Our work was insired by [3] where the Jacobi and other iterative algorithms were alied to comuting the otimal estimate in a different roblem, one where absolute measurements of random node variables (such as temerature) were available, but the node variables were correlated. In a revious aer [2], the authors considered how

2 the variance of the otimal estimator (for the roblem considered in this aer) grows with the distance of a node from the reference, and how that growth deends on the structure of the grah. A classification of grahs were obtained that determined the bounds on the variance achieved by the otimal estimate. In this aer, we roose an algorithm that asymtotically obtains the otimal estimate (when the number of iterations aroaches infinity), while simultaneously being simle, scalable, distributed and robust to communication failures. The aer is organized as follows. In section III, we describe the first distributed algorithm that imlements the Jacobi iterative method and establish its erformance bounds. In section IV we modify this algorithm to handle faulty communication links. Section V describes a modification to this algorithm to imrove its erformance. Simulations done with the resulting algorithm are resented in section VI. The aer concludes with a note on future directions in section VII. II. THE OPTIMAL ESTIMATE Consider a measurement grah G with n nodes and m edges. Let X be a vector in R (n 1)k obtained by stacking together all the unknown node variables, i.e., X := [x T 2,..., x T n ] T. Define Z := [ζ1 T, ζ2 T,..., ζm] T T R km and ɛ := [ɛ T 1, ɛt 2,..., ɛt m ]T R km. Eq. (1) can now be rewritten as follows: Z = A T b X + ɛ, A b := A b I k, (2) where A b R n 1 m is called the basis incidence matrix of G, defined as A b = [a uj ], where a uj = 1, 1 or 0, if edge j is incident on node u and directed away from it, is incident on node u and directed toward it, or is not incident on node u, resectively. A b has n 1 rows, each row corresonding to one node in G, excet the reference node. For a measurement grah with node variables x u R k, the we call the matrix A b R k(n 1) km defined in (2) the Generalized Basis Incidence Matrix. This otimal estimate ˆX with the measurement model 2 is the solution to the following system of linear equations [4]: L ˆX = b, (3) where L := (A b A T b ), b := A b Z, and P := E[ɛɛ T ] is the covariance matrix of the measurement error vector. The error covariance of the otimal estimate E[(X ˆX )(X ˆX ) T ] is equal to [4] L 1. Since the measurement errors on two different edges are assumed to be uncorrelated, P is a symmetric ositive definite block diagonal matrix with the measurement error covariances along the diagonal: P = diag(p 1, P 2,..., P m ) R km km, where P e is the covariance of the measurement reresented by the edge e E. A ath from a node to another node that does not resect the orientation of the edges is called an undirected ath. A 3 e4 e3 Fig. 1. A measurement grah G with 3 nodes and 4 edges. Node 1 is the reference. directed grah is said to be weakly connected if there is a undirected ath from any node to any other node. In this aer we consider only weakly connected grahs. Under this assumtion, the estimation error covariance always exists and is of finite norm [2], and therefore the otimal estimate ˆX is unique for a given set of measurements Z. We call the matrix L the Weighted Generalized Grounded Lalacian. As a simle examle, consider the weakly connected measurement grah G shown in figure 1. The basis incidence matrix for this grah is A b = [ ] Writing the measurement equations (1) for the four edges exlicitly shows how we get (2) from (1): ζ 1 ζ 2 ζ 3 ζ 4 = e2 I k 0 0 I k I k I k I k I k 2 e1 1 [ x2 x 3 ] For the grah in figure 1, with every measurement covariance being I k, (3) becomes [ ] [ ] [ ] 3Ik 2I k ˆx 2 ζ3 ζ 2I k 3I k ˆx = 1 ζ 4. (4) 3 ζ 4 ζ 2 ζ 3 + III. DISTRIBUTED ESTIMATION WITH PERFECT COMMUNICATION CHANNELS In order to comute the otimal estimate ˆX by solving the equations (3) directly, one needs all the measurements and their covariances (Z, P), and toology of the grah (A b ). Our goal is to comute the estimate in a distributed manner, emloying only local communication. We use the Jacobi iterative method to achieve this. A discussion of the Jacobi method can be found in standard textbooks [5], hence we refrain from describing it here. Instead we make the resentation of the algorithm self-contained. At first we assume that there are no communication failures, and design an algorithm for that scenario. In section IV, we modify the algorithm to make it robust to temorary communication failures. A node u is said to be a neighbor of another node v in a grah (V, E ) if there is an edge (u, v ) E or (v, u ) E. The set of neighbors of u in the measurement grah G are denoted by N u. We assume that after deloyment of the network, the nodes detect their neighbors and exchange their relative measurements well as the associated covariances. So every node has access to the measurements on the edges that are incident on it, whether the edge is directed to or away from it. Each node uses the measurements obtained ɛ 1 ɛ 2 ɛ 3 ɛ 4

