Mult-sensor optmal nformaton fuson Kalman flter wth moble agents n rng sensor networs Behrouz Safarneadan *, Kazem asanpoor ** *Shraz Unversty of echnology, safarnead@sutech.ac.r ** Shraz Unversty of echnology,.hasanpor@gmal.com Abstract: In ths paper, two moble agent based dstrbuted Kalman flter algorthms are proposed for optmal nformaton fuson n sensor networs. In the proposed methods, moble agents (MAs play an mportant and fundamental role n nformaton fuson. A moble agent travels among the sensors and collects the necessary nformaton. he collected nformaton s carred to a fuson centre n whch the optmal state estmates are computed. Fnally, the proposed methods are appled on a radar tracng system wth fve sensors arranged n two rngs wth some faulty sensors. erformance of the proposed methods wll be compared wth the centralzed Kalman flter. Keywords: Optmal nformaton fuson, moble agent, fault tolerance, movng agent, sensor networ, Kalman flter. 1. Introducton Smart envronments represent the next evolutonary development step n buldngs, utltes, ndustral, home, shpboard, and transportaton systems automaton. Smart envronments nclude wreless sensor networs (WSNs that are composed of nodes, each of whch has computng power and can transmt and receve messages over communcaton lns, wreless or cabled. he basc networ topologes nclude fully connected, mesh, star, rng, tree and bus [1]. Whle multple sensors measure the observatons of the same stochastc system n the sensor networ, generally we have two manners to process and fuson the measured sensor data. he frst method, whch has been proposed before, s the centralzed Kalman flter (CKF [], where all measured sensor raw data are communcated to a central ste for processng. he advantage of ths method s that t nvolves mnmal nformaton loss. owever, the centralzed Kalman flter may be unrelable and has less accuracy and stablty when there s a severe data fault. he second method, whch s proposed n ths paper, s moble agent based decentralzed Kalman flter (MABDKF n whch the nformaton from local estmators can mae the global optmal or sub optmal state estmate accordng to certan nformaton fuson crteron. he advantage of ths method s that the fuson-centre requres less communcaton capablty and memory space. Furthermore, decentralzaton leads to easy fault detecton and solaton [3]. In ths paper, two new MABDKF methods are proposed (type 1 and type, for nformaton fuson n a mult-rng sensor networ. A fuson-centre wll be used for all of the rngs. A moble agent moves among all of the nodes n the rngs, collects necessary nformaton from nodes and transmts to the fuson-centre. hen, the MABDKF s appled for optmal state estmaton. One advantage of the MABDKF method s ts fault tolerance propertes n the case of fault occurrence n the wreless sensor networ. Moble Agents (MAs have mportant role n our proposed methods. he moble agents (MAs are software programs that are executed to perform varous tass such as data gatherng, nformaton retreval, flterng or to perform some nd of computatons (processes n varous hosts whch are connected n the networ. he unque ablty of a MA s ts moblty. It can move from one place n a networ to another. hs ablty allows the ntellgence or applcaton to be moved from networ operaton center to managed devces. ere, t s appled for gatherng and processng data by movng among nodes. he benefts of moble agent consst of overcomng the lmtatons of a clent devce ncludng customzablty, hgher survvablty, asynchronous and autonomous computng, and local data access and nteroperablty [4]. So, we can classfy the
