CN103296995B - Any dimension high-order (>=4 rank) tasteless conversion and Unscented Kalman Filter method - Google Patents

Abstract

The tasteless conversion of the open a kind of any dimension high-order of the present invention (>=4 rank) and Unscented Kalman Filter method, it is possible to resolve target following, automatically control, navigate, guide and high precision nonlinear filtering estimation problem in the engineer applied field such as artificial intelligence.The key improvements of the present invention is to provide design and is applicable to any dimension and the σ point set that can accurately mate random quantity High Order Moment, and method is accurately to choose three classes 1+2N+2 altogether^NIndividual σ point (N is dimension)；Next to that provide the method carrying out the tasteless conversion of any dimension high-order based on high-order σ point set and statistically linear weighting method；And estimate that son carries out any dimension high-order Unscented Kalman Filter, recursion output system mode average and the method for estimation of covariance based on the tasteless conversion of any dimension high-order and linear optimal.The application present invention, can realize the accurate coupling below the square of any dimension gaussian random amount five rank, and can be to the coupling (can realize the accurate coupling of below nine rank squares under one-dimensional case) of six rank square non-crossing items.

Description

Any dimension high-order (>=4 rank) tasteless conversion and Unscented Kalman Filter method

Technical field

The technical field that the present invention is suitable for relate to target following, automatically control, navigate, guide, digital communication, economic system The engineer applied fields such as meter, probability inference, artificial intelligence, information fusion and fault detect, specifically by these fields For solving tasteless conversion (Uscented Transformation, UT) and the Unscented Kalman Filter that nonlinear filtering is estimated (Uscented Kalman Filtering, UKF) method is extended to any dimension high-order (>=4 rank) situation.

Background technology

In many electronic engineering fields and the engineer applied such as economic statistics and probability inference field, it is frequently run onto this The nonlinear estimation problem of sample: by statistical properties such as the average of certain random quantity known and covariances, go to estimate that it is through non-linear Random quantity statistical property after conversion；Additionally based on this non-linear estimations, to go reality according to the real-time update of observed quantity Time recurrence estimation corresponding state statistics of variables characteristic, here it is Nonlinear Filtering Problem.The core right and wrong of Nonlinear Filtering Problem Linear Estimation, the key foundation of non-linear estimations is then in the engineering Accurate Expression method to random quantity statistical property.

It is true that the optimal solution of nonlinear filtering estimation problem does not the most resolve, and As time goes on will become In Infinite-dimensional；The drastically expansion of " dimension disaster " and operand and amount of storage makes this optimal solution can not realize physically, Engineering also there is no need realize this optimal solution, and only need to do certain and approximate, namely find the suboptimum of the problems referred to above Solve.

Traditional suboptimum solution is EKF (Extended Kalman Filtering, EKF), its core The heart is based on the simple differential linearity method to nonlinear model, i.e. based on Taylor series expansion first approximation.EKF Can only achieve the 0 rank precision to random quantity Estimation of Mean and 2 rank precision of covariance estimation, its deficiency is also embodied in be needed to calculate The Jacobian matrix of model is so that implement complex, and the situation to some non-differentiabilities, EKF is weathering； The precision estimated during and system noise non-gaussian relatively strong at model nonlinear seriously reduces, and is likely to result in sending out of wave filter Dissipate.

Occur in that some exempt from differential method the most in the world, be wherein typically based on tasteless conversion (UT) with tasteless Kalman filtering (UKF) method, its core is then to use tasteless conversion to estimate random quantity average after nonlinear mapping And covariance, its basic thought is to pass through the mapping of nonlinear model with one group of σ point (sigma-point) accurately selected Pass the statistical property of random quantity, then with weighted statistical linear regression (Weighted Statistical Linear Regression, WSLR) method estimate average and the covariance of random quantity.But, typical UT and UKF for two dimension with On random quantity its can only achieve 2 rank about the estimated accuracy of average and covariance.

Generally we are the estimated accuracy tasteless conversion just referred to as high-order tasteless conversion (High Order more than Fourth-order moment UT, HOUT).According to research find, some HOUT algorithms of existing reported in literature all also exist so, such problem even , there is cross term mismatch when multidimensional is applied and even cannot be suitable for because dimension limits, therefore can not be referred to as in carelessness or error For the real tasteless conversion of high-order.

