Improved Bayesian optimization algorithm based on dynamic hyper-parameter and receiving and releasing coefficient

Abstract

The invention relates to an improved Bayesian optimization algorithm based on a dynamic hyper-parameter and a receiving and releasing coefficient. The improved Bayesian optimization algorithm is characterized by comprising the following steps: S1, calculating the receiving and releasing coefficient of a multi-dimensional parameter of a radar target signal micro-motion parameter identification system; s2, dynamic hyper-parameters are adjusted; s3, carrying out a Bayesian optimization process based on dynamic hyper-parameters and receiving and releasing coefficients; s4, obtaining an optimal solution of a multi-dimensional input parameter in a single Bayesian optimization process; s5, repeating the steps S1 to S4 until a constraint threshold requirement is met, and terminating the optimization process; the method can effectively improve the solving precision and convergence speed of a radar target signal micro-motion parameter identification system.

Description

Technical Field

[0001] The present invention relates to an improved Bayesian optimization algorithm based on dynamic hyperparameters and scaling factors, and particularly to an improved Bayesian optimization algorithm applied to a radar target signal micro-motion parameter identification system, belonging to the field of optimization and solution of multi-dimensional complex black-box system problems. Background Art

[0002] The radar target signal micro-motion parameter identification system is a multi-dimensional complex black-box system used for extracting the characteristic parameters of radar target signals in weapon equipment test and evaluation, involving multiple technologies such as target flight attitude and orbit coupling modeling, echo signal modeling of radar target motion, and radar time-frequency signal transformation processing. This system contains more than twelve-dimensional characteristic parameters, and the single evaluation time is relatively long, belonging to a multi-dimensional high-evaluation-cost system.

[0003] This system converts the identification problem of radar target signal micro-motion parameters into an optimization and solution problem based on the optimal matching degree of the time-frequency image features of radar target signals by performing simulation inversion on the multi-dimensional parameters of radar target micro-motion signals and correlating and matching them with real radar target signals.

[0004] To solve the problems of complex systems with multi-dimensional parameter spaces and no explicit analytical expressions, this system is regarded as a black-box system. The problem of the optimal matching degree based on the time-frequency image features of radar target signals is essentially a problem of solving a multi-dimensional complex black-box system.

[0005] The optimization and solution of multi-dimensional complex black-box systems are common problems faced by multiple disciplines such as communication, computer, automation, artificial intelligence, and operations research. Finding efficient optimization algorithms has become one of the main research contents of many disciplines. The classical combinatorial optimization and solution algorithms for black-box systems include the Monte Carlo method, genetic algorithm, simulated annealing algorithm, and Bayesian optimization algorithm, etc.

[0006] The Monte Carlo method realizes the optimization and solution of the system by randomly sampling points. Due to the huge multi-dimensional parameter space, when using the Monte Carlo method for radar signal micro-motion parameter identification, it is difficult to meet the requirements in terms of convergence speed and calculation accuracy.

[0007] The genetic algorithm is based on the idea of biological evolution and realizes the optimization and solution of the system after multiple iterations through initializing the population and the crossover and mutation of individuals in the population. When using the genetic algorithm in the radar target signal micro-motion parameter identification system, the optimization convergence speed is greatly affected by parameters such as the quality of the initial population, population size, and crossover and mutation strategies. In order to find the optimal solution in the multi-dimensional parameter space, the population often needs to go through a large number of iterative operations, which takes a long time.

[0008] Based on the principle of slow temperature reduction in physics, when the simulated annealing algorithm walks in the search space, it uses the Metropolis sampling criterion to adjust the probability between global optimization and local optimization, and through the physical cooling process, the random walk gradually converges to the local optimal solution. However, a large number of iterative calculations are still required during the optimization process. For the radar target signal micro-motion parameter identification system, the calculation efficiency and accuracy are also difficult to meet the requirements.

[0009] The basic principle of Bayesian optimization is based on Bayes' theorem. This algorithm is very efficient when optimizing in a low-dimensional system with high evaluation costs, but it is difficult to extend it to the optimization and solution of a multi-dimensional system with high evaluation costs. On the one hand, because the Bayesian optimization algorithm tends to global optimization when facing multi-dimensional system optimization, the algorithm performance is similar to the Monte Carlo method; on the other hand, the parameter space of the multi-dimensional system is large, and more observation points are required for solution. However, as the number of observation points increases, the time cost of a single iteration of the Bayesian optimization algorithm will gradually increase, greatly affecting the solution efficiency of the multi-dimensional system with high evaluation costs.

[0010] The radar target signal micro-motion parameter identification system belongs to a multi-dimensional system with high evaluation costs, and the time cost of a single optimization evaluation is large. It is necessary to obtain a feasible solution through a small number of iterations within a tolerable time. However, the Monte Carlo method and optimization algorithms represented by genetic algorithms and simulated annealing algorithms all require a large number of iterative operations when solving in a multi-dimensional parameter space, and the probability of obtaining a feasible solution within a tolerable time is low; the classical Bayesian optimization algorithm can greatly reduce the number of optimization solutions through a probability surrogate model when optimizing and solving a low-dimensional system with high evaluation costs, but when optimizing a multi-dimensional system with high evaluation costs, it is necessary to greatly increase the number of observation points. Since the update time of a single probability surrogate model is proportional to the cube of the number of observation points, as the number of observation points increases, the time cost of a single Bayesian optimization iteration will increase rapidly. After a certain time, the optimization speed of the Bayesian optimization algorithm will also become slower and slower, making it difficult to meet the fast convergence requirements for parameter extraction in the radar target signal micro-motion parameter identification system. Summary of the Invention

[0011] The object of the present invention is to provide an improved Bayesian optimization algorithm based on dynamic hyperparameters and scaling coefficients, which is an improved Bayesian optimization algorithm applied to the radar target signal micro-motion parameter identification system. It changes the single Bayesian optimization process to a sequential Bayesian optimization process based on the tuning of dynamic hyperparameters and scaling coefficients, which can effectively improve the solution accuracy and convergence speed of the radar target signal micro-motion parameter identification system.

