CN111191339A - Constrained multi-target intelligent optimization conversion method for solving antenna array comprehensive problem - Google Patents

Constrained multi-target intelligent optimization conversion method for solving antenna array comprehensive problem Download PDF

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CN111191339A
CN111191339A CN201911170078.7A CN201911170078A CN111191339A CN 111191339 A CN111191339 A CN 111191339A CN 201911170078 A CN201911170078 A CN 201911170078A CN 111191339 A CN111191339 A CN 111191339A
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焦儒旺
曾三友
许庆辉
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China University of Geosciences
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
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    • G06N3/004—Artificial life, i.e. computing arrangements simulating life
    • G06N3/006—Artificial life, i.e. computing arrangements simulating life based on simulated virtual individual or collective life forms, e.g. social simulations or particle swarm optimisation [PSO]
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    • H01—ELECTRIC ELEMENTS
    • H01Q—ANTENNAS, i.e. RADIO AERIALS
    • H01Q21/00—Antenna arrays or systems

Abstract

The invention discloses a constraint multi-objective intelligent optimization method for solving the comprehensive problem of an antenna array, which takes default values as a target, meanwhile, the constraint boundary is relaxed to contain the whole population in the initial stage, so that the initial population is temporarily considered to be feasible, the algorithm can focus on the balance of diversity and convergence without considering constraint conditions, at each generation, the relaxed constraint bounds are dynamically reduced, the contracted dynamic constraint bounds still contain most of the individuals in the population, therefore, most individuals in the current population can be regarded as feasible, so that a super-multi-target antenna array optimization problem with strong constraint is converted into a dynamic super-multi-target antenna array optimization problem with weak constraint or no constraint, and any existing unconstrained super-multi-target evolution algorithm can be embedded into the method to solve the super-multi-target antenna array comprehensive problem.

Description

Constrained multi-target intelligent optimization conversion method for solving antenna array comprehensive problem
Technical Field
The invention relates to the field of intelligent optimization, in particular to a constrained multi-objective intelligent optimization conversion method for solving an antenna array comprehensive problem.
Background
The objective of the antenna array synthesis problem is to find the appropriate excitation vectors and the appropriate layout of the antenna elements to produce the desired radiation pattern, which is a complex optimization problem with non-linearity, the array factor function being non-linear; multiple targets, one antenna array integrated design problem usually needs to optimize multiple targets, such as beam width, side lobe level, zero point, dynamic range of array element excitation amplitude, and the like; the antenna array comprehensive design optimization problem can be modeled into a constraint super-multi-objective optimization problem.
The problem of constrained super-multi-target array optimization is not solved well until now, mainly because in a high-dimensional multi-target space, not only the balance of diversity and convergence among multiple targets needs to be emphasized, but also the constraint conditions need to be satisfied. Most of the existing constraint hyper-multi-objective evolution algorithms firstly emphasize the satisfaction of the constraints and consider the balance of diversity and convergence after the constraints are satisfied. This easily causes the following two problems:
1) the feasibility is emphasized first, so that the population is easy to fall into a local infeasible area, and a feasible solution cannot be found by an algorithm; or converging the population to a local feasible region, but far away from the position of the constraint Pareto optimal solution set;
2) in a high-dimensional target space, a constraint hyper-multi-objective optimization problem usually has a plurality of disconnected feasible regions, and a method with preferential feasibility usually finds one or part of feasible regions and then stagnates in the local feasible region. Doing so results in a loss of diversity such that a complete Pareto optimal solution set for the target space cannot be found.
Disclosure of Invention
In view of the above, the present invention provides a constrained ultra-multi-objective intelligent optimization transformation method for solving an antenna array synthesis problem, which transforms a constrained ultra-multi-objective optimization problem into a dynamic weak constrained ultra-multi-objective optimization problem.
