CN105608251B - The BNSobol methods of helicopter fire control system precision sensitivity analysis - Google Patents
The BNSobol methods of helicopter fire control system precision sensitivity analysis Download PDFInfo
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Abstract
The present invention provides a kind of BNSobol methods of helicopter fire control system precision sensitivity analysis,The global sensitivity analysis that error source influences fire control system precision is carried out with being combined based on Bayesian network and Sobol indexes,Bayesian network is established according to priori,Pass through relevant parameter in Bayesian Estimation learning network,And then obtain the probability that fire control system precision under the conditions of each error source takes different value reaches specific grade,Finally gained probability results are handled using Sobol methods variation decomposition to obtain the sensitivity coefficient of each error source,The present invention proposes a kind of new sensitivity analysis mechanism of combination Bayesian network and Sobol index methods,Reference and theories integration are provided to carry out helicopter fire control system precision sensitivity analysis under the conditions of sample size is insufficient,Simultaneously also a kind of thinking of uncertain rapid qualitative analysis is provided for other large-scale complicated systems.
Description
Technical Field
The invention relates to the field of aviation firepower control and intelligent decision and optimization, in particular to the field of a helicopter fire control system.
Background
By virtue of its maneuverability, gunships play an increasingly important role in modern war. The uncontrolled weapons such as machine guns, aeroguns and rocket projectiles put higher demands on the precision of helicopter fire control systems, and in actual combat, the precision of the fire control systems is influenced by various factors due to the complexity of battlefield environment, attack conditions, target motion and the like. Analyzing the influence and the influence degree of the factors on the striking precision is very important for improving the performance of the fire control system. The influence of different error sources on the precision of the helicopter fire control system is researched and analyzed at home and abroad, and measures for reducing errors and improving the precision are provided. However, each error source does not act on the fire control system independently, and multiple errors exist simultaneously and have strong mutual coupling effect. Therefore, it is necessary to analyze the sensitivity of the fire control system to find out the main error sources and the influence of the interaction between the error sources on the final accuracy.
Sensitivity analysis methods can be classified into local sensitivity analysis methods and global sensitivity analysis methods according to the range of action. The local sensitivity analysis only checks the influence degree of a single attribute on the model; and the global sensitivity analysis is used for testing the total influence of a plurality of attributes on the model result and analyzing the influence of the interaction among the attributes on the model output. The model searched by the method has large input space and better robustness of the analysis result, so that the method can be used as a method for analyzing the precision sensitivity of the helicopter fire control system. Several commonly used global sensitivity assays are: regression Analysis (RA), Fourier Amplitude Sensitivity Test (FAST), Response Surface Method (RSM), Mutual Information Index (MII), Sobol index, and the like. The Sobol method is a feasible method for analyzing the sensitivity of the weapons and equipment by virtue of strong capability of analyzing the grouped input factors and the universality of requirements on linearity, monotonicity, input distribution characteristics and the like of the performance evaluation model. The Sobol method obtains the sensitivities of the parameters of 1, 2 times and higher respectively based on the thought of model decomposition, and can carry out sensitivity classification through the contribution ratio of the parameters to the output variance. When single variable or a combination of a few variables is processed, the calculation is rapid and the operability is strong. However, since it is a statistical-based method, once a combination of a large number of variables is involved, the amount of calculation is large, and the operation is difficult in practical applications. In addition, the computational analysis is carried out on the basis of a large amount of sample data, which can be difficult to apply in many fields, particularly in the military.
The Bayesian Network (Bayesian Network) can well represent the random uncertainty and the correlation of variables, and can carry out uncertainty reasoning, thereby not only realizing forward reasoning and deducing posterior probability from the prior probability, namely, the result is deduced from the reason, but also deducing the prior probability from the posterior probability by using a formula, namely, the reason is deduced from the result. The research at home and abroad respectively applies the Bayesian network to the reliability evaluation of the power system, the reliability evaluation of the mechanical system and the importance and sensitivity analysis of elements, and better results are obtained.
