CN114611416B - LS-SVM modeling method for nonlinear unsteady aerodynamic characteristics of missile - Google Patents

LS-SVM modeling method for nonlinear unsteady aerodynamic characteristics of missile Download PDF

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CN114611416B
CN114611416B CN202210511409.4A CN202210511409A CN114611416B CN 114611416 B CN114611416 B CN 114611416B CN 202210511409 A CN202210511409 A CN 202210511409A CN 114611416 B CN114611416 B CN 114611416B
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sample set
modeling
svm
aerodynamic
missile
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CN114611416A (en
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孔轶男
汪清
陈功
余婧
王波兰
伍彬
钱炜祺
段光强
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Computational Aerodynamics Institute of China Aerodynamics Research and Development Center
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Computational Aerodynamics Institute of China Aerodynamics Research and Development Center
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12—Simultaneous equations, e.g. systems of linear equations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N20/00—Machine learning
    • G06N20/10—Machine learning using kernel methods, e.g. support vector machines [SVM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention relates to a missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method, which decomposes the missile aerodynamic force into the sum of static aerodynamic force and aerodynamic force increment generated by dynamic motion, and adopts the LS-SVM modeling method to establish a dynamic aerodynamic force increment model, wherein a finite sampling point is used for approximately describing a motion process so as to reflect the influence of the motion process on the aerodynamic characteristic; solving a linear equation set of the LS-SVM by adopting a singular value decomposition method, and carrying out model training; and determining the penalty factor and the kernel width by adopting a training-checking method so as to improve the generalization performance of the LS-SVM model. The method aims to provide a missile large-attack-angle aerodynamic force modeling method based on machine learning.

Description

LS-SVM modeling method for nonlinear unsteady aerodynamic characteristics of missile
Technical Field
The invention belongs to the technical field of aerodynamics, and particularly relates to an LS-SVM (least squares-support vector machine) modeling method for nonlinear unsteady aerodynamic characteristics of a missile.
Background
The maneuverability of a large angle of attack is an important design index of modern tactical missiles. Particularly for air-launched missiles, the over-the-shoulder launching is realized by utilizing large attack angle maneuver, the combat response time can be greatly shortened, and the survival probability of the aerial carrier and the combat efficiency of the missiles are improved. No matter the missile is launched over the shoulder in the traditional way or launched over the shoulder in the novel way of self-overturning, the missile can experience a stall state with a large attack angle, and aerodynamic force presents high unsteady and strong nonlinear characteristics. At this time, the pneumatic derivative model based on the quasi-stationary assumption is not applicable, and a nonlinear unsteady pneumatic force model needs to be established.
The problem of missile nonlinear unsteady aerodynamic modeling is not reported in a published literature at present. In the prior art, the large-attack-angle unsteady aerodynamic modeling is only carried out on an airplane or a wing, and mainly comprises the following steps:
prior art 1: the method for modeling the large-attack-angle Unsteady aerodynamic force of an airplane based on a least square support vector machine proposed by Wang Qing et al (Wang Q, et al. uncertain aerodynamic modeling at high and of using the above mentioned supporting vector machines. Chinese Journal of Aeronautics, 2015, 28(3): 659-668). In the method, a finite sampling point is adopted to replace a time history and is used as an input vector of the LS-SVM, and a LS-SVM model is constructed by taking a full-scale pneumatic coefficient or a dynamic increment of the pneumatic coefficient as output; determining penalty factors using k-fold cross-checking
Figure DEST_PATH_IMAGE001
And width of nucleus
Figure 694216DEST_PATH_IMAGE002
. However, in this document, the linear system of equations in the LS-SVM is solved directly when the sample size is large (e.g., N)>5000) In time, the coefficient matrix a is numerically singular and the training algorithm cannot be performed.
Prior art 2: least square support vector machine-based wing unsteady dynamic process unsteady aerodynamic modeling method (Chen Shenlin, et al, Unstable unsteady aerodynamic modeling based on least square support vector machine with generation excitation, Chinese Journal ofAeroneautics, 2020, 33(10): 2499-2509). In the method, a finite sampling point is adopted to replace a time history and is used as an input vector of the LS-SVM, and a full-scale pneumatic coefficient is used as output to construct an LS-SVM model; determining penalty factors using leave-one-out cross-validation
Figure DEST_PATH_IMAGE003
And width of nucleus
Figure 327322DEST_PATH_IMAGE004
. However, in the document, a Cholesky decomposition method is adopted to solve a linear equation set in the LS-SVM, and when the sample size is large, the problem that the value of the coefficient matrix a is singular and the training algorithm cannot be performed also exists.
