aa r X i v : . [ c s . S Y ] M a y Bias estimation in sensor networks
Mingming Shi, Claudio De Persis, Pietro Tesi, Nima Monshizadeh
Abstract —This paper investigates the problem of estimatingbiases affecting relative state measurements in a sensor network.Each sensor measures the relative states of its neighbors and thismeasurement is corrupted by a constant bias. We analyse underwhat conditions on the network topology and the maximumnumber of biased sensors the biases can be correctly estimated.We show that for non-bipartite graphs the biases can alwaysbe determined even when all the sensors are corrupted, whilefor bipartite graphs more than half of the sensors should beunbiased to ensure the correctness of the bias estimation. If thebiases are heterogeneous, then the number of unbiased sensorscan be reduced to two. Based on these conditions, we proposesome algorithms to estimate the biases.
I. I
NTRODUCTION
The normal operation of many large scale systems relieson networks of sensors that provide information using forthe monitoring and management of the system operatingconditions [1]-[6]. However, when measuring the variablesof interest, sensors may generate unreliable results due tothe low quality of the hardware, environmental variations oradversary attacks. This introduces measurement errors, whichcan degrade the system performance and even lead to majordisruptions [5]-[11].In this paper, we consider networks in which each sensormeasures the difference between its state and that of itsneighbors and aim to characterize the conditions under whichthe biases corrupting the measurements can be estimated andprovide methods for their estimationThe problem in this paper is broadly linked to othersstudied in the literature. Given erroneous relative measure-ments, providing precise estimates of the relative states canbe considered as a complementary problem to the one ofestimating biases. Many papers [8], [12]-[18] have providedmethods for estimating the states of the sensors from noisyrelative measurements by solving linear or nonlinear leastsquare problems. These methods can not precisely estimatethe state since the least square approach has no robustness tothe measurement error and any error can make the estimationof the unknown deviate from the actual value [19].The formulation of the problem considered in this papercovers the situation where the biases are constant but witharbitrary magnitude, thus allowing for the presence of outliers.Similar problems have been addressed recently in [17], [18],where the focus is on the state estimation problem. However,neither one of the papers gives results on how the sparsityof the measurement errors affects the state estimation. On
M. Shi, C. De Persis, P. Tesi and N. Monshizadeh are with ENTEG,University of Groningen, 9747 AG Groningen, The Netherlands.Email:
[email protected], [email protected],[email protected], [email protected].
P. Tesi is alsowith DINFO, University of Florence, 50139 Firenze, Italy E-mail: [email protected] . the other hand, computing biases from relative measurementsreceived comparably less attention. The paper [20] proposedalgorithms to estimate sensor offsets in wireless sensor net-works. These methods only partially compensate the offsets.In problems that use the angle of arrival (AOA) measurements,if the local frame is unaligned with the global frame, thenthe unknown orientation of the local frame can be regardedas a bias. Ahn et al. [21], [22] use the consensus algorithmto estimate the orientation. However, similar to [20], theestimation error of their algorithms never vanish.In this paper, we reduce the bias estimation problem to thesolution of linear equations (LEs). Several algorithms havebeen devoted to the distributed solution of LEs, with focuson asynchronous implementations [23], [24], graph connec-tivity conditions [25], secure computing [26], to name a few.However, in these algorithms, each node needs to find all theentries of the vector of the unknowns, which, if employed inour problem, would require the nodes to know the networksize. Instead, we exploit a suitable sparsity condition on thebiases to ensure they can be uniquely determined, which isan important problem in compressive sensing [27]-[34], andis related to secure state estimation [5], [35]-[38].A related problem, which several papers have studied, isthe one of achieving consensus or a prescribed formation inthe presence of inconsistent or biased measurements. In [11],the authors use estimators to counteract compass mismatches,while requiring each node to measure the relative positionsof all the edges. The paper [10] addresses the rigid formationcontrol problem where the agents disagree on the prescribedinter-agent distances. For the problem considered in our paper,this method would require that for each pair of adjacent nodes,at least one of the nodes is bias-free. A similar set-up is alsoadopted in [39]. For second-order consensus, [40] proposesan adaptive compensator to prevent the state unboundednesscaused by the biases. The proposed compensator cannot makethe system achieve exact consensus. Our contribution.
Given relative state measurements that areaffected by biases, we find conditions under which the biasesare identified so that the actual relative states can be exactlyreconstructed. Similar to [1], [8], [13], [20], [21], [22], [41],we assume that biased measurements can be exchanged amongthe neighboring nodes. Differently from [17], [40], we assumethat each node has one sensor, hence the relative measurementstaken by the node are affected by the same bias. The form ofthe system of LEs to which we reduce the problem is differentfrom the one formulated in papers involving range or AOAmeasurements [11], [20]-[22]. In our problem (see Section III)the biases affect the relative state measurements, whereas forproblems involving range or AOA measurements the biasesaffect the absolute value of or the pointing of the vector of therelative measurements (distances or bearings). The LEs of the form considered in [11], [20]-[22] also appears in papers thatstudied problems of sensor synchronization [42] and multi-agent fault estimation [35].We provide conditions under which the biases are uniquelydetermined from the proposed system of LEs. Our resultsanswer the question: “what is the maximum number of sensorbiases that can be estimated from erroneous relative statemeasurements?” For non-bipartite graphs, the answer is “allthe nodes” and we provide a distributed algorithm to estimatethe biases. In the algorithm, each sensor only needs to estimateits own bias, leading to a reduction of the computationalresources and memory sizes required at each node, a solutionthat is different from those in [23]-[26].For bipartite graphs, similar to secure state estimationproblems [5], [35]-[38], we show that the biases can becorrectly computed when less than half of the sensors isbiased. Furthermore, we prove that the maximum number ofbiased sensors can be increased if the biases are heterogenous.This reduces the number of unbiased sensors to only two andimproves the results in secure state estimation. We providetwo algorithms to compute the biases. By exploiting theheterogeneous assumption and a coordinator to coordinate thesensors, the first algorithm we propose computes the biases ina finite number of steps. To remove the coordinator and makethe estimation fully distributed, in the second algorithm wesolve a relaxed ℓ -norm optimization problem as in [35], [37].We show an interesting result that the actual vector of biasesis the unique solution of the ℓ -norm optimization problemif less than half of the sensors are biased, which does notworsen the bound on the sparsity condition of the biases forthe non-relaxed problem.We also apply the bias estimation algorithms to a consensusproblem. Different from [40], we can prove that the systemachieves exact consensus. Our algorithms do not require eachnode to measure the relative states of all the edges, in contrastto [11].The rest of the paper is organized as follows. In SectionII, we introduce the notation, some general notions aboutgraphs and and few specialized results on bipartite graphs. Weformulate the problem and provide a useful lemma in SectionIII. Section IV deals with the bias estimation algorithm fornon-bipartite graphs. In Section V, we introduce the sparsitycondition on biases that ensures the correctness of the biasestimation, we provide two bias estimation algorithms andshow consensus using one of the proposed algorithms. SectionVI presents numerical experiments to validate the theoreticalfindings. II. P RELIMINARIES
