CN112016662B - Array directional diagram synthesis method based on hybrid differential evolution algorithm and weighted total least square method - Google Patents
Array directional diagram synthesis method based on hybrid differential evolution algorithm and weighted total least square method Download PDFInfo
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Abstract
The invention discloses an array directional diagram synthesis method based on a hybrid differential evolution algorithm and a weighted total least square method, which comprises the following steps: step 1: setting target parameters of a hybrid differential evolution algorithm and a weighted total least square method; step 2: the array pattern synthesis problem is written as a formula [ A ] [ I ] = [ S ], and a diagonal matrix X is added on the formula to control the weights of different angle differences of the pattern; step 3: calculating to obtain array element current I in the array by using a total least square method, substituting the array element current I into a radiation pattern formula to obtain a preliminary radiation pattern; step 4: updating and optimizing the weight matrix X by utilizing a differential evolution algorithm; step 5: calculating the fitness function value of the array antenna population through the fitness function; step 6: judging whether the fitness function value is smaller than a certain value epsilon or whether the maximum evolution algebra is reached; step 7: outputting an optimal solution, and recording a radiation pattern corresponding to the fitness value. The invention can obtain lower side lobe level and deeper zero point.
Description
Technical Field
The invention belongs to the technical field of antennas, and particularly relates to an array directional diagram synthesis method based on a hybrid differential evolution algorithm and a weighted total least square method.
Background
In the past, antenna array patterns have been of interest due to their integration into radar and communication applications. Among them, the design of conformal antenna arrays is a great challenge for researchers. Since the conformal array has a curved surface, each antenna in the array points in a different direction. Thus, conventional planar array theory does not provide a typical array radiation pattern synthesis approach thereto. In addition, in a receiving system, the radiation pattern of the antenna array sometimes requires beamforming nulls. Interference suppression can then be achieved by controlling the null position to point in the direction of the interfering signal. This presents difficulties for the optimization algorithm because the null level is well below the side and main lobes. In the field of antenna array synthesis, many analysis methods have been proposed that can implement typical array synthesis, such as chebyshev, fourier transform, and the like. The parsing method has a remarkable advantage in convergence speed, but its application is limited. And carrying out iterative solution by a Least Squares Method (LSM) to obtain a good linear array comprehensive result. However, as the conformal array synthesis faces different array element postures, the analytic method is difficult to directly apply, so that the use is limited. The Total Least Squares (TLSM) method adopts a direct method to solve the array synthesis problem, but is not suitable for processing the conformal array synthesis problem due to the existence of weights such as matrixes.
In the development of conformal array design technology, many scholars propose a conformal array synthesis algorithm based on a biomimetic optimization algorithm. Such as genetic algorithms, particle swarm optimization algorithms, differential evolution algorithms, etc. Among other things, differential Evolution (DE) algorithms are used to solve many different types of optimization problems, such as sub-array linear antenna array design, axis parallel decision trees, multi-mode optimization problems, and searching for optimal locations of battery exchange sites within a certain area.
Because the Differential Evolution (DE) algorithm is simple in principle, few in control parameters and easy to understand and realize, the method can effectively solve the problems of complex function optimization, global optimization and multi-objective optimization. While the Differential Evolution (DE) algorithm performs well in maintaining population diversity and global search capability, it converges at a slower rate for some complex problems, such as pattern synthesis for conformal antenna arrays. To synthesize complex arrays, various optimization methods have been proposed. For example, the modified NSGA-II algorithm may be used to optimize a conformal phased array antenna. A blind beam forming method based on a linear constraint minimum variance algorithm. However, some biomimetic optimization algorithms have slow convergence rates, limiting the application of these algorithms.
The current method for synthesizing the processing directional diagram has low convergence speed and poor convergence precision, and when the method is applied to a linear array, the Side Lobe Level (SLL), main lobe and null (beam width and direction) of the antenna array cannot be effectively controlled, and the Side Lobe Level (SLL) and null depth are reduced.