[0012] To achieve the above object, the solution of the present invention is implemented as follows: The improved Bayesian optimization algorithm based on dynamic hyperparameters and scaling coefficients is an improvement based on the classical Bayesian optimization algorithm, and is characterized by including the following steps:

[0013] S1. Calculate the scaling coefficient of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system;

[0014] S2. Adjust the dynamic hyperparameters: Calculate the numerical range of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system and the number of observation points in the Bayesian optimization process;

[0015] S3. Conduct the Bayesian optimization process based on the dynamic hyperparameters and the scaling coefficient;

[0016] S4. Obtain the optimal solution of the multi-dimensional input parameters for a single Bayesian optimization process;

[0017] S5. Repeat steps S1 to S4 until the optimization process is terminated when the constraint threshold requirement is met;

[0018] Split the classical Bayesian optimization algorithm into a sequential optimization process of cyclic Bayesian optimization, and represent the sequential optimization process as: {Ω1, Ω2, Ω3, …, Ω n}, where n is the number of cycles of the improved Bayesian optimization algorithm, n >= 2; the number of observation points corresponding to each cyclic Bayesian optimization is respectively {m1, m2, m3, …, m n}; the objective function values of the historical global optimal solutions corresponding to n cyclic Bayesian optimizations are respectively {B1, B2, B3, …, B p}, and the total number of global observation points corresponding to {B1, B2, B3, …, B p} is {u1, u2, u3, …, u p}, where p is the subscript of the sequential optimal solution set obtained by n cyclic Bayesian optimizations, and p is a positive integer.

[0019] The main role of the scaling coefficient in step S1 is to dynamically contract and expand the search space of the multi-dimensional input parameters to accelerate the convergence speed of the cyclic Bayesian optimization. It is obtained by coupling the principal component regression coefficient of the multi-dimensional parameters with the objective function value of the global optimal solution.

[0020] Assume that the input of the radar target signal micro-motion parameter identification system contains i_n-dimensional parameters, and the scaling coefficients corresponding to the i_n-dimensional parameters are {S1, S2, S3…S i_n}, then the calculation process of the scaling coefficient is as follows:

[0021] First, calculate the principal component regression coefficients {g1, g2, g3…g i_n} of the i_n-dimensional parameters. The principal component is a recombined form of the i_n-dimensional parameters, and the projection of the i_n-dimensional parameters in the principal component direction has the maximum variance. Assume that {d1, d2, d3…d i_n} represents the observed values of the multi-dimensional input variables of the radar target signal micro-motion parameter identification system. Using the formula D i =(di -μ i ) / σ i (i ∈ [1, i_n], μ i represents the mean of {d i}, and σ i represents the standard deviation of {d i}}. Standardize {d1, d2, d3... d i_n} to obtain new observed values {D1, D2, D3... D i_n}.

[0022] Establish a weighted sum function for the new observed values {D1, D2, D3... D i_n}

[0023] f = c1D1 + c2D2 +... c i_n D i_n , where {c1, c2, c3... c i_n} represents the weight coefficients of the i_n - dimensional parameters, To perform principal component analysis on f = c1D1 + c2D2 +... c i_n D i_n , construct q standard orthogonal direction vectors {Z1, Z2, Z3... Z q} such that the cumulative contribution rate of these q orthogonal vectors ≥ 80%. The expression of {Z1, Z2, Z3... Z q} is as follows:

[0024]

[0025] In the formula, for any i ∈ [1, q], it has

[0026] Use variance to reflect the degree of data difference, and successively calculate the {c i (Z i ∈ [Z1, Z q ) corresponding to the maximum variance of each group of Z i1 , c i2 , c i3 ... c ii_n}, and the weight matrix corresponding to the q principal components {Z1, Z2, Z3... Z q} can be obtained After that, use the least - squares method to perform regression analysis on the principal components {Z1, Z2, Z3... Z q} to obtain the principal component regression equation Among them, {z1, z2, z3... z q} represents the weights of the q principal components. Substitute into the principal component regression equation and transform it into the independent variables {D1, D2, D3... D i_nThe principal component regression equation of {} is The principal component regression coefficients {g1, g2, g3... g of the i_n-dimensional parameters can be obtained i_n}.

[0027] Secondly, couple the principal component regression coefficients {g1, g2, g3... g i_n} with the objective function value of the global optimal solution. Using the formula Normalize the principal component regression coefficients {g1, g2, g3... g i_n} to obtain the normalization coefficients After that, obtain the maximum value of the normalization coefficients Assume that the upper and lower adjustment limits of the retraction and extension coefficients {S1, S2, S3... S i_n} are [S up S down , then the initial value of the retraction and extension coefficients is

[0028] Assume that the regulation range of the objective function value of the radar target signal micro-motion parameter identification system is [0PR max , and the corresponding regulation upper and lower limits are [pr min pr max , and the objective function value corresponding to the current global optimal solution of the system is B p (p is the subscript of the optimal solution set obtained by n - cycle Bayesian optimization, p ≥ 1), then based on the objective function value B p of the adjustment coefficient After that, obtain the final result of the retraction and extension coefficients:

[0029] Where B p > PR max When, let B p = PR max .

[0030] The calculation process of step S2 is as follows:

[0031] Assume that the initial extreme value of the number of observation points is M0, and the number of observation points for a single Bayesian optimization in the n - th sequential optimization is m n , n ≥ 1 (this includes the first cycle).

[0032] There is no strict mathematical formula for the initial extreme value M0, and its value is proportional to the time cost of a single evaluation of the system. However, generally, the time cost of M0 observation values in a single update of the probability surrogate model should not be greater than the time of a single evaluation of the system.

[0033] The number of observation points m for a single Bayesian optimization n The formula is as follows:

[0034]

[0035] where u p is the total number of global observation points of the current global optimal solution, and u p-1 represents u p the total number of global observation points of the previous global optimal solution, p represents the p-th global optimal solution, p ≥ 1.

[0036] Assume that the range of the i_n-dimensional input parameters included in the radar target signal micro-motion parameter identification system is [pre1, sup1], [pre2, sup2], [pre3, sup3],.......,[pre i_n , sup i_n , and according to the scaling factors {S1, S2, S3…S i_n} in step 1, update the range of the i_n-dimensional input parameters to [max(pre1, I_B p (1) - (sup1 - pre1) * S1 / 2), min(sup1, I_B p (1) + (sup1 - pre1) * S1 / 2))], [max(pre2, I_B p (2) - (sup2 - pre2) * S2 / 2), min(sup2, I_B p (2) + (sup2 - pre2) * S2 / 2))], [max(pre3, I_B p (3) - (sup3 - pre3) * S3 / 2), min(sup3, I_B p (3) + (sup3 - pre3) * S3 / 2))],

[0037] ... where I_B p = [i_b1, i_b2, i_b3,…, i_b i_n represents the i_n-dimensional observation points corresponding to the historical global optimal objective function value B p .