Disclosure of Invention
In order to solve the problems, the invention provides a missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method, which decomposes the missile aerodynamic force into the sum of static aerodynamic force and aerodynamic force increment generated by dynamic motion, and adopts the LS-SVM modeling method to establish a dynamic aerodynamic force increment model, wherein a finite sampling point is used for approximately describing a motion process so as to reflect the influence of the motion process on the aerodynamic characteristic; solving a linear equation set of the LS-SVM by adopting a singular value decomposition method, and carrying out model training; and determining the penalty factor and the kernel width by adopting a training-checking method so as to improve the generalization performance of the LS-SVM model.
The technical scheme adopted by the invention is as follows: the LS-SVM modeling method for the nonlinear unsteady aerodynamic characteristics of the missile comprises the following steps:
s1: preparing data: calculating according to the missile aerodynamic force CFD to obtain a dynamic data table between a static aerodynamic force/aerodynamic moment coefficient, a large-amplitude pitching oscillation aerodynamic force/moment coefficient and a time course, and obtaining an input data file of pneumatic modeling after data processing; decomposing aerodynamic force into the sum of aerodynamic force increment generated by static aerodynamic force and dynamic motion by adopting a superposition principle to establish an LS-SVM model, and dividing an input data file of aerodynamic modeling into a training sample set, a checking sample set and a modeling sample set;
s2: model training: taking a training sample set as a model inputIn value, given an initial penalty factor
Figure DEST_PATH_IMAGE005
And width of nucleus
Figure 37789DEST_PATH_IMAGE006
Obtaining connection weight by singular value decomposition
Figure DEST_PATH_IMAGE007
And biasb；
S3: checking: taking a check sample set as an input value, checking the output error of the established LS-SVM model, and using a penalty factor corresponding to the current minimum output error
Figure 47334DEST_PATH_IMAGE008
And width of nucleus
Figure DEST_PATH_IMAGE009
If the value is the optimal value, if the global search is completed, the step S5 is executed, otherwise, the step S4 is executed;
s4: adjusting: adjusting penalty factors
Figure 392602DEST_PATH_IMAGE010
And width of nucleus
Figure DEST_PATH_IMAGE011
Repeatedly executing the steps S2 and S3;
s5: aerodynamic modeling: taking a modeling sample set as an input value of the LS-SVM model, and determining a penalty factor
Figure 462189DEST_PATH_IMAGE012
And width of nucleus
Figure 128794DEST_PATH_IMAGE011
Training LS-SVM model, and obtaining final connection weight value by singular value decomposition method
Figure DEST_PATH_IMAGE013
And biasbPredicting the output of a modeled sample setAnd (6) outputting the value.
Preferably, in step S1, the specific method of data processing is:
s1.1: uniformly transforming the reference points of the aerodynamic moment coefficients to the center of mass of the missile;
s1.2: calculating the angle of attack at each moment in the dynamic data table
Figure 207608DEST_PATH_IMAGE014
Angular rate of attack
Figure DEST_PATH_IMAGE015
And pitch rate
Figure 174427DEST_PATH_IMAGE016
；
S1.3: calculating dimensionless time
Figure DEST_PATH_IMAGE017
Non-dimensional angle of attack rate
Figure 149337DEST_PATH_IMAGE018
And dimensionless pitch angle rate
Figure DEST_PATH_IMAGE019
；
S1.4: and after the processing of the steps S1.1 to S1.3, intercepting one period of data to generate an input data file for pneumatic modeling.
Preferably, all input variables in the input data file for pneumatic modeling are normalized to the interval [ -1,1] with their maximum and minimum values to generate a training sample set, a check sample set, and a modeling sample set.
Preferably, in step S1, the specific method for dividing the input data file for pneumatic modeling into the training sample set, the verification sample set, and the modeling sample set includes: and selecting the pneumatic data of part of states as a verification sample set, taking the rest pneumatic data as a training sample set, and taking the modeling sample set as the sum of the training sample set and the verification sample set.
Preferably, the singular value decomposition method obtains the connection weight
Figure 834396DEST_PATH_IMAGE020
And biasbThe specific method comprises the following steps: respectively constructing matrixes by calculating kernel functions of input samples and known output samples according to the selected sample setAAndbconnection weight
Figure DEST_PATH_IMAGE021
And biasbIs a linear system of equations:
Figure 185743DEST_PATH_IMAGE022
and (3) solving the least square solution of the linear equation system by adopting a singular value decomposition method.
Preferably, the input variables of the training sample set, the checking sample set and the modeling sample set are the angle of attack at the current moment and the angle of attack at the previous n moments
Figure 496814DEST_PATH_IMAGE023
And corresponding angle of attack rate
Figure DEST_PATH_IMAGE024
And pitch rate
Figure 642625DEST_PATH_IMAGE025
(ii) a And the output variables of the training sample set, the checking sample set and the modeling sample set are dynamic aerodynamic force increment.
