Time Allocation of a Set of Radars in a Multitarget Environment

T ime Allocation of a Set of Ra dars in a Multit ar g et En vironment Emmanuel Duflos LA GIS UMR CNRS 8146 INRIA Futurs Ecole Centrale de Lille BP 46 59651 LI LLE Cedex FRANCE Email: em manuel.d uflos@ec-lille.fr Marie d e V il morin FRANCT echno parc futu ra Universié d’Artois Faculté des Sciences Ap pliquées 62400 Bethune FRANCE Email: marie.d e_vilmor in@fsa.univ-artois.fr Philippe V anheeghe LA GIS UMR CNRS 8146 INRIA Futurs Ecole Centrale d e Lille BP 46 59651 LI LLE Cedex FRANCE Email: philipp e.vanheeghe@ec-lille.fr Abstract — The qu estion tackled here is the time allocation of radars in a multitarget en vironment. At a g ive n time radars can only observ e a limited part of the space; it is theref ore necessary to mov e their axis with respect to time, in order to be able to explore the overall space facing them. S uch sensors are used to detect, to locate and to identi fy targets which are in their surrounding aerial sp ace. In this paper we focus on the detection schema when sev eral targets need to b e detected by a set of delocalized radars. Thi s work is based on th e modelling of the radar detection performances in terms of probability of detection and on th e opti mization of a criterion based on detection probabilities. Th is optimization leads to the deriv ation of allocation strategies and is made for sev eral contexts and sev eral hypotheses about the targ ets locations. Keywords: Sensor Management, Time Alloc ation, T arget Detection. I . I N T R O D U C T I O N In many a pplications sensors are nowadays a part of a multisensor system, e ach sensor bringing its complem entarity and its redund ancy to the overall system. Y ear afte r year the co mplexity and th e per formance s of many sensor s have increased leadin g to more and mo re complex multisensor systems which supply the decision center s with an in creasing amount o f data. Th is increasing com plexity also led to o ther uses fo r each sensor and therefo re for the m ultisensor systems. It is no more con sidered as a p assi ve system the role of which is ju st limited to simp le measuremen t actions; the many parameters o f each sensor and the interaction s b etween all the sensors allow to choose how th e measurem ent action must be done: the sen sors n eed to be man aged. The co mplexity of this problem is such that it is often impossible to a man to find an optimal solution ( with respe ct to the goal of the mission of the multisensor system) and multisensor manag ement strategies must be d erived. That is the reason why sensors managemen t has beco me d uring the past years an active field of re search. From a theorical poin t of view this pr oblem can be w ritten in the frame of op timal con trol and the sensor manage ment viewed as a Markov decision prob lem. Optimal solutions could therefor e be found. Unfo rtunately , the complexity is such that it is impossible in p ractice to derive these solutions. Su b- optimal solutions as w ell as altern ativ e approach es h av e then been proposed. In [1] or [2] th e au thors u se rein forcemen t learning, Q-learnin g and appr oximation func tions to d eriv e sub-optim al so lutions. I n many works the choice of the next action is based on infor mation theory an d inform ation d iv er- gence like the Rényi inf ormation divergence and th e Kullback Leibler di vergence [3], [1], [2] . In [4] Mahler propo ses to solve the problem in the frame of ran dom sets. All these works bring a po ssible solution to the sensor mana gement problem but as far as the authors know , it is often d ifficult to derive bound of p erforma nce which can be a drawback in an opera tional context. Moreover, these appro aches rarely take into accoun t the character istics of th e sensors. Th e work descr ibed in this paper pr oposes, in the frame of an ae rial patrol in charge of the de tection of p otential targets, to der i ve rad ar optimal time allocations (a part of the sensor managem ent pro blem) which allow to determine such boun ds an d which are b ased on th e mod elling of the de tection perfo rmances of a radar . It is assumed h ere that each aircr aft is equ ipped with an ESA (Ele ctronically Steered Antenn a) r adar . W e focu s on the d etection step for whic h a fixed duration T has b een allocated. Method s exist to optim ize th e detection o f a sing le target b y a single sensor and th e frame Search Theor y is dev oted to such a pro blem [5], [6]. In this pap er w e consid er a multitarget en v ironmen t an d the optimization pr ocess is led by co nsidering th e overall targets an d no t the targets on e by one. Th e pro blem then beco mes: if radars h a ve to observe P targets durin g T , h ow do they o rganize themselves to detect them in the be st possible way , i.e. how do th ey distribute