Disclosure of Invention
In the military field, because the cost problem often difficultly provides a large amount of sample data for analysis and research, the traditional sensitivity analysis method is limited, and the accuracy of an analysis result cannot be ensured.
In order to overcome the defects of the prior art, the invention creatively provides a BNSobol method based on the combination of a Bayes network and a Sobol index to carry out global sensitivity analysis of the influence of error sources on the accuracy of a fire control system, utilizes the characteristics of Bayes network inference, establishes the Bayes network according to priori knowledge, estimates and learns related parameters in the network through Bayes, further infers the probability that the accuracy of the fire control system reaches a specific level under the condition that each error source takes different values, and finally processes the probability result obtained by inference by using the concept of Sobol method variance decomposition to obtain the sensitivity coefficient of each error source.
The invention can well solve the problem of sensitivity analysis of system precision under the condition of insufficient data volume and ensure the precision of the analysis result.
The technical scheme adopted by the invention for solving the technical problems is as follows:
step 1: determining accuracy sensitivity analysis index
Decomposing the model into single parameters and functions combined among the parameters by adopting a sensitivity index defined by a Sobol index method through variance decomposition, and analyzing the importance of the parameters and the interaction effect among the parameters by calculating the influence of the variance of the single input parameter or the input parameter set on the total output variance;
the Sobol index method defines each sensitivity index as follows:
(1) the main effect, also called first-order sensitivity index, is defined asIs XiContribution of "alone" to the variance of Y, with values of [0,1]Internal;
(2) the second order interaction effect is defined asThe influence on the output is the interaction effect of the two;
step 2: bayesian network model for establishing precision sensitivity analysis of helicopter fire control system
Determining the precision grade according to the size of the error source, and establishing a naive Bayesian network:
y represents an index of accuracy evaluation, and X1,…,X4Respectively represent different error sources, and X is { X for variable set1,X2,X3,X4In which X isiValue range or state set of e.XriThe number of states of each child node, namely the number of value intervals of the error source; d ═ C1,…,CnIs a data sample, i.e. a data set or database, ClIs an event, i.e. a test case or a record of a database, here a live-fire target hit data;a parameter variable of prior probability, which represents an assumption h that the user has a knowledge state ξ and the network structure is S, i.e. Sh，XiOn the premise that the parent node set Pa has the jth state, the variable XiTaking the objective probability of the kth value,pa has a value range of { Pa1,…paqThe number of all possible states with q as Pa, i.e. the number of levels of the precision index, is recordedThen
The following three assumptions are made:
⑴ random sample D is complete, i.e., no data is lost in D;
⑵. the parameter vectors are independent of each other, namely:
⑶. the parameter vector is a Dirichle distribution, namely:
wherein, N'ijkGreater than 0 is an exponential coefficient or a super parameter of Dirichle distribution;
and step 3: precision sensitivity analysis of fire control system by BNSobol method
Firstly, carrying out Bayesian network parameter learning:
⑴. prior distribution of parameters
Wherein, N'ijk＝N'·p(Xi＝k,pa＝j|Sh,ξ)。
⑵ posterior distribution of parameters:
wherein N isijkIs satisfied in the database DAnd pa equals the number of cases of j;
the probability process of calculating in Bayes network is called Bayes inference, local conditional probability distribution function is obtained by parameter learning, and accordingly, the probability of specific precision index grade corresponding to all error source value interval combination, namely the probability of specific precision index grade can be obtainedWherein j is 1, …, q, ki＝1,…,ri；
Calculating a sensitivity index:
main effect
According to Sobol index method XiDefinition of the Main EffectThe main effect can be directly calculated by combining all probability values obtained by Bayesian network inference
V(Y)＝E(Y2)-E2(Y) (5)
Wherein n is an error source value combination subscript,is the value of the ith error source under the nth combination, and the same applies:
in the formulaIs the kth error sourceiA value ofIs except for xiThe nth value combination of other error sources;
bringing the formula (5-9) intoCan find the error source XiThe main effect of (c);
second (or second) order interaction effect
XiAnd XjThe interaction effect of the two is defined asWherein,andall can be calculated;
in the formula (x)ixj)kIs the k-th value combination of the ith and jth error sources, andis except for xiAnd xjThe nth value combination of other error sources;
bringing into the formulae (10), (11)Can obtain X by the definition formulaiAnd XjThe interaction effect of the two.
