CN111797541B - Method for quantifying coupling uncertainty and calculating global sensitivity of field line - Google Patents
Method for quantifying coupling uncertainty and calculating global sensitivity of field line Download PDFInfo
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Abstract
The invention relates to a field line coupling uncertainty quantification and global sensitivity calculation method, which combines an adaptive hyperbolic truncation scheme with a minimum angle regression method, sparsely processes a generalized chaotic polynomial, establishes a proxy model to analyze uncertainty of radiation sensitivity of a multi-conductor transmission line, and analyzes global sensitivity of an input variable by combining a global sensitivity analysis method.
Description
Technical Field
The invention relates to the field of multi-conductor transmission line radiation sensitivity, in particular to a field line coupling uncertainty quantification and global sensitivity calculation method.
Background
With the continuous development of science and technology, various electrical systems and electronic devices are widely applied to real life, resulting in the increasingly complex electromagnetic environment in which the electrical devices are located. The transmission line is used as a carrier for transmitting electric energy and signals and is an indispensable part in the electric and electronic equipment; the problem of electromagnetic compatibility of transmission lines is also becoming increasingly serious due to the complexity of the electromagnetic environment. Radiosensitivity is also gaining increasing attention from researchers as a class of important issues in electromagnetic compatibility.
Due to the possible changes of the electrical and electronic equipment and the complexity of the electromagnetic environment, the problem of the radiation sensitivity of the transmission line has strong uncertainty, and the research on the problem of the uncertainty of the radiation sensitivity of the multi-conductor transmission line is more and more important. Different uncertain input variables obey corresponding random distribution, and the analysis of the influence degree of the different uncertain input variables on the whole system is also necessary.
The traditional Monte Carlo method has high calculation cost and low calculation efficiency, does not carry out global sensitivity analysis on uncertain input variables, and cannot accurately analyze the influence degree of each uncertain input variable on the whole system.
Disclosure of Invention
The invention aims to provide a method for accurately analyzing the influence degree of each uncertain input variable on the whole system.
In order to achieve the purpose, the invention provides the following scheme:
a field line coupling uncertainty quantification and global sensitivity calculation method, the method comprising:
determining an input variable according to an electromagnetic environment and an incident field where the multi-conductor transmission line is located;
determining a distribution type corresponding to the input variable according to the input variable;
determining an orthogonal base corresponding to the input variable according to the distribution type corresponding to the input variable;
establishing a generalized chaotic polynomial model of the radiation sensitivity of the multi-conductor transmission line by combining a transmission line theory according to the orthogonal substrate of the input variable to obtain an expansion of the generalized chaotic polynomial;
performing hyperbolic truncation processing on the expanded form of the generalized chaotic polynomial by using a self-adaptive hyperbolic truncation method to obtain the hyperbolic truncated generalized chaotic polynomial;
processing the hyperbolic truncated generalized chaotic polynomial by using a minimum angle regression method to obtain a self-adaptive sparse chaotic polynomial;
calculating induced current or induced voltage of the multi-conductor transmission line radiation sensitivity by using the self-adaptive sparse chaotic polynomial to obtain probability distribution of the induced current or the induced voltage at different frequency points;
converting the self-adaptive sparse chaotic polynomial into an incremental summation form, and analyzing the global sensitivity index of the input variable by using a global sensitivity analysis method; the global sensitivity indicators include: the total sensitivity index and the first-order sensitivity index.
Optionally, the expansion of the generalized chaotic polynomial is:
wherein the content of the first and second substances,
mixed orthogonal polynomials of order n, which are multidimensional standard random variables
Infinity is the highest power of the mixed orthogonal polynomial,
and phi i Are respectively connected with
And
in response to this, the mobile terminal is allowed to perform,
coefficient of expansion being said generalized chaotic polynomial, Φ i And (xi) is the product of the base functions of the one-dimensional orthogonal polynomials corresponding to the random variables.
Optionally, the hyperbolic truncation processing is performed on the expansion of the generalized chaotic polynomial by using an adaptive hyperbolic truncation method, so as to obtain a hyperbolic truncated generalized chaotic polynomial, which specifically includes:
obtaining an expansion of the generalized chaotic polynomial;
setting the highest truncation order P of the self-adaptive hyperbolic truncation method, and then performing primary sparse processing on the expansion of the generalized chaotic polynomial to obtain the primarily sparse generalized chaotic polynomial; the ith polynomial highest truncation order p i Satisfies the following conditions:
where k is the dimension of the random variable, l k And the order of the kth dimension random variable is shown, q is a norm, and n is the maximum value of the dimension of the random variable.