[0038] The specific process of step S3 is as follows:

[0039] First, establish a probabilistic surrogate model for single-shot Bayesian optimization.

[0040] Adopt a probabilistic surrogate model based on Gaussian process. Gaussian process is a generalization of multivariate Gaussian probability distribution. Assume that the vector represents the set of observation points for single-shot Bayesian optimization, where the observation components X1, X2, X3,…X t are all i_n-dimensional observation vectors, t ≤ m n , mn is the maximum number of observed points in a single Bayesian optimization in step S2, X1, X2, X3, … X t obeys a Gaussian distribution, where N(·) represents a Gaussian distribution, represents the multivariate mean vector of, Σ = cov(X m , X n ) = (σ mn ) i_n*i_n represents the covariance matrix of, σ mn is the covariance of any two random variables X m and X n , where m, n ∈ [1, t] and m ≠ n. The probability density function of is where power(·) represents a power function and e is the natural constant.

[0041] The Gaussian process can perform regression prediction on the radar target signal micro-motion parameter identification system. In the regression model, no functional relationship between the observed variables and the target variables is established. Instead, the relationship between the target variables is directly established through the kernel function K(X, Z), and its expression is

[0042]

[0043] Here, the radial basis kernel function is used. Among them, γ is the amplitude adjustment parameter, and λ is the width parameter, which controls the radial action range.

[0044] Assume that Y is the target function value of the radar target signal micro-motion parameter identification system. Let where ε is the noise, and the probability distribution (σ ε is the variance of the noise ε), and the prior probability distribution of the function f(·) is p(f) = GP(m(*), K(*, *)), where GP represents the Gaussian process, m(*) is the mean function, and K(*, *) is the kernel function. Assume that the current dataset composed of known observed points and target function values is D 1:t = {(X1, Y1), (X2, Y2), (X3, Y3), …, (X t , Y t )}(t is the number of known observed points), and the vector to be updated is X new , then the posterior probability distribution of its target function value Y new = f(X new ) + ε is:

[0045] p(Y new ) = N(μ(X new |D 1:t), σ 2 (X new | D 1:t ))

[0046] where μ(X new | D 1:t ) is the predicted mean of the objective function value Y new , and σ 2 (X new | D 1:t ) is the variance of Y new corresponding to the new observation point X new . Thus, the probabilistic surrogate model based on the Gaussian process is established.

[0047] To infer a new observation point X 1:t from the known observation data set D new , it is necessary to establish an acquisition function. Here, the EI (expected improvement) acquisition function is selected, and its expression is:

[0048]

[0049] where f(x + ) represents the optimal solution of the current observation set; μ(x) and σ(x) represent the mean and variance of the posterior distribution respectively, and Φ(·) represents the probability density function of the standard normal distribution.

[0050] Select the next most "promising" evaluation point X t+1 (i.e., X new ) according to the maximization of the acquisition function, calculate its objective function value Y t+1 (i.e., Y new ), and add the newly obtained input-observation value (X t+1 , Y t+1 ) to the known observation set D 1:t , update the probabilistic surrogate model, and prepare for the next iteration until the number of observation points is equal to the maximum number of observations m n calculated in step S2 for single Bayesian optimization.

[0051] Step S4 mentioned above is to set the global storage variable DATA_BEST for sequential optimization. Assume that the current is the nth (n ≥ 1) sequential optimization. If the objective function value corresponding to the optimal solution of this optimization process is better than the objective function value corresponding to DATA_BEST, then update DATA_BEST to the optimal solution of the nth sequential optimization.

[0052] Step S5 mentioned above is to check whether the constraint threshold is satisfied, i.e., u n - u p > ST. Among them, u nDenote the total number of global observations after the sequential optimization of the n-th order (n is the number of cycles of Bayesian sequential optimization), and u p is the global optimal solution B obtained by the sequential optimization of the n-th order p (p represents the p-th global optimal solution, p≥1), and the corresponding total number of observations; the termination extreme value ST of sequential optimization = c*M0, generally c≥3, when u n -u p >ST, all sequential optimization processes should be terminated. If the constraint threshold is not met, repeat steps S1 to S4 to continue the sequential optimization of the radar target signal micro-motion parameter identification system.

[0053] After adopting the above scheme, the positive effects of the present invention are as follows:

[0054] 1. It avoids the problem that the total number of observations for optimization in the classical Bayesian optimization algorithm is too large, resulting in a sharp increase in time consumption;

[0055] 2. Compared with the Monte Carlo method, simulated annealing algorithm, genetic algorithm, and classical Bayesian optimization algorithm, the improved Bayesian optimization algorithm achieves a double improvement in convergence speed and solution accuracy;

[0056] 3. It improves the matching efficiency and accuracy of the radar target signal micro-motion parameter identification system, and plays an important role in promoting the application and popularization of this system in the field of weapon equipment test and evaluation. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 is a schematic flowchart of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0058] The following further describes the invention in conjunction with the drawings and embodiments: As Figure 1 shown, the improved Bayesian optimization algorithm based on dynamic hyperparameters and scaling factors is characterized by including the following steps:

[0059] S1. Calculate the scaling factors of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system;

[0060] S2. Adjust the dynamic hyperparameters: calculate the numerical range of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system and the number of observation points in the Bayesian optimization process;

[0061] S3. Conduct the Bayesian optimization process based on dynamic hyperparameters and scaling factors;

[0062] S4. Obtain the optimal solution of the multi-dimensional input parameters in a single Bayesian optimization process;

[0063] S5. Repeat steps S1 to S4 until the constraint threshold requirement is met to terminate the optimization process;

[0064] The classical Bayesian optimization algorithm is split into a sequential optimization process of cyclic Bayesian optimization, and the sequential optimization process is expressed as: {Ω1, Ω2, Ω3, …, Ω n}, where n is the number of cycles of the improved Bayesian optimization algorithm, n >= 2; the number of observation points corresponding to each cycle of Bayesian optimization is {m1, m2, m3, …, m n} respectively; the objective function values of the historical global optimal solutions corresponding to n cycles of Bayesian optimization are {B1, B2, B3, …, B p} respectively, and the total number of global observation points corresponding to {B1, B2, B3, …, B p} is {u1, u2, u3, …, u p}, where p is the subscript of the sequential optimal solution set obtained by n cycles of Bayesian optimization, and p is a positive integer.