Disclosure of Invention
In order to solve the defects existing in the prior art, the invention aims to: the array pattern synthesis method (HDE-WTLSM for short) based on the hybrid differential evolution algorithm and the weighted total least square method is provided, and has the advantages of higher convergence speed, higher convergence precision and wider application range; at the same time, lower sidelobe levels and deeper nulls can be obtained.
The technical scheme of the invention is as follows:
an array directional diagram synthesis method based on a hybrid differential evolution algorithm and a weighted total least square method comprises the following steps:
step 1: setting target parameters of a hybrid differential evolution algorithm and a weighted total least square method: population number, scaling factor, crossover probability, desired sidelobe level value and desired null depth value;
step 2: the array pattern synthesis problem is written as a formula [ A ] [ I ] = [ S ], and a diagonal matrix X is added on the array pattern synthesis problem to control weights of different angle differences of the pattern, namely: [ X ] [ A ] [ I ] = [ S ];
step 3: calculating to obtain array element current I in the array by using a total least square method, substituting the array element current I into a radiation pattern formula to obtain a preliminary radiation pattern;
step 4: updating and optimizing the weight matrix X by utilizing a differential evolution algorithm;
step 5: calculating the fitness function value of the array antenna population through the fitness function;
step 6: judging whether the fitness function value is smaller than a certain value epsilon or whether the maximum evolution algebra is reached, if not, entering the next generation and returning to the step 3, otherwise, executing the step 7;
step 7: and outputting an optimal solution, recording a radiation pattern corresponding to the fitness value, and finishing pattern synthesis.
Further, in the step 2, the matrix a is a manifold matrix of the array:
wherein a is ij Is an element of a steering vector, which can be expressed by formula (2):
k is the wave number of the electromagnetic wave in free space, which can be expressed by formula (3):
in the above formula, lambda is a wavelength; m is the total number of sampling points, M represents the number of the mth sampling point; n is the total number of array elements in the array, N representing the nth array element; θ i Representing pitch angle, f of the source j (θ i ) Is the j-th array element at theta i Radiation pattern in the direction; d, d n Is the distance between the nth array element and the first array element;
i is the array element current in the N-element array:
wherein phi is j Representing azimuth angles of the sources;
vector S is the radiation pattern required for the array:
[S] 1×M ＝[S(θ 1 ),…,S(θ i ),…,S(θ M )] T (5)
wherein I is n Is the current of the nth array element;
the formula for calculating the diagonal matrix X is:
wherein x is ii Is an element on the diagonal of the diagonal matrix X, S (θ i ) A desired radiation pattern for the array.
Further, the specific step of calculating the array element current I in the array by the overall least square method in the step 3 is as follows:
first, the least square method is used to convert [ X ]][A][I]＝[S]Become [ XA+E ]][I]＝[XS]+ε, equivalent to
Wherein E is an M×N-dimensional correction matrix, [ epsilon ]] 1×M ＝[ε(θ 1 ),…,ε(θ i ),…,ε(θ M )] T Epsilon is the error of the desired pattern from the resulting pattern;
second step, set C= [ XA|XS]，Δ＝[E|ε]，
Then->
Becomes (c+Δ) v=0;
third step, hermite conjugate matrix C Η C, decomposing the eigenvalue to obtain an eigenvector V belonging to the minimum eigenvalue S ：
Wherein t is a constant;
fourth step, according to
And calculating array element current I in the array.
Further, in the step 4, the specific steps of updating and optimizing the weight matrix X by the differential evolution algorithm are as follows:
firstly, establishing a differential evolution population for array antenna individuals in the current iteration;
second, a mutation operation is performed, i.e. 3 individuals x are randomly selected from the population p1,G ,x p2,G ,x p3,G And the individuals selected are not identical, then the three individual generated variance vectors are:
v i,G+1 ＝x p1,G +F(x p2,G -x p3,G ) (10)
wherein p1, p2, p 3E [1, NP]P1+notep2+notep3+notei; f is a scaling factor and is a real constant, satisfying F.epsilon.0, 2]The method comprises the steps of carrying out a first treatment on the surface of the F controls differential variants (x p2,G -x p3,G )；
Step three, performing crossover operation, namely crossing the mutation vector with a predetermined parent individual vector to generate a test vector u i,G Enhancing diversity of the population:
wherein r (j) is the jth evaluation of real random numbers uniformly distributed in the [0,1] range; CR is a real constant in [0,1] which represents the crossover probability;
fourth, judging the test vector u i,G Whether or not the fitness value of (a) is smaller than the parent individual vector x i,G If yes, selecting test vector u i,G As new differentially evolved population individuals, otherwise, the parent individual vector x is reserved i,G ；
And fifthly, judging whether the population quantity is reached, if not, adding 1 to the population quantity, returning to the first step, otherwise, selecting the array antenna individual with the minimum fitness value as a new array antenna individual.
