0 Introduction

Modern fighters have the ability to attack multiple targets and carry long range air－to－air missiles.Beyond visual range（BVR）air combat has been the mainstream with the development of modern fighters，where fighters are required to exchange information and attack multiple targets cooperatively［1－2］.To complete cooperative multiple target attack （CMTA），decision－making（DM）is necessary for fighters to allot targets and missiles according to the shared information［3－4］.Thus，the missile－target assignment （MTA）problem is the main part of DM when it comes to CMTA.

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There are many algorithms applied to DM problem in CMTA，such as particle swarm optimization（PSO），genetic algorithm （GA）and ant colony optimization （ACO）［5－7］.A heuristic algorithm is introduced to adaptive genetic algorithm in Ref.［8］and improves local search capability.Adaptive pseudo－parallel genetic algorithm is also considered to deal with air combat DM problem beyond visual range［9］.However，GA is not a real－time algorithm and may not work sometimes.Some intelligent algorithms are also used to solve DM problems［10－12］.In Ref.［13］，fuzzy neural network is applied to assign missiles according to the threat of enemy fighters and the bomb load of our fighters.However，it is hard to obtain practical and complex air situation data for neural network training.Considering the uncertain information in the MTA problem，grey system theory is introduced in DM problem［14］.

In this paper，an improved particle swarm optimizer（IPSO）is deduced to handle with the DM problem for CMTA in the air combat.The IPSO algorithm has stronger global searching capability by designing a new velocity learning strategy.

1 DM Problem in CMTA

1.1 Air combat situation

Air combat decision－making is based on the air combat situation.To establish the model of air combat situation，it is assumed that there are M our fighters which are marked in blue and Nenemy fighters which are marked in red.Denote our fighter set B ＝ Bi，i＝1，2，…，M｛

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｝and enemy fighter set R ＝ ｛Rj，j＝1，2，3，…，N｝.In an air combat，the situation between our fighters and enemy fighters can be illustrated in Fig.1，where LOS is the line of sight and Dijthe distance between Biand Rj.xBiand VBiare the position and velocity of Bi，respectively.εijis the bore of sight（BOS）angle of Rjto Bi.xRjand VRjare the position and velocity of Rj，respectively.εjiis the BOS angle of Bito Rj.

Fig.1 The situation between Biand Rj

Distance，BOS angle and velocity are taken into consideration as threat factors when constructing the threat function［15］.The threat function is described as a composite of all its threat factors，namely

where is the distance threat factor，the BOS angle threat factor，the velocity threat factor，andω1，ω2are non－negative weight coefficients and satisfy

Moreover，the value range of all the threat factor functions is ［0，1］.Thus，there is thij ∈

The distance threat factor can be constructed as

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where RaBis the maximum effective striking distance of missiles carried by our fighters and TrB the maximum radar tracking distance of our fight－

In the PSO algorithm，the updating formulae of the velocity and the position of each particle are given by

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The velocity threat function can be constructed as

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1.2 MTA model

Multi－fighter cooperative attack problem is aimed at optimizing target assignment for missiles carried by our fighters.According to the threat function known，multiple target assignment develops a proposal where there are more attack success and less fighter casualties.

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Assume that our fighter Bicarries Limissiles to attack enemy fighter targets.Thus，there are Z

1.3 Analysis on coordinated attack tactics

When our fighters attack enemy fighter targets，assignment rules need to be determined for our fighters.The assignment rules work so that our fighters get more benefit when attacking.

This series of constraints are used to check the solution of MTA problem and make some adjustments if necessary.The steps are as follows：

IPSO algorithm with fewer parameters not only keeps the diversity of the velocities but also does not alleviate the certainty of directing to the destination.The particles with better performance will increase their inertia movements，which expands the searching space and improves the searching speed.The particles with worse performance will increase their learning steps，which reduces the differences among the population and improves the whole performance of the population.

Then，the MTA problem is to find a solution πto minimize the equation above and accord with coordinated attack tactics.

2 Improved Particle Swarm Optimization

If the position value ckr（t＋1）isbiggerthan thetargetnumber，it is restricted in the last target.If the position value ckr（t＋1）is less than 1，it is restricted in the first target.Otherwise，all the position values are rounded down to make sure the whole positions are integer within the range.

whereλ1，λ2are the positive constants.Better attack angle results in better attack effect.

where kis a pseudo－time increment and represents iterations；Pi＝（pi1，pi2，…，piD）is the local optimal position of the ith particle；Pg ＝（pg1，pg2，…，pgD）represents the global optimal position in the swarm，here gis the index of the best particle among all the particles in the population；c1andc2 are called the cognitive and the social coefficients，respectively；rand1and rand2are two random numbers in range［0，1］.

Based on the PSO algorithm above，an improved PSO （IPSO）is presented，in which a new learning strategy is introduced in the particle velocity update equation，described as

where rand1and rand2are the random numbers in range［0，1］.χis the constriction coefficient；Pb＝［pb1，…，pbD］the particle position with better performance which is selected randomly；j the arrangement number according to the performance，here the smaller j corresponds to the better performance of the jth particle；nthe whole number of the particles in the population.

There is optimal attack effect when one of the assigned value is much larger than the other.

Thus，the IPSO algorithm flow can be described in Fig.2.