followng dfferent moble agents: pop-agent, pushagent, tnerant-agent, and cooperatng-agent. Itnerant-agent roams n a sub-networ or mult rngs sensor networ. It moves to a place, performs some tass, and then goes to another place wth ts part achevement. Itnerant-agent usually has tass consstng of many operatons, whch are related to dfferent devces dstrbuted n the networ. here are many possble applcatons for tnerant-agents. Devce search, for example, can be done wth t. he tnerant-agent travels through the networ to fnd a determned devce and reports some wanted nformaton about t. Itnerary can be determned ether statcally or dynamcally whch can be calculated ether before the agent s dspatched or whle the agent s mgratng [5]. Dynamc tnerary plannng s more flexble, and can adapt to envronmental changes (sensor ups and downs n real tme. owever, snce the tnerary s calculated on the fly, t also consumes more computaton tme and more power of the local sensor node. In addton, the moble agent (MA technology has been proposed for the management of mult rngs sensor networs and dstrbuted systems as an answer to the scalablty problems of the centralzed [6]. In ths paper, statc tnerary MAs are used n the rngs. he rest of the paper s organzed as follows: In secton, the problem formulaton s ntroduced. wo movng agent based dstrbuted Kalman flter algorthms (MABDKF wll be proposed n secton 3. In secton 4, the two proposed MABDKF algorthms are appled to a radar tracng system wth fve sensors n two rngs wth one faulty sensor. he results wll be compared wth another method. Fnally, secton 5 concludes the paper.. roblem formulaton In ths part, we assume a mult-rng sensor networ, n whch a movng agent mgrates between all nodes. he MA collects the requred nformaton from dfferent nodes and transmts them to the fuson centre so that optmal covarance and estmaton s obtaned. Consder a dscrete tme (possbly tme-varyng general model of a lnear fnte-dmensonal stochastc system wth m sensors: x = F 1 x 1 + G 1 u 1 + w 1 (1 y = x + v = 1,,..., l ( where F s state transton matrx, s observaton n vector, x s state vector and y m s plant p observaton, u 1 s a nown control nput, w -1 s the process nose and v s the measurement nose. w -1 s not necessarly whte but t should be a zero mean nose. E ( v v = R δ (4 ere, w s the process nose wth zero mean and Q covarance matrx, v s the measurement nose wth zero mean and R covarance matrx. Ew ( v = Mδ + (5 1 rocess and measurement nose are correlated wth the cross covarance gven by M δ -+1. he ntal state x( s ndependent of w and v, =1,,...,m, and E [ x( ] = μ (6 E[( x( μ ( x( μ ] = Our goal s to fnd the optmal nformaton fuson Kalman flter of the state x(t based on measurements (y,...,y 1, =1,,...,m, under followng condtons: (a Unbasedness, namely,. (b Optmalty, namely, to fnd the optmal matrx weghs, =1,,...,m, to mnmze the trace of the fuson flterng error varance,.e. mn, where the symbol tr denotes the trace of a matrx, denotes the varance of the optmal fuson flter wth matrx weghts and denotes the varance of an arbtrary fuson flter wth matrx weghts. 3. Optmal nformaton fuson Kalman flter wth moble agents 3.1. he dscrete-tme Kalman flter Dscrete-tme Kalman flter equatons are ntroduced n ths part assumng that process and measurement noses are correlated [7]. 1. Assume that the system and measurement equatons are gven by equatons (1-5.. he ntalzed values are gven by equaton (6. 3. For each tme step =1,,... the Kalman flter equatons are gven as art 1( tme update : + = F 1 1 F 1 + Q 1 K = ( ( + M 1 + 1 + R = R + 1 1 1 = ( + ( M ( R M ( M + xˆ = F 1 xˆ 1 + G 1 u 1 art ( measurment update : + xˆ = xˆ + K ( y xˆ + = ( I K ( I K + K ( M + R K M K K M = K ( (7 (8 Eww ( = Qδ (3
3.. Optmal nformaton fuson by usng moble agents (frst method ere, the frst moble agent base decentralzed Kalman flter (MABDKF1 algorthm wll be proposed for a mult-rng sensor networ. In ths method, a moble agent executes tme update (U step only once n the fuson center and then travels to all of the sensors n the rng and executes measurement update (MU step for each sensor. Fnally, optmal state estmaton and covarance matrx are obtaned. We assume that process and measurement noses are correlated. he MABDKF1 algorthm s explaned as follows: 1. he dynamc system and ntalzed values are gven by equatons (1-6.. Moble agent computes tme update (U step only once usng equaton (7 n the fuson centre. 3. Moble agent moves to all of the sensors and performs measurement update (MU step for each sensor usng equaton (8. he second & thrd steps wll repeat. In ths method, optmal estmaton s obtaned by the movng agent n the fuson center. he MABDKF1 algorthm s shown n Fg.. Xˆ, 1 1 Xˆ, Xˆ, 3 3 Xˆ, 4 4 Xˆ = Xˆ, = o 5 o 5 Fg. : MABDKF1 algorthm wth fve sensors and a movng agent for two rng sensor networ. 