Hui-xi Xing ,Hu Wu ,You Chen ,,Kun Wng

a Aeronautics Engineering College,Air Force Engineering University,Xi’an,710038,China

b School of Energy and Power Engineering,Shandong University,Jinan,250061,China

Keywords: Radar countermeasure Adaptive heuristics Adversarial effectiveness Fuzzy comprehensive evaluation Mutation operator

ABSTRACT To deal with the radio frequency threat posed by modern complex radar networks to aircraft,we researched the unmanned aerial vehicle(UAV)formations radar countermeasures,aiming at the solution of radar jamming resource allocation under system countermeasures.A jamming resource allocation method based on an improved fire fly algorithm(FA)is proposed.Firstly,the comprehensive factors affecting the level of threat and interference ef ficiency of radiation source are quanti fied by a fuzzy comprehensive evaluation.Besides,the interference ef ficiency matrix and the objective function of the allocation model are determined to establish the interference resource allocation model.Finally,A mutation operator and an adaptive heuristic are integtated into the FA algorithm,which searches an interference resource allocation scheme.The simulation results show that the improved FA algorithm can compensate for the de ficiencies of the FA algorithm.The improved FA algorithm provides a more scienti fic and reasonable decision-making plan for aircraft mission allocation and can effectively deal with the battle field threats of the enemy radar network.Moreover,in terms of convergence accuracy and speed as well as algorithm stability,the improved FA algorithm is superior to the simulated annealing algorithm(SA),the niche genetic algorithm(NGA),the improved discrete cuckoo algorithm(IDCS),the mutant fire fly algorithm(MFA),the cuckoo search and fire flies algorithm(CSFA),and the best neighbor fire fly algorithm(BNFA).

1.Introduction

With the advances in the mine field network technology,the effectiveness of the electronic countermeasures of a single platform in the air is limited.Combat capabilities can be effectively enhanced by the cooperative jamming radar network of unmanned aerial vehicle(UAV)formations.The effective usage of jammers in formations to signi ficantly reduce the target exposure is a top priority for cooperative interference.The distribution of jamming aircraft directly affects combat power.As for the improvement of radar confrontation in modern warfare,the method to quickly form an effective interference resource allocation plan or decision based on the battle field environment situation,the existing interference resources and tactical requirements[1,2]is of great practical signi ficance.For combination optimization of interference resource allocation,this investigation aims at the selection and design of optimization algorithms because the interference resource allocation scheme is implicit in a multi-parameter,multi-constrained,and non-deterministic polynomia(NP),so that the limited interference resources can be utilized scienti fically,reasonably,and ef ficiently.

Traditional methods of allocating interference resources are mostly combinational optimization methods[3,4],including the 0-1 planning method,similarity method,and dynamic programming.These approaches are suitable for addressing small-scale resource allocation issues.However,with the increasing scale of radar emitter sources and interference resources,the number of jamming combination scheme expands dramatically.These methods have inadequate real-time performance because the difficulties in obtaining better results in a short time.

Recent research indicates that the heuristic optimization methods excel in solving optimization problems.Liu Xiang et al.[5]used an improved genetic algorithm to address the problem of cooperative interference resource allocation.The two-dimensional integer coding was applied to determine the correspondence between the jammer and the radar network.The algorithm converges quickly,but it is easy to premature.In the study conducted by Xiang et al.[6],a modi fied binary particle swarm algorithm was used to solve the problem of dynamic resource allocation with the following formation interference.TheS-type function converts the particle velocity,and the particle position is updated.However,the algorithm has a complicated coding form,and high computation,leading to low search ef ficiency.Considering the impact of different power allocation schemes on the bit error rate,Shao et al.[7]utilized a simulated annealing(SA)algorithm to achieve the jamming resources allocation.However,the algorithm received inadequate solutions with a certain probability,and the convergence speed was slow.