3 initially for all future comutation. Arguably, in future internode communication, additional measurements between the same node air may become available. However, we want to comare the estimate roduced by the roosed algorithm with the otimal estimate, and the otimal estimate is defined for a given set of measurements. Therefore we assume that the measurements are given a riori and the measurement grah does not change with time. For simlicity, it is assumed that comutation time is negligible comared to communication time. SCG OPTIMAL ESTIMATOR 1 : After the deloyment of the network, the reference node initializes its estimate to 0 and never changes it. Every other nodes initializes its estimate to an arbitrary value. At the start of iteration i + 1, a node sends its most recent estimate ˆx (i) to its neighbors along with the corresonding iteration number i. It also gathers the i-th estimates of its neighbors, ˆx (i) v, v N, and then udates its estimate by solving the following k equations for ˆx (i+1) : M ˆx (i+1) = b + Pv 1 ˆx (i) v + Pv 1 ˆx (i) v, (5) where M := b := v ζ,v v + v, (6) v ζ v. (7) To understand where this udate equation came from, we note that the otimal estimate of a articular node, say V, satisfies the following equations, which are obtained simly by exanding (3) and using the definitions of M and b : M ˆx ( v ˆx v + ) Pv 1 ˆx v = b. (8) If at some iteration i, ˆx (i+1) = ˆx (i) for all V, then equations 5 and 8 become equivalent. Thus, if the estimates, V converge to anything (and in a moment we show that they do),they must converge to the otimal estimates. Note that to comute the udate ˆx (i+1), node needs M, b and the neighbors estimates ˆx (i) v, v N. The quantities M and b are comuted in the beginning from the measurements and the associated covariances on the edges that are incident on, so all the comutation needs only local information. The algorithm is summarized in table I. ˆx (i) A. Correctness and Performance Let ˆX(i) = [ˆx (i)t 2,..., ˆx (i)t n ] T be the vector of node estimates after the ith iteration of the algorithm has been comleted. One iteration is said to be comlete when all nodes udate their estimate once. The error at the ith 1 SCG: Static communication Grah Name: SCG OPTIMAL ESTIMATOR Goal: Comute estimates of node variables that aroach the otimal estimate. Initialization: x (0) 1 = 0, x (0) u is arbitrary for u V \ {1}. After deloyment, every node V \ {1} erforms: 1 Detect all neighbors N. 2 Obtain measurements ζ v, ζ v and the associated covariances P uv, P vu for every v N. Comute M and b from (6) and (7). At every iteration i, a node erforms: 3 Send ˆx (i), i to every v N, get ˆx (i) v from all v N. 4 Comute ˆx (i+1) from neighbors estimates ˆx (i) v for every v N and reviously comuted M and b, using (5). TABLE I PSEUDO-CODE FOR THE SCG OPTIMAL ESTIMATOR ALGORITHM. iteration is ˆX (i) ˆX, where ˆX is the otimal estimate. Let ε (i) := ˆX (i) ˆX / ˆX (0) ˆX. The algorithm is said to be correct if the ε (i) 0 as i for any initial condition X (0). Aart from correctness, a quantifiable measure of erformance is also desirable. In networks with energy and bandwidth constrained sensor nodes, achieving a good accuracy with as few iterations as ossible is imortant. In light of this, we use as erformance metric the number of iterations i required so that ε (i) achieves a given value ɛ. To make the analysis of correctness and erformance of the algorithm tractable, we make the following additional assumtion: Assumtion 1: Either the node variables are scalars (k = 1) and the measurement error variances are bounded, or, the node variables are vectors (k > 1) and all the measurement error covariance matrices are equal. For establishing the erformance bounds, we restrict our attention to grahs with low Fiedler value. Recall that the Fiedler value, or the Algebraic connectivity, α(g) of a grah G is the second smallest eigenvalue of the grah Lalacian. Theorem 1. Consider a weakly connected measurement grah G which satisfies Assumtion 1. Then the roosed algorithm is correct. If, in addition, α(g) << 1, for every 0 < ɛ < 1, the number of iterations N(ɛ) required so that ε (i) < ɛ, i > N(ɛ), satisfies N(ɛ) > Ω ( log ɛ ). α(g) Proof: When assumtion 1 holds, the algorithm is simly a Jacobi iteration on a non-singular M matrix [6], and the Jacobi method is correct if the sectral radius of the Jacobi iterative oerator is < 1. It follows from standard results (see Theorem in [6]) that is true in our case. For the lower bound on erformance, see [7]. Remark 1: The reason for considering grahs with low Fiedler value is that for most large ad-hoc networks, which

