CN113407909A - Tasteless algorithm for non-analytic complex nonlinear system - Google Patents

Tasteless algorithm for non-analytic complex nonlinear system Download PDF

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CN113407909A
CN113407909A CN202110801131.XA CN202110801131A CN113407909A CN 113407909 A CN113407909 A CN 113407909A CN 202110801131 A CN202110801131 A CN 202110801131A CN 113407909 A CN113407909 A CN 113407909A
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夏亦犁
石琬婷
裴文江
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Abstract

The invention discloses a tasteless algorithm for a non-analytic nonlinear complex system, which comprises the following steps: calculating second-order statistics of the input signals in the amplification form, namely amplification variance and covariance, inputting the second-order statistics into a system, and calculating sigma points; the tasteless algorithm can obtain first-order and second-order statistics of the output points corresponding to the nonlinear system through the calculation of a small number of sigma characteristic points; and by researching the relation between the error and the non-roundness, beta parameters are adjusted, and the algorithm error is reduced. The invention can obtain approximate second-order statistic of the output signal by calculating the statistical property of a small number of sigma points aiming at the condition that the input signal is a complex signal and the system function is nonlinear non-analytic, and reduce the system error by researching the relation between beta parameters and non-roundness when the signal is non-circular.

Description

Tasteless algorithm for non-analytic complex nonlinear system
Technical Field
The invention relates to the technical field of complex signal processing, in particular to a tasteless algorithm for a non-analytic complex nonlinear system.
Background
An Unscented Transform (UT) is an algorithm that handles non-linear transfer of mean and covariance. Finite parameters are used to approximate the probabilistic statistics of random quantities by determining finite sigma points. A large amount of computation and data storage can be avoided. The method is often applied to actual scenes such as automatic control, navigation, tracking, artificial intelligence, fault estimation and the like by combining with a Kalman filtering algorithm, and state estimation under a nonlinear system function is carried out.
But the traditional tasteless algorithm mainly aims at real signals, the research on the complex signals is still less, especially the research on the non-analytic mapping function is blank,
disclosure of Invention
It is therefore an objective of the claimed invention to provide a non-odor algorithm for non-analytic complex nonlinear systems to solve the above-mentioned problems.
In order to achieve the purpose, the invention adopts the following technical scheme:
a tasteless algorithm for non-analytic complex nonlinear systems, comprising the steps of:
step S1, calculating second-order statistics of the input signal in the amplification form, inputting the second-order statistics into a nonlinear system, and calculating sigma points;
step S2, according to the sigma point calculated in the step S1, a tasteless algorithm is adopted to obtain first-order and second-order statistics of the output point corresponding to the nonlinear system;
and step S3, when the input signal is non-circular, calculating the optimal beta parameter by considering the relation between the error and the non-circular degree, and reducing the system error.
Further, the step S1 specifically includes:
step S101, the input signal is a complex signal, and the complex signal and the conjugate signal thereof are combined into an amplification form:x＝[xT，xH]Twherein x is a complex signal, xTExpressed as transposes of complex signals, xHExpressed as a conjugate transpose of the complex signal;
step S102, calculating second-order statistics according to the input signal, wherein the second-order statistics comprise a covariance matrix and a pseudo covariance matrix, and the expression of the covariance matrix is as follows:
Figure BDA0003164602370000021
wherein the content of the first and second substances,
Figure BDA0003164602370000022
expressed as a covariance matrix of the input signal, RxExpressed as the covariance, P, of the input signalxExpressed as the pseudo-covariance of the input signal, i.e. as the conjugate sign
Figure BDA0003164602370000023
And
Figure BDA0003164602370000024
representing the covariance and conjugate of the pseudo-covariance, respectively.
The expression of the pseudo covariance matrix is:
Figure BDA0003164602370000025
step S103, sigma point calculation:
Figure BDA0003164602370000026
Figure BDA0003164602370000027
Figure BDA0003164602370000028
Figure BDA0003164602370000029
wherein the content of the first and second substances,
Figure BDA00031646023700000210
is a sigma point, especially
Figure BDA00031646023700000211
Is a mean point, where p ═ L is the dimension of x,
Figure BDA00031646023700000212
Is the mean value of x and lambda is a scale parameter.
Further, the step S2 specifically includes:
step S201, the weight of the tasteless algorithm is calculated as:
Figure BDA00031646023700000213
