CN110046399A - A kind of Pattern Synthesis of Antenna Array design method based on dynamic cooperative grey wolf optimization algorithm - Google Patents
A kind of Pattern Synthesis of Antenna Array design method based on dynamic cooperative grey wolf optimization algorithm Download PDFInfo
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Abstract
The present invention relates to a kind of Pattern Synthesis of Antenna Array design methods based on dynamic cooperative grey wolf optimization algorithm, belong to mobile communication antenna technical field.During per generation evolves, α wolf is being determined, β wolf, after the fitness function value of δ wolf and position, not directly to α wolf, the position of β wolf, δ wolf is averaged, but dynamic cooperative weight factor is calculated according to fitness function value, the position of these three wolves is weighted, then carries out location updating again.The innovatory algorithm more emphasizes the leading role of head wolf, but does not also ignore the cooperation between leadership wolf, is balanced preferably between global optimum and local optimum, fast and effeciently can converge to optimal solution.
Description
Technical field
The present invention relates to a kind of Pattern Synthesis of Antenna Array design methods based on dynamic cooperative grey wolf optimization algorithm, belong to
In mobile communication antenna technical field.
Background technique
In a wireless communication system, in order to tackle the interference of narrow band signal, it is desirable that generated in interference signal arrival bearing deep
Zero point is with anti-interference；Problems can be solved by Pattern Synthesis of Antenna Array.The mesh of Pattern Synthesis of Antenna Array
Be determining array antenna certain parameters, make antenna array radiation characteristic meet or as close possible to given design object.
Due in Pattern Synthesis problem objective function and constraint condition be largely in multi-parameter, non-linear, non-differentiability, even do not connect
It is continuous, so that traditional numerical optimization based on gradient search technology can not effectively acquire engineering satisfactory solution.
2014, the scholars such as Seyedali Mirjalili from Australian Griffith University were in Advances
The article of one entitled " Grey Wolf Optimizer ", Wen Zhongti have been delivered in Engineering Software magazine
Go out a kind of novel swarm intelligence algorithm --- grey wolf optimization algorithm (Grey Wolf Optimizer, abbreviation GWO), the algorithm
Wolf pack Social Grading, is divided into α wolf, β by the leader's hierarchy and hunting mechanism for simulating grey wolf pack in the Nature from top to bottom
Wolf, δ wolf, four grades of ω wolf, the hunting behavior of wolf pack mainly have 3 steps, are respectively tracking and surround close to prey, tracking
Prey and attack prey carry out chasing for prey by α, β, δ grey wolf during predation, and residue ash wolf pack ω follows first three
Person is tracked and surrounds and seize, and the position of prey is the solution of problem.GWO algorithm has very strong robustness and adaptivity, energy
Enough optimal solutions for effectively converging on problem.And Seyedali Mirjalili et al. is in paper " Grey Wolf
Optimizer " in by being solved to a large amount of benchmark test functions, it was demonstrated that GWO algorithm and particle swarm optimization algorithm
(Particle Swarm Optimization, abbreviation PSO) algorithm, differential evolution algorithm (Differential
Evolution, vehicle economy) and gravitation search algorithm (Gravitational Search Algorithm, abbreviation GSA) compared to gathering around
There is higher precision and stability, shows more excellent performance.And GWO algorithm is similar with other bionic Algorithms, group
Inside is there are the mechanism of cooperative cooperating, by the competition inside group, when each iteration, leader head wolf is campaigned for out, to instruct
The cooperative cooperating of entire wolf pack.This elite leading type algorithm has many advantages, such as that parameter is less, fast convergence rate, therefore GWO is calculated
Method has had been widely used for image segmentation, water operation, boiler combustion optimization, Object Threat Evaluation, unmanned plane since proposition
The engineering fields such as path planning solve optimization problem
All evolution algorithms based on population all exist due to the randomness or evolution person's character of algorithm itself and calculate the time
Longer, the problems such as convergence rate is slow, GWO algorithm is no exception.Both at home and abroad for the research of grey wolf algorithm all at the early-stage
And the exploratory stage, the theoretical basis of algorithm is also and immature, and is usually to increase various changes for the improvement of grey wolf optimization algorithm
The operations such as different, disturbance, these methods are capable of the convergence precision and speed of boosting algorithm to a certain extent really, but this kind changes
The complexity that algorithm is undoubtedly increased into method increases algorithm complexity while boosting algorithm precision.