Further, in the step 5, the fitness function value has a calculation formula as follows:
fitness＝U(F 0 (θ)-F d (θ))[αmax|SLL-DSLL|+βmax|NULL-DNULL|] (12)
wherein the method comprises the steps of
F 0 (θ) and F d (θ) is the obtained and desired array pattern, respectively, SLL represents the resulting sidelobe level, DSLL represents the desired sidelobe level value, NULL represents the zero pit depth optimized by the optimization algorithm; DNULL represents the desired null depth value; alpha and beta represent the weights of the sidelobe levels and the nulling depths.
The beneficial effects of the invention are as follows: according to the method, the weight matrix is introduced into the Total Least Squares Method (TLSM), and then the weight matrix is optimized by adopting the Differential Evolution (DE) algorithm, and the evolution algorithm is combined with the numerical algorithm, so that the method can be used for controlling Side Lobe Levels (SLL), main lobes and nulls (beam width and direction) of complex arrays such as conformal antenna arrays and the like, and simultaneously, the optimization speed of an array directional diagram is improved. The invention adopts the embedded directional diagram in the antenna array, thereby improving the comprehensive precision of the directional diagram.
The invention has the following characteristics:
(1) Compared with the traditional antenna array pattern synthesis method, the method provided by the invention has the advantages that the weight matrix in the total least square method is updated and optimized by utilizing the differential evolution algorithm, and the defect that the traditional array synthesis analysis method is not suitable for processing the conformal array synthesis problem is overcome.
(2) Compared with the traditional antenna array pattern synthesis method, the method has the advantages that the weight matrix is updated and optimized only by using the differential evolution algorithm, so that the time consumption is small, and the method is suitable for the large-scale array synthesis problem.
(3) Simulation results show that in the application of the combined method of the mixed differential evolution algorithm and the weighted total least square method, the convergence speed and the convergence precision are improved, and meanwhile, a lower side lobe level and a lower zero point are obtained.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a normalized array pattern of low sidelobe level synthesis;
FIG. 3 is a convergence graph of low sidelobe level synthesis;
FIG. 4 is a normalized array pattern for nulling synthesis;
FIG. 5 is a convergence curve for null synthesis;
fig. 6 is an RHCP antenna and conformal truncated cone array;
FIG. 7 is a three-dimensional radiation pattern of a uniform feed;
FIG. 8 is a two-dimensional radiation pattern of a uniform feed;
FIG. 9 is a three-dimensional radiation pattern after integration;
FIG. 10 is a two-dimensional radiation pattern after integration;
fig. 11 is a converging curve for conformal array low sidelobe level synthesis.
Detailed Description
The invention will now be further described with reference to the drawings and examples, which are not intended to limit the scope of the invention.
Example 1
As shown in fig. 1, an array pattern synthesis method based on a hybrid differential evolution algorithm and a weighted total least squares method includes the following steps:
step 1: setting target parameters of a hybrid differential evolution algorithm and a weighted total least square method: population number, scaling factor, crossover probability, desired sidelobe level value and desired null depth value;
step 2: the array pattern synthesis problem is written as a formula [ A ] [ I ] = [ S ], and a diagonal matrix X is added on the array pattern synthesis problem to control weights of different angle differences of the pattern, namely: [ X ] [ A ] [ I ] = [ S ];
step 3: calculating to obtain array element current I in the array by using a total least square method, substituting the array element current I into a radiation pattern formula to obtain a preliminary radiation pattern;
step 4: updating and optimizing the weight matrix X by utilizing a differential evolution algorithm;
step 5: calculating the fitness function value of the array antenna population through the fitness function;
step 6: judging whether the fitness function value is smaller than a certain value epsilon or whether the maximum evolution algebra is reached, if not, entering the next generation and returning to the step 3, otherwise, executing the step 7;
step 7: and outputting an optimal solution, recording a radiation pattern corresponding to the fitness value, and finishing pattern synthesis.
