CN111665502A - MODE algorithm-based MIMO radar sparse array optimization method - Google Patents
MODE algorithm-based MIMO radar sparse array optimization method Download PDFInfo
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Abstract
The invention discloses a MIMO radar sparse array optimization method based on MODE algorithm, and relates to the technical field of radar imaging systems. The method comprises the steps of completing sparse array optimization in two steps, mainly performing problem modeling aiming at sparse array optimization on the basis of completing covariance matrix establishment, solving the single-target optimization problem by using a CA + CA + DE algorithm, then using the single-target optimization as a constraint condition of the optimization, further converting the single-target optimization into a multi-target optimization problem, and finally using a multi-target MODE optimization algorithm to solve to obtain the sparse array which meets pattern matching and inhibits side lobes. The method can achieve global optimization and high convergence speed under the condition that the directional diagram is fixed.
Description
Technical Field
The invention relates to the technical field of radar imaging systems, in particular to a MIMO radar sparse array optimization method based on a MODE algorithm.
Background
The microminiature millimeter wave radar is a sensor with low cost and high performance, and has the capability of high-precision speed measurement and distance measurement. Compared with optical sensors such as a camera and a laser radar, the millimeter wave radar has the advantages of working all day long, adaptability to various rainy and snowy weather, complex environments such as sand, dust and smoke, long working distance, wide visual field and the like. And compared with the laser radar, the cost is lower. The microminiature millimeter wave radar has a wide application range, and can be applied to the fields of military affairs and civil use.
In order to realize high-resolution radar imaging, one or more radar chips are generally used to realize an array antenna, and an antenna array form of a MIMO system may be adopted in order to reduce cost. Imaging methods are studied in more detail for a variety of antenna array formats. A forward-looking radar imaging technology (SIREV) for view enhancement was developed by the germany DLR and subjected to flight tests. But the volume is large, and the device can only be used for platforms such as large helicopters. The system generally adopts the separation module to build a millimeter wave front end system, and when the array units are more, the size and the cost are high. At present, multiple 60GHz and 77GHz radar chips in the market can realize an MIMO radar system, have the capability of multi-chip cascade, and provide feasibility for realizing two-dimensional and three-dimensional radar imaging systems with low cost, small volume and high resolution. Generally, the imaging resolution of the system realized by the current system is still low, and in order to realize higher angular resolution, a method for further optimizing the radar sparse array and increasing the number of the cascaded modules can be adopted, but the size, the weight and the cost of the system are inevitably increased, and the difficulty in system development is also increased.
Disclosure of Invention
The invention aims to solve the technical problem of how to provide a MODE algorithm-based MIMO radar sparse array optimization method which can converge to global optimization under the condition of fixed directional diagram and has high convergence speed.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a MIMO radar sparse array optimization method based on MODE algorithm is characterized by comprising the following steps:
solving a single-target optimization problem of the MIMO radar sparse array;
taking single-target optimization as a constraint condition of the optimization, and further converting the single-target optimization into a multi-target optimization problem;
and solving by using a multi-objective MODE optimization algorithm to obtain a sparse array meeting the directional diagram matching and inhibiting side lobes.
The further technical scheme is as follows: consider a ULA consisting of N antennas with an antenna aperture of (N-1) × d, d referring to the array element spacing, and d ═ λ/2; introducing an auxiliary vector Ps to express the relation between array elements of a sparse array and array elements of a full array, wherein the auxiliary vector Ps is composed of binary codes expressing the positions of the array elements; when the element in the vector Ps is 0, it indicates that the position array element corresponding to the element does not exist, whereas when the element in the vector Ps is 1, it indicates that the position array element corresponding to the element exists; by definition of the auxiliary vector, the steering vector of the sparse antenna can be expressed as:
as＝Ps◎a (1)
the formula-:
as is asA matrix set of sampling points at angles;
in summary, in the sparse array design of the MIMO radar, optimal pattern approximation performance is obtained by determining the position combination of M array elements in N candidate positions.