The invention has the beneficial effects that:
⑴, various main error sources of the accuracy of the helicopter fire control system are established, and the influence of environmental factors, namely rotor downwash and random wind, is considered, so that the analysis result has more practical significance.
⑵, adopting each sensitivity index defined by Sobol index method, and using Sobol method variance decomposition idea to carry out global sensitivity analysis when in concrete implementation, so that the analysis result is more comprehensive.
⑶, by establishing a Bayesian network model for precision sensitivity analysis of the helicopter fire control system, network parameter learning is performed according to sample data, and then the required probability is inferred according to the learning result, so that the precision requirement is met, the sample data amount required by analysis is reduced, and the cost required by analysis is reduced.
In a word, the invention provides a new sensitivity analysis mechanism combining a Bayesian network and a Sobol index method, provides reference and theoretical support for developing precision sensitivity analysis of a helicopter fire control system under the condition of insufficient sample size, and provides an uncertainty rapid quantitative analysis idea for other large-scale complex systems.
Drawings
FIG. 1 is a naive Bayesian network for accuracy sensitivity analysis of a fire control system of the present invention.
FIG. 2 is a diagram of the Bayesian network inference process of the present invention, wherein Y represents an accuracy evaluation index, and X represents a precision evaluation index1,X2,X3,X4Respectively, represent different error sources.
FIG. 3 is a graph of error source relationships for the present invention.
Figure 4 is a view of the rotor downwash of the present invention.
FIG. 5 is a diagram of the results of a random wind field simulation of the present invention.
Fig. 6 is a schematic diagram of horizontal attack CCIP targeting of the present invention.
FIG. 7 is a flow chart of simulation analysis of the present invention.
FIG. 8 is a graph comparing the main effects of error sources obtained by two assays of the present invention.
FIG. 9 is a graph comparing the second order interaction effect between error sources obtained by the two assays of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
Step 1: determining error source and precision evaluation index of helicopter fire control system
And analyzing the importance of the parameters and the interaction effect among the parameters by calculating the influence of the variance of a single input parameter or an input parameter set on the total output variance by decomposing the model into a function of single parameters and the mutual combination of the parameters by using the sensitivity index defined by a Sobol index method and by using variance decomposition.
The Sobol index method defines each sensitivity index as follows:
(1) the main effect, also called first-order sensitivity index, is defined asDescribe XiContribution of "alone" to the variance of Y, with values of [0,1]In the method, each variable is subjected to sensitivity sequencing according to the magnitude of the main effect, and the larger the main effect index is, the larger the influence of the variation of the variable on the variation of the output is shown, so that the variation of the output is controlled, and the main point is to control the variation of the input with the large main effect index;
(2) the second order interaction effect is defined asThe influence of the interaction effect of the two on the output is described;
determining error sources
The main studied error sources of the present invention are as follows: the sensor (radar aiming error), inertial navigation (measurement error), hanging rack (shaking error), environment error (random wind, rotor wing wash-down flow), because the environment error can not be controlled manually, it is taken as the fixed value, other errors are all superimposed on the standard value as white noise, wherein the relationship among each error source is shown in fig. 3.
In a hovering state, because the rocket projectile is suspended below a main rotor of the helicopter, a downwash flow field is generated when the rotor rotates, and the rocket projectile passes through the flow field after leaving the launching device, so that aerodynamic force and moment acting on the rocket projectile are changed, and the initial trajectory of the rocket projectile is influenced. Therefore, the influence of the downward washing flow of the helicopter rotor on the launching trajectory of the rocket projectile cannot be ignored, and the influence is taken into account when the fire control calculation is carried out. For the aerogun, the initial speed is high, the time for passing through the flow field is short, and the size of the aerogun projectile is small, so that the effect of rotor downwash on the aerogun is small, and the rotor downwash flow field is shown in fig. 4.