Preferably, the time interval between n time angles of attack is chosen, and the dimensionless time interval between n time angles of attack is 1.35).
Preferably, dynamic aerodynamic force increment
Figure DEST_PATH_IMAGE026
WhereinCFor the pneumatic force and moment coefficients,
Figure 549401DEST_PATH_IMAGE027
，
Figure DEST_PATH_IMAGE028
is static aerodynamic forceAnd (4) the coefficient.
Preferably, in step S3, the output errors are the root mean square error RMSE and the relative root mean square error RMSE (%).
The invention has the following beneficial effects:
1) because the configuration of the airplane and the guided missile is different, the aerodynamic characteristics are different in the unsteady motion; the method is suitable for modeling the nonlinear unsteady aerodynamic force of the guided missile, the prior art only aims at the maneuvering of an airplane with a large attack angle or the dynamic movement of wings, but not aims at the maneuvering scene of the guided missile with the large attack angle, and the method has engineering practical value for modeling the nonlinear unsteady aerodynamic force of the guided missile;
2) the least square solution of a linear equation set in the LS-SVM is calculated by adopting a singular value decomposition method, LS-SVM training is carried out, the robustness of a training algorithm is greatly improved, and the method is suitable for missile nonlinear unsteady aerodynamic modeling with a large sample scale (for example, N is more than 5000);
3) the method adopts a training-checking method to carry out modeling, determines penalty factors and kernel width, and greatly reduces the calculated amount, thereby avoiding the problem of overlarge calculated amount in the prior art when the wind tunnel test/CFD calculated state number is large;
4) the invention is directed to dynamic aerodynamic augmentation
Figure 704439DEST_PATH_IMAGE029
The LS-SVM modeling is carried out, so that on one hand, the adverse effect of dynamic wind tunnel test/CFD calculation errors on static aerodynamic prediction results can be avoided, and on the other hand, the generalization performance of the aerodynamic model is improved to a certain extent; the pitch angle rate is used as one of input variables of the LS-SVM, so that the influence of pitch damping on the aerodynamic characteristics can be accurately reflected, and the prediction result is more accurate.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on the drawings without creative efforts.
FIG. 1 is a dynamic aerodynamic force increment LS-SVM model;
FIG. 2 is a flow chart of aerodynamic LS-SVM modeling;
FIG. 3 shows the data after being processed
Figure 114692DEST_PATH_IMAGE030
(center angle of attack 45 degrees, amplitude 45 degrees);
FIG. 4 shows the data after being processed
Figure 493720DEST_PATH_IMAGE030
(center angle of attack 90 degrees, amplitude 45 degrees);
FIG. 5 shows the data after being processed
Figure 622213DEST_PATH_IMAGE030
(center angle of attack 135 degrees, amplitude 45 degrees);
FIG. 6 shows the data after being processed
Figure 315363DEST_PATH_IMAGE030
(center angle of attack 90 degrees, amplitude 90 degrees);
FIG. 7 shows the result after data processing
Figure 908018DEST_PATH_IMAGE031
(center angle of attack 45 degrees, amplitude 45 degrees);
FIG. 8 shows the result after data processing
Figure 395631DEST_PATH_IMAGE031
(center angle of attack 90 degrees, amplitude 45 degrees);
FIG. 9 shows the result after data processing
Figure 808158DEST_PATH_IMAGE031
(center angle of attack 135 degrees, amplitude 45 degrees);
FIG. 10 shows the result after data processing
Figure 537955DEST_PATH_IMAGE031
(center angle of attack 90 degrees, amplitude 90 degrees);
FIG. 11 shows the result after data processing
Figure 985116DEST_PATH_IMAGE032
(center angle of attack 45 degrees, amplitude 45 degrees);
FIG. 12 shows the result after data processing
Figure 643631DEST_PATH_IMAGE032
(center angle of attack 90 degrees, amplitude 45 degrees);
FIG. 13 shows the data after being processed
Figure 12295DEST_PATH_IMAGE032
(center angle of attack 135 degrees, amplitude 45 degrees);
FIG. 14 shows the result after data processing
Figure 843985DEST_PATH_IMAGE032
(center angle of attack 90 degrees, amplitude 90 degrees);
FIG. 15 is a drawing showing
Figure 83337DEST_PATH_IMAGE030
The verification result of the LS-SVM model (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 16 is a drawing showing
Figure 975069DEST_PATH_IMAGE030
The verification result of the LS-SVM model (center attack angle 90 degrees, amplitude 45 degrees);
