Dynamic optimization algorithms for radar resource …cradpdf.drdc-rddc.gc.ca/PDFS/unc106/p534470_A1b.pdf · Dynamic optimization algorithms for radar resource management ... Contract

Dynamic optimization algorithms for radar resource management Alain Gosselin and Yongkui Wang

The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada.

Defence R&D Canada – Ottawa

Contract Report DRDC Ottawa CR 2010-232

December 2010

Alain Gosselin Yongkui Wang

Prepared By: Alain Gosselin Department of Mathematics and Computer Science Royal Military College Kingston, Ontario, K7K 7B4

Contract Number: A1410003FE Contract Project Manager: Peter Moo, Defence Scientist, 613-998-2879 CSA: Zhen Ding, Defence Scientist, 613-990-7553

The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada.

Contract Report DRDC Ottawa CR 2010-232 December 2010

Scientific Authority

Original signed by Zhen Ding

Zhen Ding

Defence Scientist

Approved by

Original signed by Doreen Dyck

Doreen Dyck

Defence Scientist, Head / Radar Systems Section

Approved for release by

Original signed by Chris McMillan

Chris McMillan

Chief Scientist, Head/Document Review Panel

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2010

DRDC Ottawa CR 2010-232 i

Typically, the scheduling of multi-function radars is based on the optimization of some pre-defined aggregate function made of all attributes of interest. By selecting a specific aggregate function these methods force the generation of a specific solution without the full awareness of the possible solution space. By introducing methods of Multicriteria Optimization [2] the proposed scheduling algorithms allow for the determination of the set of all non-dominated solutions with its associated efficient set in decision space: the space of all possible schedules. The determination of efficient sets will generate a better understanding of the solution space and could lead to a better selection of the particular schedule to be implemented in addition to allowing for the development of more efficient scheduling algorithms. This report presents a simple two-criterion scheduling model based on Multicriteria Optimization.

Typiquement, l’ordonnancement de radars multifonctions est basé sur l’optimisation d’une fonction d’agrégat prédéfinie de tous les attributs d’intérêt. En choisissant une fonction d’agrégat précise, ces méthodes forcent la génération d’une solution précise sans connaître complètement l’espace de solutions possibles. En ajoutant des méthodes d’optimisation multicritère [2], les algorithmes d’ordonnancement proposés permettent de déterminer l’ensemble de toutes les solutions non dominées et l’ensemble efficient connexe de l’espace de décisions : l’espace de tous les ordonnancements possibles. La détermination d’ensembles efficients permettra une meilleure compréhension de l’espace de solutions et pourrait donner lieu à un choix plus juste de l’ordonnancement particulier à mettre en œuvre, en plus de permettre le développement d’algorithmes d’ordonnancement plus efficients. Le présent rapport décrit un modèle d’ordonnancement simple bicritère basé sur l’optimisation multicritère.

ii DRDC Ottawa CR 2010-232

This page intentionally left blank.

DRDC Ottawa CR 2010-232 iii

The purpose of this task is to explore the implementation of new scheduling algorithms for multi-function radar (MFR). Given that new MFR integrate the functions that were typically carried out by different systems and that the complexity of the functions to be implemented are such that existing resources are insufficient to satisfy the requirements of all such functions, decisions have to be made as to which tasks of each function are to be executed in order to optimize utilization. To better understand the problem to be solved, the optimal schedule is searched in multi dimensional space in order to reflect the underlying reality of the problem. This approach redefines the problem as a multi-criteria optimization problem (MCOP).

In this first exploration of the solution of the multi-criteria optimization problem, only two criteria are utilized: the value of Tracking performed and the value of the Detection performed. The functions associated with each are called the Tracking Benefit function and the Detection Benefit function respectively. This yields a two dimensional MCOP whose efficient set (the set of points whose values are not dominated by any other) is connected. As a consequence, once a member of the set has been found, all others can be identified using local search ideas. The pair of Benefit functions can be considered as a vector valued functions from the space of all possible schedules (the decision space) to the Real plane (the criterion space). In this case, the trivial efficient schedule consisting of performing only tracking is easily determined and serves as the starting point to uncover the efficient set.

Using the Benefit functions defined for Tracking and Detection, an algorithm using local variations was implemented in Matlab to generate the efficient set for a given list of Tracking and Detection tasks. The resulting set of non-dominated points is plotted on a two-dimensional graph to illustrate the possible efficient schedules.

Typical scheduling algorithms for Radar Resource Management use a preselected aggregate function to assess the value of a particular schedule. By doing so, many possible solutions to the problem are ignored. One of the benefits of treating the Radar Resource Management problem as a MCOP is that the efficient set includes the results of every aggregate function at once. This provides for a better understanding of the problem at hand allowing for a better informed decision applied to the scheduling problem. In some cases, the understanding gained through the MCOP can yield to more efficient scheduling algorithms by allowing for the selection of a subset of efficient points easier to obtain.

