This is the published version of a paper presented at Control Conference (ECC), 2013 European.

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1 Ths s the publshed verson of a paper presented at Control Conference (ECC), 3 European. Ctaton for the orgnal publshed paper: Boem, F., Xu, Y., Fschone, C., Parsn, T. (3) Dstrbuted Fault Detecton usng Sensor Networks and Pareto Estmaton. In: (pp ). N.B. When ctng ths work, cte the orgnal publshed paper. Permanent lnk to ths verson:

2 3 European Control Conference (ECC) July 7-9, 3, Zürch, Swtzerland. Dstrbuted Fault Detecton usng Sensor Networks and Pareto Estmaton Francesca Boem, Yuzhe Xu, Carlo Fschone and Thomas Parsn Abstract In ths paper, a prelmnary novel dstrbuted fault detecton archtecture for dynamc systems usng sensor networks and a dstrbuted estmaton method based on Pareto optmzaton s proposed. The goal s to montor large-scale or dstrbuted systems by usng a sensor network where each node acts as a local estmaton agent wthout centralzed coordnaton. Probablstc detecton thresholds related to a gven rate of false alarms are derved n several dfferent scenaros as far as the measurement pattern and the nomnal dynamcs s concerned. Prelmnary smulaton results show the effectveness of the proposed fault detecton methodology. I. INTRODUCTION The scentfc nterest towards large-scale and/or dstrbuted systems has become mportant durng last decades. One of the open challenges s the quanttatve montorng of these systems wthout the help of a centralzed coordnaton agent. The condtons n whch these systems operate rely on the possblty of system faults represented by crtcal changes n the system dynamcs from the desgned behavor. Therefore, fault dagnoss, whch conssts n the detecton, solaton and dentfcaton of the fault, s a key requrement n the desgn of relable modern systems (see [] and the references cted theren). The centralzed soluton of ths problem suffers from scalablty ssues and sometmes t s even not possble when dealng wth large-scale systems. An nterestng soluton for dstrbuted fault dagnoss can be realzed by explotng sensor networks. Some works exst addressng the problem of fault dagnoss of sensor networks, such as sensor fault detectng, packet losses and energy consumpton montorng ([], [3], []), but fewer are the works usng sensor networks as a tool for dynamcal systems montorng. Classcal methods for quanttatve fault dagnoss n the state of the art deal wth the use of modelbased analytcal redundancy technques. A lot of these works requre the centralzed collecton of the nformaton obtaned from the sensor devces. Some exceptons are [5], [6], [7] and other works dealng wth dscrete-event systems ([8], [9]). Even f methods for dstrbuted estmaton already exst (see [] for a survey and [], [] as examples), lnks between dstrbuted estmaton and dstrbuted montorng are stll lackng. An excepton s [3], where a dstrbuted F. Boem s wth the Dept. of Engneerng and Archtecture, Unversty of Treste, Italy. (francesca.boem@phd.unts.t) Y. Xu and C. Fschone are wth Electrcal Engneerng and ACCESS Lnnaeus Center, KTH Royal Insttute of Technology, Stockholm, Sweden. (yuzhe@kth.se, carlof@kth.se ) T. Parsn s wth the Dept. of Electrcal and Electronc Engneerng at the Imperal College London, UK, and also wth the Dept. of Engneerng and Archtecture at the Unversty of Treste, Italy (t.parsn@gmal.com) fault detecton and solaton technque s desgned, relyng on decentralzed Kalman state-estmaton method. In fact, an mportant branch of research on dstrbuted estmaton s represented by dstrbuted Kalman flters ([]) and ther combnaton wth the dffuson mechansm ([5], [6]). In ths paper, a prelmnary study s proposed to show that the dstrbuted estmaton method presented n [7] for sensor