[0065] In step S1, the main function of the scaling coefficient is to dynamically contract and expand the search space of multi-dimensional input parameters to accelerate the convergence rate of cyclic Bayesian optimization. It is obtained by coupling the principal component regression coefficient of multi-dimensional parameters with the objective function value of the global optimal solution;

[0066] Suppose the input of the radar target signal micro-motion parameter identification system contains i_n-dimensional parameters, and the scaling coefficients corresponding to the i_n-dimensional parameters are {S1, S2, S3…S i_n}, then the calculation process of the scaling coefficient is as follows:

[0067] First, calculate the principal component regression coefficients {g1, g2, g3…g i_n} of the i_n-dimensional parameters; the principal components are a re-combination of the i_n-dimensional parameters, and the projections of the i_n-dimensional parameters in the principal component direction have the largest variance; suppose {d1, d2, d3…d i_n} represents the observed values of the multi-dimensional input variables of the radar target signal micro-motion parameter identification system. Using the formula D i =(d i -μ i ) / σ i (i ∈ [1, i_n], μ i represents the mean of {d i}, and σ i represents the standard deviation of {d i}) to standardize {d1, d2, d3…d i_n} to obtain new observed values {D1, D2, D3…D i_n};

[0068] Establish a weighted sum function of the new observed values {D1, D2, D3…D i_n}

[0069] f = c1D1 + c2D2 + … c i_n D i_n , where {c1, c2, c3…c i_n} represents the weight coefficients of the i_n - dimensional parameters. In order to perform principal component analysis on f = c1D1 + c2D2 + … c i_n D i_n , q standard orthogonal direction vectors {Z1, Z2, Z3…Z q} are constructed such that the cumulative contribution rate of these q orthogonal vectors ≥ 80%. The expression of {Z1, Z2, Z3…Z q} is as follows:

[0070]

[0071] In the formula, for any i ∈ [1, q], it has

[0072] Using variance to reflect the degree of data difference, calculate in turn the {c i (Z i ∈ [Z1, Z q ) corresponding to the maximum variance of each group of Z i1 , c i2 , c i3 …c ii_n}, and the weight matrix corresponding to the q principal components {Z1, Z2, Z3…Z q} can be obtained. After that, using the least - squares method to perform regression analysis on the principal components {Z1, Z2, Z3…Z q}, the principal component regression equation can be obtained. Among them, {z1, z2, z3…z q} represents the weights of the q principal components. Substitute into the principal component regression equation and transform it into the principal component regression equation of the independent variables {D1, D2, D3…D i_n} as The principal component regression coefficients {g1, g2, g3…g i_n} of the i_n - dimensional parameters can be obtained.

[0073] Secondly, couple the principal component regression coefficients {g1, g2, g3…g i_n} with the objective function value of the global optimal solution; use the formula to normalize the principal component regression coefficients {g1, g2, g3…g i_n}, and the normalized coefficients are obtained. Suppose the retraction and release coefficients {S1, S2, S3…S i_nThe upper and lower limits for the adjustment of {} are [S up S down , then the initial value of the retraction and release coefficient is

[0074] Assume that the regulation range of the objective function value of the radar target signal micro-motion parameter identification system is [0PR max , and the corresponding regulation upper and lower limits are [pr min pr max , and the objective function value corresponding to the current global optimal solution of the system is B p (p is the subscript of the optimal solution set obtained by n - cycle Bayesian optimization, p≥1), then based on the system objective function value B p The adjustment coefficient After that, the final result of the retraction and release coefficient is obtained:

[0075] Where B p >PR max When, let B p =PR max .

[0076] The calculation process of step S2 is as follows:

[0077] Assume that the initial extreme value of the number of observation points is M0, and the number of observation points for single - time Bayesian optimization in the n - th sequential optimization is m n , n≥1 (including the first cycle here);

[0078] There is no strict mathematical formula for the initial extreme value M0, and its value is proportional to the time cost of a single evaluation of the system. However, generally, the time cost of M0 observation values in a single update of the probabilistic surrogate model should not be greater than the time of a single evaluation of the system;

[0079] The number of observation points m for single - time Bayesian optimization n The formula is as follows:

[0080]

[0081] Where u p Is the total number of global observations of the current global optimal solution, u p-1 Represents u p The total number of global observations of the previous global optimal solution, and p represents the p - th global optimal solution, p≥1;

[0082] Assume that the i_n - dimensional input parameter range included in the radar target signal micro - motion parameter identification system is [pre1, sup1], [pre2, sup2], [pre3, sup3],;;;;;;;, [pre i_n , sup i_n, update the range of the \(i_n\)-dimensional input parameters as \([\max(\text{pre}1, I_B i_n (1) - (\text{sup}1 - \text{pre}1) * S1 / 2), \min(\text{sup}1, I_B p (1) + (\text{sup}1 - \text{pre}1) * S1 / 2))]\), \([\max(\text{pre}2, I_B p (2) - (\text{sup}2 - \text{pre}2) * S2 / 2), \min(\text{sup}2, I_B p (2) + (\text{sup}2 - \text{pre}2) * S2 / 2))]\), \([\max(\text{pre}3, I_B p (3) - (\text{sup}3 - \text{pre}3) * S3 / 2), \min(\text{sup}3, I_B p (3) + (\text{sup}3 - \text{pre}3) * S3 / 2))]\), p ...、

[0083] ...、 where \(I_B p = [i_b1, i_b2, i_b3, \ldots, i_b i_n \) represents the \(i_n\)-dimensional observation point corresponding to the historical global optimal objective function value \(B p ;

[0084] The specific process of step S3 is as follows:

[0085] First, establish a probabilistic surrogate model for single - shot Bayesian optimization;