Grey wolf optimization algorithm is a kind of elite leading type algorithm, and the searching process of original grey wolf algorithm is successively to elect α
Wolf, β wolf, δ wolf, then intragroup other individual (ω wolf packs) refers to α wolf, and β wolf, the location information of δ wolf is in target prey week
It encloses and randomly updates position X respectively1, X2, X3, the final position of next-generation wolf pack is by X1, X2, X3Average value generate.That is,
In original grey wolf algorithm, α wolf, β wolf, δ wolf be to the directive significance and influence power of ω wolf pack it is identical, this can not embody essence
The characteristics of English is led, is unfavorable for the fast convergence of algorithm.The present invention proposes a kind of new dynamic cooperative guidance mechanism, so that being in
The wolf of leadership dynamically can instruct wolf pack to advance according to its fitness function value in leadership's proportion, both guarantee to lead
Cooperative work between layer wolf, and emphasize the absolute leadership effect of a wolf, the calculating speed and precision of algorithm can be effectively improved.
Summary of the invention
It is more effectively excellent the technical problem to be solved by the present invention is to be needed for complex array Antenna measuring table problem
Change algorithm, provide a kind of Pattern Synthesis of Antenna Array design method based on dynamic cooperative grey wolf optimization algorithm, to reduce
The time is calculated, computational accuracy and efficiency are improved.
The technical scheme is that a kind of Pattern Synthesis of Antenna Array based on dynamic cooperative grey wolf optimization algorithm is set
Meter method, specific steps are as follows:
Step1: in solution space initialization population, initial evolutionary generation t=1 is set, maximum evolutionary generation is itermax；
Specifically, population scale N is provided, maximum number of iterations itermax, problem dimension D and problem feasible solution it is empty
Between, one group of initial solution X=[X is randomly generated1,X2,…,XN], wherein N is population scale, Xi=[xi1,xi2,…,xiD], i=1,
2 ... N are i-th of grey wolf individual.
Step2: the fitness function value of each grey wolf individual is calculated；
The fitness function value of each grey wolf individual is calculated by formula (1)；
In formula, PSL is Peak sidelobe level (Peak Sidelobe Level, abbreviation PSL), and TPSL is target peak pair
Valve level (Target Peak Sidelobe Level, abbreviation TPSL), NSLL are deep null level (Nulling Sidelobe
Level), the selection principle of weight coefficient α and β is: according to the directional diagram of each design section close to the degree of design object, evolving
Slow region can suitably increase weight coefficient, by testing repeatedly, determine the value of α and β, and meet alpha+beta=1.
Step3: from big to small according to fitness function value, α wolf, β wolf, δ wolf are successively selected, and its current location is denoted as
Xα,Xβ,Xδ, and the X that fitness value is bestαIt is denoted as optimal solution；
Step4: residue ω wolf pack and X are calculatedα,Xβ,XδDistance；
Remaining ω wolf pack and X are calculated by formula (2)~(4)α,Xβ,XδDistance；
Dα=| C1·Xα(t)-X(t)| (2)
Dβ=| C2·Xβ(t)-X(t)| (3)
Dδ=| C3·Xδ(t)-X(t)| (4)
In formula, t indicates that current evolutionary generation, X (t) indicate the position of contemporary wolf pack, Ci=2r1(i=1,2,3), r1∈
Random number between (0,1), C1Indicate the random perturbation to α wolf, C2Indicate the random perturbation to β wolf, C3Indicate to β wolf with
Machine disturbance.
Step5: α wolf, β wolf, the update position of δ wolf are calculated；
α wolf, β wolf, the update position of δ wolf are calculated by formula (5)~(7)；
X1=Xα-A1·Dα (5)
X2=Xβ-A2·Dβ (6)
X3=Xδ-A3·Dδ (7)
In formula, Ai=2r2A-a (i=1,2,3), r2Random number between ∈ (0,1), a are convergence factor, with generation of evolving
Several increases are gradually decremented to 0 from 2, its calculation formula is
Step6: according to the fitness function value of each grey wolf individual of leadership, the corresponding dynamic association of grey wolf individual is calculated
Make weight factor；
The dynamic cooperative weight factor of each grey wolf individual is calculated by formula (8)~(10)；
In formula, fitα、fitβAnd fitδRespectively α wolf in generation evolution, β wolf, the fitness function value of δ wolf.