The invention discloses a constraint multi-objective intelligent optimization method for solving an antenna array comprehensive problem, which comprises the following steps of:
step 1: random initialization of population individuals P0Including port excitation and array unit layout, and calculating its target value and constraint value, relaxing the original constraint boundary to make the individuals in the whole initial population reach
Figure BDA0002288442080000021
The method is feasible;
step 2: dynamic scaling
Figure BDA0002288442080000022
Constraining the boundary, and generating a filial generation population by using a differential evolution operator;
and step 3: converting a constraint super-multi-target linear array optimization problem with 4 targets into a dynamic weak constraint super-multi-target linear array optimization problem with 5 targets, wherein the newly added target is a default value target; according to
Figure BDA0002288442080000023
Constraining Pareto dominance criterion, 5 objectives are optimized simultaneously using NSGA-III algorithm, specifically according to
Figure BDA0002288442080000024
Feasibility of clustering parent and offspring
Figure BDA0002288442080000025
Feasible set sum
Figure BDA0002288442080000026
Can not be collected when
Figure BDA0002288442080000027
When the feasible set is larger than or equal to the population size N, the method uses
Figure BDA0002288442080000028
Constraint Pareto dominance criterion and reference point based elitism selection mechanism
Figure BDA0002288442080000029
Feasible setSelecting N individuals to form a next generation parent population; when merging populations
Figure BDA00022884420800000210
When the number of feasible solutions is less than the population size N, all the solutions are added
Figure BDA00022884420800000211
The feasible solutions are directly added into the next generation parent population; then, in the rest
Figure BDA00022884420800000212
In the infeasible solution, sorting according to default values from small to large, and selecting the individuals with the minimum default values to be sequentially placed into the next generation parent population until the next generation parent population is filled;
and 4, step 4: and (4) increasing the evolution environment state by 1, repeating the steps 2-3 until the maximum evolution environment state S is reached, and outputting an optimal Pareto solution set of the linear array problem.
Further, the target in step 1 is the beam width FNBW, the maximum side lobe level MSLL, the NULL point and the line source length
Figure BDA0002288442080000031
The constraint conditions are beam width and zero depth, and the linear array optimization problem model is as follows:
Figure BDA0002288442080000032
wherein the content of the first and second substances,
Figure BDA0002288442080000033
two constraints for the target vector to be optimized
Figure BDA0002288442080000034
And
Figure BDA0002288442080000035
respectively representing beam width and zero depth, optimization variables
Figure BDA0002288442080000036
Including port excitation
Figure BDA0002288442080000037
And array cell layout
Figure BDA0002288442080000038
Wherein
Figure BDA0002288442080000039
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the initial point, n is the number of antenna units, n is 14,
Figure BDA00022884420800000310
a decision space is represented in the form of,
further, the calculation formulas of the target values in step 1 are respectively as follows:
(1) the maximum sidelobe level calculation formula is as follows:
Figure BDA00022884420800000311
Figure BDA00022884420800000312
wherein the content of the first and second substances,
Figure BDA00022884420800000313
is the value of the array factor in the theta direction, λ denotes the wavelength, aiRepresenting the excitation amplitude, d, of the ith array element portiIs the distance from the ith array unit to the initial point, j represents a complex number, n represents the number of antenna array units, n is 14, MSLL is the maximum side lobe level, thetaMSLLIndicating the azimuth angle of the maximum side lobe level, maxfIs composed of
Figure BDA00022884420800000314
Maximum value of (d);
(2) the zero NULL calculation formula is:
Figure BDA00022884420800000315
wherein the content of the first and second substances,
Figure BDA0002288442080000041
is the value of the array factor in the theta direction, NULL is zero, thetaNULLThe total number of zeros is 6: +/-30 degrees, +/-32.5 degrees and +/-35 degrees, and finally taking the worst zero value in the 6 angles as a third target in the linear array optimization problem model;
total length of line source
Figure BDA0002288442080000042
Is equal to the sum of the spacings of all antenna elements.
Further, the method for relaxing the original constraint boundary in step 1 is as follows:
initially linear array population P0The maximum default value of each constraint in the set as the dynamic of the initial slack
Figure BDA0002288442080000043
Bound boundaries, i.e.
Figure BDA0002288442080000044
i＝1,2，
Figure BDA0002288442080000045
Indicating the constraint that the ith needs to be satisfied,
Figure BDA0002288442080000046
an initial dynamic constraint boundary.
Further, in the step 2, the linear array dynamic constraint boundary
Figure BDA0002288442080000047
The shrinkage of (a) adopts an exponential decreasing formula in a simulated annealing algorithm:
Figure BDA0002288442080000048
where σ is a real number close to 0, σ ═ 1e-8, i denotes the ith constraint, s is the evolving environmental state, and the parameter Q isiAnd RiFrom initial dynamic constraint boundaries
Figure BDA0002288442080000049
And the dynamic constraint boundary under the final evolution environment state S
Figure BDA00022884420800000410
Determined by the following calculation formula:
Figure BDA00022884420800000411
further, in step 3, the default value is used as a new target, and the transformed linear array optimization problem has the following form:
Figure BDA00022884420800000412
wherein the content of the first and second substances,
Figure BDA0002288442080000051
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure BDA0002288442080000052
Two constraints
Figure BDA0002288442080000053
And
Figure BDA0002288442080000054
respectively representing beam width and zero depth, optimization variables
Figure BDA0002288442080000055
Including port excitation
Figure BDA0002288442080000056
And array cell layout
Figure BDA0002288442080000057
Wherein
Figure BDA0002288442080000058
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the origin, n is the number of antenna units, n is 14,
Figure BDA0002288442080000059
representing a decision space, CMaOP(s)Representing the transformed dynamic weak constraint 5 target array optimization problem under the environment state S, wherein S represents the evolution environment state, S is 0,1,2, …, S and S are the maximum evolution environment state, epsilon1 (s)、ε2 (s)Representing dynamics of two constraints
Figure BDA00022884420800000510
The boundary of the constraint is defined,
Figure BDA00022884420800000511
the dynamic constraint boundary is contracted from the evolution environment state S being 0 to the environment state S being 1, and finally gradually contracted to the final evolution environment state S being S, and the 5 th target is newly added
Figure BDA00022884420800000512
The violation values are used to evaluate the degree of constraint violation for an individual.