Through analysis verification, the tasteless mapping algorithm of quadravalence that Julier and Uhlmann proposes jointly is not particularly suited for appointing in fact Meaning dimension, and may be only available for two dimension or three-dimensional random amount.Propose in addition with by Tenne and Singh design and entered by Xie Kai etc. The tasteless conversion of high-order (High OrderUT, HOUT) of one step development.For one-dimensional Gauss distribution, Tenne and Singh asks respectively Solution gives 5 σ points of application accurately coupling 8 rank squares, the accurate σ point set design result mating 12 rank squares of 7 σ points of application.Xie Kai Deng then, the σ point set method for designing of the tasteless conversion of high-order that Tenne and Singh proposes is made further development, be the most also Main is improved, next to that attempt by one-dimensional, it is extended to higher-dimension the method for solving of σ point set exactly.But it is this " high-order " tasteless conversion and filtering method are only applicable to one-dimensional random amount in fact, and the higher-dimension not being suitable for more than two dimension is random Amount, because it ignores cross term one-dimensional in the simple expanding course of higher-dimension.

It is true that there has been the σ point set of careful design, it both can will accurately mate the statistical property of former random quantity, simultaneously Also the random quantity statistical property after non-linear transmission can accurately be estimated according to its σ point set after nonlinear mapping.Therefore, in order to Research is given and is applicable to any dimension and can accurately mate all of non-crossing item of below certain rank square and cross term square, and high-order is tasteless The careful design of the σ point set of conversion is still its key core problem place.

Summary of the invention

The invention provides a kind of any dimension high-order (>=4 rank) tasteless conversion and Unscented Kalman Filter method, mainly Under general tasteless conversion with Unscented Kalman Filter method frame, it is extended to any dimension and can be provided at random The accurate coupling of amount High Order Moment (>=4 rank), its core is to design to give go for any dimension and can accurately mate high-order The σ point set of square (>=4 rank), this σ point set is by the basis of tasteless conversion.

The present invention concretely comprises the following steps:

(1) dynamic system model of Nonlinear Filtering Problem is set up, including state equation and observational equation, wherein system shape The dimension of state random quantity x is N_x；

(2) the σ point set corresponding with said system state random quantity x in tasteless conversion is made to beWherein σ The sum of point is p+1, and i-th σ point is x⁽ⁱ⁾, corresponding weights are w⁽ⁱ⁾, lower same.Design provides dimension equal to N_xStandard normal (its average is null vector to distribution random quantityCovariance matrix is unit battle array) σ point setShould The method for designing of σ point set should apply to any dimension and can realize the accurate coupling to random quantity High Order Moment (>=4 rank)；

(3) initiation parameter of unbalanced input wave filter, or about previous moment k-1 system mode random quantity AverageAnd covarianceAnd the system quantities measured value y of current time k_k；

(4) based onWithThe σ point set Γ that conversion is obtained by step (2)_I, generate be applicable to any dimension and can essence Really mate the σ point set on High Order Moment (>=4 rank)

(5) the application tasteless conversion of any dimension high-order respectivelyEstimate draw about 5 statistics of current time k one-step prediction: the average of system mode random quantityAnd covarianceSystem measurements with The average of machine amountAnd covarianceAnd the cross covariance of system mode and measurement random quantity

(6) in application linear minimum mean-squared error (Linear Minimum Mean Square Error, LMMSE) meaning Linear optimal estimate that son estimates the output average of system mode random quantity about current time kAnd covariance

(7) if there being new measurement to arrive, returning step (3), otherwise terminating.

Described step (2) relates to be applicable to any dimension and can accurately mate the σ point set Γ of High Order Moment (>=4 rank)_I Generation problem, its basic method for designing is:

Random quantity is distributed the most very much for N-dimensional standard, its σ point set Γ_IHave three classes 1+p=1+2N+2 altogether^NIndividual σ point, Qi Zhong Class σ point one, is directly taken as initial point, and weights are w₀；Equations of The Second Kind σ point 2N altogether_xIndividual, it is positioned on the coordinate axes of each dimension, distance initial point It is s₁, weights are as w₁；3rd class σ point is altogetherIndividual, it is positioned on the bisector of All Quardrants, its coordinate figure in each dimension For+s₂Or-s₂, and take time institute's likely value, and weights are w₂.Above-mentioned parameter w₀、w₁、w₂、s₁s₂Determination method as follows:

When N=1 (the most one-dimensional), the Simultaneous Equations of demand solution is as follows:

w₀+2Nw₁+2^Nw₂-1=0 (1)

$2w_1{s}_1^2+2^N{w}_2{s}_2^2-1=0---\left(2\right)$

$2w_1{s}_1^4+2^N{w}_2{s}_2^4-3=0---\left(3\right)$

$2w_1{s}_1^6+2^N{w}_2{s}_2^6-15=0---\left(4\right)$

$2w_1{s}_1^8+2^N{w}_2{s}_2^8-105=0---\left(5\right)$

When N >=2 (i.e. multidimensional), the Simultaneous Equations of demand solution is as follows:

w₀+2Nw₁+2^Nw₂-1=0 (6)