Step7: the position of next generation wolf pack is determined；
The position of next-generation wolf pack is determined by formula (11)；
X (t+1)=ω1X1(t)+ω2X2(t)+ω3X3(t) (11)。
Step8: judging whether algorithm reaches maximum evolution number, i.e. differentiation t >=itermax, if satisfied, algorithm terminates simultaneously
Export optimal solution Xα；Otherwise, t=t+1 returns to Step2.
The beneficial effects of the present invention are:
The present invention uses elite guiding mechanism, and group's Personal campaigns for out the α of leadership by operations such as cooperation, competitions
Wolf, β wolf, δ wolf, under the guidance of three leadership wolves, the individual in population is constantly close to target wolf, evolves by number generation,
It is finally exactly the optimal solution of algorithm searching closest to the head wolf position of prey.And in order to emphasize a wolf α wolf to the absolute of wolf pack
Leadership and directive function, the invention proposes dynamic cooperative mechanism, are being led according to the fitness function value of each wolf of leadership
Shared specific gravity in all wolf fitness function values of conducting shell, to determine that each leader wolf instructs specific gravity to wolf pack.In this way into
In the every generation changed, each leadership wolf instructs specific gravity to be determined by the fitness function value of itself, is emphasizing the absolute of a wolf
While guiding role, the effect of other two leaders is not ignored yet.Moreover, with the variation of evolutionary generation, leadership is each
The specific gravity that instructs of wolf is dynamic change, this dynamic change be conducive to algorithm done between global optimum and local optimum it is flat
Weighing apparatus, is also beneficial to algorithm fast convergence.
The present invention is used for the grey wolf optimization algorithm based on dynamic cooperative to carry out Pattern Synthesis of Antenna Array, by being poised for battle
The excitation amplitude and phase of each array element of array antenna optimize, and under the premise of main lobe width does not broaden, are formed and are lower than index
It is required that minor level and wave beam.
Grey wolf optimization algorithm is used for array of designs Antenna measuring table problem for the first time by the present invention, and for substantially grey
That when solving Pattern Synthesis of Antenna Array problem, there are convergence rates is slow for wolf optimization algorithm, is easily trapped into local optimum etc. asks
Topic, proposes the grey wolf optimization algorithm based on dynamic cooperative.Grey wolf optimization algorithm is the hierarchy stringent based on grey wolf group
With cooperation hunting mode, and generate follow the novel Swarm Intelligent Algorithm that leadership wolf hunts for one's prey.The present invention proposes
Dynamic cooperative mechanism and dynamic cooperative weight factor, each leadership wolf instructs specific gravity dynamic cooperative weight factor to wolf pack
It determines, depends primarily on the fitness function value of each wolf itself, fitness function value is more excellent, bigger to the influence power of wolf pack.
This dynamic cooperative mechanism is not ignored other while emphasizing a wolf to the absolute leadership status of wolf pack and directive function yet
The leading role of Liang Ge leadership wolf.Moreover, with the variation of evolutionary generation, the fitness function value of each wolf in leadership wolf
It is dynamic change, therefore corresponding dynamic cooperative weight factor is also dynamic change, this dynamic change is conducive to algorithm
Coordinate between global optimum and local optimum and balance, is also beneficial to algorithm and quickly and effectively converges to optimal solution.
Detailed description of the invention
Fig. 1 is step flow chart of the invention；
Fig. 2 is the directional diagram that dynamic cooperative grey wolf optimization algorithm proposed by the present invention obtains；
Fig. 3 is the directional diagram that original grey wolf optimization algorithm obtains；
Fig. 4 is the directional diagram that particle swarm optimization algorithm obtains.
Specific embodiment
With reference to the accompanying drawings and detailed description, the invention will be further described.
A kind of Pattern Synthesis of Antenna Array design method based on dynamic cooperative grey wolf optimization algorithm, during per generation evolves,
Determining α wolf, β wolf, after the fitness function value of δ wolf and position, not directly to α wolf, the position of β wolf, δ wolf is averaged,
But dynamic cooperative weight factor is calculated according to fitness function value, the position of these three wolves is weighted, is then carried out again
Location updating.