The work carried out to this point was mostly an introduction of multi-criteria optimization to the problem of Radar Resource Management. Only two simple criteria were selected out of a multitude of actual parameters. Future research should explore the problem in additional dimension and with various Benefit functions associated to each of these criteria.

iv DRDC Ottawa CR 2010-232

Furthermore, the current MCOP solves for the set of efficient point at a given point in time and additional work is required to consider the dynamical case over multiple consecutive scheduling intervals. These additions will no doubt generate a criterion space with weaker properties and more advanced solutions methods will need to be implemented.

DRDC Ottawa CR 2010-232 v

La présente tâche vise à étudier la mise en œuvre de nouveaux algorithmes d’ordonnancement pour les radars multifonction (MFR). Étant donné que les nouveaux MFR incorporent les fonctions typiquement effectuées par plusieurs systèmes, et que la complexité des fonctions à mettre en œuvre est telle qu’il est impossible de satisfaire aux exigences de toutes ces fonctions au moyen des ressources actuelles, des décisions doivent être prises en ce qui a trait aux tâches de chaque fonction qui doivent être effectuées en vue d’optimiser l’utilisation. Afin de mieux comprendre le problème, l’ordonnancement optimal est recherché dans l’espace multidimensionnel pour refléter la véritable nature du problème. Cette méthode redéfinit le problème en tant que problème d’optimisation multicritère (MCOP).

Dans ces premières recherches sur la solution au problème d’optimisation multicritère, seulement deux critères sont utilisés : la valeur de la poursuite effectuée et la valeur de la détection effectuée. Les fonctions associées à chacun de ces critères portent le nom de fonction d’avantage de poursuite et fonction d’avantage de détection, respectivement. Cela donne lieu à un MCOP à deux dimensions auquel est associé l’ensemble efficient (l’ensemble de points dont les valeurs ne sont dominées par aucune autre). Par conséquent, lorsqu’un membre de l’ensemble a été trouvé, tous les autres peuvent être déterminés au moyen d’idées de recherche locales. La paire de fonctions d’avantage peut être considérée comme une fonction à valeurs vectorielles de l’espace de tous les ordonnancements possibles (l’espace de décisions) du plan des réels (l’espace de critères). Dans ce cas, l’ordonnancement efficient évident, qui consiste à effectuer la poursuite uniquement, est facilement trouvé, et il sert de point de départ à la détermination de l’ensemble efficient.

À l’aide des fonctions d’avantage définies pour la poursuite et la détection, un algorithme utilisant des variations locales a été mis en œuvre dans MATLAB pour générer l’ensemble efficient d’une liste de tâches de poursuite et de détection. L’ensemble de points non dominés ainsi obtenu est porté sur un graphique à deux dimensions en vue de montrer les ordonnancements efficients possibles.

Les algorithmes typiques de gestion des ressources radar utilisent une fonction d’agrégat présélectionnée pour évaluer la valeur d’un ordonnancement précis. Beaucoup de solutions possibles au problème sont ainsi laissées de côté. L’un des avantages à traiter le problème de gestion des ressources radar en tant que MCOP est que l’ensemble efficient englobe les résultats de toutes les fonctions d’agrégat à la fois. Cela permet une meilleure compréhension du problème, ce qui donne lieu à une décision plus éclairée relativement au problème d’ordonnancement. Dans certains cas, la compréhension obtenue à partir du MCOP peut donner lieu à des algorithmes d’ordonnancement plus efficients en permettant le choix d’un sous-ensemble de points efficients plus faciles à obtenir.

vi DRDC Ottawa CR 2010-232

Les travaux effectués jusqu’à présent consistaient surtout en une introduction de l’optimisation multicritère pour le problème de gestion des ressources radar. Deux critères simples uniquement ont été choisis parmi de nombreux paramètres actuels. Les recherches futures devraient viser l’étude du problème dans des dimensions supplémentaires, et avec diverses fonctions d’avantage en lien avec chacun de ces critères. De plus, la méthode MCOP actuelle permet de déterminer l’ensemble de points efficients à un moment précis dans le temps, et des recherches supplémentaires sont nécessaires pour considérer le cas dynamique à des intervalles d’ordonnancement consécutifs multiples. Ces recherches vont sans doute générer un espace de critères avec des propriétés plus faibles, et des méthodes de solutions plus approfondies devront être mises en œuvre.

DRDC Ottawa CR 2010-232 vii

Abstract …….. ................................................................................................................................. i Résumé …..... ................................................................................................................................... i Executive summary ........................................................................................................................ iii Sommaire ........................................................................................................................................ v Table of contents ........................................................................................................................... vii List of figures ............................................................................................................................... viii 1 Introduction ................................................................................................................................. 1 2 Radar system functions and tasks................................................................................................ 2 3 Resource and source depletion .................................................................................................... 3

3.1 Resource definition at time k ............................................................................................. 3 3.2 Source depletion definition................................................................................................. 3

4 Task benefit ................................................................................................................................. 5 4.1 Task effectiveness ratio definition ...................................................................................... 5 4.2 Task weighting definition ................................................................................................... 5 4.3 Task benefit definition ........................................................................................................ 6

5 Mathematical model for multicriteria optimization..................................................................... 7 6 Implementation............................................................................................................................ 9

