CN108508424A - A kind of Sidelobe based on antenna array error answers weight vector optimization method - Google Patents
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Abstract
The invention belongs to radar signal processing field, discloses a kind of Sidelobe based on antenna array error and answer weight vector optimization method, include the following steps：It is even linear array to establish array antenna model, obtains the pattern function expression formula of the even linear array；According to the pattern function expression formula of the even linear array, the pattern function expression formula for considering amplitude phase error is obtained, the pattern function expression formula for considering amplitude phase error is the function expression about multiple weight vector；According to the pattern function expression formula for considering amplitude phase error, the objective cost function about multiple weight vector is established；The objective cost function is solved using differential evolution algorithm, the multiple weight vector after being optimized solves considering array amplitude phase error, and antenna side lobe level is to error more sensitive issue.
Description
Technical field
The invention belongs to Radar Signal Processing Technology field more particularly to a kind of Sidelobe based on antenna array error are multiple
Weight vector optimization method is considering array error, using differential evolution algorithm to array antenna sidelobe level and main lobe
Width optimizes, and reaches target call.
Background technology
Sidelobe is one of important technology index of radar antenna.This characteristic can not only overcome noise jamming, but also
The probability found by enemy can be reduced.If not making any weighting to the excitation of array antenna to handle, the first minor level
Theoretical value is about -13.5dB, and requirement of the phased-array radar to minor level is not achieved.During actual design antenna,
Inevitably introduce random error so that the Aperture distribution of array changes, and directly affects the performance of antenna array.At random
The introducing of error finally may appear as the range error and phase error of array each unit, therefore need to study in Random amplitude
Under the influence of phase error, the optimization problem of array antenna minor level.
Solve the problems, such as Sidelobe at present is commonly divided into analytic method and numerical method.Classical analytic method cuts ratio
Husband's integrated approach, Taylor's integrated approach etc. are avenged, this kind of method design is simple, but due to the excitation amplitude of array both ends array element
Array element difference adjacent thereto is larger, and prodigious difficulty, and the slight error of array element excitation amplitude are brought to the feed of antenna
Sidelobe level will be made to generate prodigious fluctuation.
In the evolution algorithm inside numerical method, differential evolution algorithm compared to population and genetic algorithm for, it is whole
Body performance is more excellent.In terms of coding standard, genetic algorithm uses binary coding, compares the reality of population and differential evolution algorithm
There is certain error for number encoder；In terms of parameter setting, differential evolution algorithm only needs to adjust there are two parameter, and parameter
Adjustment result is influenced little, and the parameter of population and genetic algorithm is more, and different parameters is to convergence rate and too early
Converge to being affected for Local Extremum；For higher-dimension problem, genetic algorithm convergence rate is very slow cannot even to be restrained, still
Population and differential evolution algorithm then can be very good to solve, especially differential evolution algorithm, convergent very fast and result compared with
Accurately；In terms of constringency performance, for optimization problem, relative to genetic algorithm, differential evolution algorithm and particle cluster algorithm convergence
Speed, but population is easily trapped into Local Extremum and unstable.
Invention content
In view of the above-mentioned problems, the purpose of the present invention is to provide a kind of Sidelobes based on antenna array error to answer weight vector
Optimization method solves the problems, such as that the sidelobe level of conventional method directional diagram is very sensitive to array element error.
In order to achieve the above objectives, the present invention is realised by adopting the following technical scheme.
A kind of Sidelobe based on antenna array error answers weight vector optimization method, and described method includes following steps：
Step 1, it is even linear array to establish array antenna model, obtains the pattern function expression formula of the even linear array；
Step 2, according to the pattern function expression formula of the even linear array, the pattern function for considering amplitude phase error is obtained
Expression formula, the pattern function expression formula for considering amplitude phase error are the function expression about multiple weight vector；
Step 3, according to the pattern function expression formula for considering amplitude phase error, the target generation about multiple weight vector is established
Valence function；
Step 4, the objective cost function, the multiple weight vector after being optimized are solved using differential evolution algorithm.