Specific steps are as follows:
Step1: in solution space initialization population, initial evolutionary generation t=1 is set, maximum evolutionary generation is itermax；
Specifically, population scale N is provided, maximum number of iterations itermax, problem dimension D and problem feasible solution it is empty
Between, one group of initial solution X=[X is randomly generated1,X2,…,XN], wherein N is population scale, Xi=[xi1,xi2,…,xiD], i=1,
2 ... N are i-th of grey wolf individual.
Step2: the fitness function value of each grey wolf individual is calculated；
The fitness function value of each grey wolf individual is calculated by formula (1)；
In formula, PSL is Peak sidelobe level (Peak Sidelobe Level, abbreviation PSL), and TPSL is target peak pair
Valve level (Target Peak Sidelobe Level, abbreviation TPSL), NSLL are deep null level (Nulling Sidelobe
Level), the selection principle of weight coefficient α and β is: according to the directional diagram of each design section close to the degree of design object, evolving
Slow region can suitably increase weight coefficient, by testing repeatedly, determine the value of α and β, and meet alpha+beta=1.
Step3: from big to small according to fitness function value, α wolf, β wolf, δ wolf are successively selected, and its current location is denoted as
Xα,Xβ,Xδ, and the X that fitness value is bestαIt is denoted as optimal solution；
Step4: residue ω wolf pack and X are calculatedα,Xβ,XδDistance；
Remaining ω wolf pack and X are calculated by formula (2)~(4)α,Xβ,XδDistance；
Dα=| C1·Xα(t)-X(t)| (2)
Dβ=| C2·Xβ(t)-X(t)| (3)
Dδ=| C3·Xδ(t)-X(t)| (4)
In formula, t indicates that current evolutionary generation, X (t) indicate the position of contemporary wolf pack, Ci=2r1(i=1,2,3), r1∈
Random number between (0,1), C1Indicate the random perturbation to α wolf, C2Indicate the random perturbation to β wolf, C3Indicate to β wolf with
Machine disturbance.
Step5: α wolf, β wolf, the update position of δ wolf are calculated；
α wolf, β wolf, the update position of δ wolf are calculated by formula (5)~(7)；
X1=Xα-A1·Dα (5)
X2=Xβ-A2·Dβ (6)
X3=Xδ-A3·Dδ (7)
In formula, Ai=2r2A-a (i=1,2,3), r2Random number between ∈ (0,1), a are convergence factor, with generation of evolving
Several increases are gradually decremented to 0 from 2, its calculation formula is
Step6: according to the fitness function value of each grey wolf individual of leadership, the corresponding dynamic association of grey wolf individual is calculated
Make weight factor；
The dynamic cooperative weight factor of each grey wolf individual is calculated by formula (8)~(10)；
In formula, fitα、fitβAnd fitδRespectively α wolf in generation evolution, β wolf, the fitness function value of δ wolf.
Step7: the position of next generation wolf pack is determined；
The position of next-generation wolf pack is determined by formula (11)；
X (t+1)=ω1X1(t)+ω2X2(t)+ω3X3(t) (11)。
Step8: judging whether algorithm reaches maximum evolution number, i.e. differentiation t >=itermax, if satisfied, algorithm terminates simultaneously
Export optimal solution Xα；Otherwise, t=t+1 returns to Step2.
The present invention carries out comprehensive design, design to array aerial direction figure using the grey wolf optimization algorithm based on dynamic cooperative
Object is the equidistant line array being made of 2N ideal point source.Cost is calculated in order to reduce optimized variable, reduce, the present invention adopts
With exciting current amplitude central symmetry and excitation phase is that the line array of zero (while penetrating battle array) is designed, directional diagram expression formula
As shown in formula (12).Design object are as follows: in the case where guaranteeing that array main lobe width is constant, formed in minor lobe area and be lower than desired value
Minor level, and form deep null in designated position.
Wherein, InFor the excitation amplitude of n-th of array element, the spacing of k=2 π/λ, d between array element, λ is free space wavelength,
The present invention takes d=0.5 λ, angle of the θ between directions of rays and array axis.