6.1 Task benefit for the detection function ............................................................................... 9 6.2 Task benefit for the tracking function............................................................................... 13 6.3 MODS............................................................................................................................... 26

6.3.1 )(1

kBF ................................................................................................................. 26

6.3.2 )(2

kBF ................................................................................................................. 27 6.3.3 Source depletion and k-time interval.................................................................. 27 6.3.4 MODS formulation............................................................................................. 28 6.3.5 Simulation design ............................................................................................... 29 6.3.6 Algorithm ........................................................................................................... 30

7 Conclusion................................................................................................................................. 32 References ..... ............................................................................................................................... 33 Annex A Scheduling example..................................................................................................... 35 Annex B Matlab source code ...................................................................................................... 36 List of symbols/abbreviations/acronyms/initialisms ..................................................................... 42

viii DRDC Ottawa CR 2010-232

Figure 1 Discrete beam space.......................................................................................................... 9 Figure 2 Radar beam ..................................................................................................................... 11 Figure 3 Surface intercept ............................................................................................................. 14 Figure 4 Distances to surface interception .................................................................................... 12 Figure 5 Distance cones................................................................................................................. 11 Figure 6 Distance approximation .................................................................................................. 11 Figure 7 Source depletion line....................................................................................................... 27 Figure 8 Non-dominated set example ........................................................................................... 34

DRDC Ottawa CR 2010-232 1

This report is the third deliverable in accordance with the Statement of Work for Task #1 of Dynamic Optimization Algorithms for Radar Resource Management (RRM) as agreed between RMC and DRDC Ottawa [1]. The purpose of this task is to explore the implementation of new scheduling algorithms for multi-function radar (MFR). Given that new MFR integrate the functions that were typically carried out by different systems and that the complexity of the functions to be implemented are such that existing resources are insufficient to satisfy the requirements of all such functions, decisions have to be made as to which tasks of each function are to be executed in order to optimize utilization. To further complicate the problem, the “optimum” to be achieved cannot be perfectly expressed by a scalar function. The performance of algorithm must therefore be assessed in multi dimensional space in order to reflect the underlying reality. In this case, each evaluation criteria will be considered to be a real valued function and thus, the problem can be defined as a multi-criteria optimization problem (MCOP) in n . For the purpose of this work, the order in n will be that determined by the cone nkxxxK k

nk ,,1,0|( , which makes n the

product of n copies of the real line with its usual order. The use of other positive cones will not be considered at this time but could be explored further in later work. As this report is meant to be comprehensive as to the progress achieved on the MCOP to this point, the underlying mathematical model develop in the beginning of this Task is presented with some minor refinements. For this implementation of the MCOP, it has been decided that, at a first step, only two simplified radar functions will be implemented: Tracking and Detection. The scheduling problem is one of selecting an “optimal set of tasks”. To assess the value associated with completing a task, Benefit functions are defined. In this case, two such functions were defined: Tacking Benefit and Detection Benefit functions. The MCOP obtain is well behaved in the sense that the set of non-dominated points is connected. As such, the problem is solved by using the trivial “tracking only” schedule as a starting point and disturbing the decision space slightly locally to find another point of the efficient set. This process is simply reapplied recursively until all points have been identified. The algorithm used for the solution of the MCOP has been implemented in Matlab. To illustrate the behavior of the solution, the set of non-dominated points for an example is provided in the annexes.

2 DRDC Ottawa CR 2010-232

A radar system function ( FunctionRadar _ ) consists of a finite number of functions. Every function is composed of numbered set of tasks which are of the same kind. A radar system is a string of time intervals during which one task is executed at a time. During one time sub interval, all tasks of all existing functions are considered for execution. The continuing work of the radar on the time axis can be divided as ,,,,, 110 kk ttttTime , (1) where ,2,1,1 ktt kk . kk tt ,1 is k time interval during which the radar schedules selected tasks from all existing functions. Denote 1kkk ttt as the length of the k time interval; note that kt can depend on tasks and functions being considered for scheduling. Consider a radar system function made up of N functions, NfunctionfunctionFunctionRadar ,,_ 1 . (2) In k time interval kk tt ,1 , it is )()()(

1)()( )(,,)1( k

Nkkkk functionNufunctionuFunctionRadar (3)

where

Niotherwise

existsfunctioninumberifiu k ,,1,

01

)()(

is the existence function,

Ni

task

tasktask

function

kMi

ki

ki

ki

ki

,,1,

)(,

)(2,

)(1,

)(

)(

,

intervaltimeinfunction intasksofsumthe)( kiM ki .


Given that every target appears continuously and that a radar system as defined above works over discrete time intervals, to ensure the discrete information obtained by the radar is related closelyto the continuous target, the time interval length should be limited. Clearly, the time interval is the resource of the radar to be managed and the time cost for every task in FunctionRadar _ is the source depletion. Since the characteristics of each appearing tasks are different, the resource used varies depending on these traits and, thus, source depletion depends the tasks to be scheduled.