The present invention has the characteristics that compared with prior art：
(1) compared with prior art, the present invention considering the channel error of array when establishing aerial array model.
During actual design antenna, array error is inevitably introduced.Array error can be caused by many factors, such as：
The amplitude and phase error of multiple weight vector, influence of the channel frequency response inconsistency (Channel Mismatch) to system performance, sense
Evaluated error, the quantization error of weight vector, error etc. caused by individual array elements break down, either electrical error or machinery
Foozle can finally be attributed to systematic error and random error.It generally can be relatively easily to part system error
Influence is assessed and is corrected, and the generation of random error imprevision (for example weather, temperature, the reasons such as mismachining tolerance introduce
Random error) so that be difficult to correct.Random error can make the Aperture distribution of array change, and directly affect antenna array
Performance, and the introducing of random error finally may appear as the range error and phase error of array each unit, therefore
Technical solution of the present invention optimizes weighting with the amplitude phase error of compensated array by answering weight vector to Sidelobe.
2, compared with prior art, the present invention the multiple weight vector and Chebyshev that are optimized simultaneously using amplitude and phase are added
Power, the obtained real number weight vector of analytic methods such as Taylor's weighting can only compensation magnitude error compare, obtained optimal power is for width
Degree and phase error have certain compensatory, and these analytic methods are more sensitive for the error of aerial array, a bit
Point tolerance will make antenna radiation pattern generate distortion, therefore technical solution of the present invention optimizes direction simultaneously using amplitude and phase
The algorithm of figure has good robustness to the array antenna Sidelobe synthesis containing error.
3, compared with prior art, the present invention using evolution difference algorithm, compared to the particle cluster algorithm in evolution algorithm
And genetic algorithm, there are two a parameters that differential evolution algorithm utilizes, intersects factor CR and scale factor M, there is the stronger overall situation
Search capability, convergence and stability.
Description of the drawings
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technology description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
Some embodiments of invention for those of ordinary skill in the art without creative efforts, can be with
Obtain other attached drawings according to these attached drawings.
Fig. 1 provides to obtain a kind of Sidelobe based on antenna array error for the embodiment of the present invention and answers weight vector optimization method
Flow diagram；
Fig. 2 is even linear array model schematic provided in an embodiment of the present invention；
Fig. 3 is the implementation process schematic diagram of differential evolution algorithm provided in an embodiment of the present invention；
48 array element linear array antenna directional diagrams when Fig. 4 a are provided in an embodiment of the present invention error free；
DE algorithms corresponding convergence curve when Fig. 4 b are provided in an embodiment of the present invention error free；
48 array element linear array antenna directional diagrams when Fig. 5 a are 2% error provided in an embodiment of the present invention；
DE algorithms corresponding convergence curve when Fig. 5 b are 2% error provided in an embodiment of the present invention
The linear array directional diagram of PSO and DE when Fig. 6 a are 0% error provided in an embodiment of the present invention；
DE and PSO algorithms corresponding convergence curve when Fig. 6 b are 0% error provided in an embodiment of the present invention；
The linear array directional diagram of PSO and DE when Fig. 7 a are 2% error provided in an embodiment of the present invention；
DE and PSO algorithms corresponding convergence curve when Fig. 7 b are 2% error provided in an embodiment of the present invention.
Specific implementation mode
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation describes, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts every other
Embodiment shall fall within the protection scope of the present invention.
The embodiment of the present invention provides a kind of Sidelobe based on antenna array error and answers weight vector optimization method, such as Fig. 1 institutes
Show, described method includes following steps：
Step 1, it is even linear array to establish array antenna model, obtains the pattern function expression formula of the even linear array.