The Fire fly Algorithm(FA)is an intelligent optimization algorithm inspired by the Cambridge scholar Xin-She Yang.His inspiration was attracted by the glow of fire flies[8-10].The FA algorithm not only has no requirements on objective function and constraints,but also has simple operation,few parameters,and strong robustness[11].Its research has been successfully applied to image recognition[12],path planning[13],fleet operations[14],and many other fields.However,like many other intelligent algorithms,the standard FA algorithm is combined with a slow convergence rate and low convergence accuracy at the same time.Besides,the fire fly’s movement strategy does not guarantee the diversity of the decomposition space,and it is prone to fall into a local optimum state,resulting in large deviation and weak stability in the optimization process.Lu[15]alternately used the two crossover operations and proposed a disturbance resource optimization allocation technique based on the niche genetic algorithm(NGA).Li et al.[16]added the crossover and mutation operations of the genetic algorithm into the discrete cuckoo algorithm(DCS),and proposed an improved discrete cuckoo algorithm(IDCS)to implement the interference resource allocation of the enemy-friend identi fication system.To a certain extent,both of the methods improved the optimization ability of the algorithm.

Therefore,in this study,the author draws on the ideas of the genetic algorithm and particle swarm algorithm to improve the standard FA algorithm from two aspects.Firstly,we added a mutation operator to enable individuals to achieve gene communication in different dimensions.This strategy allows for individual fire flies to retain their excellent information and update bad information in a certain dimension.Therefore,it can enhance the ability of fire flies to jump out of local optimums.As a result,the search performance of the algorithm is improved.Second,we added an adaptive heuristic guidance strategy,which regards the overall optimal position as the heuristic information and dynamically adjusts the guidance coef ficient according to the individual fire flies’position.In the early stage of the search,it has a high convergence speed and a strong global searching ability.In the later stage of the search,the change of the guidance coef ficient weakens the guidance effect of the heuristic information.It leads to the enhancement of the local re fined search capability of the algorithm.Compared with the FA algorithm,the improved FA algorithm not only increases the convergence accuracy of the optimal solution and the overall average solution,but also shortens the iteration time.

In this study,first of all,we untilized the fuzzy comprehensive evaluation based on the comprehensive factors of interfering parties.By constructing membership function,different indexes are represented.Then the threat level of the radiation source and the jamming effectiveness are evaluated,and the jamming resource allocation model is built.Finally,the improved FA algorithm is used to search for the optimal allocation scheme.The experimental results show that the improved FA algorithm achieves faster calculation speed and better interference effectiveness compared with the original algorithm.Compared with other algorithms,the quality of the overall average solution has raised,and the algorithm has a better optimization ability and stability.The improved FA algorithm provides a better allocation scheme with more excellent reliability that can inform tactical decisions of battle field commanders.Besides,this algorithm inherits the characteristics of the original one,which has fewer parameters and simple principle that is easy to implement in engineering.

The rest of the paper is organized as follows:

Section 2 presents the assessment of radiation source threat level and interference effectiveness.An improved FA is proposed to solve the interference resource allocation model.Section 3 is about simulations concerning the attribute values of the radiation source and jammer.Finally,conclusions are drawn in Section 4.

The researchers use the following notations throughout the paper.Bold lowercases and bold uppercases stand for vectors and matrices,respectively.Letters with the inferior characters refer to the element of vectors or matrices.For example,aidenotes thei-th element of the vectora,andaistands for the i-th vector of matrixA.x（t）stands for the vector at thet-th iteration.For typical combat scenarios,the researchers assume that the jamming party is a mission formation consisting ofMaircraft with jamming capabilities{R1,R2,…,RM},and the enemy is a radar network consisting ofNradars{J1,J2,…,J10}.［·］ijindicates that thej-th jammerJjcorresponds to thei-th radiation sourceRi.

2.Material and methods

2.1.Jamming resource distribution model

2.1.1.Radar emitter threat assessment

Radar emitter threat assessment is an essential prerequisite for the optimal allocation of radar jamming sources.Fuzzy multiattribute decision-making has been used to analyse the threat level of the battle field emitters[17,18].According to the characteristics of practical application,there are three prioritized factors,including the types of radar emitter,working status,and platform information[19].