the duratio n T over the spac e d irections ? The aim of th is article is to derive an o ptimal tem poral allocatio n based on the modelling of the radar detection probability and on an a priori kn owledge coming from an E SM type (Electrical Suppor t Measurement) or AEW typ e (Airb one Early W arn ing) system of su pervision. Along the study , two con texts are considered . The first one is th e ideal case: the position o f the targets are k nown and we must detect them. Of course this situation is not r ealistic but it a llows to derive some interesting results f or the second context : th e p osition of the targets are known by the mean of pro bability d ensities. After having defined th e assump tions of our stud y in the second section, w e present in the third section a mod elling of the radar detection function s. Howe ver the context o f this stu dy is m ultisensor multitarget, we start by a stud y of the optimizatio n of the detection p rocess in a m onosensor mon otarget en v ironment. Comparing to existing method s, ou r aim in th is prelimin ary work is to de riv e ana lytically an optimal strategy and the correspo nding prob ability o f detectio n. This last proba bility will b e used along th e overall pap er . The third section presents analytic results a nd a perform ance ev alu ation. The mu ltitarget en v ironmen t is tackle d in the fou rth section but we are still in a m onosensor case. Unde r th e assumption of an a priori knowledge, we pro pose an optim al tem poral allo cation. Th e allocation d eriv e d in this section u ses the results der i ved in the pr evious sections. Finally , th e last section shows how all the previous results can be used to p ropose an alloc ation strategy in the multisensor multitarget case. It is importan t to understand the needs at the orig in of the study , pro posed by Th ales Op tronics, the results of which are written o ut in this pape r . The aim was to f ound b ounds o f perfor mance for optimal allocation strategies. Therefo re, the fact of considering deterministic kn owledge abou t the targets, as it is the case in some pa rts of the paper, has sense e ven if it is not realistic in an rea l op erational context. When used or adapted in such a context, th e prop osed strategies are n o mo re optim al but we know the bou nds of perfo rmance which can b e interesting. I I . H Y P OT H E S E S The main assum ption of this article is the use o f an a priori knowledge. By this expression we mean a kn owledge ab out the situation, in particular a knowledge ab out the position s of th e targets. This assump tion is justified by the integration of the sensors in a supervision system o f th e ESM type (Electrical Suppor t Measu rement) or AEW type (Airbone Early W arnin g) f or instance. Using th ese sensors, it is possible to ob tain infor mation on the angular position s of th e targets and then to derive info rmation on their distances from the sensors. In this pap er , the a priori kn owledge is ideal or deterministic (for reason ning purpo ses) or more realistic (giv en by density pro bability functio n). W e also co nsider a 2D space; this assumption does n ot limit the gen eral char acter of the study , howev e r it red uces the calculatio ns. Finally , we suppose that the ob servation du rations are sufficiently short so th at the aircr afts can be regarded as station ary which mean s that the targets do not move out their resolutio n cell durin g the observation pr ocess. I I I . D E T E C T I O N P R O B A B I L I T Y O P T I M I Z AT I O N I N A M O N O S E N S O R M O N O TA R G E T C O N T E X T A. The radar sensor In order to establish o ptimal man agement strategies for the sensors, it is n ecessary to und erstand their o perating mod e. In particular, th e modelling of the detectio n p robability is a fundam ental basis for th e m anagemen t strategies we ar e going to define. The ra dar is an active sensor since it emits a signal which is reflected on th e target. T he radar consider ed here has an electrical scan ning. It m eans that its mecanical a xis is fixed and that it is th e direction of the analyz ing wa ve which is modified durin g the ob servation. First, w e are intere sted in the signal to noise r atio. If the sensor o bserves during T a target at the range r in a directio n which for ms an angle θ with the mechanical axis of the antenna, then the signal to n oise ratio of an ech o is equal to [ 7]: S N R = αT cos 2 ( θ ) r 4 (1) where α is an operation al parameter, which dep ends of the radar and the target (target’ s rad ar cr oss sectio n). In this paper, the targets are supposed to be similar , i.e. to have the same reflexion power , then α is constant. T his expression is established in a context where th e disturbing signal, which is supposed to be only due to the ther mal noise of th