Optimal search: a practical interpretation of information-driven sensor management

Transcription

1 Optimal search: a practical interpretation of information-driven sensor management Fotios Katsilieris, Yvo Boers and Hans Driessen Thales Nederland B.V. Hengelo, the Netherlands {Fotios.Katsilieris, Yvo.Boers, Abstract We consider the problem of schedling an agile sensor for performing optimal search for a target. A probability density fnction is created for representing or knowledge abot where the target might be and it is tilized by the proposed sensor management criteria for finding optimal search strategies. The proposed criteria are: an information-driven criterion based on the Kllback-Leibler divergence and a criterion with practical meaning, i.e. performing the sensing action that will yield the maximm probability of detecting the target. It is shown that sing the aforementioned criteria reslt in the same sensing actions when searching for a target, nder certain conditions. This reslt establishes a practical operational jstification for sing information-driven sensor management for performing search. I. INTRODUCTION The problem of performing search emerges when the available sensor resorces have to be tilized in an efficient way sch that the search for an object or a featre is sccessfl. The challenges are to find the object as soon as possible while spending as few resorces as possible. Towards this goal, sensor management criteria can be tilized. The main advantage of sing sch criteria over the simple approaches of periodic or random search is that they can take into accont any available external information and as a reslt demonstrate adaptive behavior. For instance, if the object is expected to be with higher probability in a specific region, the periodic or random search approaches wold not take this information into accont bt a careflly chosen or designed criterion wold prodce search patterns that leverage this information in order to find the object faster and/or by sing less resorces. If the external information is pdated at each iteration, like in or case, then the problem amonts to performing one-step ahead (or myopic) optimal search. Some examples where these challenges appear are: target detection [1, [2, search for wreckages and srvivors [3, [4 etc. In the robotics commnity the search problem is sally referred to as the prsit-evasion problem and has been stdied nder different assmptions and solved sing different approaches, see for example [5, [6. We consider the schedling of an agile sensor for efficiently searching for a target. A characteristic example of sch a F. Katsilieris is also a PhD stdent at the Dept. of Applied Mathematics of the Univ. of Twente, Enschede, the Netherlands. sensor is a mltifnction radar (MFR). Sch a radar has received a lot of focs from the research commnity as an attempt to schedle efficiently its tasks, one of which is to perform search for ndetected targets. In [7 the track and search fnctions of an MFR are schedled according to a threat-based criterion. For schedling search fnctions, the athors se ghost targets that dictate volme or horizon search instead of tracking radar fnctions. In [8 the revisit intervals, radar beam positions, and energy per dwell are controlled for improving track qality and energy efficiency. Especially in the case of searching, the se of negative information is sggested for pdating the predictive densities of the targets and obtaining a search pattern by searching the region where the maximm of the predictive density is located. An pdated version is [9. In [10 the athors se a search-to-track ratio that the ser has to set. According to this ratio, the sensor manager schedles the corresponding tasks of the radar. When the search task is considered, an estimate of the spatial density of previosly ndetected targets is tilized. The sensing action that maximizes the expected nmber of newly detected targets is chosen whenever a search fnction is schedled. A disadvantage of this approach is that the search-to-track ratio is ser defined and not atomatically determined by the schedling algorithm according to the optimization of a criterion. A similar schedling approach is presented in [11 where the schedling criterion sggests selecting recrsively those sensors that cover the most probability mass of the predictive density. In [12 an approach similar to ors has been proposed. An a priori probability distribtion of the target to be detected is specified by a set of discrete target position probabilities corresponding to each search beam. Immediately after the increment of search effort is applied, the target position probability density is pdated by the se of Bayes rle. The proposed soltion sggests making the next look in the search cell that will provide the maximm vale of the incremental search energy and S/N payoff ratios (target cmlative probability of detection increase divided by search effort expenditre increase) for all cells and to maximize the dty factor of each cell. In [13 the athors introdce the continos doble action parameter selection algorithm (CDAPS) which manages the 439