Radar emitter can be classi fied into seeker radar,fire control radar,airborne early warning radar and long-range warning radar.The working status generally includes search,tracking,and guidance according to the level of threat.The platform information consists of distance,speed,and azimuth,which are obtained by the active or passive detection system.

Supposing there isNradar emitter in the battle field,the threat level of radar emitteri(i=1,2,…,N)is expressed by Eq.(1):

In Eq.(1):Tirepresents the threat of thei-th radar emitter type.Sirepresents the threat to the operating state of thei-th radar emitter.Piis the platform information of thei-th radar emitter.αi（i=1，2，3）is the weight of the indicator,and satis fies∑αi=1.The researchers recorded the threat level ofNradar emitter as.Ω=［ω1，ω2，…，ωN］.

2.1.2.Jamming effectiveness evaluation

Jamming must be implemented accurately in timing,frequency alignment,airspace coverage,appropriate power,and pattern matching.The jamming effect has been considered from the following aspects:time domain,frequency domain,and energy domain,as well as the jamming patterns and radar antiinterference measures.The five factors that affect the jamming ef ficiency are selected as the evaluation indexes:jamming time,jamming frequency band,jamming power,jamming pattern,and radar anti-interference measures.The fuzzy comprehensive evaluation is used to determine the jamming effectiveness matrix.There areMjammers,and thej-th(j=1,2,…,M)jammer’s interference ef ficiency on thei-th radiation source is defined as Eq.(2).

In Eq.(2),theiandjin the index indicate that thej-th jammer interferes with thei-th radar.Itijis the interference time factor,Ifijis the interference band factor,Ipijis the interference power factor andImijis the interference pattern factor.Iaijis the factor of radar antijamming measure.βi(i=1,2,…,5)is the weight values occupied by each factor,and∑βi=1.

According to the evaluation indexes mentioned above,the fuzzy comprehensive evaluation value of the interference effectiveness of each jammer on each radiation source can be determined.Thus,the interference ef ficiency matrix ofMjammers to interfere withNradiation sources can be determined,as defined in Eq.(3).

2.1.3.Objective function

After the threat level of radar emitterΩand interference ef ficiency matrixEare obtained,the focus of the interference resource optimization is to select the best interference combination fromthe interference ef ficiency matrix so as to improve the total interference bene fit under the interference combination.Eq.(4)is the optimum allocation objective function:

Thefeff（x）is the objective optimization function andxijis a decision variable.Ifxij=1,it indicates that thej-th jammer interferes with thei-th radar emitter,and ifxij=0,it indicates that thej-th jammer does not interfere with thei-th radar emitter.

The limitations are:one jammer can only interfere with one radar emitter at a time.However,one radar emitter can be interfered with by multiple jammers,as shown in Eq.(5):

2.2.Model solving

2.2.1.Standardfirefly algorithm

The fire fly algorithm(FA)contains two parameters:brightness and attraction.Brightness refers to the current position of the individual fire fly.The distance the individual fire flies move depends on the degree of attraction In the beginning,fire flies are randomly distributed in the solution space.The fire flies in the brighter position attract the fire fly individuals around through fluorescence signals and constantly updates the position of the fire fly group to achieve the optimization goal.The speci fic mathematical optimization mechanism[20]is as follows:

The fire fly population is represented byP=｛x1，…，xi，…xN｝,and every fire fly individual represents a solution in theD-dimensional solution space.

The relative fluorescence brightness of individual fire flies is shown in Eq.(6):

In Eq.(6),I0is the maximum fluorescent brightness of fi re fl ies,γ is the light absorption coef ficient,which is generally a constant,andis the Euclidean distance between fi re fl iesxiandxj.

The attractiveness of light is shown as Eq.(7):

In Eq.(7),β0represents the degree of attraction whenr=0.

After being attracted by the brighter fire flies,the fire fly‘s position is updated.Eq.(8)represents the mathematical expression of the location update.

In Eq.(8),the step factorα∈［0，1］is a constant andRandis a uniformly distributed random factor from 0 to 1.Whenβ=0,the individual moves randomly,which means there are no other individuals around thei-th fire fly.