Mathematical model of random wind field:
because the influence mechanism of wind is extremely complex, crosswind and longitudinal wind are mainly considered in actual work, the crosswind and the longitudinal wind are considered to be subjected to normal distribution, and the covariance function of a wind field is obtained according to theoretical research and experimental tests:
longitudinal wind:
transverse wind:
a wind field simulation model according to covariance function formulas (12) and (13):
in the formula: kx(τ),Ky(τ) is a covariance function of the longitudinal wind and cross wind random quantities; sx(ω),Sy(ω) is the spectral density of the longitudinal and transverse wind random quantities; l is the variation period of the simulated wind field; w1Calculating an excessive variable of a wind field model; v is the movement speed of the projectile; wxThe wind speed is the longitudinal wind speed; wyξ is the cross wind speedx、ξyTo follow a normal distributionA random variable of (a); sigmawIs the mean square error of wind speed.
Take the random wind field when V is 100m/s as an example, sigmawThe value is 5m/s, and the simulation result of the random wind field is obtained through simulation and is shown in figure 5.
Secondly, determining an accuracy evaluation index
The accuracy evaluation index adopts circular probability deviation (CEP), and when a helicopter is attacked by weapons such as aeroguns and rocket projectiles, the helicopter usually adopts continuous shooting, namely, multiple projectiles are fired at one time, and the projectiles are scattered on a firing plane due to the influence of random errors such as random wind and hanging rack shaking. Typically, the CEP solved by the intensity rating method is relative to the center of dispersion, not relative to the targeted point. The hit precision is the dispersion degree of the bullet drop point relative to the aiming point, and the size is equal to the radius r of a circle area which takes the aiming point as the center of a circle and has the impact probability of 0.5.
Step 2: establishing helicopter fire control system model
In the accuracy analysis, C represents a series of levels of accuracy index, A1,…,AnRepresenting error sources influencing the precision, determining the precision grade according to the size of the error sources, and establishing a naive Bayes network as shown in FIG. 1 for the purpose:
in FIG. 1, Y represents an accuracy evaluation index, and X1,…,X4Respectively represent different error sources, and X is { X for variable set1,X2,X3,X4In which X isiValue range or state set of e.XriThe number of states of each child node, namely the number of value intervals of the error source; d ═ C1,…,CnIs a data sample, i.e. a data set or database, ClIs an event, i.e. a test case or a record of a database, here a live-fire target hit data;a parameter variable of prior probability, which represents an assumption h that the user has a knowledge state ξ and the network structure is S, i.e. Sh，XiOn the premise that the parent node set Pa has the jth state, the variable XiTaking the objective probability of the kth value,pa has a value range of { Pa1,…paqThe number of all possible states with q as Pa, i.e. the number of levels of the precision index, is recordedThen
The following three assumptions are made:
⑴ random sample D is complete, i.e., no data is lost in D;
⑵. the parameter vectors are independent of each other, namely:
⑶. the parameter vector is a Dirichle distribution, namely:
wherein, N'ijkGreater than 0 is an exponential coefficient or a super parameter of Dirichle distribution;
fire control principle CCIP for ground attack
The continuous computing hit Point (CCIP) aiming principle is a commonly used aiming principle when a head-up display/weapon aiming system and a comprehensive fire control system implement bombing and air-to-ground shooting.
Shown in FIG. 6 is (OXXYZ)HThe mutual positions and the motion relations of the carrier, the shot and the target in the course coordinate system. Aircraft velocity vector sum XHThe axial direction is consistent. The airborne fire control computer is based on the flying height H and airspeed V of the airborne1The performance parameters of the weapon ammunition and the attack conditions such as the wind speed U, the wind direction angle epsilon and the like continuously calculate the position of the impact point C on the ground if the projectile is projected currently, the position is displayed on a head-up display, and a pilot forms an aiming line by observing the impact point C. And aiming at a target M point by using a sight line at the projection point O and projecting, and after the falling time T of the projectile, the projectile hits the target M point.