e rada r , is modelled by a nor mal random v ariable. Once the expression of the signal to noise r atio k nown, the target detection p robability can be d eriv e d. It is mo delled as: P d = ( P f a ) 1 1+ S N R (2) with P f a the false alar m pro bability which is taken here to one false alarm per second and per resolution cell. This expres- sion is established unde r the assumption s of a fluctuating t arget and a modellin g of the received e nergy of type ”Swerling 1” [8 ]. A target doe s n ot have a regular form, theref ore the reflected en ergy varies from an impulse to a nother . The target ca n th en be co nsidered as a set of elementary reflector s the positions of which in the space are related to the target orientation . The r eturned signals are then ind ependen t and the amplitud e of th e received energy fluctuates. These targets are called ”fluctuating targets” and have been mod elled by Swerling [9 ] : they ar e called Swerling 0, S werling 1, Swerling 2, Swerling 3 and Swerling 4 . The Swerling 1 type is particularly ada pted to the c ase of th e air target d etection. B. Optimizatio n o f the d etection pr ob ability in a monosensor monotarget context According to expressions (1) and (2) it can be ea sily shown that the detectio n probability is strong ly d egraded w hen the range incr eases. Th ere are se veral m ethods to improve it. A first solu tion is to use a proc edure of "aler t and confirmatio n" [10], [ 5]. This m ethod con sists in d oing two d etection steps: the first one with a low detection thresho ld, the second one with a h igher one in ord er to eliminate false alarm s fro m the aler t step. During this second step, the emitted wa ve is adapted to the target. However , this p rocess of decomp osition in two steps needs a long in tegration time. A solution could be to inc rease it but fo r high detectio n probab ility , the slope dP d dt is small. Instead o f carry ing out o nly o ne a cquisition of th e signal during T , there fore only one detection, we propo se to acquires N elementary signals and to carr y ou t an elementary detection on each received signal, that is to realize N elementary detection s. The u se of the r adar with a different emission frequen cy at each e lementary d etection allows to obtain indepen dent detection s and allo ws the analytic deriv atio n of an optimal detection prob ability as it is shown in the following [7], [1 1]. If P de denotes the elementary detection probab ility , th en the cumulative detection pro bability is eq ual to [8 ]: P d = 1 − (1 − P de ) N (3) where P de is given by th e expression (2), f or an o bservation duration eq ual to T N . The problem is to find th e num ber N o f elementary detectio ns which op timizes th is cumulative detection pro bability . By consider ing the target’ s sign al to noise ratio far hig her than on e, it is p ossible to detail th e expression of the elementary d etection probab ility as: P de = ( P f a ) 1 S N R = exp  r 4 N ln ( P f a ) αT cos 2 ( θ )  (4) At this po int it is impor tant to un derstand what is th e meaning and the limitation of "th e signal to noise ratio is far higher than one". It m eans that it is hig h enough to m ake the appr oximation o f P de by (4), but it is not high en ough to consider that this prob ability is almost equal to one; a detection phase is therefo re necessary . W e now intro duce the con stant β as : β = r 4 ln  1 P f a  αT cos 2 ( θ ) (5) which allows us to express the prob ability ( 4) as P de = exp ( − β N ) . The cu mulative detectio n proba bility is th en equal to: P d = 1 − ex p ( N ln (1 − exp ( − β N ))) . (6) Using a classical o ptimization p rocess on this last pro babil- ity , it can be shown that it is o ptimal if N is eq ual to : N opt = γ r αT (cos ( θ )) 2 r 4 ln  1 P f a  (7) with γ r = − β N = − ln ( P de ) = ln 2 . The elementary detection pr obability P de is therefo re equal to 0 . 5 and the cumulative one is equal to : P d = 1 − exp  − T τ r  (8) with τ r = r 4 ln ( P f a ) γ r α (co s ( θ )) 2 ln (1 − exp ( − γ r )) . (9) These results show on the on e ha nd that the m odelling of the radar sensor detection functions makes it possible the elab oration of a nalytical strategies o f optimization of th e detection pr obability and, on the o ther hand , that it is possible to q uantify the perfor mances. A few rem arks abou t these results: • The o ptimal number of elementary detections is no t a natural, N opt / ∈ N . However , it does no t alter the g eneral frame of o ur metho d and allows us to calcu late optimal perfor mances which will be u sed as referen ces, like the Cramér-Rao lower boun d in estimation theory . • The assumption o f an important signal to noise r atio is a trick which allows us to write the detectio n probab ility simplier . Howe ver, the prob abilities obtained can be close to 0 . 