Further, the default expression is as follows:
Figure BDA00022884420800000513
wherein the content of the first and second substances,
Figure BDA00022884420800000514
indicates the ith requirementConstraint condition satisfied, P0Is the initial population individual.
Further, updating parent operation is carried out on the transformed dynamic weak constraint 5-target array optimization problem by adopting an NSGA-III algorithm, the dominance relation between every two individuals needs to be compared when the parent operation is updated, and the method defines
Figure BDA00022884420800000515
Constraining Pareto dominance criterion to compare relationships between two population individuals, for two individuals
Figure BDA00022884420800000516
And
Figure BDA00022884420800000517
it can be said that
Figure BDA00022884420800000518
Constraint Pareto governance
Figure BDA00022884420800000519
·
Figure BDA00022884420800000520
Is that
Figure BDA00022884420800000521
A feasible solution to
Figure BDA00022884420800000522
Is that
Figure BDA00022884420800000523
Infeasible solutions;
·
Figure BDA00022884420800000524
and
Figure BDA00022884420800000525
are all that
Figure BDA00022884420800000526
A feasible solution, however
Figure BDA00022884420800000527
Pareto rule
Figure BDA00022884420800000528
·
Figure BDA00022884420800000529
And
Figure BDA00022884420800000530
are all that
Figure BDA00022884420800000531
Cannot solve it, but
Figure BDA00022884420800000532
Is less than
Figure BDA00022884420800000533
A default value of;
further, in the step 4, the dynamic constraint boundary in the final evolution environment state S is 0, that is, the dynamic constraint boundary is the original constraint boundary, which means that the algorithm finds a constraint Pareto optimal solution set of the original constraint super-multi-target array optimization problem in the final evolution environment state, that is, the transformed dynamic weak constraint 5 target array optimization problem has the following form:
Figure BDA0002288442080000061
wherein the content of the first and second substances,
Figure BDA0002288442080000062
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure BDA0002288442080000063
Two constraints
Figure BDA0002288442080000064
And
Figure BDA0002288442080000065
respectively representing beam width and zero depth, optimization variables
Figure BDA0002288442080000066
Including port excitation
Figure BDA0002288442080000067
And array cell layout
Figure BDA0002288442080000068
Wherein
Figure BDA0002288442080000069
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the origin, n is the number of antenna units, n is 14,
Figure BDA00022884420800000610
representing the decision space, S represents the final evolution environment state, CMaOP(S)And representing the dynamic weak constraint 5 target array optimization problem in the final evolution environment state.
The technical scheme provided by the invention has the beneficial effects that:
(1) after the problem conversion, in each generation, one hyper-multi-objective evolution algorithm is used for searching the unconstrained complete Pareto front edge of the current generation as if the unconstrained hyper-multi-objective optimization problem is solved, so that the complete constrained Pareto front edge of the original problem can be found out;
(2) the satisfaction, diversity and convergence of constraint conditions are three important targets of constraint super-multi-target optimization problems, and are equally important; aiming at the most of the ultra-multi-target evolution algorithms, firstly emphasizing the satisfaction of the constraint and then considering the balance of diversity and convergence among the targets, the invention takes the default value as one target and can simultaneously consider the satisfaction of the constraint and the balance of the diversity and the convergence;
(3) in the early stage and the middle stage of the evolution, the dynamic constraint boundary of the proposed method is larger, the algorithm mainly finds the unconstrained Pareto front edge, in the later stage of the evolution, the dynamic constraint boundary is gradually reduced to 0, and under the pressure of the smaller dynamic constraint boundary, the population approaches from the unconstrained Pareto front edge to the constrained Pareto front edge, so that the population is favorable for local optimization of the population passing through the infeasible area in the early stage and the middle stage of the evolution.