2 MFR resorces by tilizing an action mechanism to select parameters for individal radar tasks. The athors show that their algorithm performs better than periodic search. The approach presented in or paper bilds on the approaches described in the literatre and the specific contribtions are: The constrction of a probability density of the ndetected target and its implementation sing a particle filter. The implementation of two sensor management criteria based on the aforementioned density: a criterion based on Kllback-Leibler divergence and a criterion based on the expected probability of detection. It is proven that the two aforementioned criteria are eqivalent, in the sense that they lead to the same sensor selection scheme, nder certain conditions. The importance of this reslt lies in the connection that is established between an information-driven criterion (whose practical meaning is difficlt to explain) and a criterion that has straightforward practical meaning, i.e. choosing the action that will yield the maximm probability of detecting the target. The rest of the paper is organized as follows. In section II the system description is given and the problem nder consideration is described. In section III the proposed soltion is presented and in section IV a graphical proof of eqivalence of the proposed sensor management criteria is given. In section V the simlation reslts are presented. Finally, in section VI the conclsions are discssed along with some open qestions. II. SYSTEM SETUP AND PROBLEM FORMULATION Consider a scenario where an agile sensor has to search for one target. This system can be described mathematically by the following (discrete time) state and measrement eqations: where s k = f(s k 1, w k 1 ) (1) { { }, no target present (2a) z k = h(s k, k, v k ), one target present (2b) s 0 p(s 0 ) (3) k = 1, 2,... is the time index s k R Ns is the state of the system at time k w k R Ns is the process noise with probability density p w (w k ) k U is the chosen sensing action, with U being the set of the available sensing actions z k R Nz is the received measrement with dimensionality N z. If there is no target, then there will be no measrement and therefore (2a) will hold. v k is the N z -dimensional measrement noise with probability density p v (v k ) s 0 is the initial state of the system with probability density p(s 0 ) the vector and possibly non-linear fnction f( ) : R Ns R Ns describes the dynamics of the system similarly, the vector and possibly non-linear fnction h( ) : R Ns R Nz relates the measrement z k to the system state s k and the sensing action k The considered problem amonts to finding the best sensing action k by maximizing a sensor management criterion V (s k, z k, ) k = arg max V (s k, z k, ) (4) and then sing it for solving the attached filtering problem of determining the posterior probability density fnction p(s k Z k, U k ) that describes where the target might be. We denote by Z k = {z 1,..., z k } the measrement history and by U k = { 1,..., k } the sensing action history. III. PROPOSED SOLUTION We propose solving the described problem by employing the recrsive Bayesian estimation approach implemented by a particle filter and performing the optimization of the criteria sing qantities of the rnning particle filter. A. Recrsive Bayesian estimation In the recrsive Bayesian estimation context, given a probability density fnction p(s k 1 Z k 1, U k 1 ), first the prediction step is performed sing the Chapman-Kolmogorov eqation: p(s k Z k 1, U k 1 ) = p(s k s k 1 )p(s k 1 Z k 1, U k 1 ) ds k 1 (5) where p(s k s k 1 ) is determined by the kinematic model of the target. Then the predictive density p(s k Z k 1, U k 1 ) is pdated with the received measrement z k sing Bayes rle p(s k Z k, U k ) = p(z k s k, k ) p(s k Z k 1, U k 1 ) p(z k Z k 1, U k ) (6) p(z k s k, k ) p(s k Z k 1, U k 1 ) (7) where p(z k s k, k ) is the likelihood fnction and p(z k Z k 1, U k ) = p(z k s k, k ) p(s k Z k 1, U k 1 ) ds k (8) is a normalizing constant which in practice does not have to be calclated if a particle filter is employed. We will se a standard SIR particle filter [14 for approximating Eqations (5) and (7) with N particles s i k and corresponding weights qk i : {s i k, q i k}, i = 1,..., N (9) sch that the approximation converges to the tre posterior distribtion p(s k Z k, U k ) as N, see [

