Multi-Sensor Control for Multi-Target Tracking Using Cauchy-Schwarz Divergence

Transcription

1 Mult-Senor Control for Mult-Target Tracng Ung Cauchy-Schwarz Dvergence Meng Jang, We Y and Lngjang Kong School of Electronc Engneerng Unverty of Electronc Scence and Technology of Chna Emal:{uoy, lngjang.ong}@gmal.com arxv: v1 [c.sy] 28 Mar 216 Abtract The paper addree the problem of mult-enor control for mult-target tracng va labelled random fnte et (RFS) n the enor networ ytem. Baed on an nformaton theoretc dvergence meaure, namely Cauchy-Schwarz (CS) dvergence whch admt a cloed form oluton for GLMB dente, we propoe two novel mult-enor control approache n the framewor of generalzed Covarance Interecton (GCI). The frt jont decon mang (JDM) method optmal and can acheve overall good performance, whle the econd ndepent decon mang (IDM) method uboptmal a a fat realzaton wth maller amount of computaton. Smulaton n challengng tuaton preented to verfy the effectvene of the two propoed approache. I. INTRODUCTION Senor networ ytem have receved tremou attenton n lat decade due to ther ucceful applcaton that range from vehcular networ to battlefeld detecton and tracng [1]. In many practcal tuaton, due to communcaton and computatonal contrant, t requred that lmted amount of enor tae rght acton. In uch cae, the problem of enor control to fnd a member of the command et that can reult n bet meaurement for flterng purpoe [2]. In general, enor control compre two underlyng component, a mult-target flterng proce n conjuncton wth an optmal decon-mang method. Mult-target flterng ha been recently nvetgated n a more prncpled way due to the pont proce theory or fnte et tattc (FISST) baed mult-target tracng methodology [3]. Among thee random fnte et (RFS) baed method, the promng generalzed labeled mult-bernoull (GLMB) flter [4], [5], or mply the Vo-Vo flter, poee ome ueful analytcal properte [6] and a cloed form oluton to the Baye mult-target flter, can not only produce trajectore formally but alo outperform the probablty hypothe denty (PD) flter [7], cardnalzed PD (CPD) flter [8] and mult-bernoull (MB) flter [9]. Another mportant component of enor control oluton a decon-mang proce, whch motly reort to optmzaton of an objectve functon and generally fall nto two categore. The frt one ta-baed approach, enor control method are degned wth a drect focu on the expected performance and the objectve functon formulated a a cot functon, example of uch cot functon nclude etmated target cardnalty varance [1], [11], poteror expected error of cardnalty and tate (PEECS) [12], [13] and optmal ub-pattern agnment (OSPA) dtance [14]. The ta-baed approach ueful n ome tuaton epecally where the objectve functon can be formulated n the form of a ngle crteron, but there a challengng problem n the cae of multple competng objectve. To olve or avod th problem, the econd one nformaton-baed approach whch trve to quantfy the nformaton content of the mult-target dtrbuton, am at obtanng uperor overall performance acro multple ta objectve and the objectve functon formulated a a reward functon. The mot common choce of reward functon are baed on ome nformaton theoretc dvergence meaure uch a KullbacCLebler (KL) dvergence [15], [16] and more generally the Rény dvergence [17] [19]. owever, a major lmtaton of utlzng KL or Rény dvergence ther gnfcant computatonal cot, and hence mot of the tme, one ha to reort to numercal ntegraton method uch a Monte Carlo (MC) method to derve analytcally reult. An alternatve nformaton dvergence meaure the Cauchy-Schwarz (CS) dvergence. Ung th meaure, oang et al provded tractable formulaton between the probablty dente of two Poon pont procee [2], later, Beard et al exted the reult to two GLMB dente [6], [21] and preented an analytc expreon, whch opened the door to enor control cheme wth GLMB Model baed on nformaton-baed approach. The CS control wth GLMB model account for target trajectore n a prncpled manner, whch not poble ung other tracng method. When the urvellance area very large or target move n complex movement, one enor wth lmted enng range (LSR) not competent to the ta of mult-target tracng, enor networ ytem and ubequent multple enor control are neceary. Inpred by the good performance acheved by enor control wth GLMB model baed on CS dvergence, where Beard et al only condered ngle enor, n th paper, we addre the problem of mult-enor control for mult-target tracng ung CS dvergence va labelled random fnte et (RFS). To be pecfc, we ue Vo-Vo flter to enure local tracng performance, and Generalzed Covarance Interecton (GCI) fuon [22] [24] to maxmze nformaton content of the mult-target dtrbuton. The ey contrbuton of th paper are two tractable approache of mult-enor control, the one optmal wth a lttle complex calculaton and the other uboptmal a a fat realzaton. Smulaton reult verfy both propoed approache can perform well n complex tuaton.