A. Notation
For a vector z ∈ R p , diag { z } represents the diagonal matrixwith the i th diagonal entry equal to the i th element of z . Wedenote by S z the support of z , which is the set of indices thatcorrespond to the nonzero entries of z , and by k z k the -normof z , which is the number of elements in S z . We let m and m denote the m -dimensional vectors with all elements equalto and , respectively. Given a matrix A , A i represents its i th row and a ij represents its element in the i th row and j thcolumn. The cardinality of a set S is denoted by | S | . For twosets S and M , we let S \ M = { x ∈ S | x / ∈ M } representthe complement of M in S . B. Graph-theoretic notions
For a network with n nodes, let its topology be representedby an undirected and connected graph G = { V, E } , with V = { , , ..., n } being the set of nodes and E ⊆ V × V be theset of edges, where { i, j } ∈ E , or equivalently, node i is aneighbour of node j , means that node i can receive informationfrom node j and vice versa. We denote the set of neighborsof node i by N i , and let d i = |N i | .The adjacency matrix A of G is defined as a ij = 1 if node j is the neighbor of node i and a ij = 0 otherwise. For anundirected graph G , we can assign arbitrary orientations tothe edges such that each edge { i, j } ∈ E has a head and atail. The edge-node incidence matrix B ∈ R m × n of G , with m = | E | , is defined as b ij = 1 if j is the head node of theedge i ∈ E and b ij = − if j is the tail node. The Laplacianmatrix L of G is an n × n matrix given by l ij = − a ij for j = i and l ii = P j ∈N i a ij = d i . Since G is undirected, itis well-known that L = B ⊤ B . The incidence matrix can bedecomposed as the head incidence matrix B + ∈ R m × n andthe tail incidence matrix B − ∈ R m × n , which are given by b + ,ij = (cid:26) , if node j is the head0 , otherwise b − ,ij = (cid:26) − , if node j is the tail0 , otherwise We also let R denote the signless edge-node incidence matrixwith r ij = | b ij | . It is easy to verify that B = B + + B − and R = B + − B − . Let d = [ d d ... d n ] ⊤ and D = diag { d } . Thematrix A + D is called the signless Laplacian matrix. When G is undirected, A + D = R ⊤ R . Hence, A + D is positivesemi-definite and all its eigenvalues µ ≤ µ ≤ · · · ≤ µ n arereal and nonnegative.A path P ij from node i to node j is a sequence of nodes andedges such that each successive pair of nodes in the sequenceis adjacent. The length of a path is the number of edges in thepath. The distance between node i and j is the length of theshortest path from i to j . We denote by D G the diameter of G , which is the maximum distance between any two nodes. C. Bipartite graphs
A graph G is bipartite if the vertex set V can be partitionedinto two sets V + and V − in such a way that no two verticesfrom the same set are adjacent. The sets V + and V − are calledthe colour classes of G and ( V + , V − ) is a bipartition of G . Fora bipartite graph, the following result holds: Theorem 1 [43]
A graph G is bipartite if and only if G hasno cycle of odd length. An algebraic characterization of bipartite graphs is providednext.
Lemma 1
An undirected and connected graph G is bipartiteif and only if the signless incidence matrix R does not havefull column rank. Moreover, if G is bipartite, then any n − columns of R are linearly independent.Proof. To prove the first part, suppose that Rv = 0 for somenonzero vector v ∈ R n . It is easy to see that | v i | = | v j | = a for every i, j ∈ V , where a > . In fact, consider any pathconnecting nodes i and j . For every pair ( r, s ) of adjacentnodes in this path we must have v r = − v s otherwise Rv = 0 .Since the graph is connected and since v must be nonzero, weobtain the claim. Thus there exists a bipartition ( V + , V − ) of G ,where the nodes corresponding to the entries of v with value a and − a are assigned to V + and V − , respectively. Converselyif G is bipartite, there exists a bipartition ( V + , V − ) of G . Byletting the elements of v corresponding to V + and V − be a and − a , respectively, with a = 0 , we have Rv = 0 , whichshows that R does not have full column rank.For the second part, we prove it by contradiction. Supposethere exist some dependent columns of R and let the indexset of these columns be S ⊂ V , with | S | ≤ n − , then thereshould exist a nonzero vector v ∈ R | S | such that R S v = 0 where R S is the matrix whose columns are those indexed by S .The latter implies the existence of a nonzero vector ˜ v , whosenonzero entries are given by v , and satisfies R ˜ v = 0 . However,from the proof of the first part, the absolute values of all theelements of ˜ v should be equal to each other. Hence, v mustbe the zero vector, which is a contradiction. (cid:4) The if and only if part of the statement above is alsoprovided in [44, Lemma 2.17]. We provide the proof here,since it is used in proving the second part of the statement aswell as in other parts of the paper.For later use, by the proof of Lemma 1, we note that Rv = n ⇐⇒ ∃ a ∈ R s . t . v i = ( a i ∈ V + − a i ∈ V − (1)for a bipartite graph with bipartition ( V + , V − ) . Lemma 2 [45]
The smallest eigenvalue of the signless Lapla-cian matrix A + D of an undirected and connected graph isequal to zero if and only if the graph is bipartite. In case thegraph is bipartite, zero is a simple eigenvalue.D. Compressed sensing In the field of compressed sensing or sparse signal recovery,one of the most important problems is how to find the sparsestsolution from the number-deficient measurements. Formally,consider the following linear equation y = F x (2)where x ∈ R n is the vector of unknown variables, y ∈ R p isthe vector of known values, and F ∈ R p × n is a matrix definingthe linear relation from x to y . It is assumed that p < n , thusequation (2) is under-determined. It is then of interest to findsolutions x such that k x k ≪ n , and in particular to seek forthe sparsest solution of (2). Let us define the set of k -sparsevectors as W k := { x ∈ R n | k x k ≤ k } . (3) The following result provides a sufficient condition underwhich the solution of (2) can be uniquely determined. Lemma 3
Given an integer s ≥ , let s ≤ p , and assumethat any matrix made of s columns of F is full column rank. If x ∈ W s is a solution of (2) , then there exists no other solutionof (2) in W s . Remark 1
Under the assumptions of the lemma, the solution x ∈ W s of (2) is also the solution to min x ∈ R n k x k s . t . y = F x, (4)that is, the sparsest solution to (2). The proof of Lemma 3descends from [28, Lemma 1]. (cid:4)
However, solving x from (2) under the assumption that k x k ≤ s is cumbersome when s is not small, as it requiresto combinatorially search for s columns of F whose spancontains y . A typical way to avoid this exhaustive search is tochange the problem into the following ℓ -norm optimizationproblem min x ∈ R n k x k (5) s . t . y = F x where y is the vector of known values in (2) and the objectivefunction and the constraint are both convex. Problem (5) can besolved by linear programming [30]. The ℓ -norm minimizationmay return a solution x ∗ different from the solution x of (2).The following definition and result characterize the relationbetween the matrix F , the equation (2) and the ℓ -normminimization problem. Definition 1 (Nullspace Property)
A matrix F ∈ R p × n issaid to satisfy the nullspace property of order s , with s beinga positive integer, if for any set S ⊂ V = { , , ..., n } with | S | ≤ s and any nonzero vector v in the null space of F , thecondition below holds k v S k < k v S c k , (6) where v S ∈ R | S | and v S c ∈ R | S c | are subvectors of v whoseelements are indexed by S and S c , respectively, and S c = V \ S . The null space property is usually difficult to verify and a morerestrictive but more conveniently checkable condition knownas restricted isometry property is considered [30, p. 8]. Yet,in the special cases that are of interest to us the null spaceproperty can be easily confirmed (cf., Theorem 7), and wewill persist with it in the sequel.