3 B. Dynamical model The state of the system is assmed to be 4-dimensional, describing the position and velocity of the target in Cartesian coordinates s k = [x k v x y k v y T R 4 (10) The following target dynamics are also assmed: where: F = s k = f(s k 1, w k 1 ) = F s k 1 + w k (11) Σ = 1 T T w k N (µ, Σ) b x T 3 /3 b x T 2 /2 0 0 b x T 2 /2 b x T b y T 3 /3 b y T 2 /2 0 0 b y T 2 /2 b y T and b x = b y are the power spectral densities of the acceleration noise in the x y direction, T is the sampling time and µ = [ T is the mean of the Gassian noise. C. Measrement model and its se in the pdate step The search for an ndetected target is considered. This implies that no measrements are received or eqivalently that the measrement z k is always an empty set (Eq. 2a) and the measrement history is a vector of empty sets. Frthermore, we assme that no false alarms are present (bt this assmption can be relaxed in a straightforward manner): Z k = {,,...} (12) Therefore, if the probability of detecting the target when performing the sensing action k is defined as P d (s k, k ) (0, 1) then the likelihood fnction becomes p(z k s k, k ) = p(z k = { } s k, k ) = 1 P d (s k, k ) (13) This form of likelihood fnction is referred to in the literatre as Negative Information, see [9. From now on z k = { } and Z k = {,,...} will be skipped in the notation for simplicity reasons and we will only write p(s k U k ) etc. Given the aforementioned simplification, the prediction step in Eq. (5) becomes: p(s k U k 1 ) = p(s k s k 1 ) p(s k 1 U k 1 ) ds k 1 (14) and the pdate step in Eq. (6) becomes: with p(s k U k ) = [1 P d(s k, k ) p(s k U k 1 ) C (15) [1 P d (s k, k ) p(s k U k 1 ) (16) C = [1 P d (s k, k ) p(s k U k 1 ) ds k (17) a normalizing constant that does not need to be calclated when a particle filter is employed. D. Sensor management criteria Or knowledge abot the location of the ndetected target is represented by a probability density fnction and conseqently, the ncertainty abot this knowledge (or the information gain by means of performing search) can be conveniently described in the information theory context. We se the expected Kllback-Leibler divergence (KLD) in order to contribte to the ongoing discssion on whether taskbased or information-driven criteria shold be sed in sensor management and what the practical interpretation of the latter is (a more elaborate discssion on this sbject can be fond in [16). The maximm expected KLD will be compared to a practical (task-based) criterion that selects the search action that will yield the maximm expected probability of detecting the target. In all the following formlas for the particle approximations it will hold that the weights of all the particles will be qk i = 1/N becase resampling is performed at every time step and that s i k, sj k p(s k U k 1 ). 1) Maximm expected Kllback-Leibler divergence: The conditional entropy is a measre of ncertainty in the information theory context that is commonly sed in filtering and sensor management applications, see [17 for example. Minimizing the conditional entropy has been shown to lead to the same sensing actions as maximizing the mtal information or maximizing the expected KLD between the posterior and the predictive density if the ordering of the argments in the evalation of the KLD is: KL(q(s) p(s)) where q(s) is the posterior density and p(s) is the predictive density [16. We choose to implement the maximm expected KLD becase its comptation is the least expensive, see the particle approximations in [18, [17. The KLD between two densities q(s) and p(s) is given by ( ) q(s) KL[q p = q(s) log ds (18) p(s) As sggested in [18 for example, the maximm expected KLD between the predictive and the simlated posterior density can be sed for choosing the most informative sensing action k. The sensor management criterion wold then be: k = arg max = arg max E Z [KL(q p) [KL(q p) (19) 441