The step 1 specifically includes：
As shown in Fig. 2, the array antenna model is set as the even linear array that is made of N number of array element, and array element spacing d is
Half-wavelength, to obtain the pattern function expression formula F (θ) of the even linear array：
Wherein, k is wave number (radian/length),ΨnPhase for n-th of array element relative to first array element
Difference, Ψn=nkd [sin (θ)-sin (θ0)], θ is arrival bearing, θ0For the beam position direction of even linear array, w is that multiple power is sweared
Amount, w=[w1, w2..., wn..., wN]T, wnFor the corresponding multiple token amount of n-th of array element, A (θ) is array steering vector matrix,
Under narrowband condition, geometry (known) and the arrival bearing (θ) of array, A (θ)=[a are only depended on1(θ), a2(θ) ..., an
(θ) ..., aN(θ)]T, an(θ) is the corresponding steering vector of n-th of array element,N is the battle array that even linear array includes
First total number, d are the array element spacing of even linear array.
Step 2, according to the pattern function expression formula of the even linear array, the pattern function for considering amplitude phase error is obtained
Expression formula, the pattern function expression formula for considering amplitude phase error are the function expression about multiple weight vector.
The step 2 specifically includes：
According to the pattern function expression formula F (θ) of the even linear array, the pattern function table for considering amplitude phase error is obtained
Up to formula Ferr(θ)：
Wherein, wherein k is wave number (radian/length),ΨnIt is n-th of array element relative to first array element
Phase difference, Ψn=nkd [sin (θ)-sin (θ0)], θ is arrival bearing, θ0For the beam position direction of even linear array, w is multiple power
Vector, w=[w1, w2..., wn..., wN]T, wnFor the corresponding multiple token amount of n-th of array element, Aerr(θ) is to consider amplitude phase error
Array steering vector matrix, Aerr(θ)=[a1err(θ), a2err(θ) ... anerr(θ) ..., aNerr(θ)]T, anerr(θ) is n-th
The steering vector of consideration amplitude phase error corresponding to array element,ΔanAnd ΔΦnIt is n-th respectively
The range error and phase error of a array element, the two mean value are 0, and variance is respectivelyWithN is the battle array that even linear array includes
First total number, d are the array element spacing of even linear array.
Step 3, according to the pattern function expression formula for considering amplitude phase error, the target generation about multiple weight vector is established
Valence function.
The step 3 specifically includes：
According to the pattern function expression formula F for considering amplitude phase errorerr(θ) establishes the target generation about multiple weight vector
Valence function is as follows：
Wherein, fitness (w) indicates the objective cost function value calculated by multiple weight vector w, expression formula min
Fitness (w)=α (SLLs(w)-SLLd)+β(θ0.5(w)-θd) meaning be ask so that α (SLLs(w)-SLLd)+β(θ0.5(w)-
θd) it is minimum when multiple weight vector w, s.t. indicates that constraints, α are the first error coefficient,β
For the second error coefficient,SLLs(w) the secondary lobe maximum level in actual direction figure, SLL are indicateds
(w)=max | wH·Aerr(θs) |, θsIndicate secondary lobe region, SLLdIndicate target side lobe levels, θ0.5(w) actual half-power is indicated
Beam angle, θdIndicate target half-power width, θ0For the beam position direction of even linear array, subscript H indicates conjugate transposition.
Step 4, the objective cost function, the multiple weight vector after being optimized are solved using differential evolution algorithm.
Assuming that array number is N, in the case of no array error, then array antenna symmetry can be utilized, to be optimized
The dimension of power be Dim=N/2；After considering array error, array antenna will lose symmetry, then the dimension for the power to be optimized is
Dim=N.
With reference to Fig. 3, the step 4 specifically includes following sub-step：
(4a) sets the dimension of the multiple weight vector w as N, represents the corresponding multiple token amount of an array element per one-dimensional, each
The amplitude value range of multiple token amount is [xamin, xamax], the phase value range of each token amount again is [xpmin, xpmax]；
Specifically, each the amplitude value range of token amount is [0.01,1] again, this range can be adjusted, for solving
Array element encourages the problem for feeding difficulty that maximum value and minimum value difference are brought greatly very much.