FIG. 17 is a drawing showing
Figure 299871DEST_PATH_IMAGE030
The verification result of the LS-SVM model (center attack angle 135 degrees, amplitude 45 degrees);
FIG. 18 is a drawing showing
Figure 935252DEST_PATH_IMAGE030
The verification result of the LS-SVM model (center attack angle 90 degrees, amplitude 90 degrees);
FIG. 19 is a drawing showing
Figure 294689DEST_PATH_IMAGE031
The verification result of the LS-SVM model (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 20 is a drawing showing
Figure 560585DEST_PATH_IMAGE031
The LS-SVM model of (center angle of attack 90 degrees, amplitude 45 degrees)
FIG. 21 is a drawing showing
Figure 435000DEST_PATH_IMAGE031
The LS-SVM model of (center attack angle 135 degree, amplitude 45 degree)
FIG. 22 is a drawing showing
Figure 546176DEST_PATH_IMAGE031
The LS-SVM model of (center angle of attack 90 degrees, amplitude 90 degrees)
FIG. 23 is a drawing showing
Figure 822437DEST_PATH_IMAGE032
The verification result of the LS-SVM model (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 24 is a drawing showing
Figure 495120DEST_PATH_IMAGE032
The verification result of the LS-SVM model (center attack angle 90 degrees, amplitude 45 degrees);
FIG. 25 is a schematic view of
Figure 60093DEST_PATH_IMAGE032
The verification result of the LS-SVM model (center attack angle 135 degrees, amplitude 45 degrees);
FIG. 26 is a schematic view of
Figure 37277DEST_PATH_IMAGE032
The verification result of the LS-SVM model (center attack angle 90 degrees, amplitude 90 degrees);
FIG. 27 is a schematic view of
Figure 371306DEST_PATH_IMAGE030
Comparing the predicted result of the LS-SVM model of (1) with CFD data (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 28 is a drawing showing
Figure 775742DEST_PATH_IMAGE030
Comparing the predicted result of the LS-SVM model of (1) with CFD data (center attack angle is 90 degrees, amplitude is 45 degrees);
FIG. 29 is a schematic view of
Figure 296854DEST_PATH_IMAGE030
Comparing the predicted result of the LS-SVM model of (1) with the CFD data (center attack angle of 135 degrees, amplitude of 45 degrees);
FIG. 30 is a drawing showing
Figure 77728DEST_PATH_IMAGE030
Comparing the predicted result of the LS-SVM model with CFD data (the central attack angle is 90 degrees, and the amplitude is 90 degrees);
FIG. 31 is a drawing showing
Figure 266264DEST_PATH_IMAGE031
Comparing the predicted result of the LS-SVM model of (1) with CFD data (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 32 is a schematic view of
Figure 44864DEST_PATH_IMAGE031
Comparing the predicted result of the LS-SVM model of (1) with CFD data (center attack angle is 90 degrees, amplitude is 45 degrees);
FIG. 33 is a drawing showing
Figure 115588DEST_PATH_IMAGE031
Comparing the predicted result of the LS-SVM model of (1) with the CFD data (center attack angle of 135 degrees, amplitude of 45 degrees);
FIG. 34 is a schematic view of
Figure 372257DEST_PATH_IMAGE031
Comparing the predicted result of the LS-SVM model with CFD data (the central attack angle is 90 degrees, and the amplitude is 90 degrees);
FIG. 35 is a schematic view of
Figure 477616DEST_PATH_IMAGE032
Comparing the predicted result of the LS-SVM model of (1) with CFD data (center attack angle 45 degrees, amplitude 45 degrees);
FIG. 36 is a schematic view of
Figure 161538DEST_PATH_IMAGE032
Comparing the predicted result of the LS-SVM model with CFD data (the central attack angle is 90 degrees, and the amplitude is 45 degrees);
FIG. 37 is a schematic view of
Figure 686935DEST_PATH_IMAGE032
Comparing the predicted result of the LS-SVM model of (1) with the CFD data (center attack angle of 135 degrees, amplitude of 45 degrees);
FIG. 38 is a schematic view of
Figure 544033DEST_PATH_IMAGE032
Comparing the predicted result of the LS-SVM model with CFD data (the central attack angle is 90 degrees, and the amplitude is 90 degrees);
Figure 707161DEST_PATH_IMAGE030
is axial force coefficient (unit: dimensionless),
Figure 624302DEST_PATH_IMAGE033
is the normal force coefficient (unit: dimensionless),
Figure 607301DEST_PATH_IMAGE034
is the pitching moment coefficient (unit: dimensionless), alpha is the angle of attack (unit: degree), CFD is the result calculated by using the computational fluid dynamics method numerical value, and mod represents the result modeled by the technology of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention develops a nonlinear unsteady aerodynamic modeling method based on a least square support vector machine (LS-SVM) aiming at the problem of modeling the large-attack-angle maneuvering aerodynamic of a missile.