Let ,2,1,0,, 11 kttttRs kkkkk be k time interval of a radar system service, we will call the time interval ,2,1,0,, 11 kttttRs kkkkk (4) the k resource of the radar and 1kkkk tttRs , ,2,1,1 ktt kk (5) the value (length) of k resource. It is clear that the value of a k resource depends on the current status: how many tasks are in existence, what these tasks are and what kinds of service providing is required for these tasks. Consequently, it is conceivable that the k resource varies with the status.

The Source Depletion function can be defined as a compound function. First, the classifying function ),(: )(

, litaskf kmi on which the time consumption function is applied:

Rlit ),(: , i.e., Rtaskft k

mi ))(( )(, (6)

Is the source depletion, where )(

,kjitask number j task in ifunction issue in k resource,

),( li the class l in ifunction ,

ik

i LlandMmNi ,,2,1,,1,,,1 )( , N the number of functions in FunctionRadar _ , iL the total number of classes in ifunction .


For convenience, we define 0)(

0,k

itask , )0,0()0(f and 0)0,0(t . (7) Note that the time consumption function is a function of ifunction and class l , and it is most likely given and fixed for given li and .


It follows from section 2 that we have to build a service standard for a radar system for task’s random appearance such that the resource requirement will be determined for given tasks. It is clear that the characteristics of each task determine the service standard of the radar system. We recommend two concepts to represent the features of every task: the task effectiveness ratio and task weighting.

Let FunctionRadarfunctiontask usu _, be task number s in ufunction and assume a

radar system works on sutask , . Let evtask , be the task transferred from task sutask , in a

FunctionRadarfunctionv _ (with possibly vu ) for the radar service. The probability of the radar system working on evtask , in next service is called the effectiveness ratio of sutask , .

Denote )( ,sutaskeff as the effectiveness ratio of sutask , . By the definition, we have

The effectiveness ratio of a task is a function of the status of the task. The maximal effective ratio is 1 and the minimal benefit is 0.

As per avove, define 0)0(eff . It follows from the facts that if the radar system spends a time on sutask , less than the source depletion of sutask , , then task effective ratio 0)( ,sutaskeff and

that if the radar system spends unlimited time to sutask , then also 0)( ,sutaskeff because there will be no subsequent service. It means that

0)0()(lim

0)0()(lim

,))((

,0))((

,

,

efftaskeff

efftaskeff

sutaskft

sutaskft

su

su hold.

Let FunctionRadarfunctiontask usu _, be a task, then the task weighting function is the

compound of a classifying function ),(: , qutaskg su and a weighting function Rquw ),(: , i.e., Rtaskgw su ))(( , (9) where sutask , number m task in ifunction ,


in space )()(

2)(

1 )()1()1( )()()( kN

kk MkMkMk RNuRuRu . We know that the number of the vertexes

in )(

)()( kiM

ik Viu is

)()(

2)()( )(

0

)()( k

i

ki

MkM

h

kik iuh

Miu . By (12), the objective space is NR . It is

clear that the objective function vector Nk

Nkkk RBFBFBFBF :,, )()(

2)(

1)( . (14)

The solution of MODS (12) is the determination of the set that is Pareto optimal, E . For x Eˆ , no other x such that )ˆ()( )()( xBFxBF kk , i.e.,

NixBFxBF ki

ki ,,2,1)ˆ()( )()( .


Consider a radar system with },{_ 21 functionfunctionFunctionRadar , where etectionfunction d1 : confirm the target to obtain the position of the target, the

value and the direction of the velocity along the beam of the radar system, and trackingfunction2 : detail the position and direction of the target. The radar system spends constant time 0t at searching target in every k resource. The beam service space of the radar is eses,b ,,,,,,,:)( 11 , shown in Figure 1.

Figure 1 Discrete beam space

The effective service space is eee ,,:)( RrRr,,p sss . denote

as the surface of .In this example, 150,30,20,0 ee ss . It means that the radar system can obtain the rough position ),,( TTT rpT during its routine searching program if a target is in , the beam is at ),( TTb . Note that the position

),,( TTT rpT is not the exact the position of the target but is a value in the discrete beam space.

Detection function of a task means that a radar system spends time to confirm the position and velocity (along the direction of the beam) of an issued target during its searching program.


Let a target be in detected by the radar system by the beam ),( TTb at position

),,( TTT rpT during its routine searching program. Assume the time when the radar system has found a target is 0,, 11 kkkkk ttttRst . Beam The moving target will move to position ),,( rp at next service for the target. By definition 3.1, we build the mathematical model of detection effective ratio to issue the probability of next the service for the target. Let kv be a value of a statistic average velocity of target:

1

1111

2

1

k

H

j

jkkk

k H

vHvv

k

, (15)

where j

kv 1 the velocity of target new issues in 1kRs serviced by the radar system, 1kH the number of targets new issue in 1kRs serviced by the radar system. The target will be in sphere with centre ),,( TTTTr and radius 00 kk Rsvr , i.e.,

00 ),,,(),,( rrrp TTTT in next service. For Sphere 0 and effective service space there are two cases: i) 0 and ii) 0 . In case i), we define the detection ratio as 1

because the target still in the effective service space in next service domain (meaning that the radar system has 100% chance to service for the target in next service). By geometry, sphere

0 delimits some sections on surface , such that if the target moves towards these sections, the target will escape space , shown in Figure 3.