3.3 Optmal nformaton fuson by usng moble agents (second method In the second moble agent base decentralzed Kalman flter (MABDKF algorthm, a moble agent collects and processes data from all of the sensors to provde a global state estmaton n the fuson centre. In MABDKF algorthm we have a moble agent (MA, a local Kalman flter at each node and a data fuson approach that s performed n the fuson center. Local observatons of the th node Y s used by the local flter to perform an optmal local estmate x. A MA cares the local ˆ estmates of dfferent nodes and also the local Kalman gan to the fuson centre. hen, a weghtng matrx s calculated usng the local estmates and also local Kalman gans. and by all local estmatons, optmal estmaton and covarance wll be obtaned. he MABDKF algorthm s explaned as follows: 1. he dynamc system and ntalzed values are gven by equatons (1-6.. Moble agent moves to all sensors and performs tme update (U step and measurement update (MU step usng equatons (7-8. hen the movng agent collects the local estmate x and Kalman ˆ gan from each sensor and carry them to the fuson centre. 3. In the fuson centre, a weghtng matrx s computed and an optmal estmaton wll be obtaned. heorem 1. Let, =1,,...,m be unbased estmates of an n-dmensonal stochastc vector x. Let the estmaton errors be, =1,,...,m. Assume that and are correlated, and the varance and cross covarance matrces are denoted by (.e. and, respectvely. hen the optmal fuson (.e. lnear mnmum varance estmator wth matrx weghts s gven as xˆo = C 11 xˆ + C xˆ + + Cmxˆ (9 m where the optmal matrx weghts C, = 1,,..., m are computed by 1 1 1 (1 C = D e( e D e where D=(,,=1,,...,m s an nm nm symmetrc postve defnte matrx,,,, and,, are both nm n matrces. he correspondng varance of the optmal nformaton fuson estmator s computed by 1 1 (11 = ( e D e o And we have the concluson o, =1,,...,m. roof. Introducng the synthetcally unbased estmator xˆ = C 11 xˆ + C xˆ + + Cmxˆm (1 where C, =1,,...,m are arbtrary matrces. From the unbasedness assumpton, we have,, =1,,...,m. ang the expectaton of both sdes of (1 yelds C 1 + C + + Cm = In (13 From (1 and (13 we have the fuson estmaton error x = x xˆ = m 1 C ( x xˆ m = = 1 C x = Let C=[C 1,C,...,C m ] so the error varance matrx of the fuson estmator s = E( xx (14 = C DC and the performance ndex J =tr( becomes (15 J = tr ( = tr ( C D C he problem s to fnd the optmal fuson matrx weghts, =1,,,m, under restrcton (13 to mnmze the performance ndex (15. Applyng the Lagrange multpler method, we ntroduce the auxlary functon (16 F = J + tr[ Λ( C e I n ] Where Λ= ( λ s an n n matrx. Set and note that D =D, we have DC + ec = (17 Combnng (17 wth (11 yelds the matrx equaton as
D e C = e Λ I n (18 where D, e, are defned above. D s a symmetrc postve defnte matrx, hence e e s non-sngular. Usng the formula of the nverse matrx [8], we have 1 1 1 1 C D e D e( e D e ( = ( ( = 1 1 e I (19 Λ n ( e D e whch yelds (1. Substtutng (1 nto (14 yelds the optmal nformaton fuson estmaton error varance matrx as (11 [3]. heorem. Based on equatons (3-6, the local Kalman flter error cross covarance between the th and the th sensor subsystems have the followng recursve form: + ( + 1 = [ I ( 1 ( 1 ] n K + + + { F ( ( + 1 F ( ( ( ( + L Q L J ( R ( J ( J ( R ( J ( + J ( M ( J ( + F ( K ( [ M ( J ( M ( L ( ] + [ J ( M ( L( M ( ] K ( F ( } [ In K ( + 1 ( + 1] ( + K ( + 1 M ( + 1 K ( + 1, where,,,k ( s the flterng gan matrx and 1,,=1,,...,m( are the flterng error cross covarance matrces between the th and th sensor subsystem, the ntal values ( = [9]. So, optmal estmaton and covarance matrces are obtaned from equatons: xˆo = C 11 xˆ + C xˆ + + Cmxˆm 1 1 o = ( e D e where,,, are the estmatons obtaned from Kalman flter. he MABDKF algorthm s llustrated n Fg. 3. 