Initialization population：It sets i and indicates that i-th in population individual, i=1,2 ... NP, each individual indicate multiple and weigh arrow
Measure a kind of value of w, j indicates the jth dimension of each individual, j=1,2 ... N, and the dimensional table of each individual is given instructions in reply one in weight vector
The corresponding multiple token amount of a array element；T indicates that t for population, enables i=1, j=1, t=0；
The jth of i-th of individual ties up value in (4b) the 0th generation population：
Wherein,Indicate the jth dimension amplitude value of i-th of individual in the 0th generation population,It indicates i-th in the 0th generation population
The jth of individual ties up phase value, xamaxAnd xaminThe upper bound and the lower bound of amplitude, x are indicated respectivelypmaxAnd xpminPhase is indicated respectively
The upper bound of position and lower bound, rand indicate the random decimal between [0,1]；
(4c) enables the value of j add 1, repeats sub-step (4b), until obtaining the N-dimensional value of i-th of individual in the 0th generation populationForm i-th of individual in the 0th generation populationAnd i-th of individual is right
The fitness function value answered
(4d) resets j=1, and the value of i is enabled to add 1, sub-step (4b) and (4c) is repeated, until obtaining the NP in the 0th generation population
The corresponding fitness function value of individual and each individual；
The value that the value of i is set to 1, j is set to 1, and it is 1 to enable the value of t；
(4e) carries out mutation operation to t for i-th of individual in population, obtains t for i-th of variation in population
Body
Wherein,T is indicated respectively for arbitrary three Different Individuals in NP of population individual, M be scale because
Son；
T is carried out crossover operation by (4f) with t for i-th of individual in population for i-th of variation individual of population, is obtained
To t for i-th of intersection individual of population
The jth that t intersects individual for i-th in population ties up valueFor：
Wherein,Indicate that t becomes simultaneously individual for i-th in populationJth tie up value,Indicate t in population
I-th of individualJth tie up value, rand indicate [0,1] between random decimal, CR indicate intersect the factor, randn (N) table
Show the random integers in [1, N] range；
The N that enables the value of j take 1,2 successively ..., obtain t for population i-th intersect individual
(4g) is by t for i-th of intersection individual of populationWith t-1 for i-th of population individual fitness value into
Row compares, and the individual for selecting fitness value smaller is as t for i-th of individual of population
Fitness () indicates fitness function；
(4h) enables the value of i add 1, repeats sub-step (4e) to (4g), until obtaining NP individuals of the t for population；
Fitness value threshold value and maximum population algebraically is arranged in (4i), obtains t for fitness value in the NP individual of population
Minimum individual is as optimum individual；
By the optimum individual if the fitness value of the optimum individual is less than or equal to the fitness value threshold value
Corresponding multiple weight vector is as the multiple weight vector after final optimization pass；Alternatively, if the value of t is more than the maximum population algebraically, incite somebody to action
The optimum individual of last generation population is as the multiple weight vector after final optimization pass；
Otherwise, the value of t is enabled to add 1, the value that the value of i is set to 1, j is set to 1, returns to sub-step (4e).
The effect of the present invention is further illustrated by following l-G simulation test：
1, simulated conditions
Array Model：Using the even linear array of N=48, array element spacingAt equal intervals according to -90 °~90 ° by azimuth
It is divided into 1801 parts, 0.1 ° is divided between angle to construct array prevalence matrix A, therefore the i-th row of matrix AThe main lobe that array antenna is arranged is oriented to θ0=0 °.