4 where q = p(s k, U k 1 ) (20) p = p(s k U k 1 ) (21) The expectation over the measrement space Z is trivial and is not shown in Eq. (19) becase of the assmption that the measrement will always be an empty set, see Eq. (2a). If we set Eq. (20) eqal to Eq. (15) and sbstitte the reslt and Eq. (21) in Eq. (18) then we obtain: 1 Pd (s k, ) KL[q p = C ( ) 1 Pd (s k, ) log p(s k U k 1 ) ds k (22) C The particle approximation of Eq. (22) is given by: KL[q p 1 N and C = N { 1 Pd (s i k, ) ( 1 Pd (s i k log, ) )} Ĉ Ĉ (23) 1 N [1 P d (s k, ) p(s k U k 1 ) ds N } {1 P d (s jk, ) = Ĉ (24) j=1 where s i k p(s k U k 1 ) 2) Maximm expected probability of detection: Even thogh the se of a criterion based on the KLD is motivated by its eqivalence to the conditional entropy (for sensor management prposes), it is not easy to explain its practical meaning. For example, how cold we describe its practical interpretation when we want to motivate or criterion choice to a radar operator? For this reason, the sage of criteria that have practical operational meaning is explored. The criterion chosen from this set of criteria sggests performing the sensing action that will yield the maximm expected probability of detecting the target. The choice of this specific criterion has been motivated by the works presented in [10, [11. Given a probability density fnction q(s) that describes where the target might be and the probability of detection fnction P d (s, ) that depends on the location of the target and the sensing action, the probability of detecting the target if we perform the action is given by: ˆP D = P d (s, ) q(s) ds (25) In the considered scenario we se the predictive density p(s k U k 1 ) in order to define a criterion that selects the sensing action k that will yield the maximm probability of detecting the target: [ k = arg max P d (s k, ) p(s k U k 1 ) ds k The particle approximation of Eq. (26) is: [ k = arg max [ arg max where s i k p(s k U k 1 ) 1 N P d (s k, ) p(s k U k 1 ) ds k N P d (s i k, ) IV. PROOF OF EQUIVALENCE OF THE TWO CRITERIA (26) (27) In the simplest case scenario, where the probability of detecting the target is constant, it can be proven that the two criteria prodce the same sensor management reslts. The sensor management reslts depend on the probability of detection and the probability mass in each sector bt not on the nmber of the sensing actions or the disk size. The mathematical proof can be fond at [19 and only a graphical explanation of the proof will be provided here. In a scenario where the probability of detection is constant, the sensor wold only have to choose the direction towards where to perform search. Becase a particle filter is sed, each direction (or sector) U will contain a certain nmber of particles n sch that N U =1 n = N. Another interpretation of n is that it represents the percentage of probability mass that is located in each sector, given the fact that all the particles have eqal weights. The particle approximations of the two criteria can then be simplified by splitting the sms in two parts: a part where the probability of detection is P d (i.e. in the chosen sector) and a part where it is zero (i.e. in all the other sectors). The KLD will then be given by: KL[q p 1 N 1 P d (s j k, ( k) 1 Pd (s i k log, ) k) N j=1 Ĉ Ĉ = 1 n U ( ) N n 1 P d 1 Pd U ( ) 1 1 log + N Ĉ Ĉ Ĉ log Ĉ... j=1 j=1 = n (1 P d ) log(1 P d ) N n P d + log(n) log(n n P d ) and the sector that maximizes Eq. (28) will be chosen. Accordingly, the second criterion can be simplified as (28) 442

5 [ 1 N k arg max P d (s i N k, ) [ 1 n = arg max P d + 1 N N [ n = arg max N P d N n 0 (29) Fig. 1 shows the behavior of the maximm probability of detection based criterion as a fnction of n for varios vales of the probability of detection. It can be easily noticed that the criterion is a monotonically increasing fnction of n for any vale of P d. This means that the sector that contains the most particles, or eqivalently the most probability mass, will be chosen. This can also be inferred by Eq. (29) becase N, P d are constants (known in advance) and therefore they do not affect the sensor management reslts. Fig. 2 shows the behavior of the KL based sensor management criterion as a fnction of n for varios vales of the probability of detection. It is easy to see that it is a monotonically increasing fnction of n for any vale of P d p to a maximm point maxkl that actally depends on P d. To be more precise, maxkl is assmed for n max (N/2, N) and the exact vale of n max depends on P d. is lower than n max for every U Therefore, if n then the two criteria are eqivalent becase