Optionally, the method for determining the norm q includes:
setting the range of the preliminary norm q as [ a, b ] and the step length as s;
setting an error threshold;
the preliminary norm q is gradually increased by the step length s and is according to the formula
Calculating out a leave-one-out cross validation error;
stopping calculation when the cross validation error is smaller than the error threshold or exceeds the maximum value of the range of the preliminary norm q to obtain the norm q; wherein epsilon LOO To leave one out cross validation errors, M (x) (i) ) For hyperbolic truncation model after setting norm q value at ith sample point x (i) Response value at Point, M PC\i (x (i) ) Is a chaotic polynomial model after P-order truncation at the ith sample point x (i) The value of the response of (a) to (b),
is the mean of the response values for the L sample points.
Optionally, the hyperbolic truncated generalized chaotic polynomial is processed by using a minimum angle regression method to obtain a self-adaptive sparse chaotic polynomial, which specifically includes:
acquiring the hyperbolic truncated generalized chaotic polynomial;
converting the hyperbolic truncated generalized chaotic polynomial into a Y ═ theta X model; wherein, theta is a set of coefficients of the hyperbolic truncated generalized chaotic polynomial, and X is a set of the hyperbolic truncated generalized chaotic polynomial;
processing the Y-theta X model by using a minimum angle regression method to obtain a coefficient of a polynomial;
and substituting the coefficients of the polynomial into a self-adaptive sparse chaotic polynomial to obtain the self-adaptive sparse chaotic polynomial.
Optionally, the calculating the induced current or the induced voltage of the multi-conductor transmission line radiation sensitivity by using the adaptive sparse chaotic polynomial to obtain the probability distribution of the induced current or the probability distribution of the induced voltage at different frequency points specifically includes:
acquiring the self-adaptive sparse chaotic polynomial;
converting the adaptive sparse chaotic polynomial into
Wherein
For the sparsely processed polynomial coefficient, phi i 'is a polynomial after sparse processing, and p' is the number of terms of the polynomial after sparse processing;
obtaining the mean value of the multi-conductor transmission line radiation sensitivity induction current or the mean value of the multi-conductor transmission line radiation sensitivity induction voltage according to the formula after the hyperbolic truncation and the generalized chaotic polynomial conversion:
wherein the content of the first and second substances,
a constant term of an expansion of the truncated generalized chaotic polynomial;
obtaining the multi-conductor transmission line radiation according to the formula after the hyperbolic truncation and the generalized chaotic polynomial conversionThe variance of the sensitivity induced current or the variance of the multi-conductor transmission line radiation sensitivity induced voltage is:
optionally, the converting the adaptive sparse chaotic polynomial into an incremental summation form, and analyzing the global sensitivity index of the input variable by using a global sensitivity analysis method specifically includes:
acquiring the self-adaptive sparse chaotic polynomial;
converting the adaptive sparse chaotic polynomial into an expansion in an incremental summation form;
carrying out recursive calculation on the expansion of the incremental summation form by using an integral method, solving the corresponding coefficient of the decomposition term, and obtaining the expansion of the incremental summation form with known coefficient;
taking variance from both sides of the known coefficient incremental summation form expansion to obtain a variance decomposition formula;
obtaining a global sensitivity index according to the variance decomposition formula; the global sensitivity index includes a sensitivity index and a total sensitivity index.
Optionally, the expansion of the incremental summation form is:
wherein, the first and the second end of the pipe are connected with each other,
optionally, the sensitivity index is:
The total sensitivity index is:
wherein S is i Is an index of first-order sensitivity.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects: the method combines the self-adaptive hyperbolic truncation method with the minimum angle regression method, sparsely processes the generalized chaotic polynomial, establishes a proxy model to analyze the uncertainty problem of the radiation sensitivity of the multi-conductor transmission line, analyzes the global sensitivity of the input variable by combining the Sobol method, and analyzes and calculates the uncertainty problem of the radiation sensitivity of the multi-conductor transmission line with lower calculation cost and higher efficiency on the premise of ensuring the calculation accuracy.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a schematic flow chart of a field line coupling uncertainty quantification and global sensitivity calculation method provided by the present invention;
FIG. 2 is a schematic diagram showing the positional relationship between a multi-conductor transmission line and an incident field;
FIG. 3 is a hyperbolic truncation diagram of two-dimensional input variables and different q-norms;
fig. 4 shows the truncation effect of hyperbolic truncation when P is 5 and n is 2;
FIG. 5 is a schematic diagram of a two-dimensional variable least angle regression method;
FIG. 6 is a 3+1 multi-conductor transmission line model;
FIG. 7 is a diagram illustrating comparison results between the calculation results of the mean and standard deviation of the conventional generalized chaotic polynomial with different truncation orders and the Monte Carlo method of 20000 orders;
FIG. 8 is a graph of the pairs of ε at different frequency points and different truncation orders LOO An influence schematic;
FIG. 9 is ε LOO An error curve;
fig. 10 is a graph illustrating the comparison result of the standard deviation of the mean of the sparse chaotic polynomial with the monte carlo method when q is 0.8 and P is 15;
FIG. 11 is a comparison result of AS-PC calculating # 2 transmission line induced current probability distribution at different frequency points;
FIG. 12 is a graph comparing an upper limit of the induced current curve with a simulation curve of 20000 Monte Carlo;
FIG. 13 is a graph comparing total sensitivity indices;
FIG. 14 is a first order sensitivity index comparison plot;
FIG. 15 shows the degree of influence of various parameters in the [10MHz,1GHz ] frequency range on the radiation sensitivity of a multi-conductor transmission line.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a field line coupling uncertainty quantification and global sensitivity calculation method, which can be used for analyzing and calculating the uncertainty of the radiation sensitivity of a multi-conductor transmission line with lower calculation cost and higher efficiency on the premise of ensuring the calculation accuracy.