2 2 II. BACKGROUND Th ecton provde bacground materal on labelled mult-target flterng, GCI fuon and Cauchy-Schwarz dvergence whch are neceary for the reult of th paper. For further detal, we refer the reader to [4], [6], [23], [24]. A. Notaton In th paper, we adhere to the conventon that ngle-target tate are denoted by the mall letter, e.g.,x, x whle multtarget tate are denoted by captal letter, e.g.,x, X. Symbol for labeled tate and ther dtrbuton/tattc (ngle-target or mult-target) are bolded to dtnguh them from unlabeled one, e.g.,x,x, π, etc. To be more pecfc, the labeled ngle target tate x contructed by augmentng a tate x X wth a label l L. Obervaton generated by ngle-target tate are denoted by the mall letter, e.g., z, and the multtarget obervaton are denoted by the captal letter, e.g., Z. Addtonally, blacboard bold letter repreent pace, e.g., the tate pace repreented by X, the label pace by L, and the obervaton pace by Z. The collecton of all fnte et of X denoted by F(X). Moreover, n order to upport arbtrary argument le et, vector and nteger, the generalzed Kronecer delta functon gven by { 1, f X = Y δ Y (X) (1), otherwe and δx denote the et ntegral [3] defned by 1 f(x)δx= f({x 1,,x n })dx 1 dx n (2) n! B. GLMB RFS n= An mportant labeled RFS the GLMB RFS [4], whch a cla of tractable model for on-lne Bayean nference [3] that allevate the lmtaton of the Poon model. Under the tandard mult-object model, the GLMB a conjugate pror that alo cloed under the Chapman-Kolmogorov equaton. Let L : X L X be the projecton L((x,l)) = l, and (X) = δ X ( L(X) ) denote the dtnct label ndcator. A GLMB an RFS on X L dtrbuted accordng to π(x) = (X) c Cw (c) (L(X))[p (c) ] X (3) where C a dcrete ndex et. The weght w (c) (L) and the patal dtrbuton p (c) atfy the normalzaton condton w (c) (L) = 1 L L c C p (c) (x,l)dx = 1 Further, a δ-glmb RFS [4], [5] wth tate pace X and (dcrete) label pace L a pecal cae of a GLMB RFS wth C = F(L) Ξ w (c) (L) = w (I,ξ) δ I (L) p (c) = p (I,ξ) = p (ξ) where Ξ a dcrete pace, ξ are realzaton of Ξ, and I denote a et of trac label. In target tracng applcaton, the dcrete pace Ξ typcally repreent the htory of trac to meaurement aocaton. A δ-glmb RFS thu a pecal cae of a GLMB RFS but wth a partcular tructure on the ndex pace whch are naturally n target tracng applcaton. The δ-glmb RFS ha denty π(x) = (X) w (I,ξ) δ I (L(X))[p (ξ) ] X (4) (I,ξ) F(L) Ξ C. Cauchy-Schwarz Dvergence Compared wth Kullbac-Lebler dvergence or Rény dvergence, whch are mot commonly ued meaure of nformaton gan, CS dvergence [6], [21] ha a mathematcal form whch more amenable to cloed form oluton. Ung the relatonhp between probablty denty and belef denty, the CS dvergence between two RFS, wth repectve belef dente φ and ϕ, gven by D CS (φ,ϕ) = ln K X φ(x)ϕ(x)δx K X φ 2 (X)δX K X ϕ 2 (X)δX (5) where K the unt of hyper-volume n X. In partcular, Cauchy-Schwarz dvergence ha a cloed form for GLMB dente, n the cae where the ndvdual target dente are Gauan mxture. For two GLMB wth belef dente φ(x) = (X) c C ψ(x) = (X) d C w (c) φ (L(X))[p(c) φ ]X (6) w (d) ψ (L(X))[p(d) ψ ]X (7) the Cauchy-Schwarz dvergence between φ and ψ gven by where D CS (φ,ψ) = ln ζ(φ,ψ) = L L [K c Cd D ζ(φ, ψ) ζ(φ,φ)ζ(ψ,ψ) (8) w (c) φ (L)w(d) ψ (L) (9) p (c) φ (x, )p(d) ψ (x, )dx]l Cloed form of the analytcal expreon ung CS dvergence combne GLMB dente and nformaton theoretc dvergence meaure hence lead to a more effcent mplementaton of enor control. D. Dtrbuted Fuon In the context of enor networ ytem wth LRS, where each enor ha a fnte feld of vew (FoV), dtrbuted fuon neceary to mae the bet ue of local dtrbuton nformaton n order to olve the hadowng effect. The GCI wa propoed by Mahler [22] pecfcally to ext FISST to enor networ ytem, whch capable to fue both Gauan and non-gauan formed mult-target dtrbuton from dfferent enor wth completely unnown correlaton.