Figure 2 Radar beam


00 ),,,( rr TTTT d

Figure 3 Surface intercept

The radar system cannot service the task in the partial spheroid where the target crosses the surface of 0 , called 6,,2,1, ii . We define the detection effective ratio as

304

3

)( 0

0

0

R

dVdV

dV

dVdVtaskefd . (16)

where 6

1ii .

In actual applications, it is not practical to calculate dV in the general cases in real-time status.

To obtain an approximate real-time solution of dV , we use the following simplifications.

The distances from the centre of sphere 0 , ),,( TTT rpT , to six surfaces of the effective space are approximated by

2

sin2 0TT1,1 rd ,

2sin2 Te

T2,1 rd ,

2

sin2 0TT1,2 rd ,

2sin2 Te

T2,2 rd , (17)

0T1,3 Rrd , Te2,3 rRd . They are shown in Figure 4.


Assume that 6,,2,1,,, jijiji is rarely empty. By (16), the detective effective ratio can approximately calculated as

304

31)(

0

0

r

dV

dV

dVdVtaskefd (18)

where

otherwise

rVr

VdV

30

30

223

2,

2

1

3

1,

j ijiVV

2,1;3,2,1,0

))(( 0,2,

20,0

, jiotherwise

rddrdrV jijiji

ji ,

2,1,3,2,1,, jid ji shown in (17).

For the detections, in the case of the example herein, a random value between 0 and 1 had been assigned to )(taskwd . By formula (10), the benefit of a detection task is )()())(()( taskefdtaskefdtaskgwdtaskbfd . (19)

Figure 4 Distances to surface interception


is at position ),,( TTT rpT and its velocity is v . We would like to find the d , that is

the directional distance of the target to surface of . Let 0,vd , should be found in the next step.


Let the target be at position ),,( TTT rpT with velocity v and

vopor T, there exists 0 such that

6

1iiVoe , where

ee1 ,:),,( RrRrorV sss ,

ee2 ,:),,( RrRrorV sse ,

ee3 ,:),,( RrRrorV sss ,

ee4 ,:),,( RrRrorV sse ,

ee5 ,:),,( RorV sss ,

ee6 ,:),,( RorV sse .

The proof is direct.

2 Let increase from 0, if ) [0,for ),() ( then0 if and },{) (such that 0 1e1e11 RRrRRr ss , ) [0,for ),() ( then0 if and },{) (such that 0 2e2e22 ss , ) [0,for ),() ( then0 if and },{) (such that 0 3e3e33 ss ,

then },,min{, 321vd is the directional distance of the target to surface of .

Proof Let },,min{0 321 and without loss of generalization, let 1 then 0, 32 . By assumptions, },{) ( e1 RRr s , ),() ( e1 s and ),() ( e1 s such that the head of

vopor Tis on the surface of . Obviously, the conclusion holds for the cases of

0and00, 321 .


321 and,

Write the position of the target as

kzjyixopTTTT

kjivv x zy vv

where TT coscosTT rx , TT sincosTT ry , TsinTT rz . Then

krjriror sinsincoscoscos

vopT

kvrjvrivr zTyTxT )sin()sincos()coscos( TTTTT , i.e.,

sinsin

sincossincoscoscoscoscos

T

TT

TT

rvrrvrrvr

zT

yT

xT

. (21)

(1) 1 It follows from (21) that

cbar )( 2 , (22) where 2222

zyx vvvva ,

)sinsincoscos(cos2 TTTTT zyxT vvvrb ,

2Trc .

Since ],R[)( s eRr , i.e., 0)(r for )0,[ . By formula (22), if 0b , 04 2bac

holds. It is simple to show that if 0b , 0)(r for ),-( holds, since 04 2bac .


We always assume that 0v , i.e., 0a . To obtain the domain of , from the derivative of (22), we have

)(

221)(

rbar . (23)

i) 0b 0)(r , )(r is monotone increasing with respect to . Since cr )0( and eRr )( , the target will reach eR . By (22), we obtain, 022

eRcba . (24) The solution of equation (24) for 0 is

a

cRabb e

2)(4 22

1,1 . (25)

ii) 0b

)(r is monotone decreasing in domain ]2

[0,a

b and monotone increasing in domain

),2

[a

b. The minimal value is 2

2

4)

2( Tra

bca

br . By the definitions of

cba and, in (22), for any 0 , with 042 abc .

a) If 22

4)

2( sR

abc

abr

By the restrictive condition sRr )( , solve equation 022

sRcba and follow Theorem 2 to obtain

a

ba

Rcabb s

22)(4 22

1,2 . (26)

b) 22

4)

2( sR

abc

abr


By condition eRr )( and a

b2

, solve 022eRcba to get

a

cRabb e

2)(4 22

1,3 . (27)

It follows from i) and ii) that the solution of 1 should be

abRcbabRcb

b

s

s

4 and04 and0

0

22

22

1,3

1,2

1,1

1 . (28)

(2) 2 By formulas (21) and (22),

cba

vrr

vr zTzT sin)(

sinsin2

TT . (29)