The further technical scheme is that the single-target optimization method comprises the following steps:
the first step is as follows: selecting a directional diagram matching design algorithm, selecting a CA algorithm to complete the directional diagram design when the array is full, and obtaining a corresponding covariance matrix R;
the second step is that: combining the CA algorithm with the differential algorithm, firstly solving the first step by using the CA algorithm, then modeling the optimization problem of the result obtained by the CA algorithm, and then solving by using the differential algorithm.
The further technical scheme is that the method of the first step comprises the following steps:
firstly, completing the matching design of a directional diagram during full array, and obtaining a waveform weighting matrix W corresponding to the matched directional diagram:
s.t.Diag(UH-U)＝pd(3)
wherein W ═ R1/2，pdMatrix for desired patterns:
pd＝[φ(θ1)φ(θ2)...φ(θk)]T(4)
where φ (θ 1) represents the sampling points of the desired pattern in the observation angle region.
The further technical scheme is that the method of the second step comprises the following steps:
setting the number of array elements and optimization parameters of the sparse array, and still optimizing the array combination of the sparse array elements by taking the approximate expected directional diagram as a target function so as to enable p → pd；
Modeling the optimization problem of the above formula to obtain a problem model as follows:
s.t.p(:,1)＝1.p(:,N)＝1
the constraint p (: 1) ═ 1 and p (: N) ═ 1 indicate that the array elements are arranged at both ends of the array in order to fix the actual aperture of the antenna, and in fact, equation (6) redefines the matching mean square error MSE between the pattern of the sparse array and the desired pattern.
The further technical solution is that the method for solving by using the difference algorithm in the second step is as follows:
suppose the decision vector is composed of a continuous space vector x in D dimension (x)1,x2,x3.......xD) Representing that a global optimal solution is searched in a space of the D-dimensional real numerical value;
1) initializing a population: establishing an initial population containing NP individuals, wherein vectors of the individuals in the population are also called target vectors; considering that the individual parameter variables are constrained by the real condition, i.e. the boundary condition, the upper boundary of the jth parameter in the D-dimensional vector is assumed to be
The lower boundary is represented as
In the evolution of generation G, the initialization of the jth parameter of the ith target vector can be written as:
in equation (7), i ═ 1., NP, j ═ 1., D ═ N-2, which means that only D ═ N-2 participates in the evolution, rand (0,1) denotes random numbers uniformly distributed in the interval (0,1), and the target vector of the G th generation can be expressed as:
2) and (3) encoding: in order to mark the position of an array element in the Ps, a real variable of DE needs to be converted into a binary variable, and an initial value of the array element is just a discrete random disturbance combination in a certain antenna aperture; this perturbation is dominated by the G-th generation target vector as follows:
ps(n)＝binarysort(x(n))T(9)
since the aperture of the array has been determined, binary sort () indicates that the real numbers are sorted before binary encoding;
3) mutation operation: for each target vector, the DE algorithm generates a variant vector using a variant operation, and there are five common and disparate strategies for the differential evolution algorithm:
wherein r1, r2, r3, r4 and r5 belong to [1, NP ], are all different, and F is within [0,2], belongs to a variation factor and is a constant value, and the scaling of the differential vector is controlled; the larger the F is, the higher the population diversity is, and the slower the convergence speed is;
4) and (3) cross operation: a common crossover strategy generally selects a binomial crossover, the mathematical expression of which can be described as:
wherein CR is [0,1 ]]Is a cross probability constant defined by a user; rand (0,1) is as defined above; k is [1, D ]]Is a randomly selected number, ensuring ui,GIn which at least one dimension is a variation vector vi,GThe result is obtained; after the cross operation, in order to ensure the validity of the solution, whether the test variables meet the boundary conditions must be judged, otherwise, the mutually different variables are randomly generated again;
5) selecting operation: and selecting the individuals entering the next generation of population by adopting a greedy algorithm, wherein the test vector competes with the target vector, the fitness function is preferably selected as the individual of the filial generation, and the mathematical expression form of the greedy selection operator can be described as follows:
where f (.) is the fitness function defined above.