5 , wh ich justifies the e laboration o f an optimal detection pro cess. In th e next section we will see how to use these results to detect several targets. I V . M O N O S E N S O R M U LT I T A R G E T E N V I RO N M E N T A. Deterministic knowledge W e consider a situation where P targets are presen t in an air spac e. The knowledge about them is such that their angular deviations θ i and th eir ran ges r i are known ∀ i ∈ { 1 , .., P } . Our goal is to detect th em with a rad ar . Furth ermore, targets are supp osed to be lo calized in different direction s of th e space, that is θ i 6 = θ j ∀ ( i, j ) ∈ { 1 , .., P } 2 . Finally , we also suppo se th at all the targets d on’t repre sent the same threat which le ads to the introd uction of weights ε i . In th ese condition s, let us ca ll t i the ob servation duratio n of the target i by the radar . Since angu lar d e viations are d ifferent, the rad ar does n ot ob serve se veral targets simutalne ously and it results in the following r elation b etween d urations t i and the total duration T : P X i =1 t i = T with t i ≥ 0 ∀ i ∈ { 1 , .., P } . (10) Durations must o bviously be positiv e, but we will see that they could be null, because of the relative po sitions of th e targets with respect to the senso r . Our aim is to maximize th e detection of all the targets. As a pro bability is always p ositi ve, maximize each of th em is equivalent to maxim ize their sum . Then we d efine the criterion: J = P X i =1 ε i P di ( t i ) (11) where ε i can b e interp reted as a pote ntial threat or p riority coefficient. This c oncept o f threat is introdu ced here in a general way and its character ization is b ehind the scope of the present pap er . T hese coefficients can for instance b e inv e rsely propo rtionnal to the distan ce [12]. P di ( t i ) is the d etection probab ility of the target i by the r adar , for a duratio n t i . If this dur ation is kn own, then we ar e in the co ntext describ ed in section II I: a monosen sor mono target context. Using previous results it is possible to write: P di ( t i ) = 1 − exp  − t i τ r i  (12) with τ r i = r 4 ln ( P f a ) γ r α (co s ( θ i )) 2 ln (1 − exp ( − γ r )) . (13) According to this last expression of the detectio n p robability , the criterion J will reach its maximum when the d urations t i tend towards infinite, which is not compatible with the temporal con straint which was de fine. Our aim is then to optimize the criterion (11) under the constraints (10): J            maximize J = P P i =1 ε i P di ( t i ) under the co nstraints    P P i =1 t i = T t i ≥ 0 ∀ i ∈ { 1 , .., P } (14) W ith such a form ulation, we face to a classical pr oblem of optim ization un der constraints which can be solved by the way of Lagr angian functions and too ls of conve x optimizatio n. Such an optimiza tion process leads to the fo llowing results. Let us introd uce the func tion x 7→ ⌊ x ⌋ + defined on R by: ⌊ x ⌋ + = x if x > 0 = 0 el se (15) and λ th e single so lution of th e fo llowing equatio n: P X i =1 τ r i T  ln  T ε i τ r i λ  + − 1 = 0 . (16) If I is the su ffi x set d efined by: I =  i ∈ { 1 , .., P }     λ < T ε i τ r i  (17) then the optimal tempo ral allocation is g i ven by : ( t i = τ r i ln  T ε i τ ri λ  if i ∈ I = 0 el se (18) Furthermo re, th e o ptimal elem entary d etection num ber for each t i , i.e f or each target, is equal to : n i = γ r αt i (cos ( θ )) 2 r 4 i ln ( P f a ) . (19) Results given in (18) dep end o n the param eter λ whic h is solution of the eq uation (16). This e quation has an analytical solution if card ( I ) = P . In th is case it can be shown from the p revious r esult that the optim al allocation o f the duratio n T between th e P targets is g i ven b y : t i = P P j =1 τ r j ln  ǫ i τ rj ǫ j τ ri  + T P P j =1 τ rj τ ri ∀ i ∈ { 1 , ..., P } . (20) A sufficient condition to obtain th is r esult is that th ere exists λ 0 > λ such th at: λ 0 < T ǫ i τ r i ∀ i ∈ { 1 , ..., P } (21) So far, we have deri ved optimal allocation s for the detection of P targets given a total dur ation. W e can remark th at a n implicite a ssumption has b een m ade: th e in finite divisibility o f the du ration T , wh ich is not the case in re ality . However , this assumption is justified in this ar ticle by the use of a rad ar . This sensor having an electro nic scan ning m ode, th e m ovement Fig. 1. Repr esentation of the dire ctions and cells of the space from a n angu lar position to another can be con sidered as instantaneou s. The major hyp othesis o f this section is the deterministic kn owledge about the situation . W e pr opose in the next section an optimal tempor al allocation in the case o f an a priori knowledge d efined by pr obability densities which correspo nds to more realistic context. B. Pr ob abilistic knowledge In this section we