networks can be useful for montorng and fault detecton purposes. More specfcally, each sensor node s a local estmaton agent and a dstrbuted estmaton flter s desgned based on Pareto optmzaton followng the very recent approach proposed n [7]. The use of central coordnaton s not requred. The advantages of usng the dstrbuted estmaton method based on Pareto optmzaton are that t guarantees to mnmze at the same tme both bas and varance of the estmaton error and, thanks to the way t s desgned, t allows to compute mean and varance of the estmaton error, thus allowng to obtan sutable thresholds for some sutable resdual sgnals wth known confdence levels and defnable rate of false alarms. Moreover, snce the dstrbuted Pareto estmator does not requre the model of the system that generates the sgnal to track, t can be appled to uncertan and even nonlnear systems, although, n ths prelmnary work, we consder manly the lnear case. We do not assume any topology of the network, unlke prevous works that requre an all-to-all topology (see [8] as example). The paper s organzed as follows. After formulatng the problem under concern n the next secton, Secton III presents the dstrbuted estmaton methodology, whereas n Secton IV the fault detecton algorthm explotng the dstrbuted estmaton tool s presented. Some extensons are consdered n Secton V and smulaton results are gven n Secton VI. Fnally, some concludng remarks n Secton VII. II. PROBLEM FORMULATION Let us frst ntroduce a few useful notatons that wll be used throughout the paper. By we denote the cardnalty of the argument and by the spectral norm of a matrx. Gven a stochastc varable x, we represent as Ex ts expected value. Fnally, by and I we denote the vector (,...,) and the dentty matrx, respectvely. Let us consder the model of the plant to be montored: x(t+) = Ax(t)+ξ(t)+β(t t )φ(x(t),u(t),t), () where x R n denotes the state vector, ξ R n s a nose wth mean µ ξ and covarance σξ, representng the dsturbances affectng the state vector, and β(t t )φ(x(t),u(t),t) descrbes the dynamcs of an addtve fault occurrng at the / 3 EUCA 93

3 unknown tme t, wth β(t t ) beng a scalar functon representng tme profle of the fault (modelng both abrupt and ncpent faults [7]). In healthy condtons, the model s x(t+) = Ax(t)+ξ(t). A sensor network s avalable to montor system () and t has to be desgned to perform a dstrbuted fault detecton task. In partcular, every node of the network observes the same state and exchanges measurements and state estmatons wth neghborng nodes n order to reach a common fault detecton decson. The communcaton network s modeled as an undrected graph G = (V, E), where N = {j V : (j,) E} {} s the set of neghbors of node V plus the node tself. It s assumed that there are no communcaton delays. We want to show that the dstrbuted estmaton method presented n [7], mnmzng bas and varance of the estmaton error smultaneously, can be extended to carry out dagnoss tasks. In the followng, we propose the fault detecton archtecture for a lnear case n whch all components of the state vector are avalable for measurement (corrupted by nose). Later on, the extenson to more general scenaros wll be brefly addressed (e.g., not all components of the state are accessble, sensors do not observe the same state, the system s not lnear, etc.). III. DISTRIBUTED MODEL ESTIMATION Let us consder a sensor network made of N nodes. For each node, wth =,...,N, we have y (t) = x(t)+v (t), () where y s the measurements vector obtaned at node and v s the zero-mean measurement nose. In ths secton, for the sake of smplcty, we assume that all sensors can measure all the state components. In Secton V, more general measurement models are proposed. Each node uses two steps to estmate the state