Because of the shortcomings of the standard FA and its limitations in solving optimization problems,domestic and foreign scholars have carried out many studies on this filed[21-27].In this paper,the FA algorithm is improved in the convergence speed and global search ability.

2.2.2.Firefly algorithm improvement

I.Mutation Operation

This paper draws on the genetic mutation operation idea of genetic algorithm.A mutation factorηis added to the FA algorithm to update the position information of some fire fly individuals during the iteration process.The roulette selection method is used to select individuals that need to be mutated.Individualxiis selected according to the probability of Eq.(9).

In terms of evolutionary fitness,individuals with higher fitness values are more likely to be selected.The researchers mutate the selected fire fly individuals according to Eq.(10).

In Eq.(10):ci（t）is the mutation intermediate ofxi（t）;the variation factorη∈［0，1］.

The researchers compared the fitness values ofci（t）andxi（t）to determine whether mutation produces new individuals.Eq.(11)shows the generation of new individuals

The mutation operation can increase the diversity of the fire fly population.When the algorithm search near the neighborhood of the optimal solution,the mutation operator will enhance the partial random search ability,thus accelerating the convergence of the population to the optimal solution.

II.Adaptive Heuristic Guidance

The individuals of the standard FA are memoryless,and move only based on the brightness information.Therefore,it is easy to lose the information of the optimal solution obtained in the iteration,and extend the iteration time,thereby reducing the search speed,or even deviating from the optimal solution.Hence,it is necessary to improve the movement mechanism of fire flies.Unlike the FA algorithm,the particles in the particle swarm algorithm[28]have memory.The historical best position of the population and the particle itself determine the speed update of the particle,so that the particle swarm quickly converges to the global optimal position.The fire fly position update equation is modi fied by adding an adaptive heuristic algorithm,which is described as Eq.(12).

In Eq.(12),ωi（t）is the adaptive guidance coef ficient,defined as Eq.(13).

tmaxis the maximum iteration time,pbest（t）andpworst（t）are the optimal and worst positions of the population at the t-th iteration,respectively.ωmaxis the maximum value of the adaptive guidance coef ficient.

The increase of the adaptive heuristics makes the fire fly individuals move toward the overall optimal position while approaching the brightest fire flies in the neighborhood.The selfadaptive dynamic adjustment of the guide coef ficientωi（t）has two main advantages.First,the initial guide coef ficient is large,and fire flies quickly move to the overall optimal position.During the iteration,as the guide coef ficient decreases,the attractiveness of the brightest fire fly gradually weakens.The fire flies move to the brightest location in the neighborhood,which enhances the accurate partial search capability.Second,the changes in the guide coef ficient are associated with the location information of fire fly individuals.Individuals with lower fitness values(The higher the value of‖pbest（t）-xi（t）‖)results in larger guide coef ficients,which accelerate them to move closer to the optimal position of the population and increase the global searching ability.

2.3.Algorithm implementation

2.3.1.Coding method

Aiming at interference resource allocation,the individuals of the population should be coded to establish a mapping relationship between jammers and radar emitters.Compared with binary coding,the real vector coding method proposed in this paper can make full use of individual characteristics and reduce the dimension problem effectively.SupposeMjammers can interfere withNradar emitter.The dimensionDof the individual fire fly represents the number of the radar emitterN.The integer part of each dimension represents the number of the radar emitter,and the decimal part is arranged in descending order.The sequence value represents the number of jammer that is interfering with the radar emitter.Individual positions are represented by random numbers in the interval[1,N+1].Because each jammer can interfere with at most one radar emitter,there is usually an inequalityM≥Nto ensure that each radar emitter is interfered with a jammer.There is an example of 6 jammers interfering with 4 radar emitters.The position of the fire fly is a random numbers in the interval[1,4+1],as shown in Fig.1,which intuitively re flects the corresponding interference relationship between the jammer and the radar.

Fig.1.Schematic diagram of encoding mapping relationship.

2.3.2.Algorithmflow

After the fire fly individual coding,the interference resource allocation can be achieved.Fig.2 is the flowchart of the interference resource allocation algorithm based on the improved FA and the speci fic implementation steps.