In order to make the aforementioned objects, features and advantages of the present invention more comprehensible, the present invention is described in detail with reference to the accompanying drawings and the detailed description thereof.
As shown in fig. 1, a flow chart of a field line coupling uncertainty quantification and global sensitivity calculation method provided by the present invention is shown, and the method specifically includes:
step 101: the input variables are determined from the electromagnetic environment in which the multi-conductor transmission line is located and the incident field.
Step 102: and determining the distribution type corresponding to the input variable according to the input variable.
Step 103: and determining the orthogonal basis corresponding to the input variable according to the distribution type corresponding to the input variable.
Step 104: and establishing a generalized chaotic polynomial model of the radiation sensitivity of the multi-conductor transmission line by combining a transmission line theory according to the orthogonal basis of the input variables to obtain an expansion of the generalized chaotic polynomial.
Step 105: and performing hyperbolic truncation processing on the expanded form of the generalized chaotic polynomial by using a self-adaptive hyperbolic truncation method to obtain the hyperbolic truncated generalized chaotic polynomial.
Step 106: and processing the hyperbolic truncated generalized chaotic polynomial by using a minimum angle regression method to obtain a self-adaptive sparse chaotic polynomial.
Step 107: and calculating the induced current or induced voltage of the multi-conductor transmission line radiation sensitivity by using the self-adaptive sparse chaotic polynomial to obtain the probability distribution of the induced current or the probability distribution of the induced voltage at different frequency points.
Step 108: converting the self-adaptive sparse chaotic polynomial into an incremental summation form, and analyzing the global sensitivity index of the input variable by using a global sensitivity analysis method; the global sensitivity indicators include: the total sensitivity index and the first-order sensitivity index.
The scheme integrally combines an adaptive hyperbolic truncation method and a minimum angle regression method, sparsely processes a generalized chaotic polynomial, establishes a proxy model to analyze the uncertainty problem of the radiation sensitivity of the multi-conductor transmission line, and combines a Sobol method to analyze the global sensitivity of an input variable.
The position relationship between the multi-conductor transmission line and the incident field is shown in FIG. 2. since the uncertainty in the multi-conductor radiation sensitivity uncertainty problem is concentrated in the incident field, the present invention determines the input variable as the input variable during the implementation processElevation angle theta, azimuth angle psi, polarization angle eta, and level amplitude E of incident plane wave 0 (ii) a l is the length of the transmission line, R is the termination impedance of the transmission line, and the number of transmission lines is S.
And (3) taking the original model as Y (xi), and carrying out expansion on the original model by using a generalized chaotic polynomial, wherein the expansion result is as follows:
wherein the content of the first and second substances,
mixed orthogonal polynomial of order n, which is a multidimensional standard random variable
Infinity is the highest power of the mixed orthogonal polynomial,
and phi i Are respectively connected with
And
in response to this, the mobile terminal is allowed to,
coefficient of expansion being said generalized chaotic polynomial, Φ i And (xi) is the product of the one-dimensional orthogonal polynomial basis functions corresponding to the random variables.
For the generalized chaotic polynomial, the following describes the processing procedure of the adaptive hyperbolic truncation method and the minimum angle regression method.
The method for performing hyperbolic truncation processing on the expanded form of the generalized chaotic polynomial by using the self-adaptive hyperbolic truncation method specifically comprises the following steps:
obtaining an expansion of the generalized chaotic polynomial;
setting the highest truncation order P of the self-adaptive hyperbolic truncation method, and then performing primary sparse processing on the expansion of the generalized chaotic polynomial to obtain the primarily sparse generalized chaotic polynomial; the ith polynomial has the highest truncation order p i Satisfies the following conditions:
where k is the dimension of the random variable, l k And the order of the kth dimension random variable is shown, q is a norm, and n is the maximum value of the dimension of the random variable.
According to the highest truncation order p i Satisfied formula it can be seen that when q is 1, p max The truncation scheme is the conventional truncation scheme, i.e. P
Meanwhile, the hyperbolic truncation scheme is to perform sparse processing on the high-order effect of the model on the basis of the traditional truncation scheme, and when q is used<1, the reserved polynomials are all positioned under a hyperbolic curve or curved surface. As shown in fig. 3, when n is 2, the norm q is 1, 0.75, and 0.5, and the hyperbolic truncation diagram corresponds to a schematic diagram, it can be intuitively seen that, for input variables with different dimensions, the hyperbolic truncation scheme can effectively reduce the influence of the model high order effect and keep the low order effect below the curve, and as the q value decreases, the penalty of the hyperbolic truncation scheme on the model high order effect is more obvious.