4. Smulaton Results A mult-sensor target tracng problem s consdered to verfy performance of the proposed algorthms (MABDKF1 and MABDKF. Consder a tracng system wth fve sensors. rocess and measurement equatons are gven as: Fg. 3: MABDKF algorthm wth fve sensors and a movng agent for two a rng sensor networ..4.35.3.5..15.1.5 5 1 15 5 3.4.35.3.5..15.1.5 In ths example, exst two rng sensor networs that the frst rng has three sensors and other two sensors. We consder that the frst sensor have been fault, so that the 5 1 15 measurement equaton s that y 1 (= 1 x(+v 1 (+f(, where f( satsfes f(= (<1; (c f(=.5 (1 ; Fg. f(= 4:Comparson ( >.he of local values flters, of Centralzed f(t shows and that MABDKF the (type1, flters :(a flterng error varance of poston S(t; (b flterng error varance of poston ; (c flterng error varance of poston.local flter of sensor 1(1; Local flter of sensor (; Local flter of sensor 3(3; Local flter of sensor 4(4; Local flter of sensor 5(5; centralzed flter cen ; decentralzed flter of type 1(decentype1; decentralzed flter of type (decentype. (a (b 1 3 4 5 1(decentype1 1(cen (decentype 1 3 4 5 6 7 8 9 1 15 1 5 1 3 4 5 3(decentype1 3(cen 3(decentype 1 3 4 5 (decentype1 (cen (decentype
1 / x = 1 x + w 1 1 1 1 y = x + v v = α w + ξ 1 = 1,, 3, 4, 5 where s the samplng perod. he state s, where, and are the poston, velocty and acceleraton of the target at tme t, respectvely. y, are the measurement sgnals and v, are the measurement noses of fve sensors, whch are correlated wth the whte Gaussan nose w wth mean zero and varance. he coeffcents α are constant scalars, and ζ, are Gaussans whte noses wth mean zeros and varance matrces, and are ndependent of w(t. Our am s to fnd the optmal nformaton fuson centralzed and decentralzed Kalman flter. In the smulaton, settng =.1; 1, 5, 8, 1, 1, 15 and then α1=.5, α=.8, α3=.4, α4=.3, α5=.6,1=[3,,], =[,3,], 3=[,,3], 4=[,,], 5=[1,,], the ntal value x =, =.1 I 3, and we tae 3 samples. frst sensor appears faulty at =1 and t s restored at =. In Fg. 4, we show covarance of local flter, Centralzed, MABDKF1 and MABDKF for fve sensors n two rngs. Consequently, the precson of the optmal fuson MABDKF1 and MABDKF are hgher than the others. hen, n Fg. 5, 6 and 7 we compare tracng of MABDKF, centralzed, MABDKF1 and states of system when the frst sensor s faulty. 14 1 1 x1 x1(decentype1 16 8 14 x1 6 1 1 8 x1cen 4 6 4-5 1 15 5 3 1 1 x (a 5 1 15 5 3 1 9 8 7 6 5 4 x x(decentype1 (a 8 6 4 xcen 3 1 5 1 15 5 3 (b - 5 1 15 5 3 (b 15 1 x3 x3(decentype1 15 1 5 x3 x3cen 5-5 -5-1 5 1 15 5 3 (c Fg. 5: he tracng performance comparson of the centralzed flter and state when the frst sensor s faulty, (a : state 1,(b : state, (c : state3. -1 5 1 15 5 3 (c Fg. 6: he tracng performance comparson of the decentralzed type1 flter and state when the frst sensor s faulty, (a : state 1,(b : state, (c :state3.
5. Conclusons In ths paper, two new methods (MABDKF1 and MABDKF1 were proposed for nformaton fuson n mult-rng sensor networs to reach the optmal state estmaton. In smulaton, we showed that the represented methods have robustness propertes when some sensors have fault. Fnally, by comparng these proposed methods wth the centralzed Kalman flter (CKF, ther promsng performance was clarfed. 14 1 1 8 6 4 x1 x1(decentype References [1] D. J. Coo and S.. Das, "Wreless sensor networs", John Wley Inc, 4. [] D. Wllner, C.B. Chang and K.. Dunn, " Kalman flter algorthm for a multsensor system",. In roceedngs of IEEE conference on decson and control, pp. 57-574, 1976. 5 1 15 5 3 1 1 8 6 x x(decentype (a [3] S.L. Sun and Z.L. Deng, " Mult-sensor optmal nformaton fuson Kalman flter ", Automatca, vol. 4, pp. 117-13, 4. 4 [4] G.J. otte, " Wreless sensor networs ", roceedng of the IEEE Informaton heory Worshop, pp. 139-14, 1998. [5] Y. Xu and. Q, " Moble agent mgraton modelng and desgn for target tracng n wreless sensor networs", Ad oc Networs, vol. 6, pp. 1-16, 8. - 5 1 15 5 3 (b x3 [6] G.Goodwn Clfford and K.S.Sn, " Adaptve Flterng redcton and control ", rentce-all Inc, Englewood Clffs, 1984. [7] D.Smon, "Optmal State Estmaton", ublshed by John Wley & Sons, Inc, 6. 15 1 5 x3(decentype [8] [9] N.S. Xu," Stochastc sgnal estmaton and system control ", Beng: Beng Industry Unversty ress, 1. B.D.O. Anderson and J.B. Moore, " Optmal flterng ", rentce-hall Englewood Clffs, NJ, Vol. 11, 1979. -5-1 5 1 15 5 3 (c Fg. 7: he tracng performance comparson of the decentralzed type flter and state when the frst sensor s faulty, (a : state 1,(b : state, (c :state3.