4 are the networks of interest here, the Fiedler value is generally quite small. For examle, the Fiedler value of a grid grah is B. Communication Cost An imortant erformance measure for a distributed algorithm is the total number of messages that have to be exchanged before a given accuracy is achieved. In sensor networks with energy constrained nodes and unreliable communication channels, this issue is articularly acute. In the SCG OPTIMAL ESTIMATOR algorithm, every air of neighboring nodes has to exchange two messages er iteration to tell each other their current estimates. Therefore it will take 2mi messages to comlete i iterations, where m is the number of edges. For comarison, it takes at most nd G messages to transmit all the measurements and node IDs to a central node that can then comute the otimal estimate directly, where n is the number of nodes and D G is the diameter of G. The diameter of a grah is the length of the longest shortest ath between any two nodes. Interestingly, communication cost of the centralized estimator can be lower than that of the distributed one, esecially if a higher accuracy is desired. IV. DISTRIBUTED ESTIMATION WITH FAULTY COMMUNICATION One desirable attribute of any distributed algorithm is robustness to communication failures, such failures being unavoidable in ractice. We modify our algorithm slightly to make it robust to such failures. The resulting algorithm is call the DCG OPTIMAL ESTIMATOR 2 algorithm. We assume for the sake of simlicity that during the estimation rocess, no nodes fail ermanently and no new nodes become active. This assumtion does not lace any restriction on emloying the algorithm when there are ermanent changes to the network. It is made solely to facilitates a fair comarison between the estimates roduced by the algorithm to the otimal one, the latter being defined only for a given set of node variables and measurements. Since a neighbor may become unavailable at any time, every node stores in its local memory the estimates of its neighbors variables recorded from the last successful communication. We denote by (ˆx v ) (i) the estimate of x v ket in s local memory at the end of the ith iteration. If the last successful communication between and v took lace during the jth iteration, j < i, then (ˆx v ) (i) = ˆx (j) v. We assume that comutation time is negligible comared to communication time. DCG OPTIMAL ESTIMATOR: Let t (i) be the local time at node in the beginning of the i+1th iteration. At this time, every node tries to communicate with all its neighbors N for τ c (i, ) seconds, after which it stos attemts at communication until the next iteration. Let N (i) N be the set of nodes is able to and get data from successfully 2 DCG: Dynamic Communication Grah during the time eriod (t (i), t (i) + τ c ). Thus, node gets its most recent estimates ˆx (i) from every v N (i) Moreover, it sends its own most recent estimate, ˆx (i), to as many nodes in N as ossible during this eriod. After the communication eriod of τ c seconds, node then udates its coy of its neighbors estimates with the recent estimates gathered: (ˆx v ) (i) ˆx v (i). For the nodes it was not able to get data from, it kees the local coies of their, v N (i) estimates unchanged: (ˆx v ) (i) (ˆx v ) (i 1) After this, node comutes its own estimate udate, ˆx (i+1) v., v N \N (i) by solving the following system of k linear equations: M ˆx (i+1) = b + Pv 1 (ˆx v) (i) +., v (ˆx v) (i). The seudo-code in Table II imlements the algorithm just described. An additional difference in the algorithm described in Table II from the first algorithm is the initialization scheme described in V. The DCG OPTIMAL ESTIMATOR algorithm in its final form uses therefore not the udate equation shown above, but (9). A. Correctness under Faulty Communication In the resence of temorary link failures, the DCG OPTIMAL ESTIMATOR algorithm executes what are known in arallel comuting literature as asynchronous Iterations. This is a well-studied roblem and the conditions under which it converges are known [8], [9]. Suose a node is at the i-th iteration, and the last successful communication between and one of its neighbors v took lace at the jth iteration, with j < i, meaning that the revious i j communications between the nodes had failed. Theorem 2. Consider a weakly connected measurement grah G which satisfies Assumtion 1. Then the DCG OPTIMAL ESTIMATOR algorithm is correct if there is a ositive integer l < such