varables: frstly, t communcates wth neghborng nodes and computes an a-posteror estmate usng the Pareto optmzaton archtecture. For each k-th component, wth k =,...,n, x (k) (t) = j N k l k,,j (t) x (k) j (t )+ j N k h k,,j (t)y (k) j (t), (3) where l k,,j (t) and h k,,j (t) are the tme-varyng weghts of the flter desgned n [7], N k s the set of neghbors of node measurng the varable x (k) plus the node tself and wth x (k) we denote the k-th component of the state estmate computed by node. The optmal values for l k,,j and h k,,j are obtaned by Pareto optmzaton that mnmzes both the bas and varance of the estmaton errors. In Secton III-A, the Pareto estmaton archtecture s dscussed n some detal. Subsequently, the second step conssts on the a-pror estmate, computed on the bass of the model of the system: ˆx (t+) = A x (t)+λ (ˆx (t) x (t)), wth < λ <. Thanks to the communcaton wth neghborng nodes n the frst step, each node can reduce ts measurement uncertanty. Therefore, t s convenent to use fltered estmates nstead of measurements, because, n ths way, we can defne less conservatve thresholds for fault detecton purposes, as explaned n Secton IV. In the followng subsecton, we go nto more detal descrbng the dstrbuted Pareto estmator. A. The Pareto estmator The dstrbuted estmaton archtecture for sensor networks s ntroduced n [7]. For the sake of smplcty, here we consder Eq. () for thek-th state component. Let us consder N k > sensor nodes, measurng a common sgnal x (k) (t) affected by addtve nose: y (k) (t) = x (k) (t)+v (k) (t), =,...,N k, where v (t) s a zero-mean whte nose. Collectng the varables n vectors, t s possble to wrte: y (k) (t) = x (k) (t)+v (k) (t). We assume the covarance matrx Σ k of v (k) to be dagonal. For the sake of smplcty and wthout loss of generalty, here we defneσ k = σk I, but all the results can be extended to the more general and realstc case Σ k = dag([σk,...,σ kn k ]) n a smple way (see [7] for detals). As already expressed n Eq. (3), each node computes an estmate x (k) (t) of the sgnal x (k) (t) as a lnear combnaton of neghborng estmates and measurements. In vector notaton, we have: x (k) (t) = L k (t) x (k) (t )+H k (t)y (k) (t), () where L k and H k are the matrces contanng tme-varyng weghts of the flter. The algorthm s ntalzed wth x j () = y (k) (), j N k. In [7], the optmal values of these weghts are derved as the soluton of a mult-objectve optmzaton problem that mnmzes at the same tme both the bas and the varance of the estmaton error e (k) (t) = x (k) (t) x (k) (t) by usng a Pareto optmzaton framework: mn ( ρ,k )V,k +ρ,k B κ,k,k,η,k s. t. (κ,k (t)+η,k (t)) =, where ρ,k, B,k = Ee (k) (t) s the bas term of the estmaton error, V,k = E(e (k) (t) Ee (k) (t)) s the varance term and we ntroduced κ,k (t) and η,k (t) that correspond to the non zero elements of the -th row of matrces L k (t) and H k (t) respectvely. The constrant s a local condton needed to guarantee the convergence propertes of the centralzed estmaton error, that are derved n [] and that hold lkewse n our case. It s equvalent to the followng global assumpton: Assumpton : We assume that (L k (t)+h k (t)) =. We derved the expressons for the bas and the estmaton error varance that can be computed n a dstrbuted way: Ee (k) (t) = κ,k (t)eǫ,k(t ) κ,k (t)δ(k) (t), where ǫ,k (t) collects the estmaton errors avalable at node for the k-th state component, ordered accordng to ther 933