assum e that we h a ve a weak knowledge of the targets p ositions. Several targets can app ear in th e same direction which is m ore realistic than in th e previous section. Thus we hav e to consider all the space and not a few directions as it was po ssible previously . The space d irections we co nsider are the ang ular field s o f view of the r adar . M oreover , since the sensor obser ves th e globality o f a d irection at the sam e time, we w an t to determin e the observation duration in this dir ection, that is the dur ation t j in the dir ection j . In or der to d o so, the observation space is samp led accordin g to the resolu tion cells of the radar as it is describ ed in the figu re 1. The s ampled space is ther efore defined by the in terval of rang es [ r min , r max ] in which the detection is realized an d the angular sector c j i.e the direction j, j ∈ { 1 , .., N d } . W e hav e seen in th e sectio n III -A that the sensor form s a set of rang e r esolution cells in each direction, i.e in each c ell c j . Then we con sider a set of sub- cells, the resolution cells, c ij , at the range r i , in the direction j, i ∈ { 1 , ..., N r } . Since a sensor observes simultaneo usly all th e targets present in the same direction, we are going to determ ine the prob ability of d etecting from on e to sev er al targets in e ach of th ese direction s. Fir st we consider the exp ression of the detection prob ability giv e n by relatio n (4) . It rep resents the detection pr obability of a target kn owing that it is at a rang e r from th e sensor . L et H k be th e event "the target k is detected ". Using the form alism of conditionn al prob abilities we can write the prob ability of detecting th e target k at a r ange r as: P ( H k , r ) = P ( H k | r ) P ( k , r ) (22) with P ( k , r ) th e target location p robability at the range r. W e assume that the target detectio n probab ility in a given cell can be ap proxima ted by the detection probab ility o f this target at the ran ge on which the cell is cen tered. T he expression o f the previous p robability can th erefore be wr itten as: P ( H k , c ij ) = P ( H k | c ij ) P ( k , c ij ) (23) where P ( k , c ij ) is ob tained by the integration of the a priori density of pro bability in the cell c ij . W e no te P ( k , c ij ) = ρ ij k . P ( H k | c ij ) is der i ved from the expre ssion (4), with th e target at the ap proxim ate rang e r i . Finally we ob tain: P ( H k , c ij ) = P dij k = ρ ij k e − δ i r 4 i t j (24) with δ i = − ln( P f a ) α (cos( θ )) 2 . Since resolution c ells are indepen dent, the detection p rob- ability of the target k in the d irection j is the sum of probab ilities in the cells of th is direction : P d j k = N r X i P dij k (25) Lastly , we deter mine the pr obability P d j of d etecting from one to sev e ral targets in a same dir ection. This prob ability is the union of p revious probabilities for k f rom one to P . The Poincaré fo rmula allows us to r ealize the calculation: P ( ∪ H i , i ∈ { 1 , .., P } ) = P n i =1 P ( H i ) − P n i,j =1 ,i 6 = j P ( H i ∩ H j ) + P n i,j,l =1 ,i 6 = j 6 = l P ( H i ∩ H j ∩ H l ) − ... (26) Our aim is to optimize this probability over the who le space. Unfortu nately , its expression ( 26) is not easily exploitab le if we want to use th e r esults described in the sectio n above. It is the reason why we prop ose to realize a par ametric mo delling of this probab ility . The mod el we used is: P d j ≃ exp  − ω j t − n j j  (27) where ω j and n j are the m odelling pa rameters in the direction j . Th ey are d etermined in o rder to minimize mean square err or criterion . Under this for mulation, the pr obability has the same pro perties as th e one given by relation (4); it is then possible to optimize it like using the framework of section III, i.e. b y a deco mposition in to a n optimal numbe r of elementary d etections. Leading the same op timization pro cess as in section III, the following r esult can b e shown. L et u s c all γ sj the u nique solution o f the equ ation : (1 − exp ( − γ sj )) ln ( 1 − exp ( − γ sj )) + n j γ sj exp ( γ sj ) = 0 (28) and M j the numb er of indepen dant detection s realized in the dir ection j durin g an time t j . If each elementary detection last t j M j then the detec tion p robability f rom one to N targets in the direction j is ma ximum when : M j,opt = γ sj t n j j ω j ! 