Step 1Objective function determination:The researchers calculate radar emitter threat assessment levels and interference effectiveness matrices.

Step 2Basic parameters initialization:The number of fire flies is NP,the maximum number of iterations isH,the maximum attraction isβ0,the light absorption coef ficient isγ,the step factor isα,the number of enemy radar emitter isN,and the number of jammers isM,M≥N.On the premise of satisfying the constraint conditions,the correspondence between the jammer and the radar emitter is initialized according to the real vector coding guidelines.

Step 3Population Initialization.At the initial moment,the fire flies are distributed randomly,and the fitness value and maximum fluorescent brightness of each fire fly individual are determined.

Step 4Crossover and mutation.After calculating the relative brightness and attraction between individuals,the current optimal fire fly is selected to generate new fire fly individuals through a cross mutation operation,and make a comparison of the two bodies.If the evolved fire fly individual has higher brightness,it replaces the original one.

Step 5Individual fire fly positions update.The direction of the movement of each fire fly is determined by comparing the brightness of each fire fly with other individuals in the neighborhood.The population position is updated according to Equation(12).The individual with the best position moves randomly.The algorithm return toStep 4.

Step6Termination conditions determination:When the maximum number of iterations or the search accuracy is reached,the fire fly position of the optimal individual is recorded as the current optimal solution,and the interference resource allocation scheme is determined.Otherwise,the algorithm returns toStep 5.

3.Discussion

3.1.Environment simulation

The simulation experiments were performed on the MATLAB R2016b platform with Intel Core i5-4210H CPU@2.90GHz,4 GB memory,and 64-bit operating system.

Fig.2.Algorithm flowchart.

According to the parameter settings of reference[5],The researchers assumed that there were 10 radar emitters{R1,R2,…,R10}and 10 jammers{J1,J2,…,J10}in the battle field environment.The parameters of each radar emitter obtained through preliminary investigation are shown in Table 1.Letα1=0.5，α2=0.4，α3=0.1,λ1=0.4，λ2=0.2，λ3=0.2,the threat of each radar emitters is calculated asΩ=［0.223，0.199，0.481，0.540，0.522，0.348，0.456，0.282，0.302，0.355］.

The properties of the jammer are shown in Table 2.Letβ1=0.1，β2=0.2，β3=0.2，β4=0.3，β5=0.2,The authors calculated the interference ef ficiency matrix when 10 jammers interfere with 10 radar emitters,as shown in Table 3.

3.2.Simulation analysis

3.2.1.Solutions of interference resource allocation

The number of fire flies isNP=40.The iterations areT=100,the light absorption coef ficientγ=1,the maximum attraction degree β0=1,and the step size factorα=0.3.

The standard FA algorithm and the improved FA algorithm are used to optimize the interference resource allocation model,maximizing the interference ef ficiency as the optimization objective.Fig.3 shows the update of the objective function value under a single optimization experiment.The objective function of interference resource allocation mentioned in this paper is a discontinuous multimodal function,and the stepped convergence curve shows the effectiveness of the FA.Both the standard FA algorithm and the improved FA algorithm can eventually converge,but the latter has a signi ficantly faster convergence speed than the former.Because in the FA algorithm,each fire fly represents an independent solution,the mutation operation and the adaptive heuristic guidance can improve the whole and the partial search capabilities.Searchability allows individuals in a population to move quickly to optimal locations and obtain better solutions earlier.

More conclusions can be drawn from Fig.3.The improved FA algorithm converges faster than the standard FA(the improved FA algorithm converges when the iteration reaches about 60 times,while the standard FA algorithm iterates to 80 times),and the latter solution can achieve better interference performance.Table 4 is the jammer allocation scheme solved by the improved fire fly algorithm.

Table 1 Emitter property sheet.

Table 2 Jammer property sheet.

Table 3 10×10 jamming effectiveness matrix.

Table 4 Jammer allocation scheme.