$2w_1{s}_1^2+2^N{w}_2{s}_2^2-1=0---\left(7\right)$

$2w_1{s}_1^4+2^N{w}_2{s}_2^4-3=0---\left(8\right)$

$2^N{w}_2{s}_2^4-1=0---\left(9\right)$

$2w_1{s}_1^6+2^N{w}_2{s}_2^6-15=0---\left(10\right)$

Described step (4) relates to be applicable to any dimension and can accurately mate the random quantity on High Order Moment (>=4 rank) x_k-1|k-1σ point setGeneration problem, its basic method for designing is:

The σ point set Γ of random quantity it is distributed the most very much based on the N-dimensional standard of design in step (2)_I, σ point here can be derived CollectionThe wherein all weights correspondent equals of the two, and the available following transformation relation of all σ points represented with matrix obtains:

${\mathrm{\Γ}}_{x_{k-1|k-1}}={\hat{x}}_{k-1|k-1}+\sqrt{P_{x_{k-1|k-1}}}{\mathrm{\Γ}}_I---\left(11\right)$

Described step (5) relates to for estimating any of random quantity statistical property method after nonlinear transformation The dimension tasteless conversion of high-orderIts basic process step is for can be summarized as: about the σ of x The generation of point set → probabilistic nonlinear transformation and transmission → about the reckoning of statistical property of y.It is exactly specifically:

1) the σ point set obtained based on step (4)Passed through nonlinear model y=g (x) (representing the state equation in step (1) or observational equation the most respectively) converts the uncertainty of transmission random quantity, obtains accordingly New σ point setWherein y⁽ⁱ⁾=g (x⁽ⁱ⁾), and weight w⁽ⁱ⁾Constant.

2) according to σ point set Γ_yAverage and the covariance of y after nonlinear transformation is calculated respectively by following relational expression, or by Γ_x With Γ_yThe cross covariance of estimation x Yu y:

$\hat{y}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}{y}^{\left(i\right)}---\left(12\right)$

$P_y=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}(y^{\left(i\right)}-\hat{y}){(y^{\left(i\right)}-\hat{y})}^T---\left(13\right)$

$P_{xy}=\underset{i=1}{\overset{N}{\Σ}}{w}^{\left(i\right)}(x^{\left(i\right)}-\stackrel{\‾}{x}){(y^{\left(i\right)}-\hat{y})}^T---\left(14\right)$

If here replacing y=g (x) i.e. to can get the average of system mode random quantity with state equationAnd covarianceIf replacing y=g (x) i.e. to can get the average of system measurements random quantity with observational equationAnd covarianceAnd The cross covariance of the two

Described step (6) relates to the average for estimating to export the system mode random quantity about current time kAnd covarianceLinear optimal estimate son, available formulae express is as follows:

${\tilde{y}}_k=y_k-{\hat{y}}_{k|k-1}---\left(15\right)$

$K_k=P_{x_{k|k-1}{y}_{k|k-1}}{P}_{y_{k|k-1}}^{-1}---\left(16\right)$

${\hat{x}}_{k|k}={\hat{x}}_{k|k-1}+K_k{\tilde{y}}_k---\left(17\right)$

$P_{x_{k|k}}=P_{x_{k|k-1}}-K_k{P}_{y_{k|k-1}}{K}_k^T---\left(18\right)$

Compared with the conventional method, its key technology is improved by designing and gives the height going for any dimension the present invention Rank (>=4 rank) tasteless alternative approach, and give any dimension high-order Unscented Kalman Filter method based on this.The present invention's Core is design to give accurately to be applicable to the σ point set of any dimension high-order (>=4 rank) tasteless conversion.Application is the present invention provide Any dimension high-order (>=4 rank) tasteless alternative approach, for general any dimension gaussian random amount through the statistics of non-linear transmission Characteristic estimating, its precision should be able to reach the accurate coupling below the square of random quantity five rank, and can realize six rank square non-crossing items Coupling (then can realize the accurate coupling of below nine rank squares for one-dimensional random amount situation).But, by convention, i.e. high-order without Taste conversion only calculates all non-crossing items of below the exponent number square represented with even number that method is capable of and cross term when name Mate completely.Therefore, any dimension high-order (>=4 rank) the tasteless conversion that the present invention provides can use UT4 for application more than two dimension Represent, then can use UT8 to represent for one-dimensional application；Corresponding Unscented Kalman Filter represents with UKF4 and UKF8 the most respectively.

It should be noted that for one-dimensional random amount, the UT8 of our research design designs with by Tenne and Singh The high-order UT method proposed and developed further by Xie Kai etc. is consistent on end product.For two-dimensional random amount, The UT4 of our research design is then consistent with the quadravalence UT algorithm jointly proposed by Julier and Uhlmann on end product , but the high-order UT method at this moment being proposed by Tenne and Singh design and being developed further by Xie Kai etc. is then because can not mate All of cross term square and inapplicable.For random quantity more than three-dimensional, then only have the UT4 of our research design the most permissible Accomplish above-mentioned matching precision, say, that be real high-order UT.