2, emulation content is analyzed
Experiment 1
In the case where excitation range is 0.01 to 1, the optimal power that is obtained using DE optimization algorithms (differential evolution algorithm)
It comparing with Chebyshev's weighting, antenna array weights are at symmetry when due to not having error, so array number is N=48, I
The dimension of power that optimizes be Dim=24；When containing error, antenna array weights do not have symmetry, so there is error
When, the dimension for the power that we optimize is Dim=48.Target maximum side petal SLLd=-50dB, target half-power beam width are
θd=3.2 °.
The parameter of DE algorithms is：Population number NP=72 intersects factor CR=0.9, mutagenic factor M=0.5.
If Fig. 4 a are under no error condition, DE algorithms can obtain the Sidelobe as Chebyshev, and edge
The secondary lobe of array is relatively low compared to Chebyshev, and Fig. 4 b are algorithmic statement figures, it can be seen that when having reached target call, algorithm is received
It holds back stopping and reaches generation, convergence is very fast.
If Fig. 5 a are to contain the array aerial direction figure of 2% amplitude phase error in array element, it may be seen that cutting
It is affected by amplitude phase error than snow husband's weighting, under 2% error condition, major lobe of directional diagram width that Chebyshev weights
It it is 3.2 °, maximum sidelobe level is -38.86dB, and the major lobe of directional diagram width obtained with DE algorithm optimizations is 3 °, maximum secondary lobe
Level is -50dB, has reached the sidelobe level of target call, and main lobe reduces 0.2 °, compared to Chebyshev's maximum secondary lobe electricity
It is flat to reduce 11.2dB.Fig. 5 b are DE algorithmic statement figures, it can be seen that when having reached target call, algorithmic statement stopping changes
In generation, what is optimized when due to containing amplitude phase error is complex-valued weights, is real number without the power to be optimized when error, so iteration
Number is big compared to for Fig. 4 b, and convergence is also relatively slow.
Experiment 2
The optimal power that DE optimization algorithms obtain compares with improved Particle Swarm Algorithm, ibid, due to antenna array weights
At symmetry, so array number is N=48, the dimension for the power that we optimize when error free is Dim=24.Have and optimizes when error
The dimension of power is Dim=48.Target side lobe levels SLLd=-50dB, target half-power width θd=3.2 °.
The parameter of DE algorithms is：Population number NP=72 intersects factor CR=0.9, mutagenic factor F=0.5.
The parameter of improved Particle Swarm Algorithm (PSO) is：Studying factors c1=2, c2=2, maximum weight value wmax=0.8,
Minimum value wmin=0.3, speed upper bound νmax=0.8, Discontinuous Factors p=0.2.
If Fig. 6 a are under no error condition, DE algorithms can obtain target Sidelobe and main lobe width, and PSO algorithms are simultaneously
Globe optimum cannot be converged to, the ability of searching optimum that the convergence curve both from Fig. 6 b can be seen that DE algorithms is stronger,
Iteration can be generated compared with the figure of merit ninety-nine times out of a hundred, and the later stage can continue convergence and only slow, until reaching target function value end
Only iteration.And PSO algorithms are easily ensnared into Local Extremum, and jump out Local Extremum and successive ignition, later stage is needed to be more difficult to
Jump out Local Extremum.
It is that the main lobe width that DE algorithms can reach optimization is θ such as Fig. 7 a under the array element error condition containing 2%0.5=
3 °, maximum sidelobe level is SLLmax=-49.1dB has reached approximately target call.And the main lobe width that PSO algorithm optimizations go out
For θ0.5=3 °, maximum sidelobe level is SLLmax=-38.81dB.DE algorithms are improved compared to PSO algorithm maximum sidelobe levels
Nearly 12.2dB.The ability of searching optimum that can be seen that DE algorithms from the convergence curve of both Fig. 7 b is stronger, although later stage convergence is slow
Slowly it but still is evolving.And PSO algorithms are easily ensnared into Local Extremum early period, and jump out Local Extremum and need successive ignition,
Later stage is more difficult to jump out Local Extremum.