From formula (29), there exists a root named z

T

vr T

1,2sin

for 0zv . If 0zv , there exits

two roots in , i.e., 0)(lim . We are not interested in case 0zv . For 0zv , replace

TsinTr by zv2,1- , derive (29) with respect to ,

)()(

b)2)(-(21

)(ddcos 2

2,1

rrav

rv zz

)()(2

b)2)(-()(22

2,12

rravrv zz

)()(2

)2()2(2

2,12,1

rrbcvabv zz

i.e.,

)cos()(2

)2()2(dd

22,12,1

rrbcvabv zz , (30)

where 20and 0],,[ eses . Further consider


)2()2()( 2,12,1 bcvabvG zz It follows from )2()( 2,1abvG z that the following results hold. i) 02 2,1ab Immediately,

0)2(0)2(

increasing monotonecreasingde monotone

-)(2,1

2,1

bcvbcv

z

z .

By 02 2,1ab and 04 2bac ,

0)4(22 22,1 bacabc .

Hence

00

increasing monotonecreasingde monotone

-)(z

z

vv

.

Since 0sin0sin T ,

a

v

cba

vr zzT sinlimsinlim2

T , (32)

we have

00

sin

sin

-)(sinincreasing monotoneT

creasingde monotoneT

z

z

zT

zT

vv

av

cr

av

cr

. (33)

It is clear that if 0zv there exist ),0( such that 0sin)(sin s a

vzesin then

there exists ),0( such that esin)(sin for 0zv .


Let 02 2,1ab and 0zv , define 2,1

2,1

2,1

2,10 2

2)2()2(

abbc

abvbcv

z

z . The relationship

of 0 , 2,1 and )0(2 2,1ab is obtained by the following analysis. It follows from

2,1

2,122,1

2,10 2)(2

abcba

,

and 02,1

22,1 cba , that

2,10

2,10

2,1

2,1

2,1

2,1

2,10

2,10

0202

0202

abab

abab

(34)

ii) 0)2( 2,1 zvab , i.e., 2,102,10

00 zz vor

v

)2()2()( 2,12,1 bcvabvG zz

= )-)(2()2()2(

)2( 02,12,1

2,12,1 abv

abvbcv

abv zz

zz .

It means that

,,,

increasingmonotoneminimum

decreasingmonotone-)(

0

0

0

iii) 0)2( 2,1 zvab , i.e., 2,102,10

00 zz vor

v

)-)(2()( 02,1abvG z such that

.,,

sinmonotonemaximum

increasingmonotone-)(

0

0

0

gdecrea


Cases ii) and iii) can be written as

a) 0zv ( 02,1 )

.,



2,10

2,10

02,10 and

(see Figure (1), there exists a solution 0)(0 2,102,1 thatsuch )

02,10

2,10

2,10

and,

decreasingmonotonemaximum


(see Figure (3), exists solution 0)( 2,102,1 thatsuch and if

existstherethenand e00 sin)(sin0 ,sin)(sin e

2,10where )

b) 0zv ( 02,1 )

.,



02,10

2,10

2,10

and

(if ee sin)(sin0sin thatsuchexiststherethena

vz see Figure

(2))

.decreasingmonotonemaximum


2,10

2,10

02,10 and

( if existstherethena

vand z )(sinsinor)(sinsin0 0e0e0

esin)(sin0 thatsuch see Figure (4))

0s . A target spends time, i.e., 0t , to reaches 0s if and only if 0zv .


Proof If a target reaches 0s , it means that there exists 01,2 such that

0)(

sinsin

2,1

2,1T2,1 r

vr zT .

Since 0)( 2,1r , 0sin 2,1T zT vr . By 0sin TTr and 01,2 , 0zv .

Inversely, if 0zv by (32) and (33), 0sinlima

vz . If 0zv , solve equation (29) to

obtain the root 0sin- T2,1

z

T

vr

.

02e . A target spends time, i.e., 0t , to reaches 02e , if and

only if the following cases hold.

0Zv )sin()(sin

0

e0

02,1

or

0Zv a

vZ)sin(

0

e

0

)sin()(sin

0

e0

0 a

b22,1

where 2,1

2,10 2

2ab

bc and

z

T

vr T

1,2sin

.

In other words

)sin()(sin

0

e0

0 or a

vvand

Z

Z

)sin(

00

e

0

or a

bvZ

2

0

2,1.

(35)


If a target spends time to reach 02e , i.e., it satisfies formula (35), then there exists

02,2 such that)(

sinsin

2,2

2,2Te r

vr zT and 2,2 , the positive solution of equation

0BA 2 C , should be

0B0,C and 0

040,00,

,24 22

2,2 AACBandBAorA

CBAACBB (36)

where 2

e2sinA zva ,

Te2 sin2sinB zT vrb ,

) because(0)sin-(sinC TeT2

e2 c .