Accompanying drawing explanation

Fig. 1 is any dimension high-order (>=4 rank) Unscented Kalman Filter method flow diagram；

Fig. 2 is that two-dimensional particles is rebuffed the bounce-back schematic diagram (particle initial position probability is oval and UT (×) the σ point with UT4 (*) Distribution)；

Fig. 3 is that the result of UT and UT4 and MC method compares figure；

Fig. 4 is the running orbit of Ballistic Target, speed and acceleration；

Fig. 5 is the RMSE comparison figure that EKF, UKF and UKF4 estimate about each component of target state；

Fig. 6 is target root-mean-square position and the comparison figure of velocity error of EKF, UKF and UKF4 estimation.

Detailed description of the invention

As it is shown in figure 1, the specific embodiment party of any dimension high-order (>=4 rank) the Unscented Kalman Filter method of present invention offer Method is as follows:

(1) set up the dynamic system model of Nonlinear Filtering Problem, including state equation and observational equation, rise for generality See, can use the most discrete or nonlinear dynamic system model representation of discretization respectively:

x_k+1=f (x_k, v_k) (19)

y_k=h (x_k, w_k) (20)

Its Chinese style (19) is state equation, and (20) are observational equation, and x_kIt is that state-noise, y are for the state vector in k moment, v Observation vector, w are observation noise, and respective dimension is respectively N_x、N_v、N_yAnd N_w, f () is state model, h () For observation model, model is it is known that at least one of which is nonlinear.For simplicity, it is assumed that noise v and w is zero The white noise of average, and have

$\∀i,j\left\{\begin{array}{c}E\[v_i{v}_j^T\]={\mathrm{\δ}}_{ij}{P}_{v_i}\\ E\[w_i{w}_j^T\]={\mathrm{\δ}}_{ij}{P}_{w_i}\\ E\[v_i{w}_j^T\]=0\\ E\[x_0{v}_i^T\]=0\\ E\[x_0{w}_i^T\]=0\end{array}\right.---\left(21\right)$

(2) design is given and is applicable to any dimension N and (is taken as the dimension N of system mode random quantity x here_x) and can realize To standard normal distribution random quantity, (average is null vectorCovariance matrix is unit battle array) accurate of High Order Moment (>=4 rank) The σ point set joinedConcrete grammar is according to following three types minute by σ point therein Other places are managed:

First kind σ point one, for initial point beWeights are w₀；

Equations of The Second Kind σ point 2N altogether_xIndividual, it is positioned on the coordinate axes of each dimension, is all s apart from initial point₁, i.e.Weights are

3rd class σ point is altogetherBeing positioned on the bisector of All Quardrants, its coordinate figure in each dimension is+s₂Or-s₂, And take time institute's likely value, and weights are

Above-mentioned parameter w₀、w₁、w₂、s₁、s₂Determination method as follows:

Work as N_xIt is that the solution of following Simultaneous Equations is time=1 (the most one-dimensional):

$w_0+2N_x{w}_1+2^{N_x}{w}_2-1=0---\left(22\right)$

$2w_1{s}_1^2+2^{N_x}{w}_2{s}_2^2-1=0---\left(23\right)$

$2w_1{s}_1^4+2^{N_x}{w}_2{s}_2^4-3=0---\left(24\right)$

$2w_1{s}_1^6+2^{N_x}{w}_2{s}_2^6-15=0---\left(25\right)$

$2w_1{s}_1^8+2^{N_x}{w}_2{s}_2^8-105=0---\left(26\right)$

Work as N_xIt is the solution of following Simultaneous Equations time >=2 (i.e. multidimensional):

$w_0+2N_x{w}_1+2^{N_x}{w}_2-1=0---\left(27\right)$

$2w_1{s}_1^2+2^{N_x}{w}_2{s}_2^2-1=0---\left(28\right)$

$2w_1{s}_1^4+2^{N_x}{w}_2{s}_2^4-3=0---\left(29\right)$

$2^{N_x}{w}_2{s}_2^4-1=0---\left(30\right)$

$2w_1{s}_1^6+2^{N_x}{w}_2{s}_2^6-15=0---\left(31\right)$

(3) initiation parameter of unbalanced input wave filter, or about previous moment k-1 system mode random quantity AverageAnd covarianceAnd the system quantities measured value y of current time k_k。

(4) based onWithThe σ point set Γ that conversion is obtained by step (2)_I, generate be applicable to any dimension and can essence Really mate the σ point set on High Order Moment (>=4 rank)Its method particularly includes:

Keep σ point set Γ_IWithIn all weights correspondent equals, all σ points represented with matrix then meet such as Down conversion relation:

${\mathrm{\Γ}}_{x_{k-1|k-1}}={\hat{x}}_{k-1|k-1}+\sqrt{P_{x_{k-1|k-1}}}{\mathrm{\Γ}}_I---\left(32\right)$