3.2.2.Comparison of different types of intelligent optimization methods

The traditional simulated annealing(SA)algorithm[29]was used to solve the UAV allocation scheme in this study.The simulation parameters are set as follows:initial temperatureT0=150,termination temperatureTmin=0.01,temperature drop coef ficient α=0.8,and the number of iterations is 8 times the number of jammers.Fig.4 indicates the change of the objective function value.According to Fig.4,when the SA algorithm iterates to about 80 times,the objective function value can stabilize around 2.85,and the objective function value fluctuates greatly.The SA algorithm accepts inferior solutions with a certain probability.Even if it can avoid converging to a local extremum,it tend to cause the algorithm to converge slowly and the final result is not the optimal solution.

To compare the performance differences with other improved optimization algorithms,the authors applied NGA,IDCS and the improved FA algorithm to the interference resource allocation model in this paper.

Parameter settings:

NGA:population size 100 and mutation probability 0.05.

IDCS:number of bird nests 20 and drop probability 0.5.

Improved FA:population size 50,light absorption coef ficient γ=1,maximum attractionβ0=1 and step size factorα=0.02.

The number of iterations is 100,and repeating 20 times.Fig.5 shows the average convergence curve.

The average number of iterations of the NGA algorithm and the IDCS algorithm is about 95,while the average number of iterations of the improved FA algorithm is 75,indicating that the improved FA algorithm has faster convergence speed.Compared with the average objective function value calculated by several algorithms,the improved FA is more ef ficient and has better optimization ability.

Fig.3.Comparison chart of changes in objective function values.

Fig.4.Simulated annealing algorithms for solving object function values.

Fig.5.Convergence curves of three types of intelligent optimization algorithms.

3.2.3.Comparison of different improvedfirefly algorithms

Fan et al.[30]designed a mutant fire fly algorithm(MFA)by combining ef ficient coding methods of external and internal coding systems to solve the flow shop scheduling problem.An improved cuckoo search and fire flies algorithm(CSFA)[31]was proposed,in which cuckoo search was used to optimize the initial population of the fire fly algorithm to accelerate the algorithm’s convergence.The performance of the CSFA algorithm was tested with standard test functions.Wu et al.[32]proposed a new fire fly algorithm,which is called the best neighbor fire fly algorithm(BNFA).It employs the best neighbor guidance strategy,where each fire fly is attracted to the optimal fire fly randomly selected in the neighborhood.Thus,the improvement measures reduce the fire fly oscillations in every attraction-induced migration stage,while increasing the probability of the guidance in a new better direction.In this paper,the above three methods including MFA,CSFA and BNFAwere applied to solve the problem model and compared with the improved FA.The population number of fire flies is 50,and the maximum number of iterations is 100.The authors conducted 20 independent repeated experiments and compared the convergence results,as shown in Fig.6.

Fig.6.Convergence curves of several improved FA.

The average fitness value of the solution obtained by the MFA algorithm is only 2.739,which is lower than other algorithms.Therefore,the special encoding method of the algorithm is dif ficult to apply to the many-to-many interference resource allocation problem.The early convergence speed of the CSFA is very fast,because it optimizes the fire fly population at the beginning,which signi ficantly improved the quality of the initial population,thus accelerating the convergence speed of fire fly to the optimal solution.However,the partial search ability is insufficient in the later stage.Therefore,the convergence accuracy is low,and it may not be able to get the best distribution solution.The interference resource allocation scheme obtained by the BNFA algorithm and the improved FA algorithm can achieve the ideal interference ef ficiency,indicating that they are equivalent in their optimization capabilities.For the former,the best neighbor fire fly individuals can overcome the shortcomings of early maturity of the algorithm,but the increase of population tend to make it more complex.In contrast,the latter converges earlier,because the fire fly’s individual mutation mechanism is added to the algorithm,which enhances the global searching ability without increasing the number of calculations.Besides,the improved FA algorithm accelerates the convergence to the optimal solution under the guidance of heuristic information in the early stage of iteration.It focuses more on the partial search in the later stage of iteration.