4 ndexes: ǫ,k = (e (k),...,e (k) ), M k < < M k, wth M k = N k denotng the number of neghbors of plus tself. Then, we compute the varance of the estmaton error: E(e (k) (t) Ee (k) (t)) = κ,k (t)γ,k(t )κ,k (t) +σ,kη,k(t)η,k (t), where Γ,k (t) = E(ǫ,k (t) Eǫ,k (t))(ǫ,k (t) Eǫ,k (t)) s the error covarance matrx. The obtaned optmzaton problem can be solved n a dstrbuted way for each node by usng Karush-Kuhn-Tucker condtons, dervng the followng soluton for a gven ρ,k : κ,k (t) = η,k (t) = ( ρ,k )σk Θ,k ( ρ,k )σk Θ, (5),k +Mk ( ρ,k )σk Θ. (6),k +Mk where Θ,k (t) = ( ρ,k )Γ,k (t ) + ρ,k Λ,k (t), wth Λ,k (t) = (Eǫ,k (t ) δ(t))(eǫ,k (t ) δ (k) (t)). In the lterature, the best value of ρ,k s obtaned by buldng the Pareto trade off curve and selectng the knee pont of ths curve, that s, choosng ρ,k such that B,k and V,k, computed wth the values κ,k (ρ,k ) and η,k (ρ,k ), are V,k = B,k.We chose to compute t locally usng the Nelder- Mead smplex algorthm (see [9]). As an alternatve, the Pareto parameter could be determned n order to defne an approprate bound on the bas. In [7] t s shown how to defne a bound on the bas by approprately settng ρ,k. B. Estmaton error analyss In the prevous subsectons, we ntroduced the two-steps estmaton method. Here, we analyze dynamcs and features of the estmaton error. By defnng e (t) = x (t) x(t) the estmaton error vector of the frst estmaton phase, whch collects the components of e (k) (t) wth k =,...,n, we can analyze the dynamcs of a resdual: r (t+) = x (t+) ˆx (t+) = x (t+) x(t+)+x(t+) ˆx (t+) = e (t+)+ax(t)+ξ(t) A x (t) λ (ˆx (t) x (t)) = λ r (t)+e (t+) Ae (t)+ξ(t). Now, we derve mean and varance of the random varable χ (t) = e (t) Ae (t )+ξ(t ). The mean s gven by µ χ (t) = Eχ (t) = Ee (t) AEe (t )+Eξ(t ) = B (t) AB (t )+µ ξ, where B (t) collects the components B,k of the bas that are computed at each step by the Pareto estmator. As far as varance s concerned, t s necessary to dstngush between two cases: f the varables are ndependent, the varance of ther sum (or dfference) s the sum of ther varances; otherwse, f the varables are correlated, then the varance of ther sum s the sum of all ther covarances: Var(X +X j ) = Var(X )+Var(X j )+Cov(X,X j ). We can assume that e and ξ are ndependent, snce e s obtaned based on the measurements only, whle e (t+) s correlated wth e (t). Therefore, we can compute the varance of χ (t) as: σ χ (t) = V (t)+a V (t )+Cov(e (t),e (t ))+σ ξ, where we denote by A the matrx collectng the element by element square and B (t) collects the components B,k of the varance that are computed at each step by the Pareto estmator. Cov(e (t),e (t )) s agan a column vector: each k-th component can be computed as κ,k (t) V k, where V k s the column vector collectng the components V,k, wth =,...,N. Snce κ (t) has non null elements only n correspondence of the neghbors of, only ths components have to be communcated. Ths expresson can be derved consderng the vector notaton and obtanng the global estmaton error for the k-th state component e (k) (t) = x (k) (t) x (k) R N : e (k) (t) =L k (t)e (k) (t )+x (k) (t)(l k (t)+h k (t) I) Snce δ (k) (t)l k (t)+h k (t)v (k) (t). Cov(e (k) (t),e (k) (t )) = E[(e (k) (t) Ee (k) (t))(e (k) (t ) Ee (k) (t ))] and rememberng that v (k) (t) has zero mean and s ndependent of e (k) (t ), we obtan Cov(e (k) (t),e (k) (t )) = L k (t)e[(e (k) (t ) Ee (k) (t )) ]. It s suffcent to select the -th row to determne the k-th element of Cov(e (t),e (t )). In the next secton, we wll see how the computed quanttes can be useful n order to derve some boundng thresholds for the estmaton resdual. IV. FAULT DETECTION BY SENSOR NETWORKS BASED DISTRIBUTED ESTIMATION In ths secton, the prevous results are exploted for fault detecton purposes. Now, we are gong to determne how to bound the resdual. If we do not know the dstrbuton of the random varable χ (t), we can use the Chebyshev nequaltes (wthout any assumpton on the dstrbuton): Pr(µ(χ ) (k) ασ(χ ) (k) χ (k) µ(χ ) (k) +ασ(χ ) (k) ) α. (8) It s possble to fnd better results f we assume to know the dstrbuton of v (t) and ξ (t). As example, let us assume v and ξ to be normally dstrbuted. Then, t s possble to show that also each k-th component of e (t) (whch s the -th component of e (k) (t), Eq. (7), where the only stochastc varable s v (k) (t)) has a Gaussan dstrbuton snce t s a lnear functon of Gaussan stochastc varables. In the normal case, the percentages are: 68,3% wth α = ; 95,5% wth α = ; 99,% wth α =,58; 99,7% wth (7) 93