they are both monotonically increasing fnctions of n for any vale of P d. This can be noticed at Fig. 1 and Fig. 2. then we have to compare the vale of KL(n, P d ) to the worst case scenario vale of KL(N n, P d ) and it actally holds that On the other hand, if n is greater than n max KL(n, P d ) > KL(N n, P d ), n (n max (P d ), N) (30) Therefore, the two criteria are still eqivalent. The claim that Eq. (30) refers to the worst case scenario can be explained by the fact that N n (0, N/2) holds. Therefore, it will also hold that KL(N n, P d ) > KL(n, P d ) (31) for any nmber of particles n that satisfies N n > n becase the KL divergence is a monotonically increasing fnction for any n (0, N/2) and for any P d. The conclsion that can be drawn is that both criteria will choose the sector that contains the highest probability mass. Eqivalently, if a particle filter approximation is sed, they will both choose to search the sector with the largest nmber of particles. A. Constant P d V. SIMULATIONS The reslts of the previos section are illstrated by performing 50 Monte Carlo simlations where the sensor has Fig. 1. The behavior of the maximm probability of detection based criterion as a fnction of n for different vales of P d. Fig. 2. The behavior of the maximm KL based criterion as a fnction of n for different vales of P d. to perform search in 8 sectors with constant P d (0, 1) for k = 1,..., 160 sec. An example of sch a scenario, where a particle filter approximates the posterior density, is depicted in Fig. 3. The sensor is located at the origin of the axes and it has to choose one of the 8 sectors for performing search. Therefore, the set of sensing actions is eqal to set of sectors (8 sectors in this example): U = {1, 2,.., 8}. Obviosly, the probability of detection in the chosen sector is P d and in all the other sectors is zero. The physical interpretation of this assmption is that we cannot detect the target in sectors that we do not look at. The density is initialized at k = 0 by niformly distribting the particles in an disk of 100 km radis. The velocities v x and v y are chosen sch that the radial speed of the targets is niformly distribted in [0, 400 m/s and they move towards the radar. This initialization process resembles the real life scenario of the moment when the sensor is trned on and there is no information abot the target s location, meaning that the target might be anywhere. For the motion model, we choose b x = b y = 2 (m/s 2 ) 2 as the power spectral densities of the acceleration noise in the x y direction and T = 1 sec as the sampling time. Frthermore, target birth is modeled at the border of the field of view of the sensor in order to take into accont the 443

6 Fig. 5. The percentage of differently ranked sensing actions as a fnction of the nmber of particles sed in the simlations. The reslts are averaged over 50 MC rns and over the dration of each simlated scenario (160 sec). Fig. 3. An example of the density that describes where the ndetected target might be. The radar has to search with constant P d < 1 an area of 100 km radis divided in 8 sectors. Fig. 6. The search pattern prodced by the KL-based criterion for a scenario with constant P d. It can be noticed that there are several repetitive sbpatterns. Fig. 4. The percentage of same chosen sensing actions as a fnction of the nmber of particles sed in the simlations. The reslts are averaged over 50 MC rns and over the dration of each simlated scenario (160 sec). fact that the target might have not entered the area yet. In the simlations, the nmber of particles is varied sch that N = (5, 10,..., 100) 10 3 in order to stdy the inflence of sing limited nmber of particles. We compare the ranking of the sensing actions (in this case sectors) and the percentage of same chosen sensing actions (top ranked sensing actions) of the two criteria. The reslts are shown in Fig. 4 and Fig. 5. Fig. 4 shows that as the nmber of particles increases, the percentage of same chosen sensing actions approaches 100%. Fig. 5 shows that the percentage of differently ranked sensing actions approaches 0% as the nmber of particles increases. Therefore, the experimental reslts spport the theoretical reslt that the two sensor management criteria are eqivalent. An interesting point is that both criteria prodce search patterns that are somehow repetitive and this becomes more obvios as the nmber of particles sed in the simlations increases. Fig. 6 shows an example of a search pattern where this phenomenon