(5) the application tasteless conversion of any dimension high-order respectivelyEstimate draw about 5 statistics of current time k one-step prediction: the average of system mode random quantityAnd covarianceSystem measurements with The average of machine amountAnd covarianceAnd the cross covariance of system mode and measurement random quantityConcrete process Step is as follows:

1) the σ point set obtained based on step (4)Passed sequentially through state equation (19) and observational equation (20) Two new σ point sets can be respectively obtainedWithThe most respective weight w⁽ⁱ⁾Constant, and

$x_{k|k-1}^{\left(i\right)}=f(x_{k-1|k-1}^{\left(i\right)},v_k)---\left(33\right)$

$y_{k|k-1}^{\left(i\right)}=h(x_{k|k-1}^{\left(i\right)},w_k)---\left(34\right)$

2) 5 statistics about current time k one-step prediction are drawn by following various estimation respectively:

${\hat{x}}_{k|k-1}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}{x}_{k|k-1}^{\left(i\right)}---\left(35\right)$

$P_{x_{k|k-1}}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}(x_{k|k-1}^{\left(i\right)}-{\hat{x}}_{k|k-1}){(x_{k|k-1}^{\left(i\right)}-{\hat{x}}_{k|k-1})}^T---\left(36\right)$

${\hat{y}}_{k|k-1}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}{y}_{k|k-1}^{\left(i\right)}---\left(37\right)$

$P_{y_{k|k-1}}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}(y_{k|k-1}^{\left(i\right)}-{\hat{y}}_{k|k-1}){(y_{k|k-1}^{\left(i\right)}-{\hat{y}}_{k|k-1})}^T---\left(38\right)$

$P_{x_{k|k-1}{y}_{k|k-1}}=\underset{i=0}{\overset{p}{\Σ}}{w}^{\left(i\right)}(x_{k|k-1}^{\left(i\right)}-{\hat{x}}_{k|k-1}){(y_{k|k-1}^{\left(i\right)}-{\hat{y}}_{k|k-1})}^T---\left(39\right)$

(6) application linear optimal estimates that son estimates the output average about the system mode random quantity of current time kWith CovarianceIt is formulated as follows:

${\tilde{y}}_k=y_k-{\hat{y}}_{k|k~1}---\left(40\right)$

$K_k=P_{x_{k|k-1}{y}_{k|k-1}}{P}_{y_{k|k-1}}^{-1}---\left(41\right)$

${\hat{x}}_{k|k}={\hat{x}}_{k|k-1}+K_k{\tilde{y}}_k---\left(42\right)$

$P_{x_{k|k}}=P_{x_{k|k-1}}-K_k{P}_{y_{k|k-1}}{K}_k^T---\left(43\right)$

(7) if there being new measurement to arrive, returning step (3), otherwise terminating.

The key improvements of the present invention is that in step (2) that design provides and is applicable to any dimension N and to standard just can realize The σ point set of state distribution random quantity High Order Moment (>=4 rank) accurately couplingEspecially There is provided the design distribution patterns of a kind of σ point；This is the follow-up tasteless conversion of any dimension high-order and the core of Unscented Kalman Filter Heart basis.

Application example 1: any dimension high-order provided by the present invention (>=4 rank) tasteless conversion is applied to solving as follows at random Amount statistical property after nonlinear transformation.Involved nonlinear transformation is:

Wherein x=[x₁ x₂]^T, y=[y] (44)

Wherein component x₁And x₂It is respectively for the one-dimensional random quantity being distributed the most very much, average and varianceWith Require now to estimate the average about random quantity yAnd varianceFirst solve the two exact value in theory.Assume random quantity The true value of each component of x can be written as

$\begin{array}{cc}{x}_1={\stackrel{\‾}{x}}_1+{\mathrm{\δ}}_{x_1}& x_2={\stackrel{\‾}{x}}_2+{\mathrm{\δ}}_{x_2}\end{array}---\left(45\right)$

WhereinFor the one-dimensional random quantity being distributed the most very much, average is 0, and variance is respectivelySubstitution formula (44) true value i.e. obtaining y is