[0086] Adopt a probabilistic surrogate model based on Gaussian process. Gaussian process is a generalization of the multivariate Gaussian probability distribution. Assume the vector represents the set of observation points for single - shot Bayesian optimization, where the observation components \(X1, X2, X3, \ldots X t are all \(i_n\) - dimensional observation vectors, \(t\leq m n , \(m n is the maximum number of observations for single - shot Bayesian optimization in step S2, \(X1, X2, X3, \ldots X t follow a Gaussian distribution, where \(N(\cdot)\) represents the Gaussian distribution, represents 's multivariate mean vector, \(\Sigma=\text{cov}(X m , X n ) = (\sigma mn ) i_n*i_n represents 's covariance matrix, \(\sigma mn is the covariance of any two random variables \(X m and \(X n ), where \(m, n\in[1, t]\) and \(m\neq n\); The probability density function is where power(·) represents the power function and e is the natural constant;

[0087] The Gaussian process can perform regression prediction on the radar target signal micro-motion parameter identification system. In the regression model, instead of establishing a functional relationship between the observed variables and the target variables, the relationship between the target variables is directly established through the kernel function K(X,Z), and its expression is

[0088]

[0089] The radial basis kernel function is used here. Among them, γ is the amplitude adjustment parameter, and λ is the width parameter, which controls the radial action range;

[0090] Assume that Y is the target function value of the radar target signal micro-motion parameter identification system, and let where ε is the noise, and the probability distribution (σ ε is the variance of the noise ε), and the prior probability distribution of the function f(·) is p(f) = GP(m(*),K(*,*)). Among them, GP represents the Gaussian process, m(*) is the mean function, and K(*,*) is the kernel function. Assume that the dataset composed of the current known observation points and the target function values is D 1:t ={(X1,Y1),(X2,Y2),(X3,Y3),…,(X t ,Y t )} t is the number of known observation points, and the vector to be updated is X new , then the posterior probability distribution of its target function value Y new =f(X new )+ε is:

[0091] p(Y new ) = N(μ(X new |D 1:t ),σ 2 (X new |D 1:t ))

[0092] where μ(X new |D 1:t ) is the predicted mean of the target function value Y new , and σ 2 (X new |D 1:t ) is the variance corresponding to the new observation point X new of Y new ; thus, the probability surrogate model based on the Gaussian process is established;

[0093] In order to obtain from the known observation dataset D 1:tEstimate a new observation point X new , it is necessary to establish an acquisition function. Here, the EI (expected improvement) acquisition function is selected, and its expression is:

[0094]

[0095] where f(x + ) represents the optimal solution of the current observation set; μ(x) and σ(x) represent the mean and variance of the posterior distribution respectively, and Φ(·) represents the probability density function of the standard normal distribution;

[0096] Select the most "promising" next evaluation point X according to the maximized acquisition function t+1 i.e., X new , calculate its objective function value Y t+1 i.e., Y new , add the newly obtained input-observation value (X t+1 , Y t+1 ) to the known observation set D 1:t , update the probability surrogate model, and prepare for the next iteration until the number of observation points is equal to the maximum number of observations m for single Bayesian optimization calculated in step S2 n ;

[0097] The step S4 is to set the global storage variable DATA_BEST for sequential optimization. Assume that the current is the nth (n≥1) sequential optimization. If the objective function value corresponding to the optimal solution in this optimization process is better than the objective function value corresponding to DATA_BEST, then update DATA_BEST to the optimal solution of the nth sequential optimization;

[0098] The step S5 is to compare whether the constraint threshold is satisfied u n -u p > ST; where, u n represents the total number of global observations after the end of the nth sequential optimization, n is the number of cycles of Bayesian sequential optimization, u p is the global optimal solution B obtained from the nth sequential optimization p , p represents the pth global optimal solution, and the total number of observations corresponding to p≥1; the termination extreme value ST of sequential optimization = c * M0, generally c≥3. When u n -u p > ST, all sequential optimization processes should be terminated. If the constraint threshold is not satisfied, repeat the steps from S1 to S4 and continue the sequential optimization of the radar target signal micro-motion parameter identification system.

[0099] The improved Bayesian optimization algorithm applied to the radar target signal micro-motion parameter identification system is improved based on the classical Bayesian optimization algorithm, which is further described below:

[0100] Example

[0101] The radar target signal micro-motion parameter identification system used includes a twelve-dimensional input parameter range: [pre1, sup1], [pre2, sup2], [pre3, sup3],....., [pre 12 , sup 12 , and the system objective function is where \(x\) represents a twelve-dimensional decision vector, \(X\) represents the decision space, and \(F\) represents the objective function.

[0102] Regarding step S1, assume that the scaling coefficients corresponding to the twelve-dimensional parameters are \(\{S1, S2, S3…S 12 \}\), then the calculation process of the scaling coefficients is as follows:

[0103] First, calculate the principal component regression coefficients \(\{g1, g2, g3…g 12 \}\). For the \(n\)th (\(n\geq2\)) cycle of the improved Bayesian optimization (for the first cycle, each parameter takes the initial value), represent the twelve-dimensional observation points of the \((n - 1)\)th cycle as a vector \(\{d1, d2, d3…d 12 \}\), and use the formula \(D i =(d i -\(\mu i ) / \(\sigma i \) (\(i\in[1, 12]\), \(\mu i \) represents the mean of \(\{d i \}\), and \(\sigma i \) represents the standard deviation of \(\{d i \}\)) to standardize \(\{d1, d2, d3…d 12 \}\) to obtain new observed values \(\{D1, D2, D3…D 12 \}\).

[0104] Establish a weighted sum function of the new observed values \(\{D1, D2, D3…D 12 \)

[0105] \(f = c1D1 + c2D2 + …c 12 D 12 \), where \(\{c1, c2, c3…c 12 \}\) represents the weight coefficients of the twelve-dimensional parameters, In order to perform principal component analysis on \(f = c1D1 + c2D2 + …c 12 D 12 \), construct \(q\) standard orthogonal direction vectors \(\{Z1, Z2, Z3…Z q \}\) such that the cumulative contribution rate of these \(q\) orthogonal vectors \(\geq80\%\), and the expression of \(\{Z1, Z2, Z3…Z q \) is as follows:

[0106]

[0107] In the formula, for any \(i\in[1,q]\), there is

[0108] Using the variance to reflect the degree of data difference, calculate in turn to make the variance of each group of \(Z\) i (\(Z\) i \(\in[Z_1,Z\) q ) reach the maximum value, and the corresponding \(\{c\) i1 , \(c\) i2 , \(c\) i3 \(\cdots c\) i12}\), and the weight matrix corresponding to \(q\) principal components \(\{Z_1, Z_2, Z_3\cdots Z\) q}\) can be obtained After that, use the least squares method to perform regression analysis on the principal components \(\{Z_1, Z_2, Z_3\cdots Z\) q}\) to obtain the principal component regression equation Among them, \(\{z_1, z_2, z_3\cdots z\) q}\) represents the weights of \(q\) principal components. Substitute into the principal component regression equation and transform it into the principal component regression equation of independent variables \(\{D_1, D_2, D_3\cdots D\) 12}\) as The principal component regression coefficients \(\{g_1, g_2, g_3\cdots g\) 12}\) of the 12-dimensional parameters can be obtained.