can be observed. A reason for the search pattern not to be totally repetitive is the randomness indced by the particle filter itself. There is no measrement-indced ncertainty becase of the assmption that the measrements indicate that no target has been detected, see sbsection III-C. B. Taking into accont external information We now consider a scenario where the target is expected to be in the 4 northern sectors with 80% probability and in the 4 sothern with 20%. All the other parameters in the simlation are the same as the ones sed in the previos example. Fig. 7 demonstrates the adaptiveness of the KL based criterion that focses on the 4 northern sectors. The task based criterion has (bt is not shown) exactly the same behavior becase the probability of detection is assmed to be constant. On the other hand, the simple approach of periodic search wastes time and resorces in sectors where the target is not expected to be fond with high probability. This is an improvement becase the target wold be detected faster if we se the presented criteria instead of periodic search since they spend more search effort on sectors with higher probability of target existence. Fig. 7. Search time per sector when the target is expected from the north with 80% probability. 444

7 C. Nonconstant P d In the case of nonconstant P d we assme that the sensor models the behavior of a mltifnction radar. Conseqently, P d depends on the radar cross-section (RCS) of the target and on its distance from the radar. The rest of the parameters of the scenario are the same, meaning that the radar has to perform search in 8 sectors and that we employ a particle filter with the same dynamical model for the target. For each particle in the sector to be searched, first the radar eqation is sed for evalating the SNR i : SNR i (db) = 10 log(p peak ) + 10 log(t plse ) + 20 log(λ) Fig. 8. The percentage of differently ranked sensing actions as a fnction of the nmber of particles sed for simlation and RCS. The reslts are averaged over 20 MC rns and over the dration of each simlated scenario (160 sec) log(rcs i ) + G tx + G rx 10 log(k Boltzman ) 10 log(t emp) F L 10 log[r 4 i (4π) 3 (32) and then the Swerling I case is sed for evalating the corresponding P d (i) [20: P d (i) = P 1/(1+SNRi) fa (33) where: r i = x 2 i + y2 i, λ = 0.03 m, P peak = 100 kw atts, T plse = sec, G tx = G rx = 35 db, k Boltzman = , T emp = 300 Kelvin, F L = 1.1 db losses, probability of false alarms P fa = and i = 1, 2,..., N. Then Eq. (19), (23) and (24) are sed for the KL based criterion and Eq. (27) for the maximm probability of detection criterion. In the experiment, the nmber of particles is varied sch that N = (5, 10,..., 100) 10 3 and the target s RCS is varied sch that RCS = [ m 2. We compare the ranking of the sensing actions (again: sectors) and the percentage of same chosen sensing actions (top ranked sectors) of the two criteria. The reslts are shown in Fig. 8 to 13. It can be noticed that as the nmber of particles and the RCS increase,the percentage of different rankings approaches 0% and the percentage of same chosen sensing actions approaches 100%. These reslts indicate that the two criteria are eqivalent for high RCS targets in this more involved scenario. Frthermore, the existence of repetitive search sb-patterns was noticed again. VI. CONCLUSIONS In the previos sections, two fndamentally different sensor management criteria for performing search for a target have been presented and actally shown to be eqivalent. This reslt has two interesting and important implications. The first implication is that a practical interpretation of an information-driven criterion, i.e. maximizing the KLD between the predictive and the posterior density, can be given in the search context. In other words, performing the search action that maximizes the KLD is the same as performing the search Fig. 9. X-view of Fig. 8. Fig. 10. Y -view of Fig. 8. Fig. 11. The percentage of same chosen sensing actions as a fnction of the nmber of particles sed for simlation and RCS. The reslts are averaged over 20 MC rns and over the dration of each simlated scenario (160 sec). 445