$\begin{array}{c}y={(x_1^2+x_2^2)}^2={x_1}^4+4{x_1}^3{\mathrm{\δ}}_{x_1}+2{x_1}^2{x_2}^2+4{x_1}^2{x}_2{\mathrm{\δ}}_{x_2}+6{x_1}^2{\mathrm{\δ}}_{x_1}^2+2{x_1}^2{\mathrm{\δ}}_{x_2}^2\\ +4x_1{x_2}^2{\mathrm{\δ}}_{x_1}+8x_1{x}_2{\mathrm{\δ}}_{x_1}{\mathrm{\δ}}_{x_2}+4x_1{\mathrm{\δ}}_{x_1}^3+4x_1{\mathrm{\δ}}_{x_1}{\mathrm{\δ}}_{x_2}^2+{x_2}^4+4{x_2}^3{\mathrm{\δ}}_{x_2}+2{x_2}^2{\mathrm{\δ}}_{x_1}^2\\ +6{x_2}^2{\mathrm{\δ}}_{x_2}^2+4x_2{\mathrm{\δ}}_{x_1}^2{\mathrm{\δ}}_{x_2}+4x_2{\mathrm{\δ}}_{x_2}^3+{\mathrm{\δ}}_{x_1}^4+2{\mathrm{\δ}}_{x_1}^2{\mathrm{\δ}}_{x_2}^2+{\mathrm{\δ}}_{x_2}^4\end{array}---\left(46\right)$

Ask its average i.e. can obtain the average of random quantity y of requirementTrue value as follows

$\begin{array}{c}{\stackrel{\‾}{y}}_t={{\stackrel{\‾}{x}}_1}^4+2{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2+6{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^2+2{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_2}^2+{{\stackrel{\‾}{x}}_2}^4\\ +2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2+6{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^2+3{\mathrm{\σ}}_{x_1}^4+2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+3{\mathrm{\σ}}_{x_2}^4\end{array}---\left(47\right)$

Seek the variance of its random quantity y the most againTrue value as follows

$\begin{array}{c}{\left({\mathrm{\σ}}_y^2\right)}_t=16{{\stackrel{\‾}{x}}_1}^6{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2+16{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^2+168{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^4+32{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2\\ +8{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_2}^4+16{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^2+144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4+128{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+\\ 144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^4+384{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^6+144{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2+64{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+48{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_2}^6\\ +16{{\stackrel{\‾}{x}}_2}^6{\mathrm{\σ}}_{x_2}^2+8{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^4+32{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+168{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^4+48{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^6+64{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2\\ +144{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+384{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^6+96{\mathrm{\σ}}_{x_1}^8+48{\mathrm{\σ}}_{x_1}^6{\mathrm{\σ}}_{x_2}^2+32{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^4+48{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^6+96{\mathrm{\σ}}_{x_2}^8\end{array}---\left(48\right)$

Any dimension high-order (>=4 rank) the tasteless conversion UT4 present invention provided below is common with by Julier and Uhlmann Method of estimation and the result of the conventional tasteless conversion UT proposed contrast.

First it is conventional tasteless conversion UT.The σ point set (totally 5 σ points and corresponding weight value) of its design is as follows:

$\begin{array}{c}{\mathrm{\Γ}}_x^{UT}={\{x^{\left(i\right)},w^{\left(i\right)}\}}_{i=0}^4=\{\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2\end{array}\right],\frac{1}{3}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1-\sqrt{3}{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2\end{array}\right],\frac{1}{6}\right\},\\ \left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2-\sqrt{3}{\mathrm{\σ}}_{x_2}\end{array}\right],\frac{1}{6}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1+\sqrt{3}{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2\end{array}\right],\frac{1}{6}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2+\sqrt{3}{\mathrm{\σ}}_{x_2}\end{array}\right],\frac{1}{6}\right\}\}\end{array}---\left(49\right)$

It is about the average of random quantity yAnd varianceEstimated result respectively as follows:

$\begin{array}{c}{\stackrel{\‾}{y}}_{UT}=\underset{i=0}{\overset{4}{\Σ}}{w}^{\left(i\right)}{y}^{\left(i\right)}=\\ {x_1}^4+2{x_1}^2{x_2}^2+6{x_1}^2{\mathrm{\σ}}_{x_1}^2+2{x_1}^2{\mathrm{\σ}}_{x_2}^2+{x_2}^4+2{x_2}^2{\mathrm{\σ}}_{x_1}^2+6{x_2}^2{\mathrm{\σ}}_{x_2}^2+3{\mathrm{\σ}}_{x_1}^4+3{\mathrm{\σ}}_{x_2}^4\end{array}---\left(50\right)$

$\begin{array}{c}{\left({\mathrm{\σ}}_y^2\right)}_{UT}=\underset{i=0}{\overset{2}{\Σ}}{w}^{\left(i\right)}{(y^{\left(i\right)}-\stackrel{\‾}{y})}^2=16{{\stackrel{\‾}{x}}_1}^6{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2+16{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^2\\ +168{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^4-24{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+8{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_2}^4+16{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^2\\ +144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4-80{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^4+216{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^6-12{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2\\ -36{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+24{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_2}^6+16{{\stackrel{\‾}{x}}_2}^6{\mathrm{\σ}}_{x_2}^2+8{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^4-24{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+168{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^4\\ +24{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^6-36{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2-12{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+216{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^6+18{\mathrm{\σ}}_{x_1}^8-18{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^4+18{\mathrm{\σ}}_{x_2}^8\end{array}---\left(51\right)$