[0109] Secondly, couple the principal component regression coefficients \(\{g_1, g_2, g_3\cdots g\) 12}\) with the objective function value of the global optimal solution. Use the formula to normalize the principal component regression coefficients \(\{g_1, g_2, g_3\cdots g\) 12}\) to obtain the normalized coefficients After that, obtain the maximum value of the normalized coefficients Assume that the upper and lower adjustment limits of the retraction and release coefficients \(\{S_1, S_2, S_3\cdots S\) 12}\) are \([S\) up \(S\) down , then the initial value of the retraction and release coefficients is

[0110] Assume that the regulation range of the system objective function value is \([0, PR\) max , and the corresponding regulation upper and lower limits are \([pr\) min \(pr\) max , and the objective function value corresponding to the current global optimal solution of the system is \(B\) p (\(p\) is the subscript of the optimal solution set obtained by \(n\) - cycle Bayesian optimization, \(p\geq1\)), then the adjustment coefficient based on the system objective function value After that, the final result of the retraction and extension coefficient is obtained: where B p > PR max When, let B p = PR max .

[0111] Regarding step S2, the number of observation points corresponding to the single evaluation time of the radar target signal micro-motion parameter identification system is set as the initial extreme value M0 (M0 = 36) of the observation points.

[0112] The number of observation points m n for single Bayesian optimization is as follows:

[0113]

[0114] where u p is the total number of global observation points of the current global optimal solution, u p-1 represents the total number of global observation points of the previous global optimal solution, p represents the p-th global optimal solution, p ≥ 1. p

[0115] According to the retraction and extension coefficients {S1, S2, S3…S 12}, update the twelve-dimensional input parameter range to [max(pre1, IB p (1)-(sup1-pre1)*S1 / 2), min(sup1, IB p (1)+(sup1-pre1)*S1 / 2))], [max(pre2, IB p (2)-(sup2-pre2)*S2 / 2), min(sup2, IB p (2)+(sup2-pre2)*S2 / 2))], [max(pre3, IB p (3)-(sup3-pre3)*S3 / 2), min(sup3, IB p (3)+(sup3-pre3)*S3 / 2))],

[0116] ... where IB p = [i_b1, i_b2, i_b3,…, i_b 12 represents the 12-dimensional observation point corresponding to the historical global optimal objective function value B p .

[0117] Regarding step S3, a probabilistic surrogate model based on Gaussian process is adopted. Assuming the vector Denote the set of observation points for single Bayesian optimization, where the observed components \(X_1, X_2, X_3, \ldots, X\) t are all \(i_n\)-dimensional observation vectors, \(t \leq m\) n , \(m\) n is the maximum number of observations for single Bayesian optimization in step S2, and \(X_1, X_2, X_3, \ldots, X\) t obeys a Gaussian distribution, where \(N(\cdot)\) represents the Gaussian distribution, represents the multivariate mean vector of, \(\Sigma=\text{cov}(X\) m , \(X\) n ) = (\sigma mn ) i_n*i_n represents the covariance matrix of, \(\sigma mn is the covariance of any two random variables \(X\) m and \(X\) n , where \(m, n \in [1, t]\) and \(m \neq n\). The probability density function of is where \(\text{power}(\cdot)\) represents the power function and \(e\) is the natural constant.

[0118] The relationship between the target variables of the radar target signal micro-motion parameter identification system is established through the kernel function \(K(X, Z)\), and its expression is

[0119]

[0120] Here, the radial basis kernel function is used, where \(\gamma\) is the amplitude adjustment parameter and \(\lambda\) is the width parameter, controlling the radial action range.

[0121] Assume that \(Y\) is the value of the target function of the radar target signal micro-motion parameter identification system, and let where \(\varepsilon\) is the noise, and the probability distribution \((\sigma ε is the variance of the noise \(\varepsilon\)), the prior probability distribution of the function \(f\) is \(p(f) = GP(m(*), K(*,*))\), where \(GP\) represents the Gaussian process, \(m(*)\) is the mean function, and \(K(*,*)\) is the radial basis kernel function. Assume that the dataset composed of the current known observation points and the target function values is \(D\) 1:t =\{(X_1, Y_1), (X_2, Y_2), (X_3, Y_3), \ldots, (X t , Y t )\}\((t\) is the number of known observation points), and the vector to be updated is \(X\) new , then the posterior probability distribution of its target function value \(Y\) new = f(X new )+\(\varepsilon\) is:[[]]

[0122] p(Y new) = N(μ(X new |D 1:t ), σ 2 (X new |D 1:t ))

[0123] where μ(X new |D 1:t ) is the predicted mean of the objective function value Y new , and σ 2 (X new |D 1:t ) is the variance of Y new corresponding to the new observation point X new . Thus, the probabilistic surrogate model based on the Gaussian process is established.

[0124] To infer a new observation point X 1:t from the known observation dataset D new , it is necessary to establish an acquisition function. Here, the EI (expected improvement) acquisition function is selected, and its expression is:

[0125]

[0126] where f(x + ) represents the optimal solution of the current observation set; μ(x) and σ(x) represent the mean and variance of the posterior distribution respectively, and Φ(·) represents the probability density function of the standard normal distribution.

[0127] According to the maximization of the acquisition function, the next most "promising" evaluation point X t+1 (i.e., X new ) is selected, and its objective function value Y t+1 (i.e., Y new ) is calculated. The newly obtained input-observation value (X t+1 , Y t+1 ) is added to the known observation set D 1:t , and the probabilistic surrogate model is updated to prepare for the next iteration until the number of observation points is equal to the maximum number of observations m n calculated in step S2 for single Bayesian optimization.