Theorem 2 [30, Theorem 2.3]
Every vector x ∈ W s is theunique solution of the ℓ -norm minimization problem (5) , with y = F x , if and only if F satisfies the null space property oforder s . We highlight the role of this theorem explictly in connectionwith the equation (2). For a given y ∈ R p , let x ∈ R n be a solution of (2). Assume that k x k ≤ s and F satisfies the nullspace property of order s , with < s < n . By Theorem 2, x isthe unique solution of (5), with y = F x . Stated directly, thereexists a unique solution x ∗ of (5), with y = F x , and it satisfies x ∗ = x . Hence, under the given condition of s -sparsity of thevector x solution of (2) and the null space property of order s of the matrix F , solving the optimization problem (5), with y = F x , univocally returns x .III. P ROBLEM FORMULATION – BIASES ESTIMATION INSENSOR NETWORKS
We consider a sensor network where each sensor is iden-tified with a node in a graph G = ( V, E ) with V the set ofnodes, | V | = n ≥ and E the set of edges. Throughout thepaper, we assume that G is connected and undirected. A statevariable x i ∈ R is associated to each node i ∈ V . Each sensor i ∈ V can measure the relative information x j − x i for all j ∈ N i .We are interested in a scenario where the measurementstaken by the sensor network may be subject to constant biases.As a result of the bias, the relative information read by thesensor i , will be modified as z ij = x j − x i + w i , ∀ j ∈ N i , (7)where w i ∈ R is an unknown constant term accounted for thebias of sensor i . In case a sensor is bias free we set w i = 0 .The presence of biases deteriorate the performance of thenetwork, and may even raise stability issues. Thus it is ofinterest to estimate the biases, and possibly counteract theireffect in the network.To formulate the problem, we first rearrange the equalitiesin (7) in a suitable vector form. After assigning arbitraryorientation to G , we collect in the vector ζ ∈ R m all themeasurements z ij for which node i ∈ V is the head of the edge i, j ∈ E , which gives ζ = − Bx + B + w , with B + denotingthe head incidence matrix. Similarly, we collect in the vector η ∈ R m all the measurements z ij for which node i ∈ V is thetail of the edge i, j ∈ E , and obtain η = Bx − B − w , where B − is the tail incidence matrix. Hence, z := (cid:20) ζη (cid:21) = (cid:20) − BB (cid:21) x + (cid:20) B + − B − (cid:21) w (8)Note that, by construction, we have z ∈ im( B ) , where B := (cid:20) − B B + B − B − (cid:21) and im( · ) denotes the column span of a matrix.For a given measurement z , we are interested in finding thebias vector w in a set W ⊆ R n of admissible biases, whichis defined more precisely later. To avoid ambiguity, we firstintroduce the definition of a solution of (8) with respect to w . Definition 2 (Solution of (8) in W ) Given z ∈ im ( B ) and aset W ⊆ R n of admissible biases, the vector w ∈ W solves (8) if there exists x ∈ R n such that (8) is satisfied with ( x, w ) =( x, w ) . In this case, we say w solves (8) in W , or w is asolution of (8) in W . The uniqueness of the solution of (8) is defined below:
Definition 3 (Unique solution of (8) in W ) A solution w of (8) in W is unique if there exists no vector w ′ , with w ′ = w ,which is a solution of (8) in W . In this case, we say w uniquelysolves (8) in W . We then formulate the problem which is of interest in thispaper.
Problem formulation.
Given the vector of biased measure-ments z ∈ im ( B ) and a set W ⊆ R n of admissible biases,find conditions under which the vector of actual sensor biases w is the unique solution of (8) in W , and design algorithmsfor estimating it.Note that W should always contain the bias vector w andby construction at least one solution to (8) exists. Determiningconditions under which the solution to (8) is unique implieswe can correctly estimate the vector of actual biases affectingthe measurements. To prove the uniqueness of the solution of(8) we will rely on a reduced form of (8) provided in thefollowing result: Lemma 4
Consider the vector of biased measurements z ∈ im ( B ) and a set W ⊆ R n of admissible biases. Consider theequality Rw = ˜ z (9) where R = B + − B − is the signless edge-node incidencematrix, ˜ z = F z , and F = [ I m I m ] is the left annihilator ofthe matrix [ − B ⊤ B ⊤ ] ⊤ . Then the following two statementshold:(i) The vector w is a solution of (8) in W if and only if w ∈ W is a solution of (9) .(ii) The vector w is the unique solution of (8) in W if andonly if w is the unique solution of (9) in W .Proof. ( i ) . (Only if) If w is a solution of (8) in W , thenpre-multiplying (8) by F leads to ˜ z = Rw . Hence w ∈ W isalso a solution of (9).(If) Since w is a solution of (9) in W , then ζ + η = Rw ,with w ∈ W . Since z ∈ im( B ) , there should exist a vector x ′ ∈ R n and w ′ ∈ R n such that z = (cid:20) − BB (cid:21) x ′ + (cid:20) B + − B − (cid:21) w ′ (10)Pre-multiplying the equality above by F leads to ˜ z = Rw ′ .Combining this with ˜ z = Rw , we have R ( w ′ − w ) = m .We continue the proof considering the following two distinctcases. Case 1. G is not bipartite. Since G is not bipartite, byLemma 1 the matrix R is full-column rank, which implies w ′ = w ∈ W . Hence w ∈ W is a solution of (8). Case 2. G is bipartite. Since G is bipartite, there shouldexist a bipartition V = { V + , V − } . Let | V + | = p , label thenodes in V such that V + = { , , ..., p } , V − = { p + 1 , ..., n } and define the orientations of the edges in such a way that the head node of each edge in E belongs to V + . Bearing in mindthe identity R ( w ′ − w ) = m above, and noting (1) we have w ′ = w + f a, f = (cid:20) p − n − p (cid:21) , (11)for some a ∈ R . Substituting this back to (10) yields z = (cid:20) − BB (cid:21) x ′ + (cid:20) B + − B − (cid:21) w + (cid:20) B + − B − (cid:21) f a. (12)To prove that w is a solution of (8) in W , in view of Definition2, we need to show that z − (cid:20) B + − B − (cid:21) w ∈ im (cid:20) − BB (cid:21) , which, by (12), reduces to (cid:20) B + − B − (cid:21) f ∈ im (cid:20) − BB (cid:21) . (13)Let B + and B − be decomposed as B + = (cid:2) ˜ B + m × ( n − p ) (cid:3) , B − = (cid:2) m × p ˜ B − (cid:3) for some matrices ˜ B + and ˜ B − . Then (13) can be written as (cid:20) ˜ B + m × ( n − p ) m × p − ˜ B − (cid:21) f ∈ im (cid:20) − ˜ B + − ˜ B − ˜ B + ˜ B − (cid:21) where we have used the fact that B = B + + B − . Notingthat ˜ B + p = − ˜ B − n − p , it is easy to verify that the aboverelationship is satisfied since (cid:20) ˜ B + m × ( n − p ) m × p − ˜ B − (cid:21) f = (cid:20) − ˜ B + − ˜ B − ˜ B + ˜ B − (cid:21) (cid:20) p n − p (cid:21) . This completes the proof of part (i). ( ii ) . We only prove the “if” part since the converse impli-cation can be shown similarly. Assume w is a unique solutionof (9) in W , then by (i), we have w is also a solution of (8) in W . Now if there exists another vector w ′ ∈ W , with w ′ = w is a solution of (8), it should also be a solution of (9) by thefirst statement. This contradicts the uniqueness assumption. (cid:4) The result of Lemma 4 will be used in some of thederivations of the main results in the sequel.To study the conditions guaranteeing the uniqueness of thesolution of (8) in W , we differentiate between bipartite andnot bipartite graphs.IV. N ON - BIPARTITE GRAPHS