5 α = 3. Therefore, we can defne tme-varyng upper and lower thresholds for the dstrbuted estmaton resduals: r (k)+ r (k) (t+) = λ r (k)+ (t)+µ (k) χ (t+)+ασ χ (k) (t), (t+) = λ r (k) (t)+µ (k) χ (t+) ασ χ (k) (t). In ths way, t s possble to defne a α-tube to whch the resdual belongs to (n a probablstc sense) n healthy condtons, wth a certan probablty dependng on α value: r (k) r (k) (t) r (k)+ (t). If the computed resdual s outsde the α-tube, then we can conclude that a fault has occurred. The rate of false-alarms depends on the chosen value of α. V. EXTENSIONS In ths secton, we extend the smple methodology presented n the prevous sectons to more complex scenaros. A. Case : not drectly measurable state We consder the case that all the sensors measure the same varables C (k) x(t), wth k =,...,p: y (t) = Cx(t)+v (t), where y R p are the measurements of the -th sensor node. We assume C s a nvertble matrx. By means of the Pareto estmaton method, sensors exchange measurementsy (k) and estmates ȳ (k) of C (k) x(t). The frst estmaton step s: ȳ (k) (t) = l k,,j (t)ȳ (k) j (t )+ h k,,j (t)y (k) j (t), j N k j N k followng the same procedure presented n Secton III-A. From ths t s possble to derve: x (t) = C ȳ (t). The second step s agan the model-based estmate: ˆx (t+) = A x (t)+λ (ˆx (t) x (t)), Estmaton error and threshold can be computed followng the same steps as n the prevous case (Secton III-B - IV). B. Case : measurements of dfferent varables In ths case, sensors may measure dfferent varables: y (t) = C x(t)+v (t). We defne for each k-th measured varable a communcaton network G k = (V k,e k ), that connects all the nodes measurng varable k. Each node communcates wth all the neghborng nodes j N k and derves from measurements the same state varables: x (k) (t) = C (k) y (t). We assume C to be nvertble and observablty of the system. The dstrbuted Pareto estmator s appled to x = x(t)+c v (t), obtanng the state estmate: x (k) (t) = j N k l k,,j (t) x (k) j (t )+ h k,,j (t) x (k) j (t). j N The second step can be appled n the same way as before: ˆx (t+) = A x (t)+λ (ˆx (t) x (t)) and the new thresholds can be obtaned by smple algebra. C. Case 3: general non-lnear uncertan model Snce the flter presented n [7] does not need to assume the model of the system, we can use t also n the case of a general non-lnear uncertan model. Ths comes at the cost of more conservatve results and stronger assumptons. Let us consder the followng non-lnear system: x(t+) = f(x(t),u(t))+η(x(t),u(t),t) y (t) = x(t)+v (t), where f descrbes the known nomnal dynamcs of the model and η represents model uncertanty. Assumpton : The uncertanty functon s bounded by a known and bounded functon: η(x(t), u(t), t) η(x(t),u(t),t) (x,u,t). As n the smpler case, measurements are frst fltered by means of the Pareto estmator and x s computed (Eq. (3)). Subsequently, the model-based estmator s used: ˆx (t+) = f( x (t),u(t))+λ (ˆx (t) x (t)). The estmaton resdual can be computed: r (t+) = λ r (t)+e (t+)+f(x,u) f( x,u)+η(t). Ths resdual can be bounded as follows: r (t+) λ r (t) + e (t+) + f(x,u) f( x,u) A threshold can then be derved for r (t) : + η(t). r (t+) = λ r (t)+ē (t+)+ f(t)+ η(t), (9) where η(t) s known by assumpton, ē (t+) can be bound smlarly as n the smple case addressed n Secton IV snce we know ts mean and