[0128] Regarding step S4, the global storage variable DATA_BEST for sequential optimization is set. Assume that the current is the nth (n≥1) sequential optimization. If the objective function value corresponding to the optimal solution of this optimization process is better than the objective function value corresponding to DATA_BEST, then DATA_BEST is updated to the optimal solution of the nth sequential optimization.

[0129] Regarding step S5, it is to check whether the constraint threshold is satisfied for u n - u p>ST. Among them, u n represents the total number of global observations after the end of the n-th sequential optimization (n is the number of cycles of Bayesian sequential optimization), u p is the global optimal solution B obtained by the n-th sequential optimization p (p represents the p-th global optimal solution, p≥1) corresponding total number of observations; the termination extreme value ST of sequential optimization = c*M0, generally c≥3, when u n -u p >ST, all sequential optimization processes should be terminated. If the constraint threshold is not met, repeat the steps from S1 to S4 and continue the sequential optimization of the radar target signal micro-motion parameter identification system.

[0130] In the performance test of a certain type of weapon system, the radar target signal micro-motion parameter identification system is used to match and identify the micro-motion characteristic parameters of the target. The Monte Carlo method, simulated annealing algorithm, genetic algorithm, Bayesian optimization algorithm, and improved Bayesian optimization algorithm are respectively used for system comparison and optimization. The calculation results of different algorithms are as follows:

[0131]

[0132] Table 1 Statistical table of matching results of radar target signal micro-motion parameter identification system

[0133] Through the comparative analysis of Table 1, it can be seen that the time cost of the improved Bayesian optimization algorithm is comparable to that of the simulated annealing algorithm, Monte Carlo method, and genetic algorithm, less than half of the time cost of the classical Bayesian optimization algorithm, and it has the smallest solution error and the best accuracy among the five algorithms, reflecting the effectiveness of the algorithm.

Claims

1. The improved Bayesian optimization algorithm based on dynamic hyperparameters and scaling factors is an improvement based on the classical Bayesian optimization algorithm, and is characterized in that Including the following steps: S1. Calculate the scaling coefficient of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system; S2. Adjust the dynamic hyperparameters: calculate the numerical range of the multi-dimensional parameters of the radar target signal micro-motion parameter identification system and the number of observation points in the Bayesian optimization process; S3. Conduct the Bayesian optimization process based on the dynamic hyperparameters and the scaling coefficient; S4. Obtain the optimal solution of the multi-dimensional input parameters in a single Bayesian optimization process; S5. Repeat steps S1 to S4 until the optimization process is terminated when the constraint threshold requirement is met; The classical Bayesian optimization algorithm is split into a sequential optimization process of cyclic Bayesian optimization, and the sequential optimization process is expressed as: {Ω1, Ω2, Ω3, …, Ω n}, where n is the number of cycles of the improved Bayesian optimization algorithm, n >= 2; the number of observation points corresponding to each cycle of Bayesian optimization is respectively {m1, m2, m3, …, m n}; the objective function values of the historical global optimal solutions corresponding to n cycles of Bayesian optimization are respectively {B1, B2, B3, …, B p}, and the total number of global observation points corresponding to {B1, B2, B3, …, B p} is {u1, u2, u3, …, u p}, where p is the subscript of the set of sequential optimal solutions obtained by n cycles of Bayesian optimization, and p is a positive integer.

2. The improved Bayesian optimization algorithm based on dynamic hyperparameters and retraction and extension coefficients according to claim 1, wherein The main role of the scaling coefficient in step S1 is to dynamically contract and expand the search space of the multi-dimensional input parameters to accelerate the convergence rate of the cyclic Bayesian optimization. It is obtained by coupling the principal component regression coefficient of the multi-dimensional parameters with the objective function value of the global optimal solution; Assume that the input of the radar target signal micro-motion parameter identification system contains \(i_n\)-dimensional parameters, and the scaling factors corresponding to the \(i_n\)-dimensional parameters are {S1, S2, S3…S i_n}, then the calculation process of the scaling factors is as follows: First, calculate the principal component regression coefficients {g1, g2, g3... g i_n} of the i_n-dimensional parameters; the principal components are a recombined form of the i_n-dimensional parameters, and the projections of the i_n-dimensional parameters in the principal component directions have the maximum variance; assume {d1, d2, d3... d i_n} represents the observed values of the multi-dimensional input variables of the radar target signal micro-motion parameter identification system. Use the formula D i = (d i - μ i ) / σ i (i ∈ [1, i_n], μ i represents the mean of {d i}, and σ i represents the standard deviation of {d i}) to standardize {d1, d2, d3... d i_n} to obtain new observed values {D1, D2, D3... D i_n}; Establish the weighted sum function f = c1D1 + c2D2 + … c i_n} for the new observations {D1, D2, D3 … D i_n D i_n , where {c1, c2, c3 … c i_n} represents the weight coefficients of the i_n-dimensional parameters, In order to perform principal component analysis on f = c1D1 + c2D2 + … c i_n D i_n , construct q standard orthogonal direction vectors {Z1, Z2, Z3 … Z q} such that the cumulative contribution rate of these q orthogonal vectors ≥ 80%, and the expression of {Z1, Z2, Z3 … Z q} is as follows: where, for any i ∈ [1, q], there is Use variance to reflect the degree of data difference, and calculate in turn to make each group of Z i (Z i ∈[Z1, Z q ) have the maximum variance corresponding to {c i1 , c i2 , c i3 …c ii_n}, and the weight matrix corresponding to q principal components {Z1, Z2, Z3…Z q} can be obtained After that, use the least squares method to perform regression analysis on the principal components {Z1, Z2, Z3…Z q} to obtain the principal component regression equation Among them, {z1, z2, z3…z q} represents the weights of q principal components. Substitute into the principal component regression equation and transform it into the principal component regression equation of independent variables {D1, D2, D3…D i_n} as The principal component regression coefficients {g1, g2, g3…g i_n} of the i_n-dimensional parameters can be obtained; Secondly, couple the principal component regression coefficients {g1, g2, g3... g i_n} with the objective function value of the global optimal solution; use the formula to normalize the principal component regression coefficients {g1, g2, g3... g i_n}, and obtain the normalization coefficient After that, obtain the maximum value of the normalization coefficient Assume that the upper and lower limits of the adjustment of the retraction and extension coefficients {S1, S2, S3... S i_n} are [S up S down , then the initial value of the retraction and extension coefficient is Suppose the regulation range of the objective function value of the radar target signal micro-motion parameter identification system is [0 PR max , and the corresponding upper and lower limits of regulation are [pr min pr max . The objective function value corresponding to the current global optimal solution of the system is B p (p is the subscript of the optimal solution set obtained by n - cycle Bayesian optimization, p≥1), then based on the objective function value B p of the adjustment coefficient after that, the final result of the retraction and release coefficient is obtained: where B p > PR max when, let B p = PR max .