Example 2
As shown in fig. 1, an array pattern synthesis method based on a hybrid differential evolution algorithm and a weighted total least squares method includes the following steps:
step 1: setting target parameters of a hybrid differential evolution algorithm and a weighted total least square method: population number, scaling factor, crossover probability, desired sidelobe level value and desired null depth value;
step 2: the array pattern synthesis problem is written as a formula [ A ] [ I ] = [ S ], and a diagonal matrix X is added on the array pattern synthesis problem to control weights of different angle differences of the pattern, namely: [ X ] [ A ] [ I ] = [ S ];
further, in the step 2, the matrix a is a manifold matrix of the array:
wherein a is ij Is an element of a steering vector, which can be expressed by formula (2):
k is the wave number of the electromagnetic wave in free space, which can be expressed by formula (3):
in the above formula, lambda is a wavelength; m is the total number of sampling points, M represents the number of the mth sampling point; n is the total number of array elements in the array, N representing the nth array element; θ i Representing pitch angle, f of the source j (θ i ) Is the j-th array element at theta i Radiation pattern in the direction; d, d n Is the distance between the nth array element and the first array element;
i is the array element current in the N-element array:
wherein phi is j Representing azimuth angles of the sources;
vector S is the radiation pattern required for the array:
[S] 1×M ＝[S(θ 1 ),…,S(θ i ),…,S(θ M )] T (5)
wherein I is n Is the current of the nth array element;
the formula for calculating the diagonal matrix X is:
wherein x is ii Is an element on the diagonal of the diagonal matrix X, S (θ i ) A desired radiation pattern for the array.
Step 3: calculating to obtain array element current I in the array by using a total least square method, substituting the array element current I into a radiation pattern formula to obtain a preliminary radiation pattern;
further, the specific step of calculating the array element current I in the array by the overall least square method in the step 3 is as follows:
first, the least square method is used to convert [ X ]][A][I]＝[S]Become [ XA+E ]][I]＝[XS]+ε, equivalent to
Wherein E is an M×N-dimensional correction matrix, [ epsilon ]] 1×M ＝[ε(θ 1 ),…,ε(θ i ),…,ε(θ M )] T Epsilon is the error of the desired pattern from the resulting pattern;
second step, set C= [ XA|XS]，Δ＝[E|ε]，
Then->
Becomes (c+Δ) v=0;
third step, hermite conjugate matrix C Η C, decomposing the eigenvalue to obtain an eigenvector V belonging to the minimum eigenvalue S ：
Wherein t is a constant;
fourth step, according to
And calculating array element current I in the array.
Step 4: updating and optimizing the weight matrix X by utilizing a differential evolution algorithm;
further, in the step 4, the specific steps of updating and optimizing the weight matrix X by the differential evolution algorithm are as follows:
firstly, establishing a differential evolution population for array antenna individuals in the current iteration;
second, a mutation operation is performed, i.e. 3 individuals x are randomly selected from the population p1,G ,x p2,G ,x p3,G And the individuals selected are not identical, then the three individual generated variance vectors are:
v i,G+1 ＝x p1,G +F(x p2,G -x p3,G ) (10)
wherein p1, p2, p 3E [1, NP]P1+notep2+notep3+notei; f is a scaling factor and is a real constant, satisfying F.epsilon.0, 2]The method comprises the steps of carrying out a first treatment on the surface of the F controls differential variants (x p2,G -x p3,G )；
Step three, performing crossover operation, namely crossing the mutation vector with a predetermined parent individual vector to generate a test vector u i,G Enhancing diversity of the population:
wherein r (j) is the jth evaluation of real random numbers uniformly distributed in the [0,1] range; CR is a real constant in [0,1] which represents the crossover probability;
fourth, judging the test vector u i,G Whether or not the fitness value of (a) is smaller than the parent individual vector x i,G If yes, selecting test vector u i,G As new differentially evolved population individuals, otherwise, the parent individual vector x is reserved i,G ；
And fifthly, judging whether the population quantity is reached, if not, adding 1 to the population quantity, returning to the first step, otherwise, selecting the array antenna individual with the minimum fitness value as a new array antenna individual.