After the single-target optimization is completed, the constraint condition that the single target is used as the multi-target optimization according to the process is solved by using a MODE algorithm, and the constraint optimization problem can be described as follows:
in the formula, f (·) is an objective function, x represents a decision variable of D dimension like DE algorithm, g (x) and h (x) respectively represent unequal constraint conditions and equal constraint conditions, q represents the number of the unequal constraint conditions, and m-q represents the number of the equal constraint conditions; in equation (13), the constraint for equal can be converted into two unequal constraints by tolerance error as follows:
it follows that the two constraints are invertible to each other;
the problem model of equation (6) is applied as an inequality constraint, and the optimization problem can be described as follows according to the mathematical model of equation (13):
in the formula f1(x) As defined in formula (6), f2(x)＝PSLL。
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: in the method, the single-target optimization is performed twice more by using the CA algorithm than before, then the modeling of the optimization problem is performed, and then the DE algorithm is adopted for solving. By adopting the DE algorithm after the CA algorithm is adopted twice, convergence to global optimization can be realized under the condition that the directional diagram is fixed. The DE algorithm is adopted for solving, the DE algorithm is simple in structure, easy to program and high in development potential, and has good function optimization performance, good algorithm stability and high convergence speed. Based on a multi-objective differential evolution algorithm, and by using several definitions of multi-objective optimization, the method is applied to the sparse array optimization of the MIMO radar. Compared with two single-target evolutionary algorithms (GA and DE), the MODE algorithm can realize parallel optimization of the pattern matching and peak sidelobe suppression performance of the sparse array, and finally the optimal sparse array meeting the pattern matching and peak sidelobe suppression is obtained.
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The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
FIG. 1 is a main flow diagram of a method according to an embodiment of the present invention;
Detailed Description
The technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways than those specifically described and will be readily apparent to those of ordinary skill in the art without departing from the spirit of the present invention, and therefore the present invention is not limited to the specific embodiments disclosed below.
The sparse array is used as an array commonly used by the MIMO radar, and can provide additional freedom for the adaptive waveform design of the MIMO radar. As shown in FIG. 1, the embodiment of the invention discloses a MIMO radar sparse array optimization method based on MODE algorithm, the method comprises two steps of completing sparse array optimization, on the basis of completing covariance matrix establishment, problem modeling is mainly performed on sparse array optimization, wherein a CA + CA + DE algorithm is used for solving the single-target optimization problem, then single-target optimization is used as a constraint condition of the optimization, and further the constraint condition is converted into a multi-target optimization problem, and finally the multi-target MODE optimization algorithm is used for solving, so that a sparse array which meets directional diagram matching and inhibits side lobes is obtained.
Consider a ULA consisting of N antennas with an antenna aperture of (N-1) × d, where d refers to the array element spacing and d ═ λ/2. Here, the auxiliary vector Ps is introduced to indicate the relationship between the array elements of the sparse array and the array elements of the full array, and is composed of binary codes indicating the positions of the array elements. When the element in the vector Ps is 0, it indicates that the position array element corresponding to the element is absent, whereas when the element in the vector Ps is 1, it indicates that the position array element corresponding to the element is present. By definition of the auxiliary vector, the steering vector of the sparse antenna can be expressed as:
as＝Ps◎a (1)
wherein ^ is Hadamard product. Similarly, the guide vector matrix of the sparse array is represented by As. By definition, a sparse array MIMO radar direction diagram can be represented as:
as is asA set of matrices at angular sampling points.
In summary, in the sparse array design of the MIMO radar, optimal pattern approximation performance is obtained by determining the position combination of M array elements in N candidate positions.