(4) The conversion method provided by the invention is relatively universal, and any existing super-multi-target evolution algorithm can be embedded into the method to solve the constraint super-multi-target optimization problem and the super-multi-target antenna array comprehensive problem.
Drawings
FIG. 1 is a flow chart of a constrained multi-objective intelligent optimization transformation method for solving an antenna array comprehensive problem according to the present invention;
FIG. 2 is an illustration of a search process of the constrained hyper-multi-objective optimization method proposed by the present invention on a two-objective constrained optimization problem.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be further described with reference to the accompanying drawings.
Referring to fig. 1 and 2, the implementation of the present invention first generates an initial population, and all individuals are infeasible solutions and far away from feasible areas due to constraints. At this time, we will relax the constraint boundary, and the relaxed constraint boundary is called
Figure BDA0002288442080000071
Constraint boundaries, encompassing the entire population, which is equivalent to the entire population being considered feasible, i.e., to
Figure BDA0002288442080000072
It is feasible, and thus it is as if an unconstrained multi-objective optimization problem were solved. After generating the offspring population, the offspring population can be updated by using an unconstrained multi-target algorithmA generation of parent population. It should be noted that when updating parents, the default value is also taken as a target, which is beneficial to the population to be pushed to the place with good target value and small default value, and the target and the constraint are both considered. In the second generation of evolution, the scaling down is a bit dynamically
Figure BDA0002288442080000073
Bound the border, but leave most individuals in the population still
Figure BDA0002288442080000081
Within the constrained boundaries, the parent is also updated using an unconstrained multi-objective evolution algorithm. And thus, iteration is continued, and when the evolution is advanced to the last generation,
Figure BDA0002288442080000082
and (3) contracting the constraint boundary to the original constraint boundary, namely returning to the original constraint optimization problem, and advancing the population to the optimal Pareto front edge, wherein the specific steps are as follows:
step 1: random initialization of population individuals P0Including port excitation and array unit layout, and calculating its target value and constraint value, relaxing the original constraint boundary to make the individuals in the whole initial population reach
Figure BDA0002288442080000083
The method is feasible;
the objective of the optimization of the antenna array design is to find the proper excitation vector and the proper layout of the antenna elements to generate the desired radiation pattern, and the present invention uses a linear array as an application example, the array comprises 28 antenna elements, the spacing between every two antenna elements is not necessarily the same, and the array has 6 NULL points: plus or minus 30 degrees, plus or minus 32.5 degrees and plus or minus 35 degrees, the zero point depth is minus 60dB, and the first zero point beam width FNBW is 8.5 degrees. In the array synthesis optimization problem, 4 targets are included: beam width FNBW, maximum side lobe level MSLL, NULL, and line source length
Figure BDA0002288442080000084
The constraint being a beamWidth and zero depth, linear array optimization problem model as follows:
Figure BDA0002288442080000085
wherein the content of the first and second substances,
Figure BDA0002288442080000086
two constraints for the target vector to be optimized
Figure BDA0002288442080000087
And
Figure BDA0002288442080000088
respectively representing beam width and zero depth, optimization variables
Figure BDA0002288442080000089
Including port excitation
Figure BDA00022884420800000810
And array cell layout
Figure BDA00022884420800000811
Wherein
Figure BDA00022884420800000812
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the origin, n is the number of antenna units, n is 14,
Figure BDA00022884420800000813
a decision space is represented in the form of,
the calculation formulas of the target values are respectively as follows:
(1) the maximum side lobe level MSLL is calculated by the formula:
Figure BDA0002288442080000091
Figure BDA0002288442080000092
wherein the content of the first and second substances,
Figure BDA0002288442080000093
is the value of the array factor in the theta direction, λ denotes the wavelength, aiRepresenting the excitation amplitude, d, of the ith array element portiIs the distance from the ith array unit to the initial point, j represents a complex number, n represents the number of antenna array units, n is 14, MSLL is the maximum side lobe level, thetaMSLLIndicating the azimuth angle of the maximum side lobe level, maxfIs composed of
Figure BDA0002288442080000094
Maximum value of (d);
(2) the zero NULL calculation formula is:
Figure BDA0002288442080000095
wherein the content of the first and second substances,
Figure BDA0002288442080000096
is the value of the array factor in the theta direction, NULL is zero, thetaNULLThe total number of zeros is 6: +/-30 degrees, +/-32.5 degrees and +/-35 degrees, and finally taking the worst zero value in the 6 angles as a third target in the linear array optimization problem model;
total length of line source
Figure BDA0002288442080000097
Is equal to the sum of the spacings of all antenna elements,
initially linear array population P0The maximum default value of each constraint in the set as the dynamic of the initial slack
Figure BDA0002288442080000098
Bound boundaries, i.e.