Below with reference to emulation experiment, invention is further explained.
1, experiment condition:
Be Intel i5-7200U in CPU, inside save as 8G DDR4, in 7 operating system of Win using MATLAB 2016 into
Row emulation.Algorithm proposed by the present invention is used for Pattern Synthesis of Antenna Array problem below.
2, experiment content:
Using algorithm flow chart shown in FIG. 1, the centrosymmetric line array of current amplitude described in formula (12) is carried out
Pattern Synthesis design, wherein λ/2 2N=20, array element spacing d=, main lobe width are 2 θ0=20 °, minor lobe reduces in order to prevent
While there is the case where main lobe broadening, required in design in the case where minor level is lower than -35dB, in first zero position
80 ° (symmetric position is 100 °) generate the deep null for being lower than -100dB, but also require to generate in 60 ° 65 ° 70 ° 75 ° of angle
Deep null lower than -100dB resists strongly disturbing purpose to reach.Algorithm proposed by the present invention and original grey wolf optimization algorithm with
And particle swarm algorithm is used for this array antenna and carries out Pattern Synthesis design, three kinds of algorithms carry out the evolution of 500 generations.
3, experimental result:
Fig. 2 is the array pattern optimized using dynamic cooperative grey wolf optimization algorithm proposed by the present invention, can be seen
Main lobe width is 20 ° out, and the maximum level in minor lobe area is -33.3320dB, and in five deep nulls, there are four be lower than -100dB
Deep null, most advanced null appears in 80 ° of position, and null level is -97.8701dB.Optimized with Fig. 3 using original grey wolf
The antenna radiation pattern that algorithm obtains is compared, it can be seen that the antenna radiation pattern performance that algorithm proposed by the present invention obtains is more
Good, highest minor level reduces 2.8274dB, and most advanced null level reduces 5.4038dB.It is excellent using population with Fig. 4
Change the result that algorithm obtains to be compared, the antenna radiation pattern highest minor level that algorithm proposed by the present invention obtains reduces
3.8852dB, most advanced null level reduce 19.2887dB.Further demonstrate the validity of algorithm proposed by the present invention.
Comparison between three can be shown in Table 1.
Table 1
In order to verify the validity of the proposed algorithm of the present invention, the grey wolf optimization algorithm based on dynamic cooperative is used for by the present invention
Pattern Synthesis of Antenna Array is carried out, is optimized by the excitation amplitude and phase to each array element of array antenna, in main lobe
Under the premise of width does not broaden, the directional diagram for meeting design object is formd.Moreover, simulation result and the optimization of original grey wolf are calculated
The simulation result of method and particle swarm optimization algorithm carries out parameter comparison, further demonstrates that proposed by the present invention based on dynamic cooperative
Grey wolf optimization algorithm for array of designs Antenna measuring table problem be a kind of very effective method.
In conclusion grey wolf optimization algorithm is applied in Pattern Synthesis of Antenna Array by the present invention for the first time, and original
The grey wolf optimization algorithm based on dynamic cooperative is proposed on the basis of grey wolf optimization algorithm, which is inhibiting the same of minor level
When in multiple designated positions form deep null, show than original grey wolf optimization algorithm and particle swarm optimization algorithm more preferably
Performance.
Dynamic cooperative grey wolf optimization algorithm proposed by the present invention, performance stablize, frame is succinct, have very strong versatility and
Portability can be applied in the optimization problem of related fields, can also be embedded into other algorithm, not only array antenna
Pattern Synthesis problem proposes new idea and method, but also has effectively expanded the application depth of grey wolf optimization algorithm and wide
Degree.
In conjunction with attached drawing, the embodiment of the present invention is explained in detail above, but the present invention is not limited to above-mentioned
Embodiment within the knowledge of a person skilled in the art can also be before not departing from present inventive concept
Put that various changes can be made.