that the number of consecutive communication failures between every air of neighboring nodes in G is less than l. Proof: This follows from theorem 4.1 in [9] which states that asynchronous iterations for a system of linear equations converge to the correct solution when the sectral radius of the iteration oerator is < 1, and the roof of theorem 1 where it was shown that when Assumtion 1 is satisfied, the sectral radius of the Jacobi iterative oerator is < 1 for the system of linear equations considered in this aer. V. IMPROVING PERFORMANCE It may be ossible to imrove the convergence rate by using other iterative techniques such as Gauss Siedel, SOR or the conjugate gradient [5] methods, or even by reconditioning, but any such imrovement will come at the cost of increased communication. For the roblem at hand, however, one feasible strategy of imroving erformance without imroving the convergence

5 rate is to reduce the initial error by choosing a better initial condition. The question is how to rovide a node with a better initial condition without requiring much more communication or comutation. We add the following modification to the DCG OPTIMAL ESTIMATOR algorithm to hel a node detect which of its neighbors have good estimates so that it may udate its estimate based only on those neighbors. After a while, all neighbors will be recognized to have good estimates. Therefore this modification makes only the initial hase of the algorithm s execution different from the reviously described algorithm. After deloyment of the network, every node other than the reference initializes a flag have estimate to 0, meaning it has no estimate of its variable. The reference node initializes its flag to 1. During every inter-udate communication, a node gets from its neighbors their current estimates and the values of their flags. If the recent communication fails, it uses the flag values gathered during the last successful communication. The node then comutes its udate based only on the estimates of those neighbors whose flags were 1 in the recently concluded communication. If any of its neighbors had its flag as 1, after comuting its own estimate it sets its own flag to 1 and never resets it. Otherwise it kees the flag at 0 and does not comute any estimate. At the beginning, only the reference has an estimate. In the first iteration, neighbors of the reference can comute estimates by adding measurements they share with the reference, and then udate their have estimate flags to 1. In the next iteration, their neighbors do the same, and so on. This way, a node can detect if global coordinate information from the reference node has reached it. With successive iteration, the accuracy increases as the algorithm converges to the otimal estimate. Let h (i) v be the value of the flag have estimate of node v at iteration i and (h v ) (i) be the local coy of this variable at node. At iteration i, let N h (i) N be the set of neighbors of V that knows have their flag values at 1, i.e, N h (i) := {u N (h u ) (i) = 1}. When N h (i) is not emty, node will be emloy the following udate equations: = b h + Pv 1 (ˆx v) (i) + Pv 1 (ˆx v) (i), (9) M h ˆx(i+1) where M h := b h := Pv 1 + v ζ v v, v ζ v. After a while, all of s neighbors will have their flags at 1, at which oint M h = M and b h = b. This modification is included in the summary of the DCG OPTIMAL ESTIMATOR algorithm resented in table II. Name: DCG OPTIMAL ESTIMATOR Goal: Comute estimates of node variables that aroach the otimal estimate in the resence of faulty communication links. Data: A rule for every node to determine its communication time-out interval τ c (seconds) at the i iteration. Initialization: x (0) 1 = 0, h (0) 1 = 1. x v(0) =, v V \ {1} and h (0) u = 0, v V \ {1}. After deloyment, every node V erforms: 1 Detect all neighbors N. 2 Obtain measurements ζ v, ζ v and the associated covariances P uv, P vu for every v N. At every iteration i + 1, during the time interval (t (i), t (i) + τ c) seconds, every node V \ {1} erforms: 3a 3b Get ˆx (i) v, h (i) v, h (i) send ˆx (i) from every v N (i), and, i to as many nodes in N. Udate local coies of neighbors estimates as IF v N (i) (ˆx v) (i) ˆx (i) v, (h v) (i) h (i) v. ELSE (ˆx v) (i) (ˆx v) (i 1), (h v) (i) (h v) (i 1). At the end of t (i) + τ c seconds, node erforms 4 IF v N s.t. (h v) (i) = 1, a) Comute b, M, then udate ˆx (i+1) using (9). b) Set h (i+1) 1. ELSE ˆx (i+1) ˆx (i), h (i+1) h (i). TABLE II PSEUDO-CODE FOR THE DCG OPTIMAL ESTIMATOR ALGORITHM WITH THE INITIALIZATION SCHEME DESCRIBED IN SEC. V. THE VARIABLE h (i) v IS THE VALUE OF THE FLAG H A V E E S T I M A T EOF NODE v AT ITERATION i. VI. SIMULATION For simulations, we ick location estimation as an alication of the general roblem described in this