(1) Unsteady aerodynamic LS-SVM model
In the attack angle plane, the missile aerodynamic force and moment are nonlinear functions of Mach number, attack angle, pitch angle rate and pitch rudder deflection angle. The method adopts a superposition principle to decompose the aerodynamic force into the sum of static aerodynamic force and aerodynamic force increment generated by dynamic motion, namely:
Figure 268090DEST_PATH_IMAGE035
in the formula:Cfor the pneumatic force and moment coefficients,
Figure 285724DEST_PATH_IMAGE036
；Mis Mach number;
Figure 577028DEST_PATH_IMAGE037
is an angle of attack;
Figure 109641DEST_PATH_IMAGE038
is the pitch rudder deflection angle;
Figure 511803DEST_PATH_IMAGE039
for non-dimensional pitch angle rates,
Figure 446261DEST_PATH_IMAGE040
；lis a reference length (projectile length);Vis the incoming flow velocity.
In the formula (1), the reaction mixture is,
Figure 642887DEST_PATH_IMAGE041
the static aerodynamic coefficient is a nonlinear function of Mach number, attack angle and pitch rudder deflection angle;
Figure 662796DEST_PATH_IMAGE042
generated for dynamic motionThe aerodynamic increment (D, which is omitted for simplicity and convenience to indicate the increment), is a non-linear functional of the motion history, which is denoted here by brackets to distinguish it from the usual function, where the effect of pitch rudder deflection is negligible.
For large-amplitude oscillation wind tunnel test or CFD calculation, the Mach number is fixed and unchanged, and the pitch rudder deflection angle
Figure 134228DEST_PATH_IMAGE043
Keeping 0, the formula (1) can be simplified to
Figure 657614DEST_PATH_IMAGE044
LS-SVM modeling in the invention aims at dynamic aerodynamic force increment
Figure 792185DEST_PATH_IMAGE042
To proceed with. Under the condition of not considering the bifurcation phenomenon, the 'memory' time of aerodynamic force to the motion process is limited, the motion process can be approximately replaced by a limited number of sampling points, and the invention adopts the attack angle at the current moment
Figure 237073DEST_PATH_IMAGE045
Angle of attack from the first n moments
Figure 43355DEST_PATH_IMAGE046
To approximate the angle of attack course, then:
Figure 624509DEST_PATH_IMAGE047
dynamic aerodynamic force increment
Figure 490834DEST_PATH_IMAGE042
The LS-SVM model of (1) is shown in fig. 1. For a given predicted sample inputxThe output of LS-SVM prediction is
Figure 157439DEST_PATH_IMAGE048
In the formula:xan input vector of the prediction sample of (a), see equation (5);yan output of the prediction samples of;
Figure 970674DEST_PATH_IMAGE049
an input vector (also called a support vector) which is a training sample, see equation (6); kernel function
Figure 468651DEST_PATH_IMAGE050
The Radial Basis Function (RBF) is used, see equation (7).
Figure 443560DEST_PATH_IMAGE051
Connection weight
Figure 659778DEST_PATH_IMAGE052
And biasbObtained by LS-SVM training.
(2) LS-SVM training algorithm
At a given penalty factor
Figure 745546DEST_PATH_IMAGE053
And width of nucleus
Figure 363609DEST_PATH_IMAGE054
In the case of (2), penalty factor in the present invention
Figure 509420DEST_PATH_IMAGE055
Given range of 3.0-7.0, core width
Figure 212933DEST_PATH_IMAGE054
Is in the range of 0.05-0.17, and can be selected in the range of the interval, for the training sample set
Figure 600927DEST_PATH_IMAGE056
Building a system matrixAAndb：
Figure 73497DEST_PATH_IMAGE057
connection weight
Figure 655788DEST_PATH_IMAGE052
And biasbIs a solution of the following linear system of equations:
Figure 784281DEST_PATH_IMAGE058
in the formula
Figure 274168DEST_PATH_IMAGE059
Training sample points for modeling problem of nonlinear unsteady aerodynamic force of missileNIn the thousands and even tens of thousands, an accurate solution cannot generally be solved. In the invention, a least square solution of the linear equation set (10) is solved by a singular value decomposition method.
For matrixASingular value decomposition:
Figure 804506DEST_PATH_IMAGE060
in the formula (I), the compound is shown in the specification,UandVis an orthogonal matrix;
Figure 88857DEST_PATH_IMAGE061
。
least squares solution of the system of linear equations (10) to
Figure 704646DEST_PATH_IMAGE062
WhereinA + To representAGeneral inverse of
Figure 998224DEST_PATH_IMAGE063
If the linear system of equations has a solution, then the least squares solution is the solution to the linear system of equations.