3. The improved Bayesian optimization algorithm based on dynamic hyperparameters and retraction and extension coefficients according to claim 1, characterized in that The calculation process of step S2 is as follows: Assume that the initial extreme value of the number of observation points is M0, and the number of observation points for a single Bayesian optimization in the n-th sequential optimization is m n , where n ≥ 1 (including the first loop); There is no strict mathematical formula for the initial extreme value M0. Its value is proportional to the time cost of a single evaluation of the system. Generally, the time cost of M0 observations in a single update of the probability surrogate model should not be greater than the time of a single evaluation of the system; The number of observation points m for single Bayesian optimization n The formula is as follows: where u p is the total number of global observation points of the current global optimal solution, and u p-1 represents u p the total number of global observation points of the previous global optimal solution, p represents the p-th global optimal solution, p ≥ 1; Suppose the i_n - dimensional input parameter range included in the radar target signal micro - motion parameter identification system is [pre1, sup1], [pre2, sup2], [pre3, sup3], ;;;;;;;;, [pre i_n , sup i_n . According to the scaling factors {S1, S2, S3…S i_n} in step 1, update the i_n - dimensional input parameter range to [max(pre1, I_B p (1)-(sup1 - pre1)*S1 / 2), min(sup1, I_B p (1)+(sup1 - pre1)*S1 / 2))], [max(pre2, I_B p (2)-(sup2 - pre2)*S2 / 2), min(sup2, I_B p (2)+(sup2 - pre2)*S2 / 2))], [max(pre3, I_B p (3)-(sup3 - pre3)*S3 / 2), min(sup3, I_B p (3)+(sup3 - pre3)*S3 / 2))],... where I_B p = [i_b1, i_b2, i_b3,…, i_b i_n represents the i_n - dimensional observation point corresponding to the historical global optimal objective function value B p ; The specific process of step S3 is as follows: First, establish a probability surrogate model for a single Bayesian optimization; A probability surrogate model based on Gaussian process is adopted. Gaussian process is a generalization of the multivariate Gaussian probability distribution. Assume the vector represents the set of observation points for a single Bayesian optimization, where the observation components X1, X2, X3, …, X t are all i_n-dimensional observation vectors, t ≤ m n , and m n is the maximum number of observation points for a single Bayesian optimization in step S2. X1, X2, X3, …, X t obeys a Gaussian distribution, where N(·) represents the Gaussian distribution, represents 's multivariate mean vector, Σ = cov(X m , X n ) = (σ mn ) i_n*i_n represents 's covariance matrix, and σ mn is the covariance of any two random variables X m and X n , where m, n ∈ [1, t] and m ≠ n; 's probability density function is where power(·) represents the power function and e is the natural constant; The Gaussian process can perform regression prediction on the radar target signal micro-motion parameter identification system. In the regression model, no functional relationship between the observed variable and the target variable is established. Instead, the relationship between the target variables is directly established through the kernel function K(X,Z), and its expression is The radial basis kernel function is used here. Among them, γ is the amplitude adjustment parameter, and λ is the width parameter, which controls the radial action range; Assume that Y is the objective function value of the radar target signal micro-motion parameter identification system, and let where ε is noise with a probability distribution (σ ε is the variance of the noise ε), and the prior probability distribution of the function f(·) is p(f) = GP(m(*), K(*,*)), where GP represents the Gaussian process, m(*) is the mean function, and K(*,*) is the kernel function. Assume that the dataset composed of the current known observation points and objective function values is D 1:t ={(X1, Y1), (X2, Y2), (X3, Y3), …, (X t , Y t )} t is the number of known observation points, and the vector to be updated is X new , then the posterior probability distribution of its objective function value Y new = f(X new ) + ε is as follows: p(Y new ) = N(μ(X new |D 1:t ), σ 2 (X new |D 1:t )) where μ(X new |D 1:t ) is the predicted mean of the objective function value Y new , and σ 2 (X new |D 1:t ) is the variance of Y new corresponding to the new observation point X new ; thus, the probabilistic surrogate model based on the Gaussian process is established; To infer a new observation point X from a known observation data set D 1:t , it is necessary to establish an acquisition function. Here, the EI (expected improvement) acquisition function is selected, and its expression is as follows: new ​ where f(x + ) represents the optimal solution of the current observation set; μ(x) and σ(x) represent the mean and variance of the posterior distribution respectively, and Φ(·) represents the probability density function of the standard normal distribution; Select the next most "promising" evaluation point X according to the maximized acquisition function t+1 i.e., X new , and calculate its objective function value Y t+1 i.e., Y new , and add the newly obtained input-observation value (X t+1 , Y t+1 ) to the known observation set D 1:t , update the probabilistic surrogate model, and prepare for the next iteration until the number of observation points is equal to the maximum number of observations m for single Bayesian optimization calculated in step S2 n .

4. The improved Bayesian optimization algorithm based on dynamic hyperparameters and retraction and extension coefficients according to claim 1, characterized in that Step S4 is to set the global storage variable DATA_BEST for sequential optimization. Assume that the current is the nth (n≥1) sequential optimization. If the objective function value corresponding to the optimal solution of this optimization process is better than the objective function value corresponding to DATA_BEST, then update DATA_BEST to the optimal solution of the nth sequential optimization; The step S5 is to compare whether the constraint threshold satisfies u n -u p >ST; where u n represents the total number of global observations after the n-th sequential optimization is completed, n is the number of cycles of Bayesian sequential optimization, and u p is the global optimal solution B obtained by the n-th sequential optimization p , p represents the p-th global optimal solution, and p≥1 corresponds to the total number of observations; the termination extreme value ST of sequential optimization = c*M0, generally c≥3, when u n -u p >ST, all sequential optimization processes should be terminated. If the constraint threshold is not satisfied, repeat the steps from S1 to S4 and continue the sequential optimization of the radar target signal micro-motion parameter identification system.

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