First, a single-target optimization for sparse array optimization of MIMO radar is required. Considering that the sparse array needs to meet the requirement of pattern matching, the optimization design of the sparse array uses a two-step processing method, which specifically comprises the following steps: the first step requires the selection of an algorithm for pattern matching design. Different algorithms have different matching effects. Although the CA algorithm is not suitable for array optimization, the side lobe level of the synthesized directional diagram is kept to be the lowest because of the super-resolution of the subspace projection under the CA algorithm, and therefore the CA algorithm is still selected to complete the design of the directional diagram when the array is full, and a corresponding covariance matrix R is obtained. In the second step, the optimization of the sparse antenna array is considered to be a highly nonlinear optimization problem, so the CA algorithm and the differential algorithm are combined in the second step, the CA algorithm is used for solving the first step, then the optimization problem modeling is carried out on the result obtained by the CA algorithm, and then the differential algorithm is used for solving.
The first step is as follows: firstly, completing the matching design of a directional diagram during full array, and obtaining a waveform weighting matrix W corresponding to the matched directional diagram:
s.t.Diag(UH-U)＝pd(3)
wherein W ═ R1/2，pdMatrix for desired patterns:
pd＝[φ(θ1)φ(θ2)...φ(θk)]T(4)
where φ (θ 1) represents the sampling points of the desired pattern in the observation angle region.
The second step is that: setting the number of array elements and optimization parameters of the sparse array, and still optimizing the array combination of the sparse array elements by taking the approximate expected directional diagram as a target function so as to enable p → pd。
Although the directional diagram of the sparse array designed based on the CA + CA method can obtain the optimal approximation performance, the array is not a true sparse array, but is a combination of an N-1 dimensional uniform linear array and a single array element at the maximum aperture.
Therefore, the problem model obtained by modeling the optimization problem of the above formula is as follows:
the constraints p (: 1) ═ 1 and p (: N) ═ 1 indicate that the array elements are arranged at both ends of the array in order to fix the actual aperture of the antenna. In fact, equation (6) redefines the matching Mean Square Error (MSE) between the directional diagram of the sparse array and the expected directional diagram, where the optimization problem of the array has been separated from the original matrix approximation problem represented by norm, and becomes a solution model with one-to-one correspondence between variables and functions.
Therefore, the global optimization algorithm becomes a good choice for effectively searching the global optimization solution. The single target evolution algorithm mainly includes two kinds, one is a Genetic Algorithm (GA), and the other is a Differential Evolution (DE) algorithm. The method adopts a DE algorithm to solve a single target.
The DE algorithm mainly comprises four steps of population initialization, Mutation (Mutation), Crossover (cross) and Selection (Selection).
Suppose the decision vector is composed of a continuous space vector x in D dimension (x)1,x2,x3.......xD) Representing that a global optimal solution is searched in a space of the D-dimensional real numerical value;
1) and initializing the population. An initial population is established comprising NP individuals, the vectors of the individuals in the population also being referred to as target vectors. Considering that the individual parameter variables are constrained by the real condition, i.e. the boundary condition, the upper boundary of the jth parameter in the D-dimensional vector is assumed to be
The lower boundary is represented as
In the evolution of generation G, the initialization of the jth parameter of the ith target vector can be written as:
in equation (7), i ═ 1., NP, j ═ 1., D ═ N-2, which means that only D ═ N-2 participates in the evolution, rand (0,1) denotes random numbers uniformly distributed in the interval (0,1), and the target vector of the G th generation can be expressed as:
2) and (5) encoding. To mark the array element position in Ps, the real variable of DE needs to be converted into a binary variable. The initial value of the array element is just the discrete random disturbance combination in a certain antenna aperture. This perturbation is dominated by the G-th generation target vector as follows:
ps(n)＝binarysort(x(n))T(9)
since the aperture of the array has been determined, binary sort () indicates that the real numbers are sorted before binary encoding;
3) and (5) performing mutation operation. For each target vector, the DE algorithm generates a variant vector using a variant operation. There are five common and disparate strategies for differential evolution:
wherein r1, r2, r3, r4 and r5 belong to [1, NP ], are all different, and F is within [0,2], belongs to a variation factor and is a constant value, and the scaling of the differential vector is controlled; the larger the F is, the higher the population diversity is, and the slower the convergence speed is;
4) and (4) performing a crossover operation. A common crossover strategy generally selects a binomial crossover, the mathematical expression of which can be described as:
wherein CR is [0,1 ]]Is a cross probability constant defined by a user; rand (0,1) is as defined above; k is [1, D ]]Is a randomly selected number, ensuring ui,GIn which at least one dimension is a variation vector vi,GThe result is obtained; after the crossover operation, in order to guarantee the validity of the solution, it is necessary to judge whether the test variables satisfy the boundary conditions. Otherwise, the mutually different variables will be randomly generated again.