The hit point C is also called an explosion point or a bullet drop point. The aiming point B is the intersection point of the aiming line and the ground, and when there is no aiming error, the aiming point is actually the hit point C.
The position of the hit point C can be determined by using the C point on the navigation coordinate system (OXYZ)HIn 3 coordinate representations, i.e.
Longitudinal range: a. theXH＝A0+UTcosε (15)
Lateral range: a. theYH＝UTsinε (16)
Perpendicular range: a. theZH＝H (17)
In the formula
A0-projectile no wind range;
t-dropping time of the projectile.
The position of the hit point C, and also the range vectorRelative course coordinate system (OXYZ)HX of (2)HTwo angles of rotation of the axis and the norm of the vector, the range vectorRelative to XHShaft (i.e. V)1Direction of) is rotated through an angle mu in a Y-Z-X mannerCH-νCH0, the following results are readily obtained according to FIG. 6, namely:
the minus sign in the formula, description in accordance with the right-hand rule, shows μCHThe angle should be negative.
Uncontrolled weapon movement model
When the mass center of an uncontrolled weapon such as an aircraft gun, a rocket projectile and the like moves, the following assumptions are comprehensively provided:
⑴, in the whole flying time of the projectile, the moving speed direction of the projectile is always coincident with the moving direction of the projectile axis, namely the nutation angle delta is approximately equal to 0, the air resistance action line passes through the mass center, and the direction is opposite to the speed direction;
⑵, the thrust P or the thrust acceleration a is set to pass through the center of mass, and the direction of the thrust P or the thrust acceleration a is the same as the speed direction, namely, the rocket projectile engine is completely ideal;
⑶, because the range is not large, the gravity acceleration can be assumed to be a constant and the direction is vertically downward;
⑷, ignoring earth curvature and Coriolis acceleration;
⑸ the pressure, temperature, humidity and specific gravity of the air are standard values at the ground, and their distribution by height is also standard;
on the assumption that: the meteorological conditions are standard meteorological conditions of an aerodrome. The gravity acceleration g is 9.806m/s2(ii) a Ground standard air pressure value h0760 mm hg; ground standard virtual temperature tau0288.4 ° K; temperature gradient G-5.862 × 10-3Degree/meter; air gas constant R29.27 m/degree; standard value gamma for ground air specific gravityON＝1.225kg/m3。
① model of aerogun movement
Aerogun mass center equation of motion:
the method can be known from aviation outer projectile technology:
J＝CHτ(y1)G(vτ)v (22)
τ＝288.4-5.862×10-3×y1(25)
initial conditions: when t is 0, x is 0, y is 0, z is 0, v0And (5) launching the initial speed for the aerogun.
By means of auxiliary equations (22) to (29) and according toAccording to the table and the corresponding initial conditions, the differential equation set can be solved by a Runge Kutta method to obtain the elements of the theoretical trajectory drop point.
In the above equation, J is the air resistance acceleration, gamma is the air specific gravity, v is the projectile velocity, tau is the virtual temperature, y is the vertical movement displacement of the aircraft gun, the direction is vertically downward, h is the air pressure, C is the trajectory coefficient, a is the sonic velocity of the height of the projectile, y is the trajectory coefficient1Is the height of the projectile from the ground. WhereinThe drag coefficient of the projectile, the value of which is related to the velocity of the projectile, can be determined fromPartial data provided by the table is solved by using Lagrange interpolation methodObtaining the resistance coefficient of the projectile at any speed.