Fig. 4 is a truncation effect of a conventional truncation order P of 5 and an input variable dimension n of 2 hyperbolic truncation, and in order to show the influence of a related sparse processing method on a model, Q is a sparse coefficient: n '/N, where N' is a polynomial expansion term after sparse processing, N is a polynomial expansion term without sparse processing, and the smaller the sparse coefficient Q is, the lower the required computation cost is, for a model with an input variable N of 4 and a hyperbolic truncation norm Q of 0.8, the polynomial term pair of the traditional truncation scheme polynomial terms with different truncation orders and the hyperbolic truncation scheme polynomial term pair is shown in table 1:
TABLE 1
As can be seen from table 1, the hyperbolic truncation scheme can effectively reduce the number of terms of the polynomial, and as the truncation order increases, the sparse coefficient also decreases, so that the hyperbolic truncation scheme can effectively perform preliminary sparse processing on the model.
The norm q is an important parameter introduced by the invention, and a specific determination method comprises the following steps:
setting the range of the preliminary norm q as [ a, b ] and the step length as s;
setting an error threshold;
the preliminary norm q is gradually increased by the step length s and is according to the formula
Calculating out a leave-one-out cross validation error;
stopping calculation when the cross validation error is smaller than the error threshold or exceeds the maximum value of the range of the preliminary norm q to obtain the norm q; wherein epsilon LOO To leave one out cross validation errors, M (x) (i) ) For hyperbolic truncation model after setting norm q value at ith sample point x (i) Response value at Point, M PC\i (x (i) ) Is a chaos polynomial model after P-order truncation at the ith sample point x (i) The value of the response of (c) to (d),
is the mean of the response values for the L sample points.
When the range of the hyperbolic truncation norm q is selected, as can be seen from fig. 3, along with the decrease of q, while punishing a high-order effect, the influence of hyperbolic truncation on a low-order effect is gradually increased, when the value of q is too small, the low-order effect of the model is punished by a hyperbolic curve, which may greatly affect the precision of the model, and if a small value is included in the selection range of the norm q, this may cause unnecessary iterative computation of the program, which may cause a waste of the computation cost, so generally when the range of q is selected, a and b are often set to 0.5 and 1, respectively, so that the norm q of the model may be computed more quickly, and the computation cost may be saved.
The chaos polynomial adopting the hyperbolic truncation scheme effectively reduces the influence of high-order effects among variables in the model, and as can be seen from table 1, compared with the generalized chaos polynomial, the polynomial term number after the hyperbolic truncation processing is reduced by more than half, but as the truncation order is increased, the polynomial term number after the hyperbolic truncation processing still has a certain sparse processing space due to the fact that the basic term number of the polynomial is very much. The minimum angle regression method is an important sparse algorithm and can effectively carry out sparse processing on the model, so that on the basis of the self-adaptive hyperbolic truncation scheme, in order to further sparsify the number of terms of the chaotic polynomial, the minimum angle regression method is adopted to carry out sparse processing on the hyperbolic truncated polynomial, the method integrates the advantages of the forward selection algorithm and the forward gradient algorithm, greedy iteration is not needed to be carried out like the forward selection algorithm, and fitting calculation can be completed only by fewer iteration times than the forward gradient algorithm.
Processing the hyperbolic truncated generalized chaotic polynomial by using a minimum angle regression method to obtain a self-adaptive sparse chaotic polynomial, which specifically comprises the following steps:
acquiring the hyperbolic truncated generalized chaotic polynomial;
converting the hyperbolic truncated generalized chaotic polynomial into a Y ═ theta X model; wherein, theta is a set of coefficients of the hyperbolic truncated generalized chaotic polynomial, and X is a set of the hyperbolic truncated generalized chaotic polynomial;
selecting independent variable X most relevant to dependent variable Y by cosine similarity method j (ii) a Independent variable X j Belonging to a set of the hyperbolic truncated generalized chaotic polynomials;
calculating Y and X j Residual γ of from X j Is advanced in the direction of theta j In this case, the residual γ is Y- θ j *X j Proceed until another variable X appears t The degree of correlation with the residual y being equal to X j Equal to the residual gamma, i.e. the residual gamma is located at X j And X t Then continues forward along the bisector until the next argument appears that is the same as the residual correlation. In short, the variable most correlated to the current residual γ is continuously searched.