(3) Penalty factor and kernel width determination
Penalty factor in LS-SVM modeling
Figure 383069DEST_PATH_IMAGE064
And width of nucleus
Figure 103901DEST_PATH_IMAGE065
The fitting accuracy and generalization performance of the LS-SVM are determined. In order to make the LS-SVM have better generalization performance, the two parameters also need to be optimized. The invention adopts a simpler training-checking method, which comprises the following steps:
step 1: selecting partial state pneumatic data as a verification sample set
Figure 206986DEST_PATH_IMAGE066
And the rest pneumatic data are used as a training sample set
Figure 976359DEST_PATH_IMAGE067
；
Step 2: for a given parameter
Figure 543606DEST_PATH_IMAGE068
And
Figure 874487DEST_PATH_IMAGE069
using training sample sets
Figure 261606DEST_PATH_IMAGE070
Performing LS-SVM training to obtain model parameters
Figure 834670DEST_PATH_IMAGE071
And
Figure 459686DEST_PATH_IMAGE072
(connection weight bias);
step 3: using a set of check samples
Figure 522320DEST_PATH_IMAGE073
Performing a check to check for root mean square error of the sample set
Figure 334418DEST_PATH_IMAGE074
As a check error;
step4 adjusting parameters
Figure 507911DEST_PATH_IMAGE053
And
Figure 721854DEST_PATH_IMAGE054
(the search range is determined according to the empirical value, and the global search is performed in the search range), in the embodiment, the penalty factor
Figure 955390DEST_PATH_IMAGE075
Given range of 3.0-7.0, core width
Figure 254784DEST_PATH_IMAGE076
The given range of (1) is 0.05-0.17, and the selection can be carried out in the range of the interval, and Step2 and Step3 are repeatedly executed until the verification error reaches the minimum value (within an allowable range).
FIG. 2 shows a flow chart of missile nonlinear unsteady aerodynamic LS-SVM modeling.
The invention is further explained by taking a nonlinear unsteady aerodynamic LS-SVM model established by using CFD (computational fluid dynamics) calculation data of certain missile layout as a specific example.
(1) Data preparation
The computational state of aerodynamic force CFD of a certain missile layout is listed in Table 1, and the computational data comprises: static aerodynamic force and moment coefficients and large amplitude pitch oscillation aerodynamic force and moment coefficient time histories.
TABLE 1 certain missile layout aerodynamic force CFD calculation state
Figure 231967DEST_PATH_IMAGE077
The provided pneumatic data are processed as follows in sequence:
a. uniformly transforming the reference points of the aerodynamic moment coefficients to the center of mass of the missile;
b. calculating angle of attack of dynamic data table
Figure 300417DEST_PATH_IMAGE078
Angular rate of attack
Figure 704854DEST_PATH_IMAGE079
And pitch rate
Figure 491544DEST_PATH_IMAGE080
The like;
c. calculating dimensionless time
Figure 272418DEST_PATH_IMAGE081
Non-dimensional angle of attack rate
Figure 693910DEST_PATH_IMAGE018
And dimensionless pitch angle rate
Figure 269248DEST_PATH_IMAGE082
。
After the processing, one period of data is intercepted, and an input data file for pneumatic modeling is generated. The static and dynamic aerodynamic properties are shown in figures 3-14. As can be seen, the longitudinal aerodynamic force and moment coefficients
Figure 808814DEST_PATH_IMAGE083
The method has the advantages that the nonlinear and unsteady characteristics are obvious, and the regularity of the hysteresis loop along with the change of parameters such as an attack angle, frequency and the like is good.
(2) Input/output variable selection
In the modeling of longitudinal aerodynamic force LS-SVM, the input variable is taken as
Figure 65483DEST_PATH_IMAGE084
、
Figure 170842DEST_PATH_IMAGE085
、
Figure 854764DEST_PATH_IMAGE086
、
Figure 678364DEST_PATH_IMAGE087
To reflect the effect of the course of the angle of attack and the instantaneous pitch angle rate on the aerodynamics, wherein
Figure 473145DEST_PATH_IMAGE088
= 1.35; output variables are measured as dynamic aerodynamic increments
Figure 433010DEST_PATH_IMAGE089
Wherein
Figure 818992DEST_PATH_IMAGE090
。
All input variables are regularized to the interval [ -1,1] with their maximum and minimum values:
Figure 801992DEST_PATH_IMAGE091
in the formula:
Figure 462780DEST_PATH_IMAGE092
(ii) a Superscript "(n)" indicates regularization; the subscripts "max" and "min" represent the maximum and minimum values, respectively, of the corresponding quantities in all sample points.