3 3 Baed on GCI, wth the aumpton that all the enor node hare the ame label pace for the brth proce, Fantacc et al propoed the GCI fuon wth labeled et flter by ue the content label. The reult nclude conenu margnalzedδ- GLMB (CMδ-GLMB) and conenu LMB (CLMB) tracng flter [23]. 1) CMδ-GLMB : Suppoe that each enor = 1,...,N provded wth an Mδ-GLMB denty π of the form π = (X) L F(L) δ L (L(X))w (L) [ ] X (1) where N the total enor number and fuon weght ω (,1), N =1 ω = 1, then the fued dtrbuton gven a follow: π = (X) δ L (L(X))w (L) [ ] X (11) where w (L) = = =1 F L=1 L F(L) w (L) =1 N ( =1 w (F) [ N =1 [ N dx ] ( ) L ω (x, ) dx =1 ( ] F p (F) (x, ) dx 2) CLMB : Suppoe that each enor = 1,...,N provded wth a LMB denty π of the form{(r (l), )} l L, where N the total enor number and fuon weght ω (,1), N =1 ω = 1, then the fued dtrbuton of the form where r (l) = = =1 π = {(r (l), )} l L (12) N =1 1 r (l) =1 N ( =1 ( r (l) (x) dx N + dx =1 ( r (l) (x) dx Conenu algorthm can fue n a fully dtrbuted and calable way the nformaton collected from the multple heterogeneou and geographcally dpered enor, and therefore have a gnfcant mpact on the etmaton performance of the tracng ytem. III. MULTI-SENSOR CONTROL USING CS DIVERGENCE In mot target tracng cenaro, the enor may perform varou acton that can maxmze the tracng obervablty, and can therefore nfluence the etmaton performance of the tracng ytem. Typcally, uch acton may nclude changng the poton, alterng the enor operatng parameter, orentaton or moton of the enor platform and o on, whch n turn affect the enor ablty to detect and trac target. In the context of enor networ ytem, where there are more than one enor watng to be deployed, the allowable control acton may ncreae exponentally and hence the control of mult-enor a hgh-dmenonal optmzaton problem. Therefore, mang control decon by manual nterventon or ome determntc control polcy whch provde no guarantee of optmalty, not a good choce. Compared wth ngle enor control, there are ome challengng problem n mult-enor control uch a aforementoned hgh-dmenonal optmzaton problem and nformaton fuon problem nduced by the meaurement collected from the multple enor. In th ecton, we ee tractable oluton for mult-enor control for mult-target tracng wth GLMB model. A. Problem Formulaton In enor networ ytem, one or more enor are the drect output of the decon-mang component of the control oluton, a uch, the focu ha tradtonally been placed on mprovng the decon-mang component. owever, the mult-target tracng component alo play a gnfcant role n the overall performance of the cheme n term of accuracy and robutne. Inpred by the veratle GLMB model whch offer good trade-off between tractablty and fdelty, n flterng tage, we ue the Vo-Vo flter [4], [5] a local enor and GCI fuon to fue the nformaton collected from the multple enor n order to acheve overall uperor performance, the procedure decrbed a follow: 1) At tme tep, wth meaurement Z = {z1,,z 2,,...,z m, } where the ubcrpt denote current tme and upercrpt denote equence number of enor, each enor node = 1,...,N locally perform predcton and update ung Vo-Vo flter, the detal can be found n [5]. 2) Implement the GCI fuon wth local poteror dtrbuton π to derve the fued dtrbuton π, the upercrpt denote fued dtrbuton. Note that one need to convert δ- GLMB poteror dtrbuton to Mδ-GLMB\LMB dtrbuton for conenu fuon method ung (11) or (12). 3) After fuon, an etmate of the object et ˆX obtaned from the cardnalty probablty ma functon and the locaton PDF ung MAP technque. A peudo-code of flterng tage gven n Algorthm 1. In control trategy, we adhere to the conventon that formulatng the enor control problem a a Partally Oberved Marov Decon Proce (POMDP) ung FISST [25] and defnng the followng notaton: π ( Z 1: ) the poteror denty for enor at tme, C the control acton pace for enor and hence the N multple enor control acton pace C = C 1 C N, the length of control horzon, the π ( Z 1: ) predcted denty at tme baed on nown meaurement from tme 1 to tme, Z 1: (c 1,...,c N ) the collecton of meaurement for