In this section, we present the results for the case when themeasurement graph G is not bipartite. A. Condition for correct bias estimation
The following result shows that w can be determineduniquely from (8) if the graph is not bipartite. Theorem 3
Consider a graph G , let z ∈ im( B ) be the vectorof biased measurements, and W = R n be the set of admissiblebiases. Then w is the unique solution of (8) in W = R n ifand only if G is not bipartite.Proof . In view of Lemma 4, we need to show that the biasvector w is the unique solution of (9) if and only if G is notbipartite. This holds since, by Lemma 1, the matrix R has fullcolumn rank if and only if G is not bipartite. (cid:4) B. Distributed bias estimation
In this section we propose a distributed algorithm to esti-mate the biases. We assume the existence of a communicationnetwork, modeled by an undirected and connected graph G c = ( V c , E c ) , through which the nodes can communicatewith each other without any imperfection. We let V c = V and E c = E .We assign to each node a bias estimation variable ˆ w i of thebias w i affecting its sensor. For each node i ∈ V , we let theestimation variable evolve as follows: ˙ˆ w i = X j ∈N i ( z ij + z ji − ˆ w i − ˆ w j ) (14)Node i uses the biased measurements z ij and z ji , and the biasestimates ˆ w i and ˆ w j . Note that the values of z ji and ˆ w j arecommunicated to node i via the link { j, i } .The following result shows exponential convergence of theestimates to the actual biases. Proposition 1
The estimate vector ˆ w generated by (14) con-verges exponentially fast to the vector w of the actual biasesif the measurement graph G is not bipartite.Proof. Denote the estimation error for the bias w i as e i =ˆ w i − w i . From (14), we have ˙ e i = ˙ˆ w i − ˙ w i = X j ∈N i ( z ij + z ji − ˆ w i − ˆ w j )= X j ∈N i ( x j − x i + w i + x i − x j + w j − ˆ w i − ˆ w j )= − X j ∈N i ( e i + e j ) (15)which in a matrix form can be expressed as ˙ e = − ( A + D ) e (16)By Lemma 2, the matrix − ( A + D ) is Hurwitz if and only if G is not bipartite. The exponential convergence of the estimationerror e then follows immediately. (cid:4) An alternative way to solve for w in (9) is to use theblock partition method of [46], [41], [17]. When applied to theproblem under investigation in this paper, the method requireseach node to estimate not only its own bias but also those ofits neighbors. In contrast, the estimation algorithm (14) onlyrequires each node to store and transmit it own estimate, henceit reduces the memory space and communication burden. C. An example of use: rejecting biases in a consensus network
In this subsection, we investigate the possibility of removingthe effect of relative state measurement biases from a con-sensus algorithm. By exploiting the bias estimation methodprovided in the previous subsection, we devise a compensatorthat asymptotically rejects the biases. To this end, let ˙ x i = X j ∈N i z ij + u ci = X j ∈N i ( x j − x i ) + d i w i + u ci , ∀ i ∈ V (17)where u ci is an additional control input available to the de-signer. Note that without a proper compensation, i.e., u ci = 0 ,solutions of (17) can be unbounded. Let u ci be given by u ci = − d i ˆ w i , ∀ i ∈ V (18)where ˆ w i is given by (14). This results in the closed-loopdynamics ˙ x i = X j ∈N i y ij − u ci = X j ∈N i ( x j − x i + w i − ˆ w i )= X j ∈N i ( x j − x i − e i ) (19)which can be written compactly as ˙ x = − Lx − De. (20)In case of a non bipartite graph, the vector of biases can beasymptotically rejected and consensus can be achieved:
Proposition 2
Let G be a non-bipartite graph. Then, solu-tions ( e, x ) of (16) , (20) , exponentially converge to the point ( e ∗ , x ∗ ) , where x ∗ ∈ im( n ) and e ∗ = 0 . If ˆ w is initialized atzero, equivalently e (0) = − w , then we have x ∗ i = ⊤ n n D ( A + D ) − w + ⊤ n x (0) n . (21) for each i ∈ V .Proof . Equation (20) can be seen as the conventionalconsensus dynamics driven by the bias estimation error. Let λ < λ ≤ λ ≤ · · · ≤ λ n be the eigenvalues of L alongwith the basis of orthonormal eigenvectors { n √ n , v , · · · , v n } .Define Λ = diag [ λ , · · · , λ n ] , U = [ n √ n U ] with U =[ v · · · v n ] and apply the state transformation z = U ⊤ x .In the new coordinates, we have ˙ z = − Λ z − U ⊤ De (22)where z is the solution of ˙ z = − ⊤ n D √ n e (23)and z [2: n ] := (cid:2) z . . . z n (cid:3) ⊤ follows ˙ z [2: n ] = − Λ z [2: n ] − U ⊤ De (24)with Λ = diag [ λ , · · · , λ n ] . By Proposition 1, if G is notbipartite then the estimation errors satisfy e ( t ) = e − ( A + D ) t e (0) , (25)from which we have z ( t ) = − ⊤ n √ n D ( A + D ) − (1 − e − ( A + D ) t ) e (0) + z (0) (26) which implies lim t → + ∞ z ( t ) = − ⊤ n √ n D ( A + D ) − e (0) + z (0) . Since Λ > , the vector z n ( t ) converges to zero exponen-tially fast. Hence, we find that x exponentially converges to c n for some c ∈ R . It is easy to see that c = 1 √ n lim t → + ∞ z ( t ) . If e (0) = − w , then c = x ∗ i given by (21), for each i ∈ V ,which completes the proof. (cid:4) Although the system with bias compensation achieves con-sensus, the exact consensus value to which the agents convergeis not predictable since it depends both on the initial stateand the bias of the sensors. For those problems where itis of primary interest to converge to the average consensus,alternatively one can first run the algorithm (14) over asufficiently large time horizon to obtain a sufficiently accurateestimate of the biases, and then directly remove the biasesfrom the measurements used in the consensus algorithm.V. B
IPARTITE GRAPHS
In this section, we consider the case where the measurementgraph G is bipartite. A. Conditions for bias estimation
For bipartite graphs, the following result gives a generalcondition that ensures that the vector of biases can be correctlyestimated from the measurement (8).