The UT4 that the application present invention provides now.Equally, the σ point set (totally 9 σ points and corresponding weight value) of its design is as follows:

$\begin{array}{c}{\mathrm{\Γ}}_x^{UT4}={\{x^{\left(i\right)},w^{\left(i\right)}\}}_{i=0}^8=\{\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2\end{array}\right],w^{\left(0\right)}\right\},\\ \left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1+s_1{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2\end{array}\right],w^{\left(1\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2+s_1{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(2\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1-s_1{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2\end{array}\right],w^{\left(3\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1\\ {\stackrel{\‾}{x}}_2-s_1{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(4\right)}\right\}\\ \left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1-s_2{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2-s_2{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(5\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1-s_2{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2+s_2{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(6\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1+s_2{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2-s_2{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(7\right)}\right\},\left\{\left[\begin{array}{c}{\stackrel{\‾}{x}}_1+s_2{\mathrm{\σ}}_{x_1}\\ {\stackrel{\‾}{x}}_2+s_1{\mathrm{\σ}}_{x_2}\end{array}\right],w^{\left(8\right)}\right\}\}\end{array}---\left(52\right)$

w⁽⁰⁾=2*21^1/2/75+22/75；w⁽¹⁾=17/150-21^1/2/ 50=w⁽²⁾=w⁽³⁾=w⁽⁴⁾；

w⁽⁵⁾=21^1/2/ 75+19/300=w⁽⁶⁾=w⁽⁷⁾=w⁽⁸⁾

s₁=(2^1/2*(21^1/2+9)^1/2)/2；s₂=(6-21^1/2)^1/2

The UT4 that the application present invention provides estimates the average about random quantity y obtainedAnd varianceEstimated value as follows:

$\begin{array}{c}{\stackrel{\‾}{y}}_{UT4}=\underset{i=0}{\overset{8}{\Σ}}{w}^{\left(i\right)}{y}^{\left(i\right)}=\\ {{\stackrel{\‾}{x}}_1}^4+2{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2+6{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^2+2{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_2}^2+{{\stackrel{\‾}{x}}_2}^4+2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2+6{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^2+3{\mathrm{\σ}}_{x_1}^4+2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+3{\mathrm{\σ}}_{x_2}^4\end{array}---\left(53\right)$

$\begin{array}{c}{\left({\mathrm{\σ}}_y^2\right)}_{UT4}=\underset{i=0}{\overset{8}{\Σ}}{w}^{\left(i\right)}{(y^{\left(i\right)}-\stackrel{\‾}{y})}^2=\\ 16{{\stackrel{\‾}{x}}_1}^6{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2+16{{\stackrel{\‾}{x}}_1}^4{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^2+168{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^4+32{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+8{{\stackrel{\‾}{x}}_1}^4{\mathrm{\σ}}_{x_2}^4+\\ 16{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2+32{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^2+144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4+128{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+144{{\stackrel{\‾}{x}}_1}^2{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^4+\\ 384{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^6+49.0455{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2+7.0273{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+48{{\stackrel{\‾}{x}}_1}^2{\mathrm{\σ}}_{x_2}^6+16{{\stackrel{\‾}{x}}_2}^6{\mathrm{\σ}}_{x_2}^2+8{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^4+\\ 32{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^2+168{{\stackrel{\‾}{x}}_2}^4{\mathrm{\σ}}_{x_2}^4+48{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^6+7.0273{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^2+49.0455{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^4+\\ 384{{\stackrel{\‾}{x}}_2}^2{\mathrm{\σ}}_{x_2}^6+85.2523{\mathrm{\σ}}_{x_1}^8-3.9636{\mathrm{\σ}}_{x_1}^6{\mathrm{\σ}}_{x_2}^2-9.9455{\mathrm{\σ}}_{x_1}^4{\mathrm{\σ}}_{x_2}^4-3.9636{\mathrm{\σ}}_{x_1}^2{\mathrm{\σ}}_{x_2}^6+85.2523{\mathrm{\σ}}_{x_2}^8\end{array}---\left(54\right)$

Compare about average true value expression formula (47) of random quantity y and the true value expression formula (48) of variance, and commonly use The estimated result (53) of UT4, (54) that UT estimated result (50), (51) and the present invention provide are visible:

The rank, true value expression formula≤4 of average, the rank, true value expression formula≤8 of variance.

Conventional UT is to the rank, estimated accuracy≤2 of average the non-crossing item that mates 4 rank, but only 4 rank intersect Item mismatch (loss)；The non-crossing item on 4 rank is also mated in rank, variance≤2, other equal mismatches.