Since the carriage on the helicopter is mobile, i.e. the carriage is rotatable, when the carriage is rotated downwards the weapon initial velocity vector and the vehicle velocity vector form the weapon high and low angle muwIn time, only the initial emission conditions need to be changed: when t is 0, x is 0, y is 0, z is 0, voy＝v0sinμw，
② rocket projectile motion model
The rocket projectile motion model is similar to that of an aerogun, and the rocket projectile motion model is in an uncontrolled state after being launched and does parabolic motion with different diving angles according to different launching conditions. However, it is different from aerogun in that the initial speed of rocket projectile launching is less than that of aerogun, but rocket projectile itself has fuel, and the motion after launching is divided into two parts of active section and passive section. The active section utilizes the thrust generated by fuel combustion to do accelerated motion, and the motion of the passive section is completely the same as that of the aircraft cannon.
In the formula:
wherein:
omega is the drug loading amount; q. q.s0The initial weight of the pill; t is tKIs the active segment time of flight; u. ofeIs the effective exhaust velocity.
v0For launching rocket projectilesInitial velocity (velocity at which the slip-rail segment is ejected), initial conditions, and solution trajectory specification methods are the same as for a flight gun. The thrust acceleration acts on the active section only, and the value of the thrust acceleration is zero in the passive section.
And step 3: precision sensitivity analysis of fire control system by BNSobol method
Firstly, carrying out Bayesian network parameter learning:
⑴. parameter prior distribution
Wherein, N'ijk＝N'·p(Xi＝k,pa＝j|Sh,ξ)。
⑵ posterior distribution of parameters:
wherein N isijkIs satisfied in the database DAnd pa equals the number of cases of j;
the probability process of calculating the desired calculation in the Bayes network is called Bayes inference, theoretically, the joint distribution can infer any desired probability in the Bayes network, the parameter learning can obtain the local conditional probability distribution function, and accordingly, the probability of the specific precision index grade corresponding to all the error source value interval combinations can be inferred, namely, the probability of the specific precision index grade corresponding to all the error source value interval combinations can be inferredWherein j is 1, …, q, ki＝1,…,ri(ii) a The reasoning method is shown in fig. 2 (the process is by means of the Bayesian toolbox of Matlab):
in fig. 2, inference evidence (evidence) is a combination of values of corresponding error sources, and class is an index for precision evaluation.
Each sensitivity index was calculated:
main effect
According to Sobol index method XiDefinition of the Main EffectAll probability values obtained by combining Bayesian network inference can be directly calculated
V(Y)＝E(Y2)-E2(Y) (5)
Wherein n is an error source value combination subscript,is the value of the ith error source under the nth combination, and the same applies:
in the formulaIs the kth error sourceiA value ofIs except for xiThe nth value combination of other error sources;
bringing the formula (5-9) intoCan find the error source XiThe main effect of (c);
second (or second) order interaction effect
XiAnd XjThe interaction effect of the two is defined asWherein,andall can be calculated;
in the formula (x)ixj)kIs the k-th value combination of the ith and jth error sources, andis except for xiAnd xjThe nth value combination of other error sources;
bringing into the formulae (10), (11)Can obtain X by the definition formulaiAnd XjThe interaction effect of the two.
The corresponding error sources and interaction effects are replaced by the following symbols:
x1-errors in the measurement of the yaw angle of the vehicle;
x2-rack random jitter error;
x3-target distance measurement error;
x4-target azimuth aiming error;
the helicopter hovering rocket projectile attacks the ground, and the simulation of other attack conditions is similar to the helicopter hovering rocket projectile attack.
The overall simulation analysis flow is shown in fig. 7.
Setting initial parameters
Table 1 simulation initial parameter set-up
TABLE 2 respective error mean square error range settings
| x1 | x2 | x3 | x4 |
| 0～0.5 | 0～0.5 | 0～2 | 0～0.5 |
(II) Sobol index sensitivity index calculation based on Monte Carlo
The mean square deviation values of the errors are uniformly distributed within the ranges listed in the table, firstly, two input matrixes A and B are generated by adopting a random sampling method, and each row in the two matrixes is a group of specific value combination of four error sources.