(3) Penalty factor and kernel width determination
Determining the optimal penalty factor through training-checking by using the pneumatic data of the following states as the checking data and the pneumatic data of the other states as the training data
Figure 480415DEST_PATH_IMAGE094
And width of nucleus
Figure 568457DEST_PATH_IMAGE095
：
Central angle of attack
Figure 551936DEST_PATH_IMAGE096
Amplitude of vibration
Figure 16415DEST_PATH_IMAGE097
Frequency of
Figure 888556DEST_PATH_IMAGE098
Central angle of attack
Figure 147499DEST_PATH_IMAGE099
Amplitude of vibration
Figure 105091DEST_PATH_IMAGE100
Frequency of
Figure 107682DEST_PATH_IMAGE101
Central angle of attack
Figure 834330DEST_PATH_IMAGE102
Amplitude of vibration
Figure 529753DEST_PATH_IMAGE103
Frequency of
Figure 240220DEST_PATH_IMAGE104
Central angle of attack
Figure 718606DEST_PATH_IMAGE105
Amplitude of vibration
Figure 627656DEST_PATH_IMAGE106
Frequency of
Figure 166085DEST_PATH_IMAGE107
Table 2 lists the LS-SVM training-verification results and errors, including the root mean square error RMSE and the relative root mean square error RMSERMSE (%) is defined by formula (18) and formula (19), respectively. The verification states are shown in FIGS. 15-18, 19-22, and 23-26, respectively
Figure 160586DEST_PATH_IMAGE108
、
Figure 442662DEST_PATH_IMAGE109
、
Figure 206219DEST_PATH_IMAGE110
Comparison of model prediction results with CFD data. As can be seen from the table and the figure, for the check state,
Figure 679664DEST_PATH_IMAGE111
and
Figure 895881DEST_PATH_IMAGE112
the model prediction result is in good accordance with CFD data, and the LS-SVM model has good generalization performance;
Figure 247228DEST_PATH_IMAGE113
the model prediction results slightly deviate from the CFD data, mainly due to pitching oscillation
Figure 802974DEST_PATH_IMAGE114
The CFD data of (1) has a large hysteresis loop, but the hysteresis loop size does not vary significantly with the oscillation frequency (see fig. 3-6), in other words, from static to low frequency dynamic motion,
Figure 11102DEST_PATH_IMAGE115
there is a "jumping" phenomenon, and the model has some difficulty describing this phenomenon.
Figure 652299DEST_PATH_IMAGE116
TABLE 2 LS-SVM training-verification results and errors
Figure 338495DEST_PATH_IMAGE117
(4) Aerodynamic modeling
All pneumatic data are used as modeling samples and determined
Figure 14327DEST_PATH_IMAGE012
And
Figure 393356DEST_PATH_IMAGE065
performing LS-SVM training to obtain weight
Figure 521849DEST_PATH_IMAGE118
And biasb。
Table 3 lists the LS-SVM modeling error, and FIGS. 27-30, 31-34, and 35-38 show the results respectively
Figure 11736DEST_PATH_IMAGE119
、
Figure 542074DEST_PATH_IMAGE120
、
Figure 29688DEST_PATH_IMAGE121
Comparison of model prediction results with CFD data. As can be seen from the table and the figures,
Figure 707794DEST_PATH_IMAGE120
and
Figure 440520DEST_PATH_IMAGE122
the model prediction result is well consistent with CFD data, and the modeling error is small;
Figure 887681DEST_PATH_IMAGE123
the model prediction result is slightly worse in conformity with the CFD data, and the modeling error is slightly larger because of the CFD data
Figure 280617DEST_PATH_IMAGE114
The hysteresis loop varying little with the oscillation frequency, and the model predicts the result
Figure 446019DEST_PATH_IMAGE108
The hysteresis loop changes obviously along with the oscillation frequency. In a general view, the established LS-SVM model can better describe the nonlinear and unsteady characteristics of the aerodynamic characteristics of the missile changing along with the attack angle.
TABLE 3 LS-SVM modeling error
Figure 215392DEST_PATH_IMAGE124
It should be understood by those skilled in the art that the above embodiments are only for illustrating the present invention and are not to be used as a limitation of the present invention, and that suitable changes and modifications of the above embodiments are within the scope of the claimed invention as long as they are within the spirit and scope of the present invention.