Step 5: calculating the fitness function value of the array antenna population through the fitness function;
further, in the step 5, the fitness function value has a calculation formula as follows:
fitness＝U(F 0 (θ)-F d (θ))[αmax|SLL-DSLL|+βmax|NULL-DNULL|] (12)
wherein the method comprises the steps of
F 0 (θ) and F d (θ) is the obtained and desired array pattern, respectively, SLL represents the resulting sidelobe level, DSLL represents the desired sidelobe level value, NULL represents the zero pit depth optimized by the optimization algorithm; DNULL represents the desired null depth value; alpha and beta represent the weights of the sidelobe levels and the nulling depths.
Step 6: judging whether the fitness function value is smaller than a certain value epsilon or whether the maximum evolution algebra is reached, if not, entering the next generation and returning to the step 3, otherwise, executing the step 7;
step 7: and outputting an optimal solution, recording a radiation pattern corresponding to the fitness value, and finishing pattern synthesis.
The advantages of the invention can be further illustrated by the following simulations:
in order to verify the effectiveness of the method, a 20-array element uniform linear array with ideal point sources was simulated. Which requires a combination of low sidelobe levels and nulls. To further test the performance of the algorithm, a conformal truncated cone array was established. The algorithm obtains an embedded pattern by using a High Frequency Structure Simulator (HFSS), and improves the comprehensive precision.
1. Linear array
To verify the performance of the HDE-WTLSM algorithm, a uniform linear array of 20 elements was used to achieve low sidelobe level and null pattern synthesis.
The fitness function is as follows:
fitness＝U(F 0 (θ)-F d (θ))[αmax|SLL-DSLL|+βmax|NULL-DNULL|]
wherein the method comprises the steps of
F 0 (θ) and F d (θ) is the obtained and desired array pattern, respectively. SLL represents the resulting side lobe level, DSLL represents the desired side lobe level, and NULL represents the zero pit depth optimized by the optimization algorithm. DNULL represents the desired null depth. Alpha and beta represent the weights of the sidelobe levels and the nulling depths.
Since a symmetrical array is used, only the amplitude of 10 elements has to be optimized. And the excitation profile is centrosymmetric with respect to the array.
(1) Low sidelobe level (SLL)
The first example is low sidelobe level synthesis. In this example, the number of individuals in each population is 150, f=0.6, cr=0.8, α=1, β=0. Desired side lobe level dsll= -50dB. The main beam width is limited to-12 deg. to 12 deg..
After optimization, the low side lobe level integrated normalized radiation pattern is shown in fig. 2, and the corresponding convergence curve is shown in fig. 3. It can be seen from the figure that the sidelobe level of the HDE-WTLSM algorithm is about 11dB lower than that of the DE algorithm, and that the convergence rate of the HDE-WTLSM algorithm is faster than that of the DE algorithm.
(2) Null sinking
The second example is null synthesis. In this example, the population is 150, f=0.6, cr=0.8, dsll= -50dB, null= -80dB. The main beam width is limited to-15 degrees to 15 degrees, the zero point width is 10 degrees, and the normalized array directional diagram of the nulling combination is shown in figure 4. The convergence curve of the nulling complex is shown in figure 5. In this example, both low sidelobe level synthesis and cross polarization suppression are achieved using the HDE-WTLSM algorithm. The HDE-WTLSM algorithm also performed better than the DE algorithm, with the results from both the first and second examples from Matlab.