1 n j . (29) The elemen tary detection pro bability is then equal to exp( γ sj ) an d the overall detection p robability to : P d j ( t j ) = 1 − exp  − t j τ j  (30) with : τ j = −  ω j γ sj  1 n j 1 ln (1 − exp ( − γ sj )) . (31) Using these last expressions for detection prob abilities, the criterion to optimize is: G            maximize G = N d P i =1 ε i P d j ( t j ) under the co nstraints    N d P i =1 t j = T t j ≥ 0 ∀ j ∈ { 1 , .., N d } (32) The resolu tion is similar as the one d escribed in the deter- ministic con text. I t lead s to th e following results. Let λ be th e unique solution o f the eq uation: N d X i =1 τ j T  ln  T ε j τ j λ  + − 1 = 0 (33) and I the suffix set defined by: I =  j ∈ { 1 , .., N d }     λ < T ε j τ j  (34) then the o ptimal tempo ral a llocation is g iv en by : ( t j = τ j ln  T ε j τ j λ  if i ∈ I = 0 el se (35) Furthermo re, th e o ptimal elem entary detectio n num ber for each t j , i.e f or each d irection, is equal to: m j =  γ sj t n j ω j  1 n j . (36) C. Simulatio ns W e consider four targets located in an aer ian space. An a priori knowledge is av ailable f or each o f them, by th e way of density probab ilities. These d ensities are defined in a cartesian frame by the mea n of two dimension nal ga ussian laws. Figure 2 illustrates th is possible scenario which co rrespon ds to th e following situa tion: • target 1 : the distribution is centered around the point (20 k m, 30 k m ) , th e stand ard deviation on each coord i- nate is eq ual to 0 . 1 k m . Fig. 2. Scenario 1: targ ets are located by distrib utions cente rs,straig ht lines re present the space dir ections • target 2 : the distrib ution is centered aro und the point (40 k m, 60 k m ) , th e stand ard d e viation on each coord i- nate is equal to 0 . 1 k m. • target 3 : the distrib ution is centered aro und the point (60 k m, 40 k m ) , th e stand ard d e viation on each coord i- nate is equal to 0 . 1 k m. • target 4 : the distrib ution is centered aro und the point (110 k m, 2 0 k m ) , the standard deviation on each coord i- nate is equal to 0 . 1 k m. According to our num erics values, space is divided into forty ang ular d irections, N d = 40 . Th e result of the op timiza- tion pro cess leads to the allo cation of table I for T = 30 ms . dir . 1..3 4 5..11 12 13..19 20 21..40 ǫ j 1 1 1 1 1 1 1 t j (ms) 0 0 0 23 , 47 0 6,53 0 m j 0 0 0 1,31 0 2,60 0 P d j 0 0 0 0,59 0 0,97 0 T ABLE I Optimal tempora l allocati on withou t weights As we can see o nly tw o directio ns are ef f ectiv ely consider ed in the tempo ral allocation. T his is due to the allocation processus which r ealizes a glo bal optimization of the detection probab ility . In fact, the four th target is too distant fro m the others and from the sensor and it would spent too much time to detect it. Th is time would be allocated to th e d etriment of the other targets. It is therefore b etter no t to observe it. Coefficients ǫ j which appear in table I are weights which introdu ce ponde rations on the impo rtance of the target. They are all equal to one becau se the ponde ration - or priority - notion was not taken into accou nt initially . In table II the result of the o ptimization proce ss is written ou t when such taps are considered . As it can be seen in this table, th e sensor spend s more time in th e impor tant direction with respe ct to this taps. It resu lts in an increasing in the detectio n prob ability . V . M U LT I S E N S O R M U LT I T A R G E T E N V I RO N M E N T A. Intr o duction In this section we suppose that P radar s are used to detect N targets, each sen sor re alizing a d etection. The problem here dir. 1 .. 3 4 5 .. 11 12 13 .. 19 20 21 .. 40 ǫ j 0 0 , 07 0 0 , 18 0 0 , 74 0 t j ( ms ) 0 0 0 21 , 08 0 8 , 92 0 P d j 0 0 0 0 , 56 0 0 , 99 0 T ABLE II Optimal tempora l allocation with weights is more com plex th an previously since we are loo king for group ing r adars in order to op timize th e d etection p rocess. The que stions to solve are therefor e: • which criterio n do we want to optim ize? • how c an we realize such a regrouping in a dynamical way? • how long a group a sen sor must ob serve a po tential target? • how d o we assign a group to a direction of observation? This problem being tackled here in the deterministic context, the answer to the first qu estion is theref ore r ather easy since it is the same as in the previous sections : the criterion to optimize is the sum of the detection pr obability : C = N t X k =1 p k (37) with N t the target nu mber and p k the detection prob ability of target k . In the following each gro up of sensors will be called a pseudo senso r . The d etection of each pseudo sensor is d eriv ed from the fusion of the de tection of e ach its sensor s. W e ch oose the fusion law OR wh ich is usually used in the de tection theory . The metho d pr oposed to an swer to these questions is an heuristic based on the results o f the p revious section s. This heuristic is b roken up into two phases : 1) The initial phase where first peudo sensor s are consti- tuted and where a first time allocation is made; this phase is based o n the results of previous sectio ns. 2) The plan ification p hase where the used of th e senso rs is planified over th e time of analy sis T from th e allocation