4 4 Algorthm 1: Flterng Procedure INPUT: π 1{X Z 1: 1},Z OUTPUT: π {X Z 1:}, π for = 1 : N do local predcton local update π {X Z 1:} GCI( π {X Z 1:}) π MAP( π ) ˆX enor that would be oberved from tme 1 up to wth executed control acton (c 1,...,c N ) C at tme, note that c C a vector compoed of all poble acton what a enor can tae, uch a changng drecton of movement, velocty, power and o on. We ue CS dvergence a reward functon at the control horzon whch meaured between the predcted and poteror mult-target denty: R(c 1,...,c N ) = D CS ( π predcton, π update ) (13) then the optmal control acton decded by maxmng the expected value of the reward functon R(c 1,...,c N ) over the allowable acton pace C: (ĉ 1,...,ĉ N ) = arg max (c 1,...,c N) C EAP(R(c 1,...,c N )) (14) Note that the above expected reward not avalable to analytc oluton, o we reort to Monte Carlo ntegraton, EAP(R(c 1,...,c N )) 1 M M R (j) (c 1,...,c N ) (15) j=1 where M denote the number of ample. Alo for th reaon, we prefer CS dvergence whch provde a cloed-form oluton wth GLMB model to calculate R (j) (c 1,...,c N ), can allevate the de effect nduced by the Monte Carlo technque (15). In what followng we detal the degn of predcted dtrbuton and poteror dtrbuton n (13) and preent two mult-enor control approache. B. Mult-Senor Control Strategy JOINT DECISION MAKING ALGORITM In order to mae the bet ue of enor networ and overall collected nformaton, we propoe an optmal mult-enor control approach, referred to jont decon mang (JDM) algorthm. In th method, the flterng tage performed a decrbed n Algorthm 1, the fued denty π wll be ued for mult-target ample n order to olve the hadowng effect of ngle enor wth LSR and to compute the predcted denty at the of the control horzon. The pecfc procedure are a follow: 1) Mult-target Sample: At decon tme tep, draw a et Ψ S of M mult-target ample from fued dtrbuton π, t manly degned for dervng numercal analytcal reoluton of CS reward functon. 2) Peudo-Predcton: Compute the predcted denty at the of the control horzon π, whch wll be later ued a one term of computng CS dvergence, by carryng out repeated predcton tep of Vo-Vo flter, wthout traget brth or death, for th reaon, we ue the term peudo-predcton. 3) Generate predcted deal meaurement (PIMS): For each enor = 1,...,N and each mult-target ample X (j) Ψ S, generatng PIMS Z 1: (c,x (j) ) wth current control acton c C baed on ntal predcted trajectory n ample X (j), more detal n [21], [26]. 4) Run Vo-Vo Flter Recuron: Run each Vo-Vo flter wth ntal local poteror dtrbuton π {X Z 1: } ung PIMS Z 1: (c,x (j) ) to get the peudo updated dtrbuton π {X Z 1:, Z 1: (c,x (j) )}, we wll ue the term flter to denote Vo-Vo flter recuron [5]. 5) GCI Fuon: For mult-enor, for each poble control acton combnaton (c 1,...,c N ) C, perform the GCI fuon wth peudo updated dtrbuton π {X Z 1:, Z 1: (c,x (j) )} to get the fued peudo updated dtrbuton π (c 1,...,c N,X (j) ), t wll be later ued a another term of computng CS dvergence. 