Fig. 5 is a schematic diagram of the principle of the minimum angle regression method using two-dimensional input variables as an example, which specifically includes:
1. finding the variable X most related to the dependent variable Y through cosine similarity 1 。
2. Let variable X 1 Move in the current direction until another variable X appears 2 So that Y-X 1 *θ 1 Can divide X equally 1 And X 2 I.e. X 2 Correlation with residual gamma and X 1 The residual gamma is the same as the correlation degree of the residual gamma, and the residual gamma is positioned at X 1 And X 2 On the bisector of (2), in X, as illustrated in FIG. 4 1 Is moved by theta along the current direction 1 At time of multiplication, the residual γ at this time is Y- θ 1 *X 1 And gamma bisects X 1 And X 2 Angle of (e), at this time theta 1 I.e. variable X 1 The coefficient of (a).
3. Similarly to the second step, the process continues to proceed along the direction of the angular bisector obtained in the step 2 until a new independent variable X appears 3 The same degree of correlation with the residual gamma.
4. Updating the coefficients and moving eligible arguments from the candidate set to the active set.
5. Repeating the steps until all the variables are iterated completely, namely all important polynomials are selected, and the obtained theta is the coefficient of the polynomials. And substituting the coefficients of the polynomial into a self-adaptive sparse chaotic polynomial to obtain the self-adaptive sparse chaotic polynomial.
After an AS-PC agent model of the multi-conductor transmission line radiation sensitivity is established, the mean value and the variance of induced current or induced voltage are calculated, and the self-adaptive sparse chaotic polynomial is converted into
Wherein
For the sparsely processed polynomial coefficient, phi i 'is a polynomial after sparse processing, and p' is the number of terms of the polynomial after sparse processing;
obtaining the mean value of the multi-conductor transmission line radiation sensitivity induction current or the mean value of the multi-conductor transmission line radiation sensitivity induction voltage according to the formula after the hyperbolic truncation and the generalized chaotic polynomial conversion:
wherein the content of the first and second substances,
the constant term is the expanded constant term of the truncated generalized chaotic polynomial;
obtaining the variance of the multi-conductor transmission line radiation sensitivity induction current or the variance of the multi-conductor transmission line radiation sensitivity induction voltage according to the formula after the hyperbolic truncation is carried out, wherein the formula after the conversion of the generalized chaotic polynomial is as follows:
according to the self-adaptive sparse chaotic polynomial, the global sensitivity index of each input variable is calculated by combining a global sensitivity analysis method, and the influence degree of each input variable on the system is quantized, wherein the specific process comprises the following steps:
acquiring the self-adaptive sparse chaotic polynomial;
converting the adaptive sparse chaotic polynomial into an expansion in an incremental summation form;
carrying out recursive calculation on the expansion of the incremental summation form by using an integral method, solving the corresponding coefficient of the decomposition term, and obtaining the expansion of the incremental summation form with known coefficient;
taking variance from both sides of the expansion of the known incremental summation form of the coefficient at the same time to obtain a variance decomposition formula;
obtaining a global sensitivity index according to the variance decomposition formula; the global sensitivity index includes a sensitivity index and a total sensitivity index.
The expansion of the incremental summation form is:
wherein the content of the first and second substances,
then, the sensitivity index is:
wherein
The total sensitivity index is:
wherein S is i Is an index of first-order sensitivity.
In order to more intuitively show the technical effect that the present invention can achieve, the following takes the number S of the transmission lines as 3 as an example, and further details the specific scheme of the present invention.
A 3+1 multi-conductor transmission line model with the ground as a reference conductor is established as shown in fig. 6, the length l of the transmission line is 1m, the radius r of the transmission line is 0.4mm, the distance between the transmission lines is 1cm, the height h from the ground is 2cm, and the loads of the source end and the load end of the transmission line are both 50 Ω.
Setting random input variables as an elevation angle theta, an azimuth angle psi, a polarization angle eta and a level amplitude E of incident plane waves, enabling the elevation angle theta to obey uniform distribution on an interval [0,0.5 pi ], enabling the azimuth angle psi to obey uniform distribution on the interval [ -pi, pi ], enabling the polarization angle eta to obey uniform distribution on the interval [0,2 pi ], enabling the level amplitude E to obey normal distribution with the mean value of 1V/m and the standard deviation of 0.2V/m, and selecting corresponding orthogonal bases according to different distributions obeyed by the random input variables to construct a generalized chaotic polynomial model.
Under the conventional truncation scheme, the current I is induced at the far end of the second transmission line in FIG. 6 R2 For example, let the truncation order P be 5, 10, 15, 20, respectively, the mean and standard deviation of the generalized chaotic polynomial model of the multi-conductor transmission line radiation sensitivity are calculated, and the frequency range of the incident field is set to [10MHz,1GHz []The calculated mean and standard deviation were compared with the calculation results of the 20000 MC methods, and the comparison results are shown in fig. 7.