realized du ring th e initial phase. B. The in itial pha se The allo cation proce ss a t initial time can b e split into three steps: • Step 1 : Computa tion of the detection proba bilities of the P sensors K i with i ∈ { 1 , ..., P } . Knowing the ran ges between the sensors and the targets and the ob servation duration T , it is possible, accor ding to results of section IV -A, to derive an optima l allocation over the d uration T for all the targets and the associated dete ction probab ili- ties. This step allows the use of each sensor in an op timal way at initial time. • Step 2 : From the p robabilities found at the step 1, compute the detection probabilities o f each targets for all the possible p seudo-senso rs. The aim of this step is to use the set of sensors in the be st possible way at initial time. At this step th e d ata f usion law is OR. T o ensure that none o f the sensors will be useless, the detection probab ility of each pseudo-senso r is com puted with the shortest ob servation dur ation. For two sensors A an d B it come s to com pute : P ( d A , t A , d B , t B ) = P d ( d A , min ( t A , t B )) + P d ( d B , min ( t A , t B )) − P d ( d A , min ( t A , t B )) P d ( d B , min ( t A , t B )) (38) • Step 3 : Deter mination of the allocatio n which max imize the criterion C . Let us illustrate with an example th e different steps of the method. C. Exa mple of initial a llocation Let us consider thr ee sensors an d three targets denote d by C n , n ∈ { 1 , 2 , 3 } . The tab le III giv es th e distances b etween the senso rs and the targets. For the sake of simplicity , angles θ are supposed to b e nu ll. distance sensor − tar g et ( in k m ) K 1 K 2 K 3 C 1 45 26 52 C 2 51 45 25 C 3 50 33 41 T ABLE III Distance s sensors-tar gets (in km) 1) Step 1: co mputation of the detection pr ob abilities: L et us con sider a dur ation T = 5 ms allocated to the d etection phase. The optimal time alloc ation at initial time is written out in tab le IV and the corr espondin g detection p robabilities in the table V. tempor al alloc ation ( in s ) K 1 K 2 K 3 C 1 2 . 5807 1 . 1702 0 . 9224 C 2 1 . 0109 1 . 8768 1 . 1462 C 3 1 . 4084 1 . 9530 2 . 9314 T ABLE IV Optimal tempora l allocation (in ms) detectionpr obabilities K 1 K 2 K 3 C 1 0 . 4814 0 . 9309 0 . 1233 C 2 0 . 1444 0 . 3797 0 . 9532 C 3 0 . 2095 0 . 8206 0 . 6612 T ABLE V Detect ion pr obabilities obtained from the data of the tables III and IV 2) Step 2: pseu do-sensors a nd detection pr ob abilities: Let us denote P the numb er o f sensor s, then the number of pseudo- sensors is S = 2 P − 1 . If K 1 , K 2 and K 3 are the sen sors, th e pseudo- sensors are: K 1 , K 2 , K 3 , K 1 − K 2 , K 1 − K 3 , K 2 − K 3 et K 1 − K 2 − K 3 . The detectio n probabilities obtained by using the method describ ed in the pr evious paragra ph ar e g i ven in the table VI. K 1 K 2 K 1 - K 2 K 3 K 1 - K 3 K 2 - K 3 K 1 - K 2 - K 3 C 1 0.481 0.931 0.949 0.123 0.307 0.893 0.916 C 2 0.144 0.380 0.338 0.953 0.943 0.965 0.956 C 3 0.210 0.821 0.771 0.661 0.530 0.913 0.864 T ABLE VI Detect ion pr obabilities ass ociat ed to the pseudo-sensor s 3) Step 3: determina tion of the o ptimal allocatio n: Using the results of section V -C.2 it is easy to compu te th e value o f the criter ion ( 37) for each possible allocation T a rget - Pseudo- sensor . The list below g i ves the results of th e co mputation f or a few pseud o-sensors. The allo cation of a p seudo-sensor to a target is rep resented by an arrow . • K 1 → C 1 , K 2 → C 3 et K 3 → C 2 : 0 . 48 14 + 0 . 8206 + 0 . 9532 = 2 . 2552 , • K 1 → C 3 , K 2 → C 2 et K 3 → C 1 : 0 . 2095 + 0 . 3797 + 0 . 1233 = 0 . 7125 , • K 2 → C 1 , K 1 − K 3 → C 3 : 0 . 93 09 + 0 . 5300 = 1 . 4609 , • K 3 → C 1 , K 1 − K 2 → C 2 : 0 . 12 33 + 0 . 3384 = 0 . 4617 , • K 1 − K 2 − K 3 → C 1 : 0 . 9156 , • K 1 − K 2 − K 3 → C 2 : 0 . 9555 , If we consider all the po ssible a llocations, the maximu m is obtained with the allocatio n wh ich corresp onds to the allocation of the sensor K 1 with the target C 1 , the a llocation of the sensor K 2 with the target C 3 and the allocation of the sensor K 3 with the target C 2 . This allocatio n, is su ch that all the targets are o bserved and a ll the sen sors are used. It is interesting to remar k that it is n ot the neare st sensor to a target which is used f or its dete ction. This is beca use the context is not th e optimization o f th e d etection p erform ances of each individual sensor but a con text of globa l op timization. D. Sen sor planifica tion over T At this step, the initial allo cation is realized. It is now necessary to build a planning of the use of the sensors from this initial alloca tion. If we analy ze the initial allocatio n processus, we can see that this allo cation is made for a given time interval resulting f rom the optimization proce ss of