varance, f(t) = max x S [f(x,u) f( x,u)], where S s the set where x x = e s such that µ e ασ e e µ e +ασ e. Eq. (9) s a conservatve threshold and by choosng an approprate value of α guarantees that no false alarms occur. VI. SIMULATION RESULTS In ths secton, some prelmnary smulaton results are presented. In [7], t s explaned how to mplement the Pareto estmator and, more specfcally, how to derve the estmates of the needed quanttes; moreover, a detaled computaton complexty analyss of the dstrbuted estmaton algorthm s presented. As n [3], n order to show the effectveness of the proposed approach, we consder a network of N = sensor nodes montorng a system representng a movng object on a plane. We decded to use the same 935

6 Plot of Wreless Sensors Network Estmates, Resduals and Thresholds: component Locaton n y drecton Locaton n x drecton Sgnal Resduals Lower thresholds Upper thresholds Pareto estmates Model based estmates Tme [s] Fg.. The consdered communcaton network. Fg.. Estmates, Resduals and Thresholds for state component. smulaton example n order to allow a qualtatve comparson to the performances of the method based on dstrbuted Kalman flters. The network s obtaned by dstrbutng the nodes randomly over a squared area of sze N/ and by lettng two nodes communcate f ther relatve dstance s lower than N. The consdered network can be seen n Fg.. The dynamcs of the nomnal system can be represented as: x(t+) = Ax(t)+ξ(t), where δ δµ δ m m A = δ, δµ m where δ =.s s the samplng tme, m =.75kg s the mass of the vehcle and µ =.5 s the frcton coeffcent; the process nose ξ(t) s a zero-mean Gaussan nose wth σξ = dag(,...,). As n [3], the state vector s ntalzed as x() = col[,,.,,,.] and two types of addtve faults are consdered. The frst conssts n the sgn nverson of the force gven by one of the actuators along the vertcal drecton; the second causes the actuator governng the movement along the horzontal drecton to get stuck. Here we present the case that fault occurs at tme t = s. After t, x(t+) = Ax(t)+ξ(t)+Φ x(t), wth Φ = δ δ m δ m. It s worth notng that the fault functon could be nonlnear snce our approach does not requre any assumpton. The measurement nose s v WGN(,Σ), where Σ = dag(,,,,,) s the same for all sensor nodes. Fg. shows the sgnals and the measurements collected by sensor nodes. The parameter α s set to. There s a trade-off between the reducton of false alarms rate and the detecton tme. Smaller s α, sooner the fault wll be detected, but the false alarm rate ncreases. On the other hand, f α grows, the false alarm rate s smaller, but the fault s detected later. In Fg. 3, the fault s detected for the fourth and ffth state component. In Fg., the fourth component s analyzed: all nodes can detect the fault n less than.6s. The proposed methodology has been tested wth dfferent topology networks and we obtaned smlar results. VII. CONCLUDING REMARKS In ths prelmnary paper, t was shown that the dstrbuted estmator proposed n [7] can be exploted n the task of detectng faults by sensor networks. Several dfferent scenaros were consdered and probablstc tme-varyng thresholds for fault detecton have been devsed. Future research efforts wll be frst devoted to provde a detaled theoretcal analyss on fault detectablty, as well as complexty. Extensve smulaton results wll be provded and a quanttatve comparson to the methods avalable n the lterature wll be carred out. Moreover, the proposed method wll be extended to the case n whch correlatons are present among dfferent state components. Fnally, the case of dstrbuted nterconnected systems wll be addressed. REFERENCES [] F. Boem, R. Ferrar, T. Parsn, and M. Polycarpou, Dstrbuted fault detecton and solaton of contnuous-tme nonlnear systems, European Journal