Figure BDA0002288442080000099
i＝1,2，
Figure BDA00022884420800000910
Indicating the constraint that the ith needs to be satisfied,
Figure BDA00022884420800000911
an initial dynamic constraint boundary.
Step 2: dynamic scaling
Figure BDA00022884420800000912
Constraining the boundary, and generating a filial generation population by using a differential evolution operator;
linear array dynamic constraint boundaries
Figure BDA00022884420800000913
The shrinkage of (a) adopts an exponential decreasing formula in a simulated annealing algorithm:
Figure BDA00022884420800000914
where σ is a real number close to 0, σ ═ 1e-8, i denotes the ith constraint, s is the evolving environmental state, and the parameter Q isiAnd RiFrom initial dynamic constraint boundaries
Figure BDA00022884420800000915
And the dynamic constraint boundary under the final evolution environment state S
Figure BDA0002288442080000101
Determined by the following calculation formula:
Figure BDA0002288442080000102
and step 3: converting a constraint super-multi-target linear array optimization problem with 4 targets into a dynamic weak constraint super-multi-target linear array optimization problem with 5 targets, wherein the newly added target is a default value target; according to
Figure BDA0002288442080000103
Constraining Pareto dominance criterion, 5 objectives are optimized simultaneously using NSGA-III algorithm, specifically according to
Figure BDA0002288442080000104
Feasibility of clustering parent and offspring
Figure BDA0002288442080000105
Feasible set sum
Figure BDA0002288442080000106
Can not be collected when
Figure BDA0002288442080000107
When the feasible set is larger than or equal to the population size N, the method uses
Figure BDA0002288442080000108
Constraint Pareto dominance criterion and reference point based elitism selection mechanism
Figure BDA0002288442080000109
Selecting N individuals from the feasible set to form a next generation parent population; when merging populations
Figure BDA00022884420800001010
When the number of feasible solutions is less than the population size N, all the solutions are added
Figure BDA00022884420800001011
The feasible solutions are directly added into the next generation parent population; then, in the rest
Figure BDA00022884420800001012
In the infeasible solution, sorting according to default values from small to large, and selecting the individuals with the minimum default values to be sequentially placed into the next generation parent population until the next generation parent population is filled;
taking the default value as a new target, the transformed linear array optimization problem has the following form:
Figure BDA00022884420800001013
wherein the content of the first and second substances,
Figure BDA00022884420800001014
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure BDA00022884420800001015
Two constraints
Figure BDA00022884420800001016
And
Figure BDA00022884420800001017
respectively representing beam width and zero depth, optimization variables
Figure BDA00022884420800001018
Including port excitation
Figure BDA00022884420800001019
And array cell layout
Figure BDA00022884420800001020
Wherein
Figure BDA00022884420800001021
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the origin, n is the number of antenna units, n is 14,
Figure BDA0002288442080000111
representing a decision space, CMaOP(s)Representing the transformed dynamic weak constraint 5 target array optimization problem under the environment state S, wherein S represents the evolution environment state, S is 0,1,2, …, S and S are the maximum evolution environment state, epsilon1 (s)、ε2 (s)Representing dynamics
Figure BDA0002288442080000112
The boundary of the constraint is defined,
Figure BDA0002288442080000113
the dynamic constraint boundary is contracted from the evolution environment state S being 0 to the environment state S being 1, and finally gradually contracted to the final evolution environment state S being S, and the 5 th target is newly added
Figure BDA00022884420800001125
The violation values are used to evaluate the degree of constraint violation for an individual.
The default expression is as follows:
Figure BDA0002288442080000114
wherein the content of the first and second substances,
Figure BDA0002288442080000115
representing the i-th constraint, P, to be satisfied0Is the initial population individual.
Updating parent operation on the transformed dynamic weak constraint 5-target array optimization problem by adopting an NSGA-III algorithm, wherein the dominance relation between every two individuals needs to be compared when updating the parent operation
Figure BDA0002288442080000116
Constraining Pareto dominance criterion to compare relationships between two population individuals, for two individuals
Figure BDA0002288442080000117
And
Figure BDA0002288442080000118
it can be said that
Figure BDA0002288442080000119
Constraint Pareto governance
Figure BDA00022884420800001110
·
Figure BDA00022884420800001111
Is that
Figure BDA00022884420800001112
A feasible solution to
Figure BDA00022884420800001113
Is that
Figure BDA00022884420800001114
Infeasible solutions;
·
Figure BDA00022884420800001115
and
Figure BDA00022884420800001116
are all that
Figure BDA00022884420800001117
A feasible solution, however
Figure BDA00022884420800001118
Pareto rule
Figure BDA00022884420800001119
·
Figure BDA00022884420800001120
And
Figure BDA00022884420800001121
are all that
Figure BDA00022884420800001122
Cannot solve it, but
Figure BDA00022884420800001123
Is less than
Figure BDA00022884420800001124
A default value of;
and 4, step 4: and (4) increasing the evolution environment state by 1, repeating the steps 2-3 until the maximum evolution environment state S is reached, and outputting an optimal Pareto solution set of the linear array problem.