2. Conformal truncated cone array
To verify the optimization of the algorithm to the complex problem, a conformal truncated cone array antenna was established, which is 39 right circularly polarized (RHCP) antennas. In the embodiment, the single-port circularly polarized array element is adopted to reduce half of unknown quantity. The structure of the antenna and array is shown in fig. 6. The upper radius of the cone is 280mm, and the lower radius is 360mm. The height of the cone is 400mm. The center operating frequency is 1.575GHz. The algorithm is used to achieve low sidelobe level and low cross polarization synthesis.
The fitness function is as follows
fitness＝U(F 0 (θ)-F d (θ))[αmax|SLL-DSLL|+βmax|CROSS-DCROSS|]
Where CROSS represents the CROSS polarization level optimized by the HDE-WTLSM algorithm. DCROSS represents the desired cross-polarization level. Alpha and beta represent the weights of the sidelobe levels and the cross-polarization levels.
In this example, the number of individuals in each population is 100, f=0.6, cr=0.8, α=1, β=0. Desired side lobe level dsll= -25dB. The main beam width is limited to-18 deg. to 18 deg., the desired cross polarization level DCROSS = -23dB.
Unlike planar arrays, when a conformal truncated cone array is fed uniformly, there is a depression in the z-direction of the array radiation pattern due to polarization effects. As shown in fig. 7 and 8. The angle between co-polarizations of the antennas is different due to the different orientations of the antennas in the conformal array. The cross-polarization results are identical. This illustrates that polarization is the most difficult problem in conformal antenna array synthesis. If polarization is not considered in conformal antenna array synthesis, many unexpected results will be produced.
The HDE-WTLSM algorithm is then used to reduce the effects of cross polarization in the array. At the same time, it is desirable to have a low sidelobe level result in the main polarization direction. After synthesis, the simulation results show one main lobe in the z-direction for co-polarization. And the cross-polarization level is reduced by about 20dB. The results of fig. 7-10 were obtained from the HFSS, which included an active pattern for each antenna. The simulation examples above demonstrate that the algorithm can function in both ideal point source arrays and actual conformal antenna arrays. The convergence curves for the DE algorithm and the HDE-WTLSM algorithm for the conformal array are shown in FIG. 11. It can be seen from the figure that the convergence rate of the HDE-WTLSM algorithm is faster.
As can be seen from the simulation examples, the method can be applied to the problem of low side lobe level and null pattern synthesis of a 20-element linear array, and can also optimize a conformal truncated cone array to obtain low side lobe level and cross polarization level synthesis. Experiments also show that the HDE-WTLSM algorithm is a very effective algorithm and has the advantage of high convergence rate.
The parts of this embodiment not described in detail and the english abbreviations are common general knowledge in the industry and can be found on the internet, and are not described here.
Claims (5)
1. The array directional diagram synthesis method based on the hybrid differential evolution algorithm and the weighted total least square method is characterized by comprising the following steps:
step 1: setting target parameters of a hybrid differential evolution algorithm and a weighted total least square method: population number, scaling factor, crossover probability, desired sidelobe level value and desired null depth value;
step 2: the array pattern synthesis problem is written as a formula [ A ] [ I ] = [ S ], and a diagonal matrix X is added on the array pattern synthesis problem to control weights of different angle differences of the pattern, namely: [ X ] [ A ] [ I ] = [ S ]; wherein A is manifold matrix of the array, S is radiation pattern required by the array; x is a weight matrix;
step 3: calculating to obtain array element current I in the array by using a total least square method, substituting the array element current I into a radiation pattern formula to obtain a preliminary radiation pattern;
step 4: updating and optimizing the diagonal matrix X by utilizing a differential evolution algorithm;
step 5: calculating the fitness function value of the array antenna population through the fitness function;
step 6: judging whether the fitness function value is smaller than a certain value epsilon or whether the maximum evolution algebra is reached, if not, entering the next generation and returning to the step 3, otherwise, executing the step 7;
step 7: and outputting an optimal solution, recording a radiation pattern corresponding to the fitness function value, and finishing pattern synthesis.