of Control, vol. 5-6, pp. 63 6,. [] F. Koushanfar, M. Potkonjak, and A. Sangovann-Vncentell, Onlne fault detecton of sensor measurements, n Sensors, Proceedngs of IEEE, vol., oct. 3, pp Vol.. [3] J. Chen, S. Kher, and A. Soman, Dstrbuted fault detecton of wreless sensor networks, n Workshop on Dependablty ssues n wreless ad hoc networks and sensor networks. ACM, 6, pp [] Y. Lu, K. Lu, and M. L, Passve dagnoss for wreless sensor networks, Networks, IEEE/ACM Transactons on, vol. 8(), pp. 3,. [5] N. Lechevn and C. Rabbath, Decentralzed detecton of a class of non-abrupt faults wth applcaton to formatons of unmanned arshps, Control Systems Technology, IEEE Transactons on, vol. 7, no., pp. 8 93, 9. [6] S. Stankovc, N. Ilc, Z. Djurovc, M. Stankovc, and K. Johansson, Consensus based overlappng decentralzed fault detecton and solaton, Control and Fault-Tolerant Systems Conf., pp ,. [7] R. Ferrar, T. Parsn, and M. Polycarpou, Dstrbuted fault detecton and solaton of large-scale dscrete-tme nonlnear systems: An adaptve approxmaton approach, Automatc Control, IEEE Transactons on, vol. 57, no., pp. 75 9,. [8] P. Baron, G. Lampert, P. Poglano, and M. Zanella, Dagnoss of large actve systems, Artfcal Intellgence, vol., no., pp ,

7 Measurements: component Tme [s] Measurements: component Measurements: component Tme [s] 5 Measurements: component Tme [s] Measurements: component Tme [s] Tme [s].... Measurements: component Tme [s] Fg.. The tracked sgnal represented by the thck blue curve and the measurements realzed by the N = nodes, wth dfferent colors for each node. 8 6 Resduals and Thresholds: component Sgnal Resduals Lower thresholds Upper thresholds.5.5 Resduals and Thresholds: component Tme [s] Resduals and Thresholds: component Tme [s] Resduals and Thresholds: component Tme [s] Tme [s] Resduals and Thresholds: component Tme [s] Resduals and Thresholds: component Tme [s] Fg. 3. Smulaton results for each state component. Resduals and thresholds of all the nodes are represented n the same graph. [9] Y. Wang, T. Yoo, and S. Lafortune, Dagnoss of dscrete event systems usng decentralzed archtectures, Dscrete Event Dynamc Systems, vol., pp , 7. [] F. Garn and L. Schenato, A survey on dstrbuted estmaton and control applcatons usng lnear consensus algorthms, n Networked Control Systems. Sprnger,, vol. 6, pp [] A. Speranzon, C. Fschone, K. Johansson, and A. Sangovann- Vncentell, A dstrbuted mnmum varance estmator for sensor networks, IEEE Journal on Selected Areas n Communcatons, vol. 6, no., pp. 69 6, 8. [] F. Cattvell and A. Sayed, Dffuson lms strateges for dstrbuted estmaton, Sgnal Processng, IEEE Transactons on, vol. 58, no. 3, pp. 35 8,. [3] E. Franco, R. Olfat-Saber, T. Parsn, and M. Polycarpou, Dstrbuted fault dagnoss usng sensor networks and consensus-based flters, n Decson and Control, 5th IEEE Conf. on, 6, pp [] R. Olfat-Saber, Dstrbuted kalman flterng for sensor networks, n Decson and Control, 6th IEEE Conf. on, 7, pp [5] R. Carl, A. Chuso, L. Schenato, and S. Zamper, Dstrbuted kalman flterng based on consensus strateges, IEEE Journal on Selected Areas n Communcatons, vol. 6, no., pp , 8. [6] R. Olfat-Saber, Kalman-consensus flter : Optmalty, stablty, and performance, n Decson and Control, 8th IEEE Conf. on, held jontly wth 8th Chnese Control Conference, 9, pp [7] F. Boem, Y. Xu, C. Fschone, and T. Parsn, A dstrbuted estmaton method for sensor networks based on pareto optmzaton, n Decson and Control, Proc. of 5th Conf. on,, pp [8] W. Chung and J. Speyer, A general framework for decentralzed estmaton, n Proc. Amercan Control Conference, vol., 995, pp [9] J. Lagaras, J. Reeds, M. Wrght, and P. Wrght, Convergence propertes of the nelder-mead smplex method n low dmensons, 998. [Onlne]. Avalable: 937