Claims (7)
1. a kind of Pattern Synthesis of Antenna Array design method based on dynamic cooperative grey wolf optimization algorithm, it is characterised in that specific
Step are as follows:
Step1: in solution space initialization population, initial evolutionary generation t=1 is set, maximum evolutionary generation is itermax；
Step2: the fitness function value of each grey wolf individual is calculated；
Step3: from big to small according to fitness function value, α wolf, β wolf, δ wolf are successively selected, and its current location is denoted as Xα,
Xβ,Xδ, and the X that fitness value is bestαIt is denoted as optimal solution；
Step4: residue ω wolf pack and X are calculatedα,Xβ,XδDistance；
Step5: α wolf, β wolf, the update position of δ wolf are calculated；
Step6: according to the fitness function value of each grey wolf individual of leadership, the corresponding dynamic cooperative power of grey wolf individual is calculated
Repeated factor；
Step7: the position of next generation wolf pack is determined；
Step8: judging whether algorithm reaches maximum evolution number, i.e. differentiation t >=itermax, if satisfied, algorithm terminates and exports
Optimal solution Xα；Otherwise, t=t+1 returns to Step2.
2. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that the Step1 specifically: provide population scale N, maximum number of iterations itermax, problem dimension D and
One group of initial solution X=[X is randomly generated in the solution space of problem1,X2,…,XN], wherein N is population scale, Xi=[xi1,
xi2,…,xiD], i=1,2 ... N are i-th of grey wolf individual.
3. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that: calculate the fitness function value of each grey wolf individual in the Step2 by formula (1)；
In formula, PSL is Peak sidelobe level, and TPSL is target peak minor level, and NSLL is deep null level, and α and β are power
Weight coefficient.
4. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that: calculate remaining ω wolf pack and X in the Step4 by formula (2)~(4)α,Xβ,XδDistance；
Dα=| C1·Xα(t)-X(t)| (2)
Dβ=| C2·Xβ(t)-X(t)| (3)
Dδ=| C3·Xδ(t)-X(t)| (4)
In formula, t indicates that current evolutionary generation, X (t) indicate the position of contemporary wolf pack, Ci=2r1(i=1,2,3), r1∈ (0,1)
Between random number, C1Indicate the random perturbation to α wolf, C2Indicate the random perturbation to β wolf, C3Indicate disturbing at random to β wolf
It is dynamic.
5. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that: calculate α wolf, β wolf, the update position of δ wolf in the Step5 by formula (5)~(7)；
X1=Xα-A1·Dα (5)
X2=Xβ-A2·Dβ (6)
X3=Xδ-A3·Dδ (7)
In formula, Ai=2r2A-a (i=1,2,3), r2Random number between ∈ (0,1), a is convergence factor, with evolutionary generation
Increase from 2 and is gradually decremented to 0, its calculation formula is
6. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that: calculate the dynamic cooperative weight factor of each grey wolf individual in the Step6 by formula (8)~(10)；
In formula, fitα、fitβAnd fitδRespectively α wolf in generation evolution, β wolf, the fitness function value of δ wolf.
7. the Pattern Synthesis of Antenna Array design side according to claim 1 based on dynamic cooperative grey wolf optimization algorithm
Method, it is characterised in that: determine the position of next-generation wolf pack in the Step7 by formula (11)；
X (t+1)=ω1X1(t)+ω2X2(t)+ω3X3(t) (11)。
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| CN111142553A (en) * | 2019-12-11 | 2020-05-12 | 北京航空航天大学 | Unmanned aerial vehicle cluster autonomous task allocation method based on biological predation energy model |
| CN111787543A (en) * | 2020-06-10 | 2020-10-16 | 杭州电子科技大学 | 5G communication system resource allocation method based on improved wolf optimization algorithm |
| CN111914427A (en) * | 2020-08-10 | 2020-11-10 | 哈尔滨工程大学 | Multi-constraint rectangular array sparse optimization method based on area normalization strategy |
| CN111914427B (en) * | 2020-08-10 | 2022-08-02 | 哈尔滨工程大学 | Multi-constraint rectangular array sparse optimization method based on area normalization strategy |
| CN112488283A (en) * | 2020-12-11 | 2021-03-12 | 湖北工业大学 | Improved multi-target grey wolf optimization algorithm |
| CN112488283B (en) * | 2020-12-11 | 2024-03-22 | 湖北工业大学 | Improved multi-objective gray wolf optimization algorithm implementation method |
| CN115248591A (en) * | 2021-12-28 | 2022-10-28 | 齐齐哈尔大学 | UUV path planning method based on hybrid initialization Hui wolf particle swarm algorithm |
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