Note C3Exchanging the 3 rd column of the matrix B with the 3 rd column of the matrix A to obtain a matrix; note C-3The 3 rd column of matrix a is replaced by the 3 rd column of matrix B.
Similarly, can define C1,C2,C4,C-1,C-2,C-4And C1,2,C-1,-2And the like. The matrixes are used as error source data, are superposed on standard values and are brought into a simulation model, and then an output vector of the model, namely the accuracy index CEP, can be obtained. Remember yA,yB,yCRespectively, corresponding output column vectors of the corresponding input matrix.
The following estimates can be obtained from the monte carlo method:
note the book
The sensitivity index is estimated according to the following formula:
error source x3Index of main effect ofEstimation of (2):
error source x1And x3Second order interaction effect index ofEstimation of (2):
by adopting the method, error source data with different groups of numbers are randomly sampled for many times, and are substituted into model simulation calculation, so that the sensitivity ordering of each error source on the impact of rocket projectile hitting precision can be obtained. It was found that the analysis results gradually converged when the data volume approaches 5000 groups, and the calculation results of the main effect and the second-order interaction effect of each error source obtained by analyzing 5000 groups of data and 500 groups of data are listed here:
main effect ordering of error sources under data of table 35000
| Error source | Main effect | Sorting |
| x1 | 0.179 | 1 |
| x2 | 0.037 | 4 |
| x3 | 0.05 | 3 |
| x4 | 0.158 | 2 |
TABLE 45000 second-order interaction effects between error sources for data set
| x1 | x2 | x3 | x4 | |
| x1 | -- | 0.109 | 0.217 | 0.155 |
| x2 | 0.109 | -- | 0.182 | 0.179 |
| x3 | 0.217 | 0.182 | -- | 0.366 |
| x4 | 0.155 | 0.179 | 0.366 | -- |
Table 5500 data sets for error source main effect ranking
| Error source | Main effect | Sorting |
| x1 | 0.208 | 3 |
| x2 | 0.598 | 1 |
| x3 | 0.598 | 1 |
| x4 | 0.503 | 2 |
Second order interaction effects between error sources under data of table 6500
(III) sensitivity index calculation based on BNSobol method
And (3) carrying out input and output discretization, namely discretizing each error source value into the following intervals:
TABLE 7 value intervals of each error source
The accuracy index is divided into two levels according to the size of CEP:
TABLE 8 accuracy index rankings
On the basis of the above processing, different sets of simulation sample data are randomly extracted for multiple times to serve as real target practice data, parameter learning is carried out by means of a MatLab Bayesian network Toolbox (Bayesian networks Toolbox, BNT), and the result is gradually converged when the data volume is larger than 400. The results of analysis of 500 sets of sample data are taken here, and the parameters of the network are shown in the following table:
TABLE 9 network parameter learning result Table
In table thetaijkThe value of (A) represents the error source xiAnd when the kth value interval is reached, the CEP reaches the probability of each accuracy grade of the jth. It can be seen from the table that the values of the parameters in the network do not exhibit a monotonically increasing or decreasing relationship, which indicates that there is an interaction coupling effect between the error sources in the raw data. The hit precision under 256 error source value combinations can be obtained by reasoning according to the parameters obtained by learning, and then each sensitivity index can be obtained.
The results of two precision sensitivity analysis methods are compared to show that:
⑴, comparing the analysis results with the results in FIG. 8 and FIG. 9, it can be seen that the results obtained by the two analysis methods are not completely identical, both ①. when multiple error sources act on the fire control system simultaneously, the measurement error of the aircraft yaw angle and the target azimuth aiming error have important influence on the shooting accuracy of the rocket projectile and are main error sources, ②. strong second-order interaction effect exists between the error sources, wherein the interaction effect between the measurement error of the aircraft yaw angle and the target azimuth aiming error is most significant.