5) And (6) selecting operation. DE employs a greedy algorithm to select individuals for entry into the next generation population. The test vector will compete with the target vector and the fitness function is preferably selected as the individual of the offspring. The mathematical expression form of the greedy selection operator can be described as:
where f (.) is the fitness function defined above.
Inputting: the method comprises the following steps of waveform covariance matrix R, population size NP, the number M of sparse array elements and the number N of full array elements.
Step 1: initializing, generating i, Gx by using the formula (1-8);
step 2: encoding, generating Ps using the formula (1-9);
and step 3: calculating a fitness function value for f (x);
and 4, step 4: the different strategies are shown in the formula (1-10).
And 5: crossing, as shown in formulas (1-11);
step 6: selecting, as shown in formula (1-12);
and 7: and (4) the optimal individual is saved, the temporary population is evaluated, and the optimal fitness function and the corresponding individual are selected.
And (3) outputting: ps，best→As→p＝As HRAsIf not: go back to step 2 until exiting the iteration bar
The piece is satisfied.
After the single-target optimization is completed, solving is carried out by using a MODE algorithm according to the constraint condition of the process in which the single target is used as the multi-target optimization. The constraint optimization problem can be described as follows:
in the formula, f (·) is an objective function, x represents a decision variable of D dimension like DE algorithm, g (x) and h (x) respectively represent unequal constraint conditions and equal constraint conditions, q represents the number of the unequal constraint conditions, and m-q represents the number of the equal constraint conditions; in equation (13), the constraint of equality can be converted into two inequality constraints by tolerance (or tolerance) as follows:
it follows that the two constraints are invertible to each other.
To further suppress peak sidelobe levels, as described in the previous subsection, which is applied as an inequality constraint to the problem model of equation (6), the optimization problem can be described as follows, based on the mathematical model of equation (13):
in the formula f1(x) As defined in formula (6), f2(x)＝PSLL。
Based on the above, the implementation steps of the MODE algorithm-based MIMO radar sparse array optimization are described as follows:
inputting: a waveform covariance matrix R when the array is full, a population size NP, the number M of sparse array elements,
the number N of array elements of the full array;
step 1: initialization, generating NP random solutions using equation (7), using
Generating NP reverse solutions;
step 2: encoding, using equation (9) to generate 2NP solution sets;
and step 3: calculating fitness values, f1(x) as defined in equation (6), f2(x) ═ psll (x), evaluating fitness function values for 2NP solution sets, selecting NP feasible solutions using fast non-dominated sorting, FES ═ NP, storing them in existing population pop _ c.
And (3) starting a cycle: FES → 1 to Max _ Iteration
Beginning cycle i → 1 to NP
And 4, step 4: different from each other, three random and different individuals xr1, xr2 and xr3 are selected.