The dynamic constraint boundary in the final evolution environment state S is 0, which is the original constraint boundary, meaning that the algorithm finds the constrained Pareto optimal solution set of the original constrained super-multi-target array optimization problem in the final evolution environment state, that is, the transformed dynamic weak constrained 5-target array optimization problem has the following form:
Figure BDA0002288442080000121
wherein the content of the first and second substances,
Figure BDA0002288442080000122
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure BDA0002288442080000123
Two constraints
Figure BDA0002288442080000124
And
Figure BDA0002288442080000125
respectively representing beam width and zero depth, optimization variables
Figure BDA0002288442080000126
Including port excitation
Figure BDA0002288442080000127
And array cell layout
Figure BDA0002288442080000128
Wherein
Figure BDA0002288442080000129
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the origin, n is the number of antenna units, n is 14,
Figure BDA00022884420800001210
representing the decision space, S represents the final evolution environment state, CMaOP(S)And representing the dynamic weak constraint 5 target array optimization problem in the final evolution environment state.
The experimental results are as follows:
when the above antenna array synthesis problem is solved, the maximum evolution environment state of the 7 comparison algorithms is set to be S-5000. Because the theoretical optimal Pareto front of the antenna array problem is unknown, the non-dominated solution sets obtained by 7 algorithms, namely feasible solutions, are integrated to perform non-dominated sorting, and the finally obtained non-dominated solution set is used as the approximate Pareto front of the array optimization problem and is used for calculating the IGD index. For this array optimization problem, we first randomly sample 1000000 individuals in the decision space, and calculate their adaptive values respectively, so as to obtain the number of individuals that can be solved. The probability of feasibility of the problem was found to be less than 1E-4, as measured by dividing the number of feasible solutions by 1000000. It can be seen that its feasible region is particularly small and therefore a strongly constrained optimization problem.
TABLE 1 IGD mean values of the array synthesis problem for the proposed method and 6 advanced algorithms
Algorithm A-NSGA-III C-MOEA/D I-DBEA C-RVEA PPS C-AnD The invention
Mean value NaN 1.02e+01 4.59e+00 NaN 8.84e-01 NaN 4.92e-01
Table 1 shows the average value of IGD indexes of 30 runs when 7 algorithms solve the problem, NaN represents that the corresponding algorithm cannot continuously find a feasible solution, AnD the IGD value is smaller, which indicates that the result is better, AnD it is obvious from table 1 that 3 algorithms a-NSGA-III, C-rvaa AnD C-andd cannot continuously find a feasible solution in 30 runs because the 3 algorithms all adopt a constraint processing mode of feasible solution priority, AnD the algorithm is easily trapped in a local infeasible area by paying attention to only feasibility but not to useful information of a target. The method provided by the invention has obvious advantages in 4 other algorithms which can find feasible solutions. Compared with C-MOEA/D, I-DBEA and PPS, the Pareto optimal solution set found by the invention has better distribution and convergence, which can be reflected on the average value of IGD. While the other three algorithms, especially C-MOEA/D and I-DBEA, can also find feasible solutions, they find only a small part on the Pareto frontier, and the distribution is poor. In the design of the antenna, a multi-objective method can provide more choices for decision makers, and design schemes required by the decision makers can be selected according to the preferences of the decision makers, so that the distribution is important when the multi-objective optimization is solved. In summary, the above experimental results show that the method provided by the present invention is superior to the comparison algorithm in terms of processing constraints and quality of found solutions when solving the array synthesis problem, and the effectiveness of the method provided by the present invention is proved.