From the comparison results of fig. 7, the induced current I is radiosensitized for a multi-conductor transmission line R2 The mean and standard deviation of the low-order truncation is not good under the traditional truncation scheme, but the mean and standard deviation calculation result is closer to the calculation result of the MC as the truncation order increases. Then selecting epsilon at three frequency points of 50MHz, 80MHz and 100MHz aiming at different truncation orders LOO For comparison, as shown in fig. 8. As can be seen in FIG. 7, the lower order truncations are e.g. 5 th order and 10 th order ∈ LOO Relatively large, the calculation is not perfect, and the 15 th order truncated epsilon LOO Epsilon with truncation of order 20 LOO Lower and substantially consistent, it is proved that the accuracy of the calculation is higher as the truncation order P increases, which indicates that the 15 th order truncation can reach higher accuracy already in the case of the conventional truncation.
The expansion of the generalized chaotic polynomial in the 15-order conventional truncation scheme has 3876 terms, so that the generalized chaotic polynomial still has a large sparse processing space, and in order to further reduce the calculation cost and improve the calculation efficiency, the adaptive hyperbolic truncation scheme introduced in the foregoing is adopted to process the polynomial. With an induced current I at 50MHz R2 For example, let the selection interval of hyperbolic truncated norm q be [0.5, 1%]Step length s of 0.05,. epsilon LOO The threshold value is 0.05, and epsilon shown in FIG. 9 is calculated LOO The error curve of (2). Table 2 provides the different norms for P15q corresponding sparse coefficient:
TABLE 2
| q | 0.5 | 0.55 | 0.6 | 0.65 | 0.7 | 0.75 | 0.8 | 0.85 | 0.9 | 0.95 | 1 |
| N’ | 263 | 358 | 480 | 662 | 873 | 1133 | 1476 | 1893 | 2404 | 3095 | 3876 |
| Q | 6.79％ | 9.24％ | 12.4％ | 17.1％ | 22.5％ | 29.2％ | 38.1％ | 48.8％ | 62.0％ | 79.9％ | 1 |
As can be seen from fig. 9, when q is 0.8, epsilon LOO The threshold is reached and it can be seen that when q is reached>At 0.8, epsilon LOO Is small, and combines the sparse coefficient Q in the table with respect to Q>0.8 in case of ε LOO Almost the same time, the sparse coefficient of q 0.8 is the smallest, and the calculation cost is the lowest, so that q 0.8 meets the expectation of the present invention, and can complete the preliminary sparse processing on the polynomial. The invention proceeds to further analytical studies with q-norm set to 0.8.
After the selection of the hyperbolic truncated norm q is completed, according to the introduction in the foregoing, the present invention combines the minimum angle regression method in the foregoing to perform deeper sparse processing on the model. In the foregoing, it has been proved that under the conventional truncation scheme, better accuracy can be achieved when P ═ 15, and in order to verify that when hyperbolic truncation is combined with the minimum angle regression method, P ═ 15 can also achieve ideal accuracy, induced currents at 50MHz, 80MHz and 100MHz are taken as examples, and for different applications, the induced currents at different frequencies are taken as examplesEpsilon of truncation order LOO For comparison, as shown in fig. 10. It can be seen that when q is 0.8, P is 15 and P is 20 epsilon compared to the calculation of the low order truncation LOO The change is small and basically consistent, and the model adopted by the invention can achieve ideal precision when q is 0.8 and P is 15. In order to further verify the accuracy of the combination of the hyperbolic truncation and the minimum angle regression method, the invention calculates the probability distribution of the radiation sensitivity induced current of the multi-conductor transmission line at the frequency points of 50Mhz, 80Mhz and 100Mhz, and calculates the probability distribution curve of the radiation sensitivity induced current of the multi-conductor transmission line by combining the hyperbolic truncation and the minimum angle regression method, as shown in fig. 11, from the result of the probability distribution curve in the graph, it can be seen that the method of combining the hyperbolic truncation and the minimum angle regression method can effectively calculate the induced current I at different frequency points on the premise of ensuring the calculation accuracy R2 The calculation time of the method combining the curved truncation and the minimum angle regression method, the calculation time of the generalized chaotic polynomial and the sparse coefficient Q are further analyzed, as shown in table 3:
TABLE 3
| Calculating time/s | Sparse coefficient Q | |
| AS-PC | 6.39 | 3.25％ |
| Generalized chaotic polynomial | 13.69 | 100％ |
| 20000 times of MC | 2001.54 | nan |
The CPU main frequency of the computer used by the invention is 2.3GHz, the operating memory is 8GB, and table 3 shows that compared with the computation time of 20000 Monte Carlo methods, the computation time of the generalized chaotic polynomial is greatly shortened, but the hyperbolic truncation and the minimum angle regression method are combined to further effectively compress the computation time, so that the computation cost is saved while the computation precision is ensured, and the computation efficiency is improved. The calculation results are combined, and the hyperbolic truncation and minimum angle regression method adopted by the method can effectively make the generalized chaotic polynomial sparse and accurately calculate the probability distribution of the multi-conductor transmission line radiation sensitivity induced current at different frequency points. In practical applications, the induced current of the radiation sensitivity of the multi-conductor transmission line is limited in the interval [ mu-3 sigma, mu +3 sigma]Can be used as an induced current I R3 Approximate upper and lower limits, with high quantile confidence (e.g., 0.95, 0.99, 0.995, etc.) may be used to demonstrate that μ -3 σ may be effective as the sense current I R2 But this is not within the scope of the present invention's study, which was validated in conjunction with the calculation of 20000 monte carlo method, as shown in fig. 12: the upper limit mu +3 sigma obtained by hyperbolic truncation and minimum angle regression method calculation well covers most Monte Carlo simulation curves, and proves that the mu +3 sigma can be effectively used as the upper limit of the variation range of the induced current.