Generating variant individuals uiComprises the following steps:
Ui,G+1＝xtb,G+F(xr2,G+xr3,G) (17)
and 5: crossover, generating variant Vi using formula (1-11), FES ═ FES + 1;
step 6: pareto governs, and Vi governs Xi. If yes, in the advanced population pop _ c, replacing Xi with Vi, and adding the Xi to the advanced population pop _ a; if not, adding Xi into pop _ a;
and 7: generating NP feasible solutions by using a fast non-dominated sorting method, and storing the NP feasible solutions in pop _ c in an advanced population; where p iss,bestRepresents the best individual with respect to f1 (x);
and 8: judging whether the maximum iteration times is reached, if so, terminating the evolution, and leading the best individual p at the moments,bestAs a solution output; if not, continuing to return to the step 4;
finishing;
finishing;
and (3) outputting:
if not, returning to the step 2 until the iteration condition is met.
At this point, the array optimization of the virtual array is complete.
In the method, the single-target optimization is performed twice more by using the CA algorithm than before, then the modeling of the optimization problem is performed, and then the DE algorithm is adopted for solving. By adopting the DE algorithm after the CA algorithm is adopted twice, convergence to global optimization can be realized under the condition that the directional diagram is fixed. The DE algorithm is adopted for solving, the DE algorithm is simple in structure, easy to program and high in development potential, and has good function optimization performance, good algorithm stability and high convergence speed. Based on a multi-objective differential evolution algorithm, and by using several definitions of multi-objective optimization, the method is applied to the sparse array optimization of the MIMO radar. Compared with two single-target evolutionary algorithms (GA and DE), the MODE algorithm can realize parallel optimization of the pattern matching and peak sidelobe suppression performance of the sparse array, and finally the optimal sparse array meeting the pattern matching and peak sidelobe suppression is obtained.
Claims (7)
1. A MIMO radar sparse array optimization method based on MODE algorithm is characterized by comprising the following steps:
solving a single-target optimization problem of the MIMO radar sparse array;
taking single-target optimization as a constraint condition of the optimization, and further converting the single-target optimization into a multi-target optimization problem;
and solving by using a multi-objective MODE optimization algorithm to obtain a sparse array meeting the directional diagram matching and inhibiting side lobes.
2. The MIMO radar sparse array optimization method based on MODE algorithm of claim 1, wherein:
consider a ULA consisting of N antennas with an antenna aperture of (N-1) × d, d referring to the array element spacing, and d ═ λ/2; introducing an auxiliary vector Ps to express the relation between array elements of a sparse array and array elements of a full array, wherein the auxiliary vector Ps is composed of binary codes expressing the positions of the array elements; when the element in the vector Ps is 0, it indicates that the position array element corresponding to the element does not exist, whereas when the element in the vector Ps is 1, it indicates that the position array element corresponding to the element exists; by definition of the auxiliary vector, the steering vector of the sparse antenna can be expressed as:
as＝Ps◎a (1)
the formula-:
as is asA matrix set of sampling points at angles;
in summary, in the sparse array design of the MIMO radar, optimal pattern approximation performance is obtained by determining the position combination of M array elements in N candidate positions.
3. The MODE algorithm-based MIMO radar sparse array optimization method of claim 1, wherein the single target optimization method is as follows:
the first step is as follows: selecting a directional diagram matching design algorithm, selecting a CA algorithm to complete the directional diagram design when the array is full, and obtaining a corresponding covariance matrix R;
the second step is that: combining the CA algorithm with the differential algorithm, firstly solving the first step by using the CA algorithm, then modeling the optimization problem of the result obtained by the CA algorithm, and then solving by using the differential algorithm.
4. The method for optimizing the sparse array of the MIMO radar based on the MODE algorithm as claimed in claim 3, wherein the method of the first step is as follows:
firstly, completing the matching design of a directional diagram during full array, and obtaining a waveform weighting matrix W corresponding to the matched directional diagram:
s.t.Diag(UH-U)＝pd(3)
wherein W ═ R1/2，pdMatrix for desired patterns:
pd＝[φ(θ1)φ(θ2)...φ(θk)]T(4)
where φ (θ 1) represents the sampling points of the desired pattern in the observation angle region.