The features of the embodiments and embodiments described herein above may be combined with each other without conflict. The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (9)

1. A constraint multi-objective intelligent optimization method for solving an antenna array comprehensive problem is characterized by comprising the following steps:
step 1: random initialization of population individuals P0Including port excitation and array unit layout, and calculating its target value and constraint value, relaxing the original constraint boundary to make the individuals in the whole initial population reach
Figure FDA0002288442070000011
The method is feasible;
step 2: dynamic scaling
Figure FDA0002288442070000012
Constraining the boundary, and generating a filial generation population by using a differential evolution operator;
and step 3: converting a constraint super-multi-target linear array optimization problem with 4 targets into a dynamic weak constraint super-multi-target linear array optimization problem with 5 targets, wherein the newly added target is a default value target; according to
Figure FDA0002288442070000013
Constraining Pareto dominance criterion, 5 objectives are optimized simultaneously using NSGA-III algorithm, specifically according to
Figure FDA0002288442070000014
Feasibility will father and sonCombining the populations into
Figure FDA0002288442070000015
Feasible set sum
Figure FDA0002288442070000016
Can not be collected when
Figure FDA0002288442070000017
When the feasible set is larger than or equal to the population size N, the method uses
Figure FDA0002288442070000018
Constraint Pareto dominance criterion and reference point based elitism selection mechanism
Figure FDA0002288442070000019
Selecting N individuals from the feasible set to form a next generation parent population; when merging populations
Figure FDA00022884420700000110
When the number of feasible solutions is less than the population size N, all the solutions are added
Figure FDA00022884420700000111
The feasible solutions are directly added into the next generation parent population; then, in the rest
Figure FDA00022884420700000112
In the infeasible solution, sorting according to default values from small to large, and selecting the individuals with the minimum default values to be sequentially placed into the next generation parent population until the next generation parent population is filled;
and 4, step 4: and (4) increasing the evolution environment state by 1, repeating the steps 2-3 until the maximum evolution environment state S is reached, and outputting an optimal Pareto solution set of the linear array problem.
2. The constrained hyper-multi-objective intelligence for solving antenna array synthesis problems as recited in claim 1The optimization method is characterized in that the 4 targets in the step 1 are beam width FNBW, maximum side lobe level MSLL, zero NULL and line source length
Figure FDA00022884420700000113
The constraint conditions are beam width and zero depth, and the linear array optimization problem model is as follows:
Figure FDA0002288442070000021
wherein the content of the first and second substances,
Figure FDA0002288442070000022
two constraints for the target vector to be optimized
Figure FDA0002288442070000023
And
Figure FDA0002288442070000024
respectively representing beam width and zero depth, optimization variables
Figure FDA0002288442070000025
Including port excitation
Figure FDA0002288442070000026
And array cell layout
Figure FDA0002288442070000027
Wherein
Figure FDA0002288442070000028
Figure FDA0002288442070000029
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the initial point, n is the number of antenna units, n is 14,
Figure FDA00022884420700000210
a decision space is represented.
3. The method for constrained multi-objective intelligent optimization for solving antenna array synthesis problem according to claim 2, wherein the target values in step 1 are calculated by the following formulas:
(1) the maximum sidelobe level calculation formula is as follows:
Figure FDA00022884420700000211
Figure FDA00022884420700000212
wherein the content of the first and second substances,
Figure FDA00022884420700000213
is the value of the array factor in the theta direction, λ denotes the wavelength, aiRepresenting the excitation amplitude, d, of the ith array element portiIs the distance from the ith array unit to the initial point, j represents a complex number, n represents the number of antenna array units, n is 14, MSLL is the maximum side lobe level, thetaMSLLIndicating the azimuth angle of the maximum side lobe level, maxfIs composed of
Figure FDA00022884420700000214
Maximum value of (d);
(2) the zero NULL calculation formula is:
Figure FDA00022884420700000215
wherein the content of the first and second substances,
Figure FDA00022884420700000216
is the value of the array factor in the theta direction, thetaNULLThe total number of zeros is 6: 30 °, ± 32.5 ° and+/-35 degrees, and finally taking the worst zero value in the 6 angles as a third target in the linear array optimization problem model;
total length of line source
Figure FDA0002288442070000031
Is equal to the sum of the spacings of all antenna elements.
4. The method for intelligent optimization of constraint and multi-objective for solving antenna array synthesis problem as claimed in claim 1, wherein the method for relaxing original constraint boundary in step 1 is as follows:
initially linear array population P0The maximum default value of each constraint in the set as the dynamic of the initial slack
Figure FDA00022884420700000310
Bound boundaries, i.e.
Figure FDA0002288442070000032
i＝1,2，
Figure FDA0002288442070000033
Indicating the constraint that the ith needs to be satisfied,
Figure FDA0002288442070000034
an initial dynamic constraint boundary.