8 ACKNOWLEDGMENTS The research leading to these reslts has received fnding from the EU s Seventh Framework Program nder grant agreement n o The research has been carried ot in the MC IMPULSE project: The athors wold also like to acknowledge Edson Hiroshi Aoki (Univ. of Twente) for his insightfl comments. Fig. 12. X-view of Fig. 11. Fig. 13. Y -view of Fig. 11. action that will yield the maximm expected probability of detecting the target. The second implication is that the criterion which is based on the highest probability of detection not only has practical meaning bt it is also comptationally less expensive to implement, see Eq. (23) and (27). In fact, Eq. (29) means that the implementation of the criterion boils down to jst performing a particle cont for determining n, since N, P d are constant and known in advance. From the previos two paragraphs one can conclde that even when it is possible to explain what it means practically to maximize the KLD, its se is not always jstified. For instance, in the presented examples the KLD has a higher comptational complexity than the intitive task-based criterion. Some topics that we wold like to explore in the ftre are: We wold like to compare or approach to other approaches, sch as the one presented in [13, in terms of both search reslts and comptational efficiency. Another interesting topic is to explore the behavior of the described criteria in mltitarget scenaria where external information is also available. Becase the compared criteria appear to give different reslts for low RCS targets, we can see that the criteria are not in general eqivalent when the probability of detection varies and this phenomenon worths frther attention. REFERENCES [1 J. M. Danskin, A helicopter verss sbmarine search game, Operations Research, vol. 16, no. 3, pp , [2 B. O. Koopman, The Theory of Search. III. The Optimm Distribtion of Searching Effort, Operations Research, vol. 5, no. 5, pp , Oct [Online. Available: [3 T. M. Kratzke and J. R. Frost, Search and resce optimal planning system, in Proceedings of the 13th International Conference on Information Fsion, vol. 1, 2010, pp [4 L. Stone, C. Keller, T. Kratzke, and J. Strmpfer, Search analysis for the nderwater wreckage of Air France Flight 447, in Proceedings of the 14th International Conference on Information Fsion, vol. 1, 2011, pp [5 I. Szki and M. Yamashita, Searching for a mobile intrder in a polygonal region, SIAM Jornal on Compting, vol. 21, no. 5, pp , [6 B. P. Gerkey, S. Thrn, and G. Gordon, Visibility-based prsitevasion with limited field of view, in International Jornal of Robotics Research, 2004, pp [7 F. Bolderheij and P. Van Genderen, Mission driven sensor management, in Proceedings of the 7th International Conference on Information Fsion, [8 W. Koch, On adaptive parameter control for phased-array tracking, in Proceedings of Signal and Data Processing of Small Targets, [9, On exploiting negative sensor evidence for target tracking and sensor data fsion, Information Fsion, vol. 8, no. 1, pp , [10 K. White, J. Williams, and P. Hoffensetz, Radar sensor management for detection and tracking, in Proceedings of the 11th International Conference on Information Fsion, 2008, pp [11 Y. Boers, H. Driessen, and L. Schipper, Particle filter based sensor selection in binary sensor networks, in Proceedings of the 11th International Conference on Information Fsion, 2008, pp [12 D. Matthiesen, Efficient beam scanning, energy allocation, and time allocation for search and detection, in IEEE International Symposim on Phased Array Systems and Technology (ARRAY), oct. 2010, pp [13 A. Charlish, K. Woodbridge, and H. Griffiths, Agent based mltifnction radar srveillance control, in Proceedings of the IEEE Radar Conference (RADAR), vol. 1, 2011, pp [14 B. Ristic, S. Arlampalam, and N. Gordon, Beyond the Kalman filter. Artech Hose, [15 X.-L. H, T. Schon, and L. Ljng, A basic convergence reslt for particle filtering, IEEE Transactions on Signal Processing, vol. 56, no. 4, pp , april [16 E. Aoki, A. Bagchi, P. Mandal, and Y. Boers, A theoretical look at information-driven sensor management criteria, in Proceedings of the 14th International Conference on Information Fsion, vol. 1, 2011, pp [17 Y. Boers, H. Driessen, A. Bagchi, and P. Mandal, Particle filter based entropy, in Proceedings of the 13th International Conference on Information Fsion, [18 A. Docet, B. Vo, C. Andrie, and M. Davy, Particle filtering for mlti-target tracking and sensor management, in Proceedings of the 5th International Conference on Information Fsion, 2002, pp [19 F. Katsilieris and Y. Boers, Optimal search: a practical interpretation of information-driven sensor management, Department of Applied Mathematics, University of Twente, Enschede, Memorandm 1979, March [20 M. I. Skolnik, Introdction to Radar Systems, 3rd ed. McGraw-Hill Science/Engineering/Math,