⑵, for example, when the helicopter fire control system is analyzed with sufficient data, i.e., 5000 sets of data are provided, the Sobol method can analyze to obtain correct conclusions, and when the data is not sufficient, i.e., only 500 sets of data are provided, the analysis result of the traditional Sobol method becomes inaccurate, in military, especially target practice, each set consumes a lot of capital, which is rare per se, BNB only analyzes 4 error sources, and when the error sources are increased or the model is more complex, the sample amount required by the Sobol method is increased extremely rapidly, so that the method is obviously difficult to apply, but when only 500 sets of data are analyzed, the result similar to the traditional Sobol exponential method is applied, so that the cost required by the analysis is greatly reduced.
Claims (1)
1. A BNSobol method for precision sensitivity analysis of a helicopter fire control system is characterized by comprising the following steps:
step 1: determining accuracy sensitivity analysis index
Decomposing the model into single parameters and functions combined among the parameters by adopting a sensitivity index defined by a Sobol index method through variance decomposition, and analyzing the importance of the parameters and the interaction effect among the parameters by calculating the influence of the variance of the single input parameter or the input parameter set on the total output variance;
the Sobol index method defines each sensitivity index as follows:
(1) the main effect, also called first-order sensitivity index, is defined asIs XiContribution of "alone" to the variance of Y, with values of [0,1]Internal;
(2) the second order interaction effect is defined asThe influence on the output is the interaction effect of the two;
step 2: bayesian network model for establishing precision sensitivity analysis of helicopter fire control system
Determining the precision grade according to the size of the error source, and establishing a naive Bayesian network:
y represents an index of accuracy evaluation, and X1,…,X4Respectively represent different error sources, and X is { X for variable set1,X2,X3,X4In which X isiValue range or state set of e.XriThe number of states of each child node, namely the number of value intervals of the error source; d ═ C1,…,CnIs a data sample, i.e. a data set or database, ClIs an event, i.e. a test case or a record of a database, here a live-fire target hit data;a parameter variable of prior probability, which represents an assumption h that the user has a knowledge state ξ and the network structure is S, i.e. Sh，XiOn the premise that the parent node set Pa has the jth state, the variable XiTaking the objective probability of the kth value,pa has a value range of { Pa1,…paqThe number of all possible states with q as Pa, i.e. the number of levels of the precision index, is recordedThen
The following three assumptions are made:
⑴ random sample D is complete, i.e., no data is lost in D;
⑵. the parameter vectors are independent of each other, namely:
⑶. the parameter vector is a Dirichle distribution, namely:
wherein, N'ijkGreater than 0 is an exponential coefficient or a super parameter of Dirichle distribution;
and step 3: precision sensitivity analysis of fire control system by BNSobol method
Firstly, carrying out Bayesian network parameter learning:
⑴. prior distribution of parameters
Wherein, N'ijk＝N'·p(Xi＝k,pa＝j|Sh,ξ)；
⑵ posterior distribution of parameters:
wherein N isijkIs satisfied in the database DAnd pa equals the number of cases of j;
the probability process of calculating in Bayes network is called Bayes inference, local conditional probability distribution function is obtained by parameter learning, and accordingly, the probability of specific precision index grade corresponding to all error source value interval combination, namely the probability of specific precision index grade can be obtainedWherein j is 1, …, q, ki＝1,…,ri；
Calculating a sensitivity index:
main effect
According to Sobol index method XiDefinition of the Main EffectThe main effect can be directly calculated by combining all probability values obtained by Bayesian network inference
V(Y)＝E(Y2)-E2(Y) (5)
Wherein n is an error source value combination subscript,is the value of the ith error source under the nth combination, and the same applies:
in the formulaIs the kth error sourceiA value ofIs except for xiThe nth value combination of other error sources;
bringing the formula (5-9) intoCan find the error source XiThe main effect of (c);
second (or second) order interaction effect
XiAnd XjThe interaction effect of the two is defined asWherein,andall can be calculated;
in the formula (x)ixj)kIs the k-th value combination of the ith and jth error sources, andis except for xiAnd xjThe nth value combination of other error sources;
bringing into the formulae (10), (11)Can obtain X by the definition formulaiAnd XjThe interaction effect of the two.
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