One of ordinary skill in the art will appreciate that：Realize that all or part of step of above method embodiment can pass through
The relevant hardware of program instruction is completed, and program above-mentioned can be stored in computer read/write memory medium, which exists
When execution, step including the steps of the foregoing method embodiments is executed；And storage medium above-mentioned includes：ROM, RAM, magnetic disc or CD
Etc. the various media that can store program code.
The above description is merely a specific embodiment, but scope of protection of the present invention is not limited thereto, any
Those familiar with the art in the technical scope disclosed by the present invention, can easily think of the change or the replacement, and should all contain
Lid is within protection scope of the present invention.Therefore, protection scope of the present invention should be based on the protection scope of the described claims.
Claims (5)
1. a kind of Sidelobe based on antenna array error answers weight vector optimization method, which is characterized in that the method includes such as
Lower step：
Step 1, it is even linear array to establish array antenna model, obtains the pattern function expression formula of the even linear array；
Step 2, according to the pattern function expression formula of the even linear array, the pattern function expression for considering amplitude phase error is obtained
Formula, the pattern function expression formula for considering amplitude phase error are the function expression about multiple weight vector；
Step 3, according to the pattern function expression formula for considering amplitude phase error, the target cost letter about multiple weight vector is established
Number；
Step 4, the objective cost function, the multiple weight vector after being optimized are solved using differential evolution algorithm.
2. a kind of Sidelobe based on antenna array error according to claim 1 answers weight vector optimization method, feature
It is, the step 1 specifically includes：
The even linear array that the array antenna model is made of N number of array element is set, and array element spacing d is half-wavelength, to
To the pattern function expression formula F (θ) of the even linear array：
Wherein, k is wave number,ΨnPhase difference for n-th of array element relative to first array element, Ψn=nkd [sin
(θ)-sin(θ0)], θ is arrival bearing, θ0For the beam position direction of even linear array, w is multiple weight vector, w=[w1, w2...,
wn..., wN]T, wnFor the corresponding multiple token amount of n-th of array element, A (θ) is array steering vector matrix, A (θ)=[a1(θ), a2
(θ) ..., an(θ) ..., aN(θ)]T, an(θ) is the corresponding steering vector of n-th of array element,N is even linear array
Including array element total number, d be even linear array array element spacing.
3. a kind of Sidelobe based on antenna array error according to claim 1 answers weight vector optimization method, feature
It is, the step 2 specifically includes：
According to the pattern function expression formula F (θ) of the even linear array, the pattern function expression formula for considering amplitude phase error is obtained
Ferr(θ)：
Wherein, k is wave number,ΨnPhase difference for n-th of array element relative to first array element, Ψn=nkd [sin
(θ)-sin(θ0)], θ is arrival bearing, θ0For the beam position direction of even linear array, w is multiple weight vector, w=[w1, w2...,
wn..., wN]T, wnFor the corresponding multiple token amount of n-th of array element, Aerr(θ) is the array steering vector matrix for considering amplitude phase error,
Aerr(θ)=[a1err(θ), a2err(θ) ... anerr(θ) ..., aNerr(θ)]T, anerr(θ) is consideration width corresponding to n-th of array element
The steering vector of phase error,ΔanAnd ΔΦnIt is the range error of n-th of array element respectively
And phase error, N are the array element total number that even linear array includes, d is the array element spacing of even linear array.
4. a kind of Sidelobe based on antenna array error according to claim 1 answers weight vector optimization method, feature
It is, the step 3 specifically includes：
According to the pattern function expression formula F for considering amplitude phase errorerr(θ) establishes the target cost letter about multiple weight vector
Number is as follows：
Wherein, fitness (w) indicates the objective cost function value calculated by multiple weight vector w, expression formula min fitness
(w)=α (SLLs(w)-SLLd)+β(θ0.5(w)-θd) meaning be ask so that α (SLLs(w)-SLLd)+β(θ0.5(w)-θd) minimum
When multiple weight vector w, s.t. indicate constraints, α be the first error coefficient,β is second
Error coefficient,SLLs(w) the secondary lobe maximum level in actual direction figure, SLL are indicateds(w)=max
|wH·Aerr(θs) |, θsIndicate secondary lobe region, SLLdIndicate target side lobe levels, θ0.5(w) actual Half Power Beamwidth is indicated
Degree, θdIndicate target half-power width, θ0For the beam position direction of even linear array, subscript H indicates conjugate transposition.