Figure BDA00031646023700000214
wherein, Wi (m)And Wi (c)Weight coefficients respectively representing mean value and covariance calculation, L is input dimension, alpha and beta respectively represent scale parameters, alpha is used for controlling the distance between a sigma point and the mean value, and the value of alpha is generally smaller and is about 10-3β includes a part of the input distribution information, and when the distribution is gaussian, the value is generally 2.
In step S202, the statistic of the output y is calculated as:
Figure BDA00031646023700000215
Figure BDA00031646023700000216
Figure BDA0003164602370000031
Figure BDA0003164602370000032
wherein the content of the first and second substances,
Figure BDA0003164602370000033
respectively input and output, in particular
Figure BDA0003164602370000034
Is a mean value point,
Figure BDA0003164602370000035
Is the mean, R 'of the output'yAnd P'yCovariance and pseudo-covariance of y, respectively.
Further, in the step S3, the optimal beta parameter includes a parameter β that minimizes a pseudo covariance errorPAnd a parameter beta that minimizes the covariance errorR；
The parameter betaPThe expression of (a) is:
Figure BDA0003164602370000036
wherein the content of the first and second substances,H xxin order to be a Hessian matrix,
Figure BDA0003164602370000037
for the i-th derivative to the non-linear function,
Figure BDA0003164602370000038
the parameter betaRThe expression of (a) is:
Figure BDA0003164602370000039
the invention has the beneficial effects that:
for an analytic nonlinear system, the algorithm used by the method is basically consistent with the result of actual output after the Monte Carlo algorithm, the mean value, the variance and the covariance of the output can be accurately calculated, and the method has better accuracy compared with the traditional augmentation algorithm.
Drawings
FIG. 1 is a schematic diagram illustrating the effect of changing the non-circular coefficient on the systematic error in example 2 when β is 2.
Fig. 2 shows an example 2 in which f ═ x is analytically converted when a complex circle signal (non-circularity ρ ═ 0) is obtained2The results of the traditional augmented UT algorithm and the algorithm are shown in the figure.
Fig. 3 shows the case where the complex non-circular signal (non-circularity ρ ═ 0.6) is analytically transformed into f ═ x in example 22The results of the traditional augmented UT algorithm and the algorithm are shown in the figure. .
FIG. 4 is a flow chart of an algorithm for a tasteless algorithm for a non-analytic complex nonlinear system in example 2.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, 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 some, but not all, embodiments of the present invention. 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.
Example 1
The embodiment provides a tasteless algorithm for an unresolved complex nonlinear system, comprising the following steps:
step 101, forming an amplification form from the complex signal and its conjugate signalx＝[xT，xH]T
102, calculating second-order statistics, wherein the covariance is as follows:
Figure BDA0003164602370000041
the pseudo covariance is:
Figure BDA0003164602370000042
the augmented covariance matrix and the augmented pseudo covariance both include the covariance R of the original xxAnd pseudo covariance PxAnd (4) information.
Step 103, sigma point calculation:
Figure BDA0003164602370000043
Figure BDA0003164602370000044
Figure BDA0003164602370000045
Figure BDA0003164602370000046
can be abbreviated as:
Figure BDA0003164602370000047
wherein sigmaiThe ith column root mean square matrix representing the augmented covariance or augmented pseudocovariance.
Wherein the system function of the non-analytic complex system is related to x and its conjugate, and for the non-analytic complex nonlinear system, the Taylor expansion is:
Figure BDA0003164602370000048
Figure BDA0003164602370000051
wherein Hxx、
Figure BDA0003164602370000052
H’xx、HxxAll are Hessian matrices, defined as:
Figure BDA0003164602370000053
the mean value is:
Figure BDA0003164602370000054
the sigma point Taylor expansion of the UT algorithm is:
Figure BDA0003164602370000055
Figure BDA0003164602370000056
in step 201, the weight of the Unscented Transform (UT) is calculated as:
Figure BDA0003164602370000057
Figure BDA0003164602370000058
step 202, substituting weights, the statistics of the output y can be calculated as:
Figure BDA0003164602370000059
Figure BDA00031646023700000510
Figure BDA00031646023700000511
Figure BDA00031646023700000512
the approximate mean of the UT algorithm is calculated as:
Figure BDA00031646023700000513
odd positive and negative signs cancel each other
Figure BDA0003164602370000061
Wherein
Figure BDA0003164602370000062
So that there are
Figure BDA0003164602370000063
Note that in the actual operation of the UT the function is still considered as an analytic function, and the derivation for the complex number is introduced into the formula only by the augmented vector, so that in averaging, the pseudo-covariance is still brought in
Figure BDA0003164602370000064
Sigma point of (d):
Figure BDA0003164602370000065
to be calculated, not as used in some documents
Figure BDA0003164602370000066