In this paper,to more directly re flect the performance of the algorithm,after 100 Monte Carlo simulations,different algorithms are applied to the interference resource allocation model,and each indicator of the objective function value is calculated.The averages values obtained by several algorithms are 93.9%,92.5%,87.8%,93.5%,93.5%,92.2%,93.2%and 93.9%of the maximum objective function,respectively.Because interference resource allocation is an NP-hard problem,the algorithm cannot guarantee an optimal solution every time,but can only find a better solution in a relatively short time.The objective function of the improved FA algorithm has a higher average value and smaller variance,which indicates that the algorithm’s optimization results have higher credibility and stronger stability.Table 5 lists detailed information.

Table 5 Algorithmic comparison.

3.2.4.Adaptability verification of improved FA algorithm

In the paper,the authors simulated a collaborative confrontation scenario.In the collaborative confrontation scenario,an attribute table of 3 radar emitters and 3 jammers are given by Ref.[33],the radar threat coef ficient and jamming effectiveness matrix of 5-to-5 are given by Ref.[34],and the attribute table and jamming ef ficiency matrix of 7 jammers against 7 radar emitters are given by Ref.[35].The FA algorithm and the improved FA algorithm were used to search for the interference resource allocation schemes at different scales.Fig.7(a),Fig.8(a),Fig.9(a)shows the convergence curve of the running objective function.

According to the simulation results,the improved FA algorithm is superior to the FA algorithm in convergence speed and accuracy.Also,with the increasing of radar networking scale,the improved FA algorithm shows higher adaptability.

Then,the difference between the convergence optimal value and the global optimal value is calculated through 100 independent experiments.Fig.7(b),Fig.8(b),Fig.9(b)give the error statistics of the FA algorithm,and Fig.7(c),Fig.8(c),Fig.9(c)provide the error statistics of the improved FA algorithm.It is evident that the improved FA algorithm has less convergence errors and stronger stability.The information in Table 6 shows a more intuitive comparison of the algorithm performance.With higher complexity of the problem,the optimization ability of both algorithms declines.In three cases,the improved FA algorithm has a higher optimal convergence probability.Especially when there are 7 radar emitters,the optimal convergence probability can still be maintained at 52%.In addition,With the increase of the radar confrontation scale,the search time of the improved FA algorithm is shorter than the standard FA algorithm.When there are 7 radar emitters,the search time of the standard FA algorithm is close to 20s,while the improved one is always around 10s.

Fig.7.Simulation of 3 jammers against 3 radiation sources.

Fig.8.Simulation of 5 jammers against 5 radiation sources.

Fig.9.Simulation of 7 jammers against 7 radiation sources.

Table 6Comparison of adaptability of FA algorithm and improved FA algorithm.

4.Conclusions

This paper focuses on the radar jamming decision methods of formation UAV against radar network systems.Based on a fuzzy comprehensive evaluation,a threat level evaluation model is established with the radar emitter type,working status,and platform information as indicators.The interference ef ficiency matrix is further determined from the perspective of time,frequency,energy,interference patterns,and radar anti-jamming measures.The optimization of interference resources is also transformed into the optimization of allocation schemes to obtain the maximum overall interference ef ficiency.The improved FA can make up for the shortcomings of the standard FA algorithm.On the premise of the random movement of individuals,performing mutation operation on good individuals effectively improves the possibility of the algorithm to jump out of the partial optimal solution.Besides,the guidance of heuristic information overcomes the disadvantage of the random movement direction without contribution to the optimal solution,which leads to the slow convergence speed.The results show that the authors’method has certain advantages in search ef ficiency and convergence accuracy.Therefore,the interference resource allocation strategy formed in the system countermeasures is more credible.

This study explores the cooperative jamming method of UAV formation from the perspective of group operations.Based on the theory and practice,this paper evaluates the threat level of enemy radar emitters through multiple indicators.It establishes a scienti fic interference resource optimization model,which overcomes the subjectivity and hysteresis of arti ficial experience and has certain reference value.

Intelligent electronic interference decision-making can provide a guarantee for the aircraft to effectively attack the enemy and gain the initiative on the battle field.The improved adaptive interference resource allocation method proposed by the authors is still worth exploring in the future application research.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgements

The authors express sincere appreciation to the anonymous referees for their helpful comments to improve the quality of the paper.