Claims (9)

1. The LS-SVM modeling method for the nonlinear unsteady aerodynamic characteristics of the missile is characterized by comprising the following steps of:
s1: preparing data: a dynamic data table between static aerodynamic force/aerodynamic moment coefficients of the missile, large-amplitude pitching oscillation aerodynamic force/moment coefficients and time histories is obtained by wind tunnel tests or CFD calculation, and an input data file of pneumatic modeling is obtained after data processing; decomposing aerodynamic force into the sum of aerodynamic force increment generated by static aerodynamic force and dynamic motion by adopting a superposition principle to establish an LS-SVM model, and dividing an input data file of aerodynamic modeling into a training sample set, a checking sample set and a modeling sample set;
s2: model training: taking a training sample set as a model input value, and giving an initial penalty factor
Figure 98952DEST_PATH_IMAGE001
And width of nucleus
Figure 270171DEST_PATH_IMAGE002
Obtaining connection weight value by singular value decomposition method
Figure 631882DEST_PATH_IMAGE003
And biasb；
S3: checking: taking a check sample set as an input value, checking the output error of the established LS-SVM model, and using a penalty factor corresponding to the current minimum output error
Figure 546748DEST_PATH_IMAGE004
And width of nucleus
Figure 677515DEST_PATH_IMAGE005
If the value is the optimal value, if the global search is completed, the step S5 is executed, otherwise, the step S4 is executed;
s4: adjusting: adjusting penalty factors
Figure 285214DEST_PATH_IMAGE006
And width of nucleus
Figure 71905DEST_PATH_IMAGE007
Repeatedly executing the steps S2 and S3;
s5: aerodynamic modeling: taking a modeling sample set as an input value of the LS-SVM model, and determining a penalty factor
Figure 587200DEST_PATH_IMAGE008
And width of nucleus
Figure 775736DEST_PATH_IMAGE009
Training LS-SVM model, and obtaining final connection weight value by singular value decomposition method
Figure 351073DEST_PATH_IMAGE010
And biasbAnd predicting the output value of the modeling sample set.
2. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 1, characterized in that: in step S1, the specific method of data processing is:
s1.1: uniformly transforming the reference points of the aerodynamic moment coefficients to the center of mass of the missile;
s1.2: calculating the angle of attack at each moment in the dynamic data table
Figure 625060DEST_PATH_IMAGE011
Angular rate of attack
Figure 944046DEST_PATH_IMAGE012
And pitch rate
Figure 485623DEST_PATH_IMAGE013
；
S1.3: calculating dimensionless time
Figure 231862DEST_PATH_IMAGE014
Non-dimensional angle of attack rate
Figure 727566DEST_PATH_IMAGE015
And dimensionless pitch angle rate
Figure 53505DEST_PATH_IMAGE016
；
S1.4: and after the processing of the steps S1.1 to S1.3, intercepting one period of data to generate an input data file for pneumatic modeling.
3. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 2, characterized in that: after all input variables in the input data file of the pneumatic modeling are regularized to an interval [ -1,1] by using the maximum value and the minimum value of the input variables, a training sample set, a check sample set and a modeling sample set are generated.
4. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 1, characterized in that: in step S1, the specific method for dividing the input data file for pneumatic modeling into the training sample set, the verification sample set, and the modeling sample set is as follows: and selecting the pneumatic data of part of states as a verification sample set, taking the rest pneumatic data as a training sample set, and taking the modeling sample set as the sum of the training sample set and the verification sample set.
5. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 1, characterized in that: singular value decomposition method for obtaining connection weight
Figure 13371DEST_PATH_IMAGE017
And biasbThe specific method comprises the following steps: respectively constructing matrixes by calculating kernel functions of input samples and known output samples according to the selected sample setAAndbconnection weight value
Figure 868194DEST_PATH_IMAGE017
And biasbIs a linear system of equations:
Figure 179090DEST_PATH_IMAGE018
and (3) solving the least square solution of the linear equation system by adopting a singular value decomposition method.
6. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 1, characterized in that: the input variables of the training sample set, the checking sample set and the modeling sample set are attack angle at the current moment and attack angles at the previous n moments
Figure 43141DEST_PATH_IMAGE019
And corresponding angle of attack rate
Figure 795196DEST_PATH_IMAGE020
And pitch rate
Figure 148817DEST_PATH_IMAGE021
(ii) a And the output variables of the training sample set, the checking sample set and the modeling sample set are dynamic aerodynamic force increment.
7. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 6, characterized in that: the dimensionless time interval between n time angles of attack is chosen to be 1.35.
8. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 7, characterized in that: dynamic aerodynamic force increment
Figure 619113DEST_PATH_IMAGE022
WhereinCFor the pneumatic force and moment coefficients,
Figure 818013DEST_PATH_IMAGE023
，
Figure 690154DEST_PATH_IMAGE024
in order to be a static aerodynamic coefficient,
Figure 214676DEST_PATH_IMAGE025
in order to be the axial force coefficient,
Figure 408153DEST_PATH_IMAGE026
as a function of the normal force coefficient,
Figure 676324DEST_PATH_IMAGE027
is the pitch moment coefficient.
9. The missile nonlinear unsteady aerodynamic characteristic LS-SVM modeling method according to claim 1, characterized in that: in step S3, the output errors are the root mean square error RMSE and the relative root mean square error RMSE (%).
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