Fig.2 IPSO algorithm flow

3 Realization of IPSO for Multi－target Collaborative Combat Decision－Making

Every possible optimal solution is seen as a particle in PSO.The adaptive value of particle needs to be calculated in every position.It is reasonable for the adaptive value to be defined as objective optimization function to get the updating velocity and direction for every particle.Based on the MTA model established above，a set of missile－target assignment is dealt with as partial swarm after updating.moptimal MTA proposals correspond to mparticles in the particle swarm.Every particle is in the searching space of Zdimension.The position vector of the kth particle in the current iteration is defined as

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where k＝1，2，…，m，Zthe sum of missiles，and ckrthe position of the kth particle in the rth dimension.ckrbelongs to 1 N ＿red［

］and N＿red is the sum of enemy fighter target.

The velocity of the kth particle is given by

where vkrsatisfies vkr∈ ［－1＋NN－1］.

If the kth particle has the best fitness in the current iteration，it is defined as the local optimal solution and noted as

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If all of the particle have the best fitness in the current iteration，it is defined as the global optimal solution and noted as

The updating formulae of the velocity and the position of each particle based on IPSO are given by

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In the PSO algorithm，each particle is treated as a potential solution in D－dimensional space.The position of the ith particle is represented by a D－dimensional vector Xi ＝ （xi1，xi2，…，xiD），and the velocity of the ith particle can also be represented by aD－dimensional vector Vi＝（vi1，vi2，…，viD）.

It is essential to restrict velocity vector in a certain range to make sure that position vector is not updated too fast

According to the coordinated attack tactics above，more constraint conditions are taken into consideration.Each missile can only attack one enemy fighter target.Each target is attacked twice at most.The Boolean value of the missile is constrained as

It is supposed that each missile of our fighters can attack only one enemy fighter target.One enemy fighter is attacked by two missiles at most.It is essential to declare constraints on Xrj

Step 1 Denote a set Awhich includes all the values need to be changed.If the same position value exists in the position vectorπkmore than twice，two of them are chosen randomly and others are saved in set A.

Step 2 Denote two sets S0and S1.S0includes targets in set ［1 N＿red］whichhavenot appearedinthesolutionbefore.S1includes targets in set ［1 N＿red］that have appeared in the solution only once.

Step 3 Make some adjustments to set A.Assume that the value of the positionckrneeds to be changed and the updated position value is csc.s should belongs to ｛S 0S1｝.The principle of choosing targets is given by

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where d （csckr ）is the distance between ckrand cs.Then，the element ckris removed from set A.

Step 4 Update the two sets S0and S1.If there is cs ∈S0，cswould be saved in S1and re－moved fromS0.If there is cs ∈S1，the elements in S0and S1would not be changed.

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Step 5 Repeat Steps 3，4until set A becomes a null set.

4 Simulation Experiment of IPSO for CMTA

Assume that our fighters Band enemy fighters Rare in a BVR air combat.Our fighters Badopt CMTA strategy.In this simulation，there are four our fighters and each fighter has four missiles.Thus，the number of the missiles to attack the enemy fighter targets is 16.The velocity of our fighters is 300m/s.The effective striking distance of missiles carried by our fighters is 70km.The maximum tracking range of our fighters is 120km.There are fourteen enemy fighter targets.The velocity of enemy fighters is 300 m/s.The effective striking distance of missiles carried by our fighters is the same as that carried by the enemy fighters.The maximum tracking range of our fighters is the same as that of the enemy fighters.In a random scenario，our fighters and enemy fighters aviate face to face.The air combat situation is shown in Fig.3.

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Fig.3 The air combat situation

Then，the IPSO algorithm designed above is used to present a DM proposal of MTA problem in CMTA.The traditional PSO algorithm is also simulated here to compare with the IPSO algorithm.The constriction coefficientχis set to be 1.The assignment of all the missiles is

Fig.4illustrates the DM proposal of MTA problem.Based on the IPSO algorithm，the missiles carried by our fighter 1attack enemy fighters 2，8，7and 3.The missiles carried by our fighter 2attack enemy fighters 5，6，1and 5.The missiles carried by our fighter 3attack enemy fighters 13，10，12and 10.The missiles carried by our fighter 4attack enemy fighters 1，13，14 and 13.The repeated numbers imply that these enemy fighters threaten our fighters too much and are attacked twice as a result.Some enemy fighters are not attacked because their threat values do not reach the threat threshold value.With the traditional PSO algorithm employed，the missiles carried by our fighter 1attack enemy fighters 2，3，3and 7.The missiles carried by our fighter 2attack enemy fighters 1，1，14and 5.The missiles carried by our fighter 3attack enemy fighters 13，13，8and 8.The missiles carried by our fighter 4attack enemy fighters 10，10，5and 6.The IPSO algorithm based DM proposal of MTA problem makes full use of the missiles and destroys more threats.

Fig.4 Results of DM for MTA

Fig.5shows the fitness of iteration process.The fitness can decreased to 4.391 5when using the IPSO algorithm，while the fitness is 4.568 8 with the traditional PSO algorithm. What′s more，the DM proposal with the IPSO algorithm is faster than that with the PSO algorithm due to the less iterations when using the IPSO algorithm.

Fig.5 Fitness of iteration process

5 Conclusions

DM problem for MTA in an air combat is solved by a new improved PSO algorithm which is parametric simple but effective and efficient.The IPSO algorithm is used to minimize fitness function constructed by threat value.Coordinated attack tactics is considered to adjust DM proposal to reach better strike effect.It exhibits better performance to CMTA in an air combat with the IPSO algorithm compared with the traditional PSO algorithm.

Acknowledgement

This work was jointly granted by the Science and Technology on Avionics Integration Laboratory and the Aeronautical Science Foundation of China （No.2016ZC15008）.

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