(3) 3

By formula (21), xTTTyTTT vrvrc coscos)sincos(tan ,

TTTTTTxy crrvcv sincostancoscos)tan( . For 0tan xy vcv ,

xy

TTTTTT

vvccrr

tansincostancoscos

(35)

or

vv

r

xy

TTT

sincos)sin(cos

(36)

The conditions of 0are

0sincos

0)sin(vv xy

T (37)


or

0sincos

0)sin(vv xy

T (38)

Let s and e respectively, we have

sxsy

TsTT

vvr

sincos)sin(cos

1,3 , (39)

exey

TeTT

vvr

sincos)sin(cos

2,3 . (40)

It follows from (35) that

yTTT

xTTT

vrvrc

sincoscoscos

tan

22

)sincos()coscos()sincos()())(tan1(

yTTT

yxTTTxyTTT

vrvvrvvr

ddc

22

)sincos()sincos(cos)())(tan1(

yTTT

TxTyTT

vrvvr

ddc

If TxTy vv sincos then )( is monotone increasing, if TxTy vv sincos then )(

is monotone decreasing and if xTyT vv sincos then T)( that same the case

0tan xy vcv . Combine (37) and (38), obtain if TxTy vv sincos , then eT , i.e, 0)sin( T , thus

0sincossincos

)sin(cos3 exey

exey

TeTT vvifvv

r, (41)

and if TxTy vv sincos , then sT , i.e, 0)sin( T , thus

0sincossincos

)sin(cos3 sxsy

sxsy

TsTT vvifvv

r (42)


)(2

kBF

Let ),,,(,),,,,(),,,,( 2222211111 nnnnn vrtravrtravrtraTra be the target set

enrolled in the k time interval, where )(2

kMn is the number of targets enrolled in k time interval, ),,( iii r is the position of target i and iv is velocity of target i . Position and velocity both are obtained in the 1k time interval.

00

0

)()()()(1

21

2)(2 n

ntaskeftitaskbftiBFn

ii

ki

n

i

kk , (46)

where nitaskeft i ,,2,1),( shown in formula (20).

Let time consumption or source depletion for every detection and tracking be 1t and

2t respectively, and 1k and 2k be the selected detection targets and tracking targets, then the length of k resource is 22110 tktktRsk . (47) Let },,,{ 21 mddd and },,,{ 21 n be a distance set and a time set obtained from detection

and tracking in section 5.1 and 5.2, where targetfor )min( , i dd kji , 2,1;3,2,1 kj .

Order them as },,,{ 21 mddd and },,,{ 21 n , where mddd 21

and n21 . The following restriction conditions are necessary:

mittvdi ,,1,10

0

,

nitti ,,1,20 ,

1022110

,,1, kittktkvdi ,

202211 ,,1, kittktki . Write them as

0221110

1 ,min ttktkvdRsk , (48)


where 02110

1 ,min tttvd

.

2k

2

0

ttRsk h

kRsttktkL 02211:

1

0

ttRsk

f g 1k

Figure 7 Source depletion line

By inequality (24), we can see in Figure 4 that 21 and kk could be any point in the area bounded by line segments fh, hg and gf. In order to obtain maximal 21 and kk , the following inequality holds. ),min()( 2102211 ttttktkRsk , (49)

where 10

1 ,minvdRsk .

The MODS of formula (20) becomes )(

2)(

1)( ,max""max"" kkk BFBFBF (50)

km

kmi

ki Rstaskmftt

2

1i

M

1

)(,

)(0

(k)i

))((tosubject ,

),min()( 2102211 ttttktkRsk ,


where

(k)i

)()(

,)( M,,2,1;2,1,

0)(0

))(( liotherwise

mttasklft

kiik

lik

i ,

0t searching source depletion of the radar system,

10

1 ,minvdRsk

m

1

)(22

m

1

)(11 ,

i

k

i

k kk ,

},,,{ 21 mddd is the order of },,,{ 21 mddd ,

},,,{ 21 n is the order of },,,{ 21 n .

Basic information

scond007.0beameachofTimeDwell scond405.60t

007.041t 007.0102t smv /4000

}15,,2,1:2)1({ iiaElevation i ,

}61,,2,1:302)1({ iiAzimuth i , }1201,,2,1:30000100)1({ iirRadius i ,

}91,,2,1:10010)1({ iivVelocity i , }72,,2,1:5)1({ iiAngle i ,

. The targets issued for detection

),,(det,),,,(det),,,(det 22221111d

mdm

dmm

dddddd rrrDet ,

where kk

j mjElevationa ,,2,1,)( ,

kk

j mjAzimuth ,,2,1,)( ,

kk

j mjRadiusr ,,2,1,)( ,


IkkttttktkRskk k ,),,min()(:),( 1210221121

10. Draw benefit Figure for ))(),(( 21 kfkf

11. Print detailed information of all targets

12. Take out serviced detection targets and reduce time and distance for remain targets

13. 1ii , go to 3


During this phase of the project, the scheduling problem for multi-function radar was successfully defined as an Multicriteria Optimization Problem. Although this was a simple application with ony two simplified functions, the results obtained clearly illustrate the set of possible scheduling solutions for a given set of parameters. An example with graphical output is given in Annex A.