6) Compute Each Reward: Compute CS reward functon for each control acton combnaton and each ample ung (8), R (j) (c 1,...,c N ) = D CS ( π, π (c 1,...,c N,X (j) )) (16) after the computaton of (16) for all ample n et Ψ S, we then compute the expected value of the reward functon R(c 1,...,c N ) = EAP(R (j) (c 1,...,c N )) 1 M R (j) (c 1,...,c N ) M j=1 (17) 7) Jont Decon Mang: Maxmze the expected value of the reward functon R(c 1,...,c N ) over the allowable acton pace C ung (14). A peudo-code of above control tage hown n Algorthm 2. Note that n the JDM algorthm, GCI fuon ha been ued both n flterng tage and CS control tage, am at maxmzng obervaton nformaton content and overall CS dvergence, to enure multple enor move n drecton where the overall performance atfyng. Moreover, n order to reduce the computaton burden of the JDM algorthm, whch manly nduced by allowable control acton combnaton wth computaton complexty O( C 1 C N ), one can reort to mportance amplng technque, more detal n [27]. INDEPENDENT DECISION MAKING ALGORITM We alo propoe another uboptmal mult-enor control approach, referred to ndepent decon mang (IDM) algorthm. In th method, the flterng tage ame but the control tage mplfed a a fat mplementaton. In partcular, the GCI fuon only performed n flterng tage and each enor mae control decon ndepently n control tage, whch enable parallel executon of the control

5 5 Algorthm 2: JDM Procedure INPUT: π, π {X Z1:},C OUTPUT: (ĉ 1,...,ĉ N) Mult-target Sample: π Ψ S = {X (1),...X (M) } Peudo-Predcton: for ter = +1 : + do π π for = 1 : N do for each c C do for each X (j) Ψ S do Generate PIMS: X (j) Z 1:(c,X (j) ) Run Vo-Vo Flter Recuron: flter( π {X Z1:}, Z 1:(c,X (j) )) π {X Z1:, Z 1:(c,X (j) )} GCI Fuon: for each (c 1,...,c N) C do for each X (j) Ψ S do GCI( π 1 {X Z1:, 1 Z 1:(c,X (j) )},..., π N {X Z1:, N Z 1:(c N N,X (j) )}) π (c 1,...,c N,X (j) ) Compute Each Reward: D CS( π, π (c 1,...,c N,X (j) )) R (j) (c 1,...,c N) EAP(R (j) (c 1,...,c N)) R(c 1,...,c N) Jont Decon Mang: arg max (R(c 1,...,c N)) (ĉ 1,...,ĉ N) (c 1,...,c N ) C tep, and therefore the computaton complexty of allowable control acton reduced to O( C C N ). A peudocode of IDM algorthm hown n Algorthm 3. Note that the fued dtrbuton π ued n mult-target ample and peudo-predcton, whch can enure obervablty n control tage o that avod mang myopc decon. A comparon between JDM algorthm and IDM algorthm wth two enor llutrated n Fg. 1. GCI { 1: } { 1: } Z 2 + 1: { 1:} { 1: } GCI Peudo-Update Z 1 + 1: Peudo-Update Peudo-Predcton Z 1 + 1: + Z 2 + 1: JDM algorthm 1 { Z1:, Z 1: } 1 X } { Z1:, Z 1: } 2 X } + + IDMalgorthm Peudo-Update Peudo-Predcton Peudo-Update GCI 1 { Z1:, Z 1:, } 1 X } 2 { Z1:, Z 1:, } 2 X } ĉ 1 ĉ 2 ( cˆ,cˆ 1 2) Fg. 1. A comparon between JDM algorthm and IDM algorthm wth two enor. Algorthm 3: IDM Procedure INPUT: π, π {X Z 1:},C OUTPUT: (ĉ 1,...,ĉ N) Mult-target Sample: π Ψ S = {X (1),...X (M) } Peudo-Predcton: for ter = +1 : + do π π for = 1 : N do for each c C do for each X (j) Ψ S do Generate PIMS: X (j) Z 1:(c,X (j) ) Run Vo-Vo Flter Recuron: flter( π {X Z 1: }, Z 1:(c,X (j) )) π {X Z 1:, Z 1:(c,X (j) )} Compute Each Reward: D CS( π, π {X Z 1:, Z 1:(c,X (j) )}) R (j) (c ) EAP(R (j) (c )) R (c ) Decon Mang on Each Senor: arg max c C (R (c )) ĉ IV. SIMULATION RESULTS AND DISCUSSION In th ecton, the two propoed mult-enor control approache are appled to the problem of mult-target tracng wth two enor wth LSR. Wth both method, local flter are Vo-Vo flter, the fuon method choen a CMδ-GLMB and fuon weght of each enor ω 1,ω 2 are both choen a.5. The nematc target tate a vector of planar poton and velocty x = [t x, ṫ x, t y, ṫ y, ] T and the ngle-target tate pace model lnear Gauan accordng to tranton denty f 1 (x x 1 ) = N(x,F x 1,Q ) wth parameter F = [ ] I2 I 2,Q 2 I = σv [ 4 4 I I I 2 2 I 2 where I n and n denote the n n dentty and zero matrce repectvely, = 1 the amplng perod, σ v = 5m/ 2 the tandard devaton of the proce noe. In the context of mult-enor control, we conder the followng enor model that the meaurement a well a the detecton probablty a functon of dtance between target and enor tate. The enor meaurement are noy vector of polar poton of the form z = [ arctan( t y, y, t x, x, ) (tx, x, ) 2 +(t y, y, ) 2 ]+w (x,u ) where u = [ x, y, ] denote enor poton. w (x,u ) N( ;,R ) the meaurement noe wth covarance R = dag(σ 2 θ,σ2 r) n whch the cale of range and bearng noe are σ r = σ +η r x u 2 and σ θ = θ +η θ x u, the parameter σ = 1m, η r = m 1, θ = π/18rad ]