5. The method for optimizing the sparse array of the MIMO radar based on the MODE algorithm as claimed in claim 4, wherein the method of the second step is as follows:
setting the number of array elements and optimization parameters of the sparse array, and still optimizing the array combination of the sparse array elements by taking the approximate expected directional diagram as a target function so as to enable p → pd；
Modeling the optimization problem of the above formula to obtain a problem model as follows:
the constraint p (: 1) ═ 1 and p (: N) ═ 1 indicate that the array elements are arranged at both ends of the array in order to fix the actual aperture of the antenna, and in fact, equation (6) redefines the matching mean square error MSE between the pattern of the sparse array and the desired pattern.
6. A MODE algorithm based MIMO radar sparse array optimization method as claimed in claim 5, wherein the method for solving using the difference algorithm in the second step is as follows:
suppose the decision vector is composed of a continuous space vector x in D dimension (x)1,x2,x3.......xD) Representing that a global optimal solution is searched in a space of the D-dimensional real numerical value;
1) initializing a population: establishing an initial population containing NP individuals, wherein vectors of the individuals in the population are also called target vectors; considering that the individual parameter variables are constrained by the real condition, i.e. the boundary condition, the upper boundary of the jth parameter in the D-dimensional vector is assumed to be
The lower boundary is represented as
In the evolution of generation G, the initialization of the jth parameter of the ith target vector can be written as:
in equation (7), i ═ 1., NP, j ═ 1., D ═ N-2, which means that only D ═ N-2 participates in the evolution, rand (0,1) denotes random numbers uniformly distributed in the interval (0,1), and the target vector of the G th generation can be expressed as:
2) and (3) encoding: in order to mark the position of an array element in the Ps, a real variable of DE needs to be converted into a binary variable, and an initial value of the array element is just a discrete random disturbance combination in a certain antenna aperture; this perturbation is dominated by the G-th generation target vector as follows:
ps(n)＝binarysort(x(n))T(9)
since the aperture of the array has been determined, binary sort () indicates that the real numbers are sorted before binary encoding;
3) mutation operation: for each target vector, the DE algorithm generates a variant vector using a variant operation, and there are five common and disparate strategies for the differential evolution algorithm:
wherein r1, r2, r3, r4 and r5 belong to [1, NP ], are all different, and F is within [0,2], belongs to a variation factor and is a constant value, and the scaling of the differential vector is controlled; the larger the F is, the higher the population diversity is, and the slower the convergence speed is;
4) and (3) cross operation: a common crossover strategy generally selects a binomial crossover, the mathematical expression of which can be described as:
wherein CR is [0,1 ]]Is a cross probability constant defined by a user; rand (0,1) is as defined above; k is [1, D ]]Is a randomly selected number, ensuring ui,GIn which at least one dimension is a variation vector vi,GThe result is obtained; after the cross operation, in order to ensure the validity of the solution, whether the test variables meet the boundary conditions must be judged, otherwise, the mutually different variables are randomly generated again;
5) selecting operation: and selecting the individuals entering the next generation of population by adopting a greedy algorithm, wherein the test vector competes with the target vector, the fitness function is preferably selected as the individual of the filial generation, and the mathematical expression form of the greedy selection operator can be described as follows:
where f (.) is the fitness function defined above.
7. The MIMO radar sparse array optimization method based on MODE algorithm of claim 6, wherein:
after the single-target optimization is completed, solving the constraint condition of the single-target as multi-target optimization by using a MODE algorithm according to the process, wherein the constraint optimization problem can be described as follows:
in the formula, f (·) is an objective function, x represents a decision variable of D dimension like DE algorithm, g (x) and h (x) respectively represent unequal constraint conditions and equal constraint conditions, q represents the number of the unequal constraint conditions, and m-q represents the number of the equal constraint conditions; in equation (13), the constraint for equal can be converted into two unequal constraints by tolerance error as follows:
it follows that the two constraints are invertible to each other;
the problem model of equation (6) is applied as an inequality constraint, and the optimization problem can be described as follows according to the mathematical model of equation (13):
in the formula f1(x) As defined in formula (6), f2(x)＝PSLL。
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