2. The method for combining array patterns based on a hybrid differential evolution algorithm and a weighted total least squares method according to claim 1, wherein the method comprises the following steps: in the step 2, the matrix a is a manifold matrix of the array:
wherein a is ij Is an element of a steering vector, which can be expressed by formula (2):
in the above formula, lambda is a wavelength; m is the total number of sampling points, M represents the number of the mth sampling point; n is the total number of array elements in the array, N representing the nth array element; θ i Representing pitch angle, f of the source j (θ i ) Is the j-th array element at theta i Radiation pattern in the direction; d, d n Is the distance between the nth array element and the first array element;
i is the array element current in the N-element array:
wherein phi is j Representing the azimuth of the jth source;
vector S is the radiation pattern required for the array:
[S] 1×M ＝[S(θ 1 ),…,S(θ i ),…,S(θ M )] T (5)
wherein I is n Is the current of the nth array element;
the formula for calculating the diagonal matrix X is:
3. The method for combining array patterns based on a hybrid differential evolution algorithm and a weighted total least squares method according to claim 1, wherein the method comprises the following steps: the specific steps for obtaining the array element current I in the array by the calculation of the total least square method in the step 3 are as follows:
first, the least square method is used to convert [ X ]][A][I]＝[S]Become [ XA+E ]][I]＝[XS]+ε, equivalent to
Wherein E is an M×N-dimensional correction matrix, [ epsilon ]] 1×M ＝[ε(θ 1 ),…,ε(θ i ),…,ε(θ M )] T Epsilon is the error of the desired pattern from the resulting pattern;
second step, set C= [ XA|XS]，Δ＝[E|ε]，
Then->
Becomes (c+Δ) v=0;
third step, hermite conjugate matrix C H C, decomposing the eigenvalue to obtain an eigenvector V belonging to the minimum eigenvalue S ：
Wherein t is a constant;
fourth step, according to
And calculating array element current I in the array.
4. The method for combining array patterns based on a hybrid differential evolution algorithm and a weighted total least squares method according to claim 1, wherein the method comprises the following steps: and 4, the specific steps of updating and optimizing the diagonal matrix X by the differential evolution algorithm are as follows:
firstly, establishing a differential evolution population for array antenna individuals in the current iteration;
second, a mutation operation is performed, i.e. 3 individuals x are randomly selected from the population p1,G ,x p2,G ,x p3,G And the individuals selected are not identical, then the three individual generated variance vectors are:
v i,G+1 ＝x p1,G +F(x p2,G -x p3,G ) (10)
wherein p1, p2, p 3E [1, NP]P1+notep2+notep3+notei; f is a scaling factor and is a real constant, satisfying F.epsilon.0, 2]The method comprises the steps of carrying out a first treatment on the surface of the F controls differential variants (x p2,G -x p3,G )；
Step three, performing crossover operation, namely crossing the mutation vector with a predetermined parent individual vector to generate a test vector u i,G Enhancing diversity of the population:
wherein r (j) is the jth evaluation of real random numbers uniformly distributed in the [0,1] range; CR is a real constant in [0,1] which represents the crossover probability;
fourth, judging the test vector u i,G Whether or not the fitness value of (a) is smaller than the parent individual vector x i,G If yes, selecting test vector u i,G As new differentially evolved population individuals, otherwise, the parent individual vector x is reserved i,G ；
And fifthly, judging whether the population quantity is reached, if not, adding 1 to the population quantity, returning to the first step, otherwise, selecting the array antenna individual with the minimum fitness value as a new array antenna individual.
5. The method for combining array patterns based on a hybrid differential evolution algorithm and a weighted total least squares method according to claim 1, wherein the method comprises the following steps: and 5, calculating the fitness function value according to the following formula:
fitness＝U(F 0 (θ)-F d (θ))[αmax|SLL-DSLL|+βmax|NULL-DNULL|] (12)
wherein the method comprises the steps of
F 0 (θ) and F d (θ) is the obtained and desired array pattern, respectively, SLL represents the resulting sidelobe level, DSLL represents the desired sidelobe level value, NULL represents the zero pit depth optimized by the optimization algorithm; DNULL represents the desired null depth value; alpha and beta represent the weights of the sidelobe levels and the nulling depths. />
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