the step 1 and from the limitatio n on the time of o bservation introd uced in the step 2. This allocatio n is therefore not valid over the all time interval T . W e pr opose th e f ollowing p lanification of the sensor use over T based on the re sults of the optimization process carr ied o ut durin g the step 1 of th e initial allocation. • rule 1 : the alloc ation sen sor-target is called into q uestion as soon as one of the du rations o f observation of the "activ e" cou ples is fin ished: the sen sor or the sensor s concern ed must be allocated to an other target. An ac ti ve couple is an allocation of a sensor , or pseudo -sensor to a target. • rule 2 : if a po nderation of th e target has bee n achieved, the sensors are allocated to th e target whic h have the highest po nderation we igths. • rule 3 : if the are no po nderation , th e sensors are allocated to the target wh ich need s the lowest observation tim e different from zero . These times are th ose computed at the the step 1 of th e initial allocation . Thus, by given pr iority to the short d urations, the detection perform ances of the observed targets will be op timized in the case where some operation al constrain ts abort the d etection pr ocess. E. Exemp le of pla nification W e conside r in this example the situation used in th e section V -C with all the pon deration taps equa l to one. The alloc ation sensors-targets has been d etermined in the section V -C. T he duration du ring which this allo cation is e ffecti ve is determin ed by rule 1 . It corr esponds to th e minimum obser vation duration of the targets b y the selected sensors. Con sidering the results of table I V, th e a llocation is then c alled into q uestion at the end of the time t obs = 1 . 1 462 ms . During this du ration, the sensor K 3 has obser ved th e target C 2 in an op timal way , in the sense o f the d etection p erforma nces, and it then needs to be dir ected towards a nother target. The other targets have also been observed dur ing t obs . Th e observation of C 2 by K 3 being finished the allo cation given by table IV must be modified to make ap pear th at K 3 will no more observe C 2 and that for the oth er a ffectation targets have still being o bserved du ring t obs . Th e result is given is written out in the table VII. tempor al alloc ation ( in k m ) K 1 K 2 K 3 C 1 1 . 4345 1 . 1702 0 . 9224 C 2 1 . 0109 1 . 8768 0 C 3 1 . 4084 0 . 8068 2 . 9314 T ABLE VII T emporal allocat ion after an observation durati on t obs Using the r ule 3, th e sensor K 3 is now oriented towards the target, which need the lowest o bservation duration in order to reach op timal detectio n perfor mances. H ere, it is the target C 1 . T he sensor K 2 remains affected to th e o bservation o f th e target C 3 , then the p seudo-sensor K 1 − K 3 is used for the observation of the target C 1 . This algorithm is iter ated till th e time t = T is reached. The resulting allocation is repr esented in the figur e 3. Th e target C 1 will h a ve been observed during 3 . 12 32 ms , the target C 2 during 3 . 56 55 ms an d th e target C 3 during 4 . 8845 m s . Their d etection p robabilities are respectively 0 . 96 86 , 0 . 9751 and 0 . 9 520 . The sum o f these probab ilities is 2 . 8957 . It it is greater than 2 . 70 75 whic h is the sum which would have be en obtained if the allocation established initially h ad been used dur ing all the duration T . V I . C O N C L U S I O N This paper presents method s to manag e th e time allocation of radars over a set of targets. In a first part a method to optimize the detection process of targets is propo sed. It is based on the mo delling o f the detection proba bility of a target. This firt result is th en used to propo se optimal time allo cations in the mon osensor m ultitarget c ase. T wo o perational contexts are considere d : a determin istic con text whe re th e position Fig. 3. Represe ntation of the planning of the utilization of the sensors during the durat ion T of the target are k nown and a p robabilistic context where the knowledge of the p osition of the target is represented by probab ility density fun ctions. W e showed that the probab ilistic context can be solved using the results of the deterministic one. These results have then bee n u sed to propo se an heu ristic fo r the planificatio n of a set radar the mission of which is to detect targets : we ar e then in the multisensor multitarget case. The planification h as been pro posed in the d eterministic c ontext and still n eed to be gener alized to the p robabilistic context. V I I . A C K N O W L E D G E M E N T The au thors would like to tha nk Mich el Prenat from Thales Optronics for its imp ortant contribution to this work. R E F E R E N C E S [1] C. Kreucher , D. Blatt , A. Hero, and K. 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