Theorem 4
Consider a bipartite graph G , if a vector w solves (8) in W k , with k = ⌊ n − ⌋ , then it uniquely solves (8) in W k .Proof. Since G is bipartite, by Lemma 1, any submatrix of R with n − columns has full column rank. Hence, by Lemma3, if there exists a solution w ∈ W k of (9), then it is unique in W k . The proof ends by noticing that if w is a unique solutionof (9) in W k then it is the unique solution of (8) in W k (seeLemma 4). (cid:4) To ensure uniqueness of the solution in (8), approximatelyhalf of the sensors are required to be bias free by Theorem 4.Next, we introduce rather mild restrictions on the admissibleset of biases W in order to obtain more relaxed conditions onthe number of bias free sensors. Definition 4 (i) The set W hk , with ≤ k ≤ n , of hetero-geneous k -sparse bias vectors is the set of all vectors w ∈ W k such that their nonzero entries are differentfrom each other, namely w i = w j for any i, j ∈ V with w i = 0 and w j = 0 .(ii) The set W ak , with ≤ k ≤ n , of absolutely heterogeneous k -sparse bias vectors is the set of all vectors w ∈ W k such that their nonzero entries in absolute value aredifferent from each other, namely | w i | 6 = | w j | for any i, j ∈ V with w i = 0 and w j = 0 . Note that we have W ak ⊂ W hk ⊂ W k , for each k =2 , , . . . , n . Theorem 5
Consider a bipartite graph G ,(i) If there exists w that solves (8) in W hn − , then it uniquelysolves (8) in W n − .(ii) If there exists w that solves (8) in W an − , then it uniquelysolves (8) in W n − .Proof. Noting Lemma 4, we work with equation (9) to proveuniqueness of the solution. ( i ) We prove this part by contradiction. Suppose there existsanother solution w ′ = w of (9), satisfying w ′ ∈ W n − . Then R ( w − w ′ ) = 0 . (27)By (1) this implies that w = w ′ + f a , where f is given by(11) and a ∈ R .Let S w and S w ′ be the support of w and w ′ . If V \ ( S w ∪S w ′ ) is nonempty, i.e, there exists at least one index i ∈ V suchthat w i − w ′ i = 0 , then a = 0 . This implies w = w ′ and leadsto a contradiction. If S w ∪ S w ′ = V , we have that S w \ S w ′ =( S w ∪ S w ′ ) \ S w ′ = V \ S w ′ should have at least elementssince k w ′ k ≤ n − . However, this would imply that thereexist at least three distinct indices i, j, k ∈ S w \ S w ′ , such thateach one of w i , w j , w k is either equal to a or − a , with a = 0 .Hence, at least two elements in the set { w i , w j , w k } mustbe the same, which contradicts the heterogeneity assumption w ∈ W hn − . This completes the proof of uniqueness for part(i). ( ii ) Suppose by contradiction that there exists anothersolution w ′ = w of (9), satisfying w ′ ∈ W n − . Analogous tothe proof of a ) , if V \ ( S w ∪ S w ′ ) is nonempty, then w = w ′ ,while if S w ∪S w ′ = V , the set S w \S w ′ has at least elementssince k w k ≤ n − . This would imply that there exist at leasttwo distinct indices i, j ∈ S w \ S w ′ , such that each one of w i and w j is equal to either a or − a , with a = 0 . This resultsin | w i | = | w j | , thus contradicting the absolute heterogeneityassumption w ∈ W an − . This completes the proof. of (8). (cid:4) Thus, focusing the attention on the class of heterogeneousbiases in the sense of Definition 4 considerably increases thenumber of allowable biased sensors.