5. a kind of Sidelobe based on antenna array error according to claim 1 answers weight vector optimization method, feature
It is, the step 4 specifically includes following sub-step：
(4a) sets the dimension of the multiple weight vector w as N, and the corresponding multiple token amount of an array element, each multiple power are represented per one-dimensional
The amplitude value range of scalar is [xamin, xamax], the phase value range of each token amount again is [xpmin, xpmax]；
Initialization population：It sets i and indicates that i-th in population individual, i=1,2 ... NP, each individual indicate multiple weight vector w's
A kind of value, j indicate the jth dimension of each individual, j=1,2 ... N, and the dimensional table of each individual is given instructions in reply an array element in weight vector
Corresponding multiple token amount；T indicates that t for population, enables i=1, j=1, t=0；
The jth of i-th of individual ties up value in (4b) the 0th generation population
Wherein,Indicate the jth dimension amplitude value of i-th of individual in the 0th generation population,Indicate i-th of individual in the 0th generation population
Jth tie up phase value, xamaxAnd xaminThe upper bound and the lower bound of amplitude, x are indicated respectivelypmaxAnd xpminThe upper of phase is indicated respectively
Boundary and lower bound, rand indicate the random decimal between [0,1]；
(4c) enables the value of j add 1, repeats sub-step (4b), until obtaining the N-dimensional value of i-th of individual in the 0th generation populationForm i-th of individual in the 0th generation populationAnd i-th of individual is right
The fitness function value answered
(4d) resets j=1, and the value of i is enabled to add 1, repeats sub-step (4b) and (4c), until obtaining the NP in the 0th generation population
Body and the corresponding fitness function value of each individual；
The value that the value of i is set to 1, j is set to 1, and it is 1 to enable the value of t；
(4e) carries out mutation operation to t for i-th of individual in population, obtains t for i-th of variation individual in population
Wherein,T is indicated respectively for arbitrary three Different Individuals in the NP individual of population, and M is scale factor；
T is carried out crossover operation by (4f) for i-th in population individual and t for i-th of variation individual of population, obtains the
I-th intersection individuals of the t for population
The jth that t intersects individual for i-th in population ties up valueFor：
Wherein,Indicate t for i-th of variation individual in populationJth tie up value,Indicate t for i-th in population
IndividualJth tie up value, rand indicate [0,1] between random decimal, CR indicate intersect the factor, randn (N) indicate
Random integers in [1, N] range；
The N that enables the value of j take 1,2 successively ..., obtain t for population i-th intersect individual
(4g) is by t for i-th of intersection individual of populationCompared for the fitness value of i-th of individual of population with t-1
Compared with the individual for selecting fitness value smaller is as t for i-th of individual of population
Wherein, fitness () indicates fitness function；
(4h) enables the value of i add 1, repeats sub-step (4e) to (4g), until obtaining NP individuals of the t for population；
Fitness value threshold value and maximum population algebraically is arranged in (4i), and it is minimum for fitness value in the NP individual of population to obtain t
Individual as optimum individual；
If the fitness value of the optimum individual is less than or equal to the fitness value threshold value, the optimum individual is corresponded to
Multiple weight vector as the multiple weight vector after final optimization pass；, will be last alternatively, if the value of t is more than the maximum population algebraically
The optimum individual of generation population is as the multiple weight vector after final optimization pass；
Otherwise, the value of t is enabled to add 1, the value that the value of i is set to 1, j is set to 1, returns to sub-step (4e).
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