6 6 Y Coordnate (m) and η θ = m 1. The probablty of target detecton n each enor ndepent and of the form P D (x,u ) = N( x u ;,σ D ) N(;,σ D ) where σ D = 1m control the rate at whch the detecton probablty drop off a the range ncreae. Moreover, the urvval probablty P S, =.98, the number of clutter report n each can Poon dtrbuted wth λ c = 25. Each clutter report ampled unformly over the whole urvellance regon. The enor platform move wth contant velocty but tae coure change at pre-pecfed decon tme. The allowable control acton for each enor C = [ 18, 15,...,,...,15,18 ], the number of ample ued to compute the expected reward M = 4, the dealed meaurement are generated over a horzon length of = 5, wth amplng perod T = 2. The tet cenaro cont of 4 target, the enor eep tll durng frt 1 and mae frt decon at 1 o the econd decon at 2, thrd decon at 3, then reman on that coure untl the of the cenaro at tme 4. The regon and trac are hown n Fg. 2. Y Coordnate (m) urvve from 9 to 35 urvve from 1 to 4 Ground Truth enor1 enor2 urvve from 1 to 4 urvve from 16 to X Coordnate (m) Fg. 2. Target trajectore condered n the mulaton experment. The tart/ pont for each trajectory denoted, repectvely, by. The ndcate ntal enor poton Ground Truth enor1 enor2 decon tme X Coordnate (m) (a) Y Coordnate (m) Ground Truth enor1 enor2 decon tme X Coordnate (m) Fg. 3. (a) Trac output from a typcal run baed on IDM algorthm. (b) Trac output from a typcal run baed on JDM algorthm. Fg. 3 (a) and (b) how a ngle run to exhbt the typcal control behavour baed on IDM algorthm and JDM algorthm, repectvely. A t can be een, both control method (b) C S Dvergence C S Dvergence enor1 Coure change (deg) enor2 Coure change (deg) (a) C S Dvergence Senor 1 ( 6,3) (b) 6 Senor Fg. 4. (a) Reward curve at the tme of the econd decon (2) baed on IDM algorthm. (b) Reward curve at the tme of the econd decon (2) baed on JDM algorthm. OSPA (m)(c=1,p=2) random acton IDM algorthm JDM algorthm Tme Step Fg. 5. Comparon of OSPA error returned by randomed control acton, IDM algorthm and JDM algorthm. The plotted reult are the average of 1 Monte Carlo run. can mae proper decon that enor move cloe to the target. To be more pecfc, we denote the control acton choen by enor 1 and enor 2 by a vector (θ 1,θ 2 ), at the frt decon tme 1, two control method mae ame decon ( 3,3 ), at the econd decon tme 2, the IDM algorthm tae ( 3, ) whle the JDM algorthm tae ( 6,3 ). Fg. 4 (a) and (b) how the CS dvergence at the econd decon (2) of IDM algorthm and JDM algorthm, repectvely. Thee reult mean that compared wth the IDM algorthm, each enor controlled by JDM algorthm not greedy to oberve all target, but rather a vew of the whole pcture to mae the amount of nformaton content of fued denty larger. Fg. 5 how the comparon of OSPA error averaged over 1 Monte Carlo run among randomed control acton, IDM algorthm and JDM algorthm. A t hown, both control method can acheve better performance than randomed control trategy and the JDM algorthm preferable. Moreover, when the tuaton more complex uch a much more target or enor, the performance dfference between JDM algorthm and IDM algorthm wll ncreae and the randomed control trategy may collape. V. CONCLUSION In th paper, we addre the problem of mult-enor control for mult-target tracng va labelled random fnte et (RFS)

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