B. Distributed bias computation with coordinator
In this subsection we focus on algorithms for computing theactual vector of biases w . We propose the use of a coordinatorthat delegates the computation of the biases to the nodeswhile organising the execution of their commands. Comparedto a centralized solution, the distributed computation with acoordinator eases the analysis and does not require to knowthe network topology.We consider the case when w ∈ W an − and use the resultestablished in Theorem 5 (ii). When w ∈ W an − , there exist Note the following two identities: |S w ∪ S w ′ | = |S w | + |S w ′ | − |S w ∩S w ′ | and |S w ′ | = |S w ∩ S w ′ | + |S w ′ \ S w | . Replacing the right-hand sideof the second identity into the first one, we obtain |S w ∪ S w ′ | = |S w | + |S w ′ \ S w | , or |S w ′ \ S w | = |S w ∪ S w ′ | − |S w | . Since |S w ∪ S w ′ | = n and |S w | ≤ n − , we obtain |S w ′ \ S w | ≥ , as claimed. at least two (bias free) nodes i, j ∈ V , i = j , satisfying w i = w j = 0 . The essence of the algorithm here is to find such abias free pair. To this end, some additional notation is needed.For a pair of nodes i, j ∈ V with i = j , let P ij be a pathconnecting them, namely P ij = { k , k , . . . , k d ij } , with k = i , k d ij = j , and d ij the length of the path. Moreover, wecollect the measurements that are indexed by P ij as Z ij := z k k + z k k z k k + z k k ... z k dij − k dij + z k dij k dij − . (28)Finally, we let e d ij = (cid:2) ( − d ij − ( − d ij − . . . ( − ( − (cid:3) ⊤ . (29)We then have the following result: Proposition 3
Consider a bipartite graph G , let w be thevector of biases and assume that w ∈ W an − . For a given pairof nodes i, j ∈ V , with i = j , and a path P ij connecting them,we have:(i) I ij := e ⊤ d ij Z ij = 0 if and only if w i = w j = 0 , i.e., thepair i, j ∈ V is bias-free.(ii) If w i = 0 , then I ij = w j .(iii) I ik ℓ = − I ik ℓ − + ( z k ℓ − k ℓ + z k ℓ k ℓ − ) for ℓ ∈ { , ..., d ij } ,where I ik ℓ , I ik ℓ − are defined similarly to I ij .Proof. (i) By (7), the vector Z ij equals Z ij = w k + w k w k + w k ... w k dij − + w k dij (30)from which I ij = e ⊤ d ij Z ij = P d ij ℓ =1 ( − d ij − ℓ ( w k ℓ − + w k ℓ )= ( − d ij − w k + w k dij = ( − d ij − w i + w j . (31)Noting w ∈ W an − , we find that e ⊤ Z ij = 0 if and only if w i = w j = 0 , as claimed.(ii) By (31), we immediately obtain that I ij = w j if w i = 0 .(iii) The conclusion is straightforward to obtain by thedefinition of I ij and (29). (cid:4) From Proposition 3 (i), no matter along which path thequantity I ij is computed, the identity I ij = 0 holds if andonly if the pair i, j ∈ V is bias-free. Hence, I ij is an indicatorof whether or not a pair of nodes are bias free. In addition,by Proposition 3 (iii), if node k ∈ N j knows I ij , then itcan compute I ik . In turn, by Proposition 3 (ii), if w i = 0 ,then the variable I ik equals the bias w k . Based on Proposition3, searching the bias free nodes and solving the bias can beconcurrently carried out by the nodes in a distributed fashioncoordinated by a coordinator. The idea is to let the coordinatormake n − selections of a candidate bias-free node i and letthe other nodes j compute the variables I ij with respect to theselected node. As soon as a zero I ij is observed at a node j , Algorithm 1:
Coordinator
Data:
Set of nodes V and counter T ; Initialize: T := 0 ; for i = 1 : n − do Inform all the nodes in V to start the Node pair teststage in Algorithm 2;Inform node i that it is selected and nodes j ∈ V \ i that they need to calculate and send back thevariable I ij to the coordinator; T = T + 1 ;Once Node pair test stage is completed by all thenodes, receive I ij and t j from all j ∈ V \ i ;Compute T = T + max j ∈ V \{ i } { t j } ; if there exists one I ij = 0 then Stop the for iteration; end ifend for
Inform all the nodes to start the
Bias computing stage ; Algorithm 2:
Node j Data:
Set of neighbors N j , measurement data { z jk + z kj } k ∈N j and counter t j ; if informed to start the Node pair test stage then /* Node pair test stage */if node j is selected in iteration i , i.e. j = i then Set the auxiliary variable I jj = 0 and t j = 1 ;Send ( I jj , t j ) to all k ∈ N j ;Stop accepting data from the neighbors; else Once ( I ik , t k ) , for some k ∈ N j , are received,pick any one of ( I ik , t k ) and compute I ij := − I ik + ( z jk + z kj ) , t j = t k + 1 ;Send ( I ij , t j ) to all k ∈ N j and the coordinator;Stop accepting data from the neighbors; end ifif informed to start the Bias computing stage then /* Bias computing stage */ w j = I ij ; end if then that node informs the coordinator to terminate the search.At this stage, every node has computed the value of its biasvia the indicator variable, namely I ij = w j .The commands executed by the coordinator are summarizedin Algorithm 1, whereas the commands executed by the nodesare listed in Algorithm 2. Algorithm 2 comprises two stages, the node pair test stage , in which the coordinator and thenodes cooperate to check whether or not a given pair of nodesis bias free, and the bias computing stage during which thebiases are explicitly computed. In Algorithm 2, we assume thateach node has access to the data { z ij + z ji } j ∈N i , which canbe achieved by letting all the nodes collect the measurementsfrom their neighbors, before running Algorithms 1 and 2.To measure the number of executed instructions requiredby the algorithms to terminate the computation, we introduce counters that store integer values. In Algorithm 1, the se-quence of actions by the coordinator consisting of informingnode i that it has been selected, and asking nodes j ∈ V \ i to calculate and send back the variable I ij is considered asone instruction, which increases the counter T by unit.The single action of informing all the nodes to start the Biascomputing stage , is regarded as another instruction, and againresults in an increase of T by unit. In Algorithm 2, ateach iteration i , the variable t j , j ∈ V , stores the number ofinstructions executed from the moment that node i is selectedby the coordinator till when j computes I ij . The counters t j , j ∈ V , are communicated to the coordinator and usedto update the counter T , which therefore contains the totalnumber of instructions executed before the bias free node pairis found. Note that the counters are only introduced to storethe number of instructions needed for the computation of thesolution, as formalized in Theorem 6, but do not play any rolein the computation of the solution itself.The following result summarizes the properties of the algo-rithms: Theorem 6