Through the calculation and analysis, the hyperbolic truncation and minimum angle regression method adopted by the invention can quickly and accurately calculate related statistical characteristic parameters (such as mean standard deviation probability distribution) and the like in the uncertainty problem of the radiation sensitivity of the multi-conductor transmission line on the premise of ensuring the calculation precision. Next, in order to further analyze the influence degree of different input variables on the model in the multi-conductor transmission line radiosensitivity system, the present invention will be analyzed and studied in combination with the global sensitivity calculation method described in the foregoing.
The response current of the right end of the No. 2 transmission line at 50MHz is taken as an index, the total sensitivity index and the first-order sensitivity index comparison graph of each input variable are obtained by calculation, as shown in fig. 13 and fig. 14, and it can be seen from the two graphs that the total sensitivity index and the first-order sensitivity index calculated based on the method of the invention are basically consistent with the result calculated by 20000 times of MC method, the influence degree of each different input variable on the model is also consistent, the total sensitivity index and the first-order sensitivity index of the polarization angle eta are both kept at a higher level, which are important factors influencing the radiation sensitivity of the multi-conductor transmission line at the frequency point, and the sensitivity index of the elevation angle theta is lower, and the whole influence degree on the model is not large.
By combining the analysis, the method adopted by the invention has higher calculation speed and higher efficiency, and proves that the method adopted by the invention is effective in calculating the total sensitivity index and the first-order sensitivity index of the radiation sensitivity of the multi-conductor transmission line.
In order to more intuitively represent the influence degrees of different random variables at different frequency points on the whole model, the total sensitivity index of the influence degrees of each parameter on the multi-conductor transmission radiation sensitivity at the frequency band [10MHz,1GHz ] is calculated, as shown in fig. 15, it can be seen that the influence degree of the level amplitude E is kept at a lower level on the frequency band [10MHz,1GHz ], and the influence on the whole model is not large, although the influence at the higher frequency band, such as [900MHz,1GHz ] is increased, the influence degree of the polarization angle η at the same-frequency band is far greater than the level amplitude E, when the frequency is higher than 200MHz, the influence degree of the elevation angle θ on the model is also increased, the azimuth angle ψ becomes an important factor influencing the model at about 200MHz, but the influence degree at the high-frequency band is greatly reduced.
By combining the above analysis, in practical engineering application, when the position and frequency range of the radiation source in the surrounding environment are known, the polarization angle of the radiation source received by the system should be considered when designing an electrical system in the environment, so as to reasonably and effectively adjust the position distribution of the transmission line and avoid the occurrence of unnecessary electromagnetic compatibility problems.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.
Claims (7)
1. A method for field line coupling uncertainty quantification and global sensitivity calculation, the method comprising:
determining an input variable according to an electromagnetic environment and an incident field where the multi-conductor transmission line is located;
determining a distribution type corresponding to the input variable according to the input variable;
determining an orthogonal base corresponding to the input variable according to the distribution type corresponding to the input variable;
establishing a generalized chaotic polynomial model of the radiation sensitivity of the multi-conductor transmission line by combining a transmission line theory according to the orthogonal basis of the input variable to obtain an expansion of the generalized chaotic polynomial;
performing hyperbolic truncation processing on the expanded form of the generalized chaotic polynomial by using a self-adaptive hyperbolic truncation method to obtain the hyperbolic truncated generalized chaotic polynomial;
processing the hyperbolic truncated generalized chaotic polynomial by using a minimum angle regression method to obtain a self-adaptive sparse chaotic polynomial;
calculating induced current or induced voltage of the multi-conductor transmission line radiation sensitivity by using the self-adaptive sparse chaotic polynomial to obtain probability distribution of the induced current or the probability distribution of the induced voltage at different frequency points;
converting the self-adaptive sparse chaotic polynomial into an incremental summation form, and analyzing the global sensitivity index of the input variable by using a global sensitivity analysis method; the global sensitivity indicators include: a total sensitivity index and a first-order sensitivity index;
the hyperbolic truncation processing is performed on the expanded form of the generalized chaotic polynomial by using a self-adaptive hyperbolic truncation method to obtain the hyperbolic truncated generalized chaotic polynomial, and the method specifically comprises the following steps:
obtaining an expansion of the generalized chaotic polynomial;
setting the highest truncation order P of the self-adaptive hyperbolic truncation method, and then performing primary sparse processing on the expansion of the generalized chaotic polynomial to obtain the primarily sparse generalized chaotic polynomial; highest truncation order p of ith polynomial i Satisfies the following conditions:
where k is the dimension of the random variable, l k The order of a random variable of the kth dimension is defined, q is a norm, and n is the maximum value of the dimension of the random variable;
the method for determining the norm q comprises the following steps:
setting the range of the preliminary norm q as [ a, b ] and the step length as s;
setting an error threshold;
the preliminary norm q is gradually increased by the step length s and is according to the formula
Calculating out a leave-one-out cross validation error;
when the cross-validation error is less than the error threshold or exceeds the maximum of the range of the preliminary norm qStopping calculation to obtain the norm q; wherein epsilon LOO To leave one out cross validation errors, M (x) (i) ) For hyperbolic truncation model after setting norm q value at ith sample point x (i) Response value at Point, M PC\i (x (i) ) Is a chaotic polynomial model after P-order truncation at the ith sample point x (i) The value of the response of (c) to (d),
the average value of the response values of L sample points is shown, L is the number of the sample points, and N is the number of polynomial expansion terms which are not subjected to sparse processing.