5. The method as claimed in claim 1, wherein in step 2, the linear array dynamic constraint boundary is defined
Figure FDA0002288442070000035
The shrinkage of (a) adopts an exponential decreasing formula in a simulated annealing algorithm:
Figure FDA0002288442070000036
where σ is a real number close to 0, σ ═ 1e-8, i denotes the ith constraint, s is the evolving environmental state, and the parameter Q isiAnd RiFrom initial dynamic constraint boundaries
Figure FDA0002288442070000037
And the dynamic constraint boundary under the final evolution environment state S
Figure FDA0002288442070000038
Determined by the following calculation formula:
Figure FDA0002288442070000039
6. the method as claimed in claim 1, wherein in the step 3, if the default value is used as the new added target, the transformed linear array optimization problem has the following form:
Figure FDA0002288442070000041
wherein the content of the first and second substances,
Figure FDA0002288442070000042
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure FDA0002288442070000043
Two constraints
Figure FDA0002288442070000044
And
Figure FDA0002288442070000045
respectively representing beam width and zeroPoint depth, optimization variable
Figure FDA0002288442070000046
Including port excitation
Figure FDA0002288442070000047
And array cell layout
Figure FDA0002288442070000048
Wherein
Figure FDA0002288442070000049
Figure FDA00022884420700000410
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the initial point, n is the number of antenna units, n is 14,
Figure FDA00022884420700000411
representing a decision space, CMaOP(s)Representing the transformed dynamic weak constraint 5 target array optimization problem under the environment state S, wherein S represents the evolution environment state, S is 0,1,2, …, S and S are the maximum evolution environment state, epsilon1 (s)、ε2 (s)Representing dynamics of two constraints
Figure FDA00022884420700000412
The boundary of the constraint is defined,
Figure FDA00022884420700000413
the dynamic constraint boundary is contracted from the evolution environment state S being 0 to the environment state S being 1, and finally gradually contracted to the final evolution environment state S being S, and the 5 th target is newly added
Figure FDA00022884420700000414
The violation values are used to evaluate the degree of constraint violation for an individual.
7. The method of claim 6, wherein the default expression is as follows:
Figure FDA00022884420700000415
wherein the content of the first and second substances,
Figure FDA00022884420700000416
representing the i-th constraint, P, to be satisfied0Is the initial population individual.
8. The method for intelligent optimization of constraint multi-objective for solving antenna array comprehensive problem according to claim 1, characterized in that NSGA-III algorithm is adopted to perform updating parent operation on the transformed dynamic weak constraint 5-objective array optimization problem, the domination relationship between every two individuals needs to be compared when updating parent operation, the invention defines
Figure FDA0002288442070000051
Constraining Pareto dominance criterion to compare relationships between two population individuals, for two individuals
Figure FDA0002288442070000052
And
Figure FDA0002288442070000053
it can be said that
Figure FDA0002288442070000054
Constraint Pareto governance
Figure FDA0002288442070000055
·
Figure FDA0002288442070000056
Is that
Figure FDA0002288442070000057
A feasible solution to
Figure FDA0002288442070000058
Is that
Figure FDA0002288442070000059
Infeasible solutions;
·
Figure FDA00022884420700000510
and
Figure FDA00022884420700000511
are all that
Figure FDA00022884420700000512
A feasible solution, however
Figure FDA00022884420700000513
Dominating
Figure FDA00022884420700000514
·
Figure FDA00022884420700000515
And
Figure FDA00022884420700000516
are all that
Figure FDA00022884420700000517
Cannot solve it, but
Figure FDA00022884420700000518
Is less than
Figure FDA00022884420700000519
The default value of (a).
9. The method according to claim 1, wherein in the step 4, the dynamic constraint boundary in the final evolution environment state S is 0, which is the original constraint boundary, meaning that the algorithm finds the constrained Pareto optimal solution set of the original constrained super-multi-objective array optimization problem in the final evolution environment state, that is, the transformed dynamic weak constrained 5-objective array optimization problem has the following form:
Figure FDA00022884420700000520
wherein the content of the first and second substances,
Figure FDA00022884420700000521
for the target vector needing optimization, the beam width FNBW, the maximum side lobe level MSLL, the zero NULL and the line source length are included
Figure FDA00022884420700000522
Two constraints
Figure FDA00022884420700000523
And
Figure FDA00022884420700000524
respectively representing beam width and zero depth, optimization variables
Figure FDA00022884420700000525
Including port excitation
Figure FDA00022884420700000526
And array cell layout
Figure FDA00022884420700000527
Wherein
Figure FDA00022884420700000528
Figure FDA00022884420700000529
DjIs the distance between the j-1 th array element and the j-th array element, D1Is the distance from the first antenna unit to the initial point, n is the number of antenna units, n is 14,
Figure FDA00022884420700000530
representing a decision space, CMaOP(S)And representing the dynamic weak constraint 5 target array optimization problem in the final evolution environment state S.
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