Consider a bipartite graph G , with its diametergiven by D G , let w be the vector of biases and assume that w ∈ W an − . If the coordinator uses Algorithm 1 and the nodesAlgorithm 2, then a bias free node can be identified in T instructions and the vector of biases w can be reconstructedin T + 2 instructions with T ≤ ( n − D G + 2) .Proof. At iteration i , with i = 1 , , . . . , n − , the coordinatorselects node i and informs all the nodes to start the node pairtest stage (see Algorithm 1). We first focus on the node pairtest stage.According to Algorithm 1, if node i ∈ V \ { n } is selected,the coordinator informs all the nodes k ∈ V , and T isincreased by . According to Algorithm 2, when node i receives the message from the coordinator that it has beenselected, it sets I ii = 0 and t i = 1 , and sends them to all theneighbors j ∈ N i . The instructions executed from the instantwhen node i has been informed that it has been selected tothe instant when nodes i computes I ii are regarded as one andit is set t i = 1 .When the node j ∈ N i receives ( I ii , t i ) = (0 , , itcomputes I ij = − I ii + ( z ij + z ji ) = I ij and t j = t i + 1 = 2 ,then sends ( I ij , t j ) to the coordinator and its neighbors. Hence t j = 2 actions are executed from the instant when node i isinformed to have been selected to the instant when node j computes I ij . Let D max i be the maximum of the distancesof node i to all other nodes in V . Consequently, each node j ℓ , which is at a distance ℓ ∈ { , , ..., D max i } from node i ,receives ( I ij ℓ − , t j ℓ − ) , with t j ℓ − = ℓ , from some neighbor j ℓ − , which is at a distance ℓ − from node i . The node j ℓ computes t j ℓ = t j ℓ − + 1 = ℓ + 1 and, in view of Proposition3 (iii), we have I ij ℓ = − I ij ℓ − + ( z j ℓ − j ℓ + z j ℓ j ℓ − ) . (32)All the nodes j ℓ then send ( I ij ℓ , t j ℓ ) , with t j ℓ = ℓ + 1 , to theirneighbors and the coordinator. Hence t ℓ = ℓ + 1 instructionsare executed from the instant when node i is informed it has been selected to the instant when node j ℓ computes I ij ℓ .By this analysis, after node i has been informed at iteration i = 1 , , ..., n − , t D max i = max j ∈ V \{ i } { t j } = D max i + 1 instructions are executed before the coordinator receives I ij from all j ∈ V \{ i } . Hence at each iteration i = 1 , , ..., n − , T is increased of at most to check where the extra +1 comesfromSince I ij and I ji can be used interchangeably, the coordi-nator obtains all I ij for i, j ∈ V , i = j , in at most n − iterations. By the assumption w ∈ W an − and Proposition 3,there always exists an iteration i = 1 , , ..., n − and a node j ∈ V \ { i } such that I ij = 0 . Hence the bias free node pairshould be found in T ≤ ( n − D G + 2) steps.We then consider the bias computing stage. This occursif the coordinator received I ij = 0 at iteration i for some j ∈ V \ { i } . Then each node k ∈ V enters this stage and itconcludes that the computed quantity I ik is the bias w k . Asa matter of fact, since I ij = 0 , then w i = 0 , by Proposition3 (i), and this actually implies that I ik = w k if k = i , byProposition 3 (ii). For k = i , we note that I ii was set equal tozero in the node pair test stage, and therefore I ii = w i = 0 .To complete the computation of the number of executedinstructions, we note that by Algorithm 2 one more instructionis needed to let the coordinator inform all the nodes that i isbias free and another instruction to let the nodes compute thebiases. (cid:4) A few remarks are in order:- In case w ∈ W an \ W an − , so that the assumption w ∈ W an − in Theorem 6 is not satisfied, then I ij = 0 will not beobserved at any node, and the coordinator infers that thereis no pair of bias-free nodes.- In Algorithm 1, the coordinator is only responsible forcoordinating the nodes, namely initializing each iteration,whereas all computations are performed at the nodes in adistributed fashion. Moreover, note that the coordinator doesnot need to know the topology of the network, apart fromthe node set V .- Another method to compute the vector of biases when w ∈ W an − is to combinatorially search the pair of nodesthat is bias-free, as in [5], [47]. Specifically, for each pair ofindices i, j ∈ V , with i = j , one could look for a solutionof the modified equation Rw ( i,j ) = ˜ z , where w ( i,j ) is avector whose entries i and j are set to zero. If a solutionto this modified equation exists, then by construction itsatisfies the sparsity condition k w ( i,j ) k ≤ n − , and byTheorem 5 (ii), it will be equal to the vector of actualbiases. Hence, the determination of the vector of biases w satisfying (8) is reduced to considering the n ( n − / systems of equations and check if each of these equationsadmit a solution. Note however that such an approach wouldrequire that the unit carrying out the combinatorial searchhas access to the network topology and possesses enoughcomputational power. C. Distributed bias estimation without coordinator
In the previous section we assumed the existence of acoordinator that supervises the nodes checking the conditions of Proposition 3. In this section, we seek a method thatestimates the biases in a distributed manner without resortingto a coordinator. We show that this is achievable providedthat we restrict the class of admissible biases. To this end, bySubsection II-D and equation (9), we consider the following ℓ -norm minimization problem min w ∈ R n k w k (33) s . t . R w = ˜ z, where ˜ z is the vector of known values appearing in (9). Asmentioned in Section II-D, solving the ℓ -norm minimizationproblem may yield a solution that is different from the vectorof actual biases w . The sparsity condition under which thesolution of (33) coincides with w is provided in the followingtheorem. Theorem 7
For a bipartite graph G , the vector of biases w isthe unique solution of the ℓ -norm minimization problem (33) if the number of biased sensors is not greater than ⌊ n − ⌋ , i.e., w ∈ W ⌊ n − ⌋ .Proof. Since the graph is bipartite, then (1) holds. Hence,inequality (6) in this case is given by X i ∈ S, | S | = s | v i | < X j ∈ S c | v j |⇐⇒ s | a | < ( n − s ) | a | , a = 0 (34)which is satisfied if and only if s < n . Hence, the matrix R satisfies the null space property of order s , with s = ⌊ n − ⌋ .Therefore, by (9), Theorem 2 and the discussion followingit, if the vector of biases w in (9) satisfies w ∈ W ⌊ n − ⌋ , thenthere exists a unique solution of the optimization problem (33),with ˜ z = Rw , and it is equal to w . (cid:4) This theorem shows that for bipartite graphs, the ℓ -normminimization does not decrease the maximum number ofallowed biased sensors obtained in Theorem 4. On the otherhand, in the case where the vector of biases w belongs tothe set of heterogeneous biases W hn − or W an − consideredin Theorem 5, examples can be found where the solution ofthe ℓ -norm minimization problem does not give the correctbias estimation. Hence, below, we only discuss the solution of(33) for the case of bipartite graphs with a number of biasedsensors as characterized in Theorem 7.The ℓ -norm optimization problem (33) can be solveddirectly in a distributed manner by the methods in [48], [34]. Inthis paper, we reformulate it as a linear programming problemas [33] min η ∈ R n ⊤ n η (35) s . t . Hη = ˜ z, η ≥ where η is the decision variable and H = (cid:2) R − R (cid:3) . (36)Under the sparsity condition in Theorem 7, if η ∗ is the solutionof (35), the vector of biases can be computed as w = (cid:2) I n − I n (cid:3) η ∗ (37) The linear programming problem above can be solved byvarious distributed methods available in the literature, see e.g.[49], [50], [51]. In particular, using the result of [51], the biasestimation algorithm takes the form ˆ w = (cid:2) I n − I n (cid:3) η ˙ η i = ( f i ( η, λ ) , if η i > { , f i ( η, λ ) } if η i = 0 , i ∈ V ˙ λ = Hη − ˜ z (38)with f ( η, λ ) = − n − H ⊤ ( λ + Hη − ˜ z ) , (39)and where λ ∈ R m is the dual variable and the initial conditionsatisfies η i (0) ≥ for all i ∈ V .For this algorithm, we have the following result: Proposition 4