2. The field line coupling uncertainty quantization and global sensitivity calculation method of claim 1, wherein the expansion of the generalized chaotic polynomial is:
wherein the content of the first and second substances,
mixed orthogonal polynomials of order n, which are multidimensional standard random variables
Infinity is the highest power of the mixed orthogonal polynomial,
and phi i Are respectively connected with
And
in response to this, the mobile terminal is allowed to,
coefficient of expansion being said generalized chaotic polynomial, Φ i And (xi) is the product of the base functions of the one-dimensional orthogonal polynomials corresponding to the random variables.
3. The field line coupling uncertainty quantification and global sensitivity calculation method according to claim 1, wherein the hyperbolic truncated generalized chaotic polynomial is processed by using a minimum angle regression method to obtain an adaptive sparse chaotic polynomial, specifically comprising:
acquiring the hyperbolic truncated generalized chaotic polynomial;
converting the hyperbolic truncated generalized chaotic polynomial into a Y ═ theta X model; wherein, theta is a set of coefficients of the hyperbolic truncated generalized chaotic polynomial, and X is a set of the hyperbolic truncated generalized chaotic polynomial;
processing the Y-theta X model by using a minimum angle regression method to obtain a coefficient of a polynomial;
and substituting the coefficients of the polynomial into the self-adaptive sparse chaotic polynomial to obtain the self-adaptive sparse chaotic polynomial.
4. The field line coupling uncertainty quantification and global sensitivity calculation method according to claim 1, wherein the calculating of the induced current or induced voltage of the multi-conductor transmission line radiation sensitivity using the adaptive sparse chaotic polynomial to obtain the probability distribution of the induced current or the probability distribution of the induced voltage at different frequency points specifically comprises:
acquiring the self-adaptive sparse chaotic polynomial;
converting the adaptive sparse chaotic polynomial into
Wherein
To be sparsePolynomial coefficient after treatment, Φ' i Is a polynomial after sparse processing, and p' is the number of terms of the polynomial after sparse processing;
obtaining the mean value of the multi-conductor transmission line radiation sensitivity induction current or the mean value of the multi-conductor transmission line radiation sensitivity induction voltage according to the formula after the hyperbolic truncation and the generalized chaotic polynomial conversion:
wherein, the first and the second end of the pipe are connected with each other,
a constant term of an expansion of the truncated generalized chaotic polynomial;
obtaining the variance of the multi-conductor transmission line radiation sensitivity induction current or the variance of the multi-conductor transmission line radiation sensitivity induction voltage according to the formula after the hyperbolic truncation and the generalized chaotic polynomial conversion:
5. the field line coupling uncertainty quantification and global sensitivity calculation method according to claim 1, wherein the transforming the adaptive sparse chaotic polynomial into an incremental summation form, and analyzing the global sensitivity index of the input variable by using a global sensitivity analysis method specifically comprises:
acquiring the self-adaptive sparse chaotic polynomial;
converting the self-adaptive sparse chaotic polynomial into an expansion in an incremental summation form;
carrying out recursive calculation on the expansion of the incremental summation form by using an integral method, solving the corresponding coefficient of the decomposition term, and obtaining the expansion of the incremental summation form with known coefficient;
taking variance from both sides of the expansion of the known incremental summation form of the coefficient at the same time to obtain a variance decomposition formula;
obtaining a global sensitivity index according to the variance decomposition formula; the global sensitivity index includes a sensitivity index and a total sensitivity index.
6. The method for field line coupling uncertainty quantification and global sensitivity calculation of claim 5 wherein the expansion of the incremental summation form is:
wherein the content of the first and second substances,
7. the method of claim 6, wherein the sensitivity indicators are:
The total sensitivity index is:
wherein S is i Is the first-order sensitivity index.
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