The use of the augmented input thus enables the UT algorithm to still guarantee the first-second order accuracy for non-analytic functions.
Step 301, the UT algorithm pseudo covariance is calculated as:
Figure BDA0003164602370000067
wherein due to
Figure BDA0003164602370000071
And is
Figure BDA0003164602370000072
The fourth order can be written as:
Figure BDA0003164602370000073
for the actual covariance, the mean of the taylor-expanded odd terms at the mean is all 0, while the odd terms and higher orders are controlled in UT by a coefficient α close to 0, while the actual result of the second order is consistent with the result of the UT calculation.
And the error term of the fourth order is
Figure BDA0003164602370000074
When in useH xxIf the matrix is diagonal, then a minimum error can be obtained with a beta of 2, however, when the matrix is diagonalH xxWhen the diagonal matrix is off, i.e. f is not resolved, simply defining β as 2, the error of the previous term cannot be removed.
E.g. when f ═ x2＝xx*，
Figure BDA0003164602370000075
E[δx T H xxδxδx T H xxδx]＝4E[δx*δxδx*δx]＝4(2R2+PP*)
Figure BDA0003164602370000076
When β is 1+ ρ*And UT is consistent with the calculation result of the Monte Carlo method, wherein rho is a non-circular coefficient,
Figure BDA0003164602370000077
step 302, for UT covariance, should be brought in
Figure BDA0003164602370000078
Point calculation covariance, calculated as the optimum value of β
Figure BDA0003164602370000081
E.g. when f ═ x2＝xx*Is provided with
Figure BDA0003164602370000082
When β is ρ*+ρ+ρ*UT is consistent with the calculation result of Monte Carlo method at-1.
Example 2
Referring to fig. 1-4, the present embodiment also provides a non-odor algorithm for non-analytic complex nonlinear systems, which can be used for, but not limited to, mean and second order statistic analysis after complex nonlinear analytic transformation, and also for real and analytic function transformation. For the non-analytic function, the embodiment takes f ═ x tint2＝xx*For example, the influence of input non-roundness on system errors is analyzed, and the calculation of the algorithm after the parameters are optimized is shownThe accuracy of the calculation. For the analytic function, for convenience of comparison, the present example uses a complex number to undergo analytic transformation of f ═ x2For example, the superiority of the algorithm of the present invention is shown by the error result.
As shown in fig. 4, the algorithm of the present invention includes the following steps:
step S1, calculating second-order statistics of the input signal in the amplification form, namely amplification variance and covariance, inputting the second-order statistics into a system, and calculating sigma points;
step S2, the tasteless algorithm can obtain first-order and second-order statistics of the corresponding output points of the nonlinear system through the calculation of a small number of sigma characteristic points;
and step S3, adjusting beta parameters by researching the relation between the error and the non-roundness, and reducing the error of the algorithm.
The calculated sigma point satisfies the following equation:
Figure BDA0003164602370000083
Figure BDA0003164602370000084
Figure BDA0003164602370000085
Figure BDA0003164602370000086
the weight of the Unscented Transform (UT) is calculated as:
Figure BDA0003164602370000087
Figure BDA0003164602370000088
the statistic of output y can be calculated as:
Figure BDA0003164602370000091
Figure BDA0003164602370000092
Figure BDA0003164602370000093
Figure BDA0003164602370000094
parameter beta that minimizes the pseudo-covariance errorPSatisfy the requirement of
Figure BDA0003164602370000095
Parameter beta that minimizes covariance errorRSatisfy the requirement of
Figure BDA0003164602370000096
Specifically, there are different optimal β parameters for different nonlinear functions, but all relate to the non-circularity of the signal itself. As shown in fig. 1, the non-circularity of the input affects the error of the system, which is why the optimal parameters found in many papers are not constant. As shown in Table 1, the effect of non-circularity is balanced, and the error of the system is greatly reduced after parameters changing along with the input non-circularity are adopted.
TABLE 1
Figure BDA0003164602370000097
For the analytic nonlinear system, as shown in fig. 2 and fig. 3, the result of the algorithm used by the invention is basically consistent with the result of the actual output after the Monte Carlo algorithm, and the mean value, the variance and the covariance of the output can be accurately calculated. Has better accuracy than the prior augmentation algorithm.
The invention is not described in detail, but is well known to those skilled in the art.
The foregoing detailed description of the preferred embodiments of the invention has been presented. It should be understood that numerous modifications and variations could be devised by those skilled in the art in light of the present teachings without departing from the inventive concepts. Therefore, the technical solutions available to those skilled in the art through logic analysis, reasoning and limited experiments based on the prior art according to the concept of the present invention should be within the scope of protection defined by the claims.