aer. The node variable x u is node u s osition in 2-d Euclidean sace. 200 nodes were randomly laced in a 1 1 area (figure 2) and airs of nodes that are within a range of 0.11 took measurements of each others relative ositions. Every measurement was corruted[ by Gaussian noise ] with a covariance matrix of P o = A single set of measurements were used for all the simulations; the locations estimated by the otimal estimator are also shown in Figure 2. The Fiedler value of this grah was comuted to be Figure 3 comares the effect of different initializations on the accuracy achieved by the DCG OPTIMAL ESTIMA- TOR. The initialization described in V erforms better than fixed initialization (all ositions initialized to 0) or random initialization (initial ositions chosen from a uniform distribution). The error at the i-th iteration as a fraction of the

6 PSfrag relacements ˆX (i) ˆX / ˆX 10 1 no failure f = f =0.10 f =0.20 Fig. 2. A sensor network with 200 nodes randomly distributed in a unit square area. The edges of the measurement grah are shown as line segments connecting the true nodes ositions. The little squares are the ositions estimated by the (centralized) otimal estimator. The reference node is at (0, 0) Iteration number, i Fig. 4. Estimation error with iteration number with link failures. Three different failure robabilities are comared with the case of no link failure. PSfrag relacements ˆX (i) ˆX / ˆX flagged fixed random Iteration number, i Fig. 3. Comarison different initialization schemes. Flagged refers to the initialization scheme described in section V. It took 9 iterations for all the nodes to have their have estimate flags to 1, hence errors are shown only after i = 9 for that case. otimal estimate is shown to make it indeendent of the unit of length used. To simulate the algorithm with faulty communication, the following model of link failure was adoted. Every link fails indeendently of other links, and during every iteration it fails with a robability f that is constant for all links. Thus, the time instants that a articular link fails forms a sequence of Poisson oints. Figure 4 shows three different error histories for three different failure-robabilities: f = 0.05, 0.1 and 0.2. In all the cases, the initialization scheme described in section V was used. The error trends show the algorithm converging to the otimal estimate even with link failures. As exected, though, higher failure rates degrade erformance. VII. SUMMARY AND FUTURE WORK We have imlemented a modified version of the Jacobi iterative scheme to comute the otimal estimator; the resulting algorithm is robust to link failures, distributed, scalable and simle. The main drawback of the roosed algorithms is the otentially large number of iterations that maybe required to achieve a desired accuracy when the measurement grah has a low algebraic connectivity (theorem 1). We imroved the convergence of the algorithm by emloying a articular initialization scheme. Other ways to reduce the number of iterations or messages needed to achieve a given accuracy will be exlored in future research. Another imortant issue is that of security, when one or more node estimates may be maniulated by a hostile arty. Making the algorithm robust to security threats and extending it to be able to comute the variance of the estimate are some of the avenues for future research. REFERENCES [1] R. Kar, J. Elson, D. Estrin, and S. Shenker, Otimal and global time synchronization in sensornets, Center for Embedded Networked Sensing,Univ. of California, Los Angeles, Tech. Re., [2] P. Barooah and J. P. Hesanha, Estimation from relative measurements: Error bounds from electrical analogy, in Proceedings of the 2nd International Conference on Intelligent Sensing and Information Processing, Chennai, India., January [3] V. Delouille, R. Neelamani, and R. Baraniuk, Robust distributed estimation in sensor networks using the embedded olygon algorithms, in Third International Worksho on Information Processing in Sensor Networks (IPSN), Aril [4] J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications and Control, A. V. Oenheim, Ed. Prentice Hall P T R, [5] G. H. Golub and C. F. van Loan, Matrix Comutations, 3rd ed. The John Hokins University Press, [6] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, ser. Comuter Science and Alied Mathematics. Academic Press, [7] P. Barooah and J. P. Hesanha, Distributed estimation from relative measurements in sensor networks, University of California, Santa Barbara, Tech. Re., Setember [8] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed comutation: numerical methods. Englewood Cliffs, NJ, USA: Prentice-Hall, Inc., [9] A. Frommer and D. Szyld, On asynchronous iterations, Journal of Com. Al. Math., 123, , 2000.