The estimate ˆ w generated by the algorithm (38) , (39) converges asymptotically to the vector of biases w if G is bipartite and w ∈ W ⌊ n − ⌋ .Proof . This result follows directly from [51, Proposition IV.4]noting that the linear program (35) has a unique solution. (cid:4) Remark 2
Similarly to Subsection IV-C, one could use theestimate ˆ w generated by the algorithm (38), (39) in thecompensator (18) to reject the effect of the biases and achieveconsensus. In fact, the consensus dynamics (20) driven by theestimation error e continues to be valid and an analysis similarto the one in Proposition 2 can be carried out. In the case ofbipartite graphs, however, we cannot provide the estimate ofthe new consensus value, due to the lack of the exponentialconvergence of the estimation error. (cid:4) For the problem at hand, the algorithm (38) has someadvantages when compared with possible alternatives, suchas the one provided by the recent paper [48], where a newdistributed algorithm for solving the ℓ -norm minimizationproblem with linear equality constraints is proposed. However,in this method each node needs to reconstruct all the elementsof the solution of the ℓ -norm minimization problem, whichimplies that each node stores and communicates a vector withthe same dimension as the (unknown) solution. Moreover, animplicit requirement for the method in [48] is that each agentmust know the number of columns of the coefficient matrix,which translates to knowing the network size in our setting.In the method given by (38), on the other hand, each nodereconstructs only one element of w by communicating suitablevariables with its neighbor. The latter is done without relyingon any global information including the size of the network. Remark 3
Resorting to different formulations of the ℓ -normminimization problem, one can obtain variations of the algo-rithm (38) with different features. For instance, (33) can bereformulated as min w ∈ R n k w k (40) -2 -1.5 -1 -0.5 0 0.5 1 1.5-1-0.500.511.5 Fig. 1. Non-bipartite graph with 10 nodes s . t . R ⊤ R w = R ⊤ ˜ z, where R ⊤ R = A + D is the signless Laplacian matrix(see Section II-B). We can transform the above into a linearprogram analogous to (35). Then, one can write a distributedalgorithm similar to (38) for which the variable λ is nowdefined on the nodes, and thus has n elements. However, H in (36) becomes [ R ⊤ R − R ⊤ R ] . The term H ⊤ Hη in (39)requires each node i to collect not only η j , for j ∈ N i , but also η k , for k ∈ N j , which is a two-hop information. On the otherhand, in (38) each node only needs the decision variables andthe dual variables of its neighbors. (cid:4) VI. N
UMERICAL S IMULATIONS
In this section, we provide numerical simulations to illus-trate the results for bias estimation and compensation for bothnon-bipartite graph and bipartite graph.
A. Non-bipartite graphs
We consider a network with nodes and each node takesa sensor. The associated graph is non-bipartite and given byFig. 1. The initial state x i (0) and the bias w i of each node aregenerated randomly within the intervals [ − , and [ − , ,respectively. A specific example is given as below x (0) = [ − .
280 3 .
983 5 . − . − . , − . − .
419 3 .
508 8 .
073 8 . ⊤ w = [0 . − .
479 0 . − . − . − . − .
709 0 . − .
853 0 . ⊤ We simulate the consensus dynamic (17) with the biasestimator (14) and the bias compensator (18), where the initialcondition for the bias estimate is ˆ w = . The simulationresult is provided in Fig. 2, where Fig. 2(a) and 2(c) show thesystem state evolution without bias compensation and withbias compensation, respectively, and Fig. 2(b) shows the biasestimation error e . As can be seen in Fig. 2(a), if the biases arenot compensated, the nodes will not achieve exact consensusand the state of each node x i drifts away under the influenceof the measurement biases. On the contrary, using the biasestimator (14) and the compensator (18), the bias error e vanishes and all x i variables converge to the same finite value. time -15-10-50510 x (a) time -0.8-0.6-0.4-0.200.20.40.60.81 e (b) time -50510 x (c)Fig. 2. Bias estimation and consensus evolution for a non-bipartite graph. (a) State evolution of the consensus dynamics (17) without bias compensation; (b)bias estimation error e generated by bias estimator (14); (c) state evolution of the consensus dynamics (17) with bias compensator (18). time -2-1.5-1-0.500.511.52 e (a) time -50510 x (b)Fig. 3. Bias estimation and consensus evolution for a node bipartite graphwith biased sensors, hence the condition of Theorem 7 is violated. Thenodes apply the bias estimator (38) and the bias compensator (18). (a) Biasestimation error; (b) state evolution. B. Bipartite graphs
Now, we consider a bipartite graph, which is obtained fromthe graph in the last subsection removing the edge { , } .Theinitial state of the system is the same as the one in the previoussubsection.We first show that if more than ⌊ n − ⌋ sensors of nodes arebiased, the ℓ minimization (33) may fail to find the vector ofthe actual biases w for bipartite graphs. We assume that thesensors of the first five nodes are biased and w = [1 .
076 0 .
326 1 .
713 0 . − .
932 0 0 0 0 0] ⊤ (41)We simulate the consensus dynamics (17) with the bias estima-tor (38) and the bias compensator (18). The initial conditionsfor η and λ are set to zero. The result is given in Fig. 3, fromwhich one can see that the entries of the bias estimation error e converge to two values with the same absolute value butopposite signs, thus the biases are not correctly estimated andconsensus is not achieved.We then let the sensor of the fifth node also to be unbiased,namely the last six entries of w in (41) are all zero. Thecondition of Theorem 7 is now satisfied. The result is depictedin Fig. 4, which shows that the bias estimation error decaysto zero and the system achieves consensus.VII. C ONCLUSION
In this paper, we studied the problem of estimating thebiases in sensor networks from relative state measurements, time -2-1.5-1-0.500.5 e (a) Bias estimation error e time -50510 x (b) State evolution with compensationFig. 4. Bias estimation and consensus for a node bipartite graph with biased sensors, hence the condition of Theorem 7. The nodes apply the biasestimator (38) and the bias compensator (18). (a) Bias estimation error; (b)state evolution. with an application to the problem of consensus with biasedrelative state measurement. Without any sparsity constraint onthe biases, we show that the biases can be accurately estimatedif and only if the graph is non-bipartite. For bipartite graphs,we show that the biases can be uniquely determined from themeasurements if less than half of the sensors is biased. Thenumber of biased sensors can be increased when the biasesare heterogeneous, i.e., different from each other, or absolutelyheterogeneous, i.e., with absolute values different from eachother. For both non-bipartite and bipartite graphs, we proposedistributed methods to compute the biases.The problem considered in this paper can be further investi-gated. First, if the sensors are affected by noise in addition tobiases, one could study how noise impacts the accuracy of theestimation of the biases [47]. Second, the result for bipartitegraphs could be also used for problems where the range andangle of arrival measurements are affected by biases [11], [20]-[35]. R EFERENCES[1] V. Kekatos and G. B. Giannakis, “Distributed robust power system stateestimation,”
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