Claims (4)

1. A tasteless algorithm for use in a non-analytic complex nonlinear system, comprising the steps of:
step S1, calculating second-order statistics of the input signal in the amplification form, inputting the second-order statistics into a nonlinear system, and calculating sigma points;
step S2, according to the sigma point calculated in the step S1, a tasteless algorithm is adopted to obtain first-order and second-order statistics of the output point corresponding to the nonlinear system;
and step S3, when the input signal is non-circular, calculating the optimal beta parameter by considering the relation between the error and the non-circular degree, and reducing the system error.
2. The tasteless algorithm for an unresolved complex nonlinear system according to claim 1, wherein the step S1 specifically comprises:
step S101, the input signal is a complex signal, and the complex signal and the conjugate signal thereof are combined into an amplification form:x＝[xT,xH]Twherein x is a complex signal, xTExpressed as transposes of complex signals, xHExpressed as a conjugate transpose of the complex signal;
step S102, calculating second-order statistics according to the input signal, wherein the second-order statistics comprise a covariance matrix and a pseudo covariance matrix, and the expression of the covariance matrix is as follows:
Figure FDA0003164602360000011
wherein the content of the first and second substances,
Figure FDA0003164602360000012
expressed as a covariance matrix of the input signal, RxExpressed as the covariance, P, of the input signalxExpressed as the pseudo-covariance of the input signal, i.e. as the conjugate sign
Figure FDA0003164602360000013
And
Figure FDA0003164602360000014
representing the covariance and conjugate of the pseudo-covariance, respectively.
The expression of the pseudo covariance matrix is:
Figure FDA0003164602360000015
step S103, sigma point calculation:
Figure FDA0003164602360000016
Figure FDA0003164602360000017
Figure FDA0003164602360000018
Figure FDA0003164602360000019
wherein the content of the first and second substances,
Figure FDA00031646023600000110
in order to be the sigma point of the signal,
Figure FDA00031646023600000111
is a mean point, where p ═ L is the dimension of x,
Figure FDA00031646023600000112
Is the mean value of x and lambda is a scale parameter.
3. The tasteless algorithm for an unresolved complex nonlinear system according to claim 2, wherein the step S2 specifically comprises:
step S201, the weight of the tasteless algorithm is calculated as:
Figure FDA0003164602360000021
Figure FDA0003164602360000022
wherein, Wi (m)And Wi (c)Weight coefficients respectively representing mean and covariance calculations, L being an input dimension, α and β respectively representing scale parameters, α being 10-3Beta contains part of input distribution information, and when the distribution is Gaussian distribution, the value is 2;
in step S202, the statistic of the output y is calculated as:
Figure FDA0003164602360000023
Figure FDA0003164602360000024
Figure FDA0003164602360000025
Figure FDA0003164602360000026
wherein the content of the first and second substances,
Figure FDA0003164602360000027
respectively are the statistic points of input and output,
Figure FDA0003164602360000028
is a mean value point,
Figure FDA0003164602360000029
Is the mean, R 'of the output'yAnd P'yCovariance and pseudo-covariance of y, respectively.
4. The tasteless algorithm for an unresolved complex nonlinear system of claim 3, wherein in step S3, the optimal beta parameter comprises a parameter β that minimizes a pseudo-covariance errorPAnd a parameter beta that minimizes the covariance errorR；
The parameter betaPThe expression of (a) is:
Figure FDA00031646023600000210
wherein the content of the first and second substances,H xxin order to be a Hessian matrix,
Figure FDA00031646023600000211
for the i-th derivative to the non-linear function,
Figure FDA00031646023600000212
the parameter betaRThe expression of (a) is:
Figure FDA0003164602360000031
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