Typical scheduling algorithms for Radar Resource Management use a preselected aggregate function to assess the value of a particular schedule. By doing so, many possible solutions to the problem are ignored. One of the benefits of treating the Radar Resource Management problem as a MCOP is that the efficient set includes the results of every aggregate function at once. This provides for a better understanding of the problem at hand allowing for a better informed decision to be applied to the scheduling problem. In some cases, the understanding gained through the MCOP can yield to more efficient scheduling algorithms by allowing for the selection of a subset of efficient points easier to calculate. The work carried out to this point was mostly an introduction of multi-criteria optimization to the problem of Radar Resource Management. Only two simple criteria were selected out of a multitude of actual parameters. Future research should explore the problem in additional dimension and with various Benefit functions associated to each of these criteria. There are a number of interesting Benefit functions that can be explored and studying this resource management problem as a MCOP promises to illustrate the effect of considering these criteria for scheduling. Furthermore, the current MCOP solves for the set of efficient point at a given point in time and additional work is required to consider the dynamical case over multiple consecutive scheduling time intervals. These additions will no doubt generate a criterion space with weaker properties and more advanced solutions methods will need to be implemented.


[1] . Statement of Work – Dynamic Optimization Algorithms for Radar Resource Management, Task #1, 2009.

[2] . Multicriteria Optmization, Springer, Berlin, 2005.

[3] Optimal target tracking with restless bandits, Elsevier Digital Signal Processing, , 479 – 487, 2006.


This page intentionally left blank.


The scheduling program based on the material presented in this report was coded in Matlab (see Annex B) and executed and produced the following non-dominated sets using the listed parameters.

The following parameters were used: = 6.4050 s (the searching time) = 0.0280 s (the detection time) = 0.0700 s (the tracking time)

= 200 (the targets in function detection)

=100 (the targets in tracking function) =400 m/s (average velocity at the beginning)

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

40

Detection Benefit

Benefit Curves

Time-BenefitMaximal-Benefit

Tracking Benefit

Figure 8 Non-dominated set example








DND Department of National Defence

DRDC Defence Research & Development Canada

DRDKIM Director Research and Development Knowledge and Information Management

R&D Research & Development

(Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified) 1. ORIGINATOR (The name and address of the organization preparing the document.

Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor's report, or tasking agency, are entered in section 8.)

Department of Mathematics and Computer Science Royal Military College

2. SECURITY CLASSIFICATION (Overall security classification of the document including special warning terms if applicable.)

UNCLASSIFIED

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in parentheses after the title.) Dynamic optimization algorithms for radar resource management

4. AUTHORS (last name, followed by initials – ranks, titles, etc. not to be used) A. Gosselin and Y. Wang

5. DATE OF PUBLICATION (Month and year of publication of document.)

December 2010

6a. NO. OF PAGES (Total containing information, including Annexes, Appendices, etc.)

5

6b. NO. OF REFS (Total cited in document.)

3 7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report,

e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)

Contract Report

8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development – include address.)

Defence R&D Canada – Ottawa 3701 Carling Avenue Ottawa, Ontario K1A 0Z4

9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.)

11as02

9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.)

A1410003FE

10a. ORIGINATOR'S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.)

10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.) DRDC Ottawa CR 2010-232

11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)

Unlimited

12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement audience may be selected.)) Unlimited

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

Typically, the scheduling of multi-function radars is based on the optimization of some pre-defined aggregate function made of all attributes of interest. By selecting a specific aggregatefunction these methods force the generation of a specific solution without the full awareness ofthe possible solution space. By introducing methods of Multicriteria Optimization [2] theproposed scheduling algorithms allow for the determination of the set of all non-dominatedsolutions with its associated efficient set in decision space: the space of all possible schedules.The determination of efficient sets will generate a better understanding of the solution space andcould lead to a better selection of the particular schedule to be implemented in addition toallowing for the development of more efficient scheduling algorithms. This report presents asimple two-criterion scheduling model based on Multicriteria Optimization.

Typiquement, l’ordonnancement de radars multifonctions est basé sur l’optimisation d’unefonction d’agrégat prédéfinie de tous les attributs d’intérêt. En choisissant une fonctiond’agrégat précise, ces méthodes forcent la génération d’une solution précise sans connaîtrecomplètement l’espace de solutions possibles. En ajoutant des méthodes d’optimisationmulticritère [2], les algorithmes d’ordonnancement proposés permettent de déterminerl’ensemble de toutes les solutions non dominées et l’ensemble efficient connexe de l’espace dedécisions : l’espace de tous les ordonnancements possibles. La détermination d’ensemblesefficients permettra une meilleure compréhension de l’espace de solutions et pourrait donnerlieu à un choix plus juste de l’ordonnancement particulier à mettre en œuvre, en plus depermettre le développement d’algorithmes d’ordonnancement plus efficients. Le présent rapportdécrit un modèle d’ordonnancement simple bicritère basé sur l’optimisation multicritère.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Radar resource management, scheduling, multicriteria optimization

Documents

Dynamic optimization algorithms for radar resource …cradpdf.drdc-rddc.gc.ca/PDFS/unc106/p534470_A1b.pdf · Dynamic optimization algorithms for radar resource management ... Contract