And UT8 provided by the present invention to all cross terms on the rank, estimated accuracy≤4 of average and non-crossing item (and can Join 6 rank non-crossing items), the most i.e. achieve the coupling completely to Estimation of Mean；All cross terms on rank, variance evaluation precision≤4 With non-crossing item mate 6 rank non-crossing items, other equal mismatches, but not lacuna, i.e. total item are equal.

Below odd ordered moment item is considered the most completely, because symmetrical, so all of odd ordered moment item is all 0.If meter And odd ordered moment item, the most above-mentioned precision about various methods of estimation is stated can also add 1 rank.

Application example 2: considering further that a two-dimensional particles is rebuffed rebound model (as shown in Figure 2), its mathematical expression is as follows:

$x_{k+1}=g(x_k,t)=\left\{\begin{array}{c}{\&lsqb;x_1,x_2+v_{0}t\&rsqb;}^T,x_1\&GreaterEqual;-0.5\\ {\&lsqb;x_1,|x_2+v_{0}t|\&rsqb;}^T,x_1<-0.5\end{array}\right.,P_{x_k}=\left[\begin{array}{cc}1& 0.81\\ 0.81& 1\end{array}\right]---\left(55\right)$

It is whereinFor particle in the statistics position of initial time k, and by its covariance square Battle array is visible, there is statistic correlation, v between its two-dimensional coordinate₀For Particles Moving speed；After requiring estimation particle elapsed time t The position statistical property of subsequent time k+1, i.e. estimate x_k+1Average and variance.

The UT4 and 10 provided by common UT, the present invention the most respectively³Monte Carlo (MC) method of individual point is estimated Meter x_k+1Average and variance.Fig. 2 gives the probable ellipse (one times of standard deviation probable range) of particle initial position, and The σ point distribution situation that UT (× legend) and UT4 (* legend) uses.For the ease of display, what Fig. 3 was given is according to x_k+1Average The particle obtained with variance is in the distance estimations of k+1 moment and zero and standard deviation.Error in figure is with the knot of MC method Fruit obtains for reference.Being clear to, the UT4 estimated accuracy of research design is higher.

Application example 3: this is one five dimension example, is also more classical radar target tracking example, i.e. applies radar Reenter atmospheric Ballistic Target to implement to follow the tracks of to one.Target with high speed and height atmospheric reentry, suffered power in Non-linear, there are three kinds: most importantly aerodynamic drag, it is relevant with the flight speed of target, and depending mainly on object height non-thread Property change；Also has gravity；In addition being random vibration, wherein latter two impact is less.Fig. 4 gives the operation of a certain Ballistic Target Track, speed and acceleration change situation.Visible, along with the increase of atmospheric density in motor process, aerodynamic drag impact aggravation, Target presents huge negativeacceleration, and speed die-offs, the almost vertical drop of final trajectory target.The most true due to aerodynamic drag Qualitative make this tracking problem increasingly difficult, therefore can list pneumatic attribute in state equation as a state component.Definition shape State vectorWherein x₁And x₂For target location coordinate, x₃And x₄Two speed for target travel are divided Amount, x₅For pneumatic property parameters；State-noiseMeasurement vector y_k=[r_k, φ_k]^T；Observation is made an uproar Soundσ_r=3m, σ_φ=6 °.It is positioned at (x_ob1, x_ob2)=(-20,6394) Observation radar oblique distance and the azimuth measurement information of target are provided with the data transfer rate of 10Hz.The DSS model of this tracking problem is such as Under:

x_k+1=F_xkx_k+M_vv_k (56)

y_k=h (x_k)+w_k (57)

Wherein related definition or be expressed as follows:

$F_{xk}=\left[\begin{array}{ccccc}1& 0& T_s& 0& 0\\ 0& 1& 0& T_s& 0\\ T_s{G}_k& 0& 1+T_s{D}_k& 0& 0\\ 0& T_s{G}_k& 0& 1+T_s{D}_k& 0\\ 0& 0& 0& 0& 1\end{array}\right],M_v=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ T_s& 0& 0\\ 0& T_s& 0\\ 0& 0& T_s\end{array}\right]---\left(58\right)$

D_k=β_kexp((R₀-R_k)/H₀) (59)

$G_k=-G_{m0}/R_k^3---\left(60\right)$

β_k=β₀exp(x_5k) (61)

R_k=(x_1k ²+x_2k ²)^1/2 (62

V_k=(x_3k ²+x_4k ²)^1/2 (63)

r_k=((x_1k-x_ob1)²+(x_2k-x_ob2)²)^1/2+w_rk (64)

φ_k=tan^-1((x_2k-x_ob2)/(x_1k-x_ob1))+w_φk (65)

Convolution (56) and (58) are visible, and state equation here is time-variant nonlinear, and observational equation is also non-linear.Its Middle β₀=-0.59783, H₀=13.406, G_m0=3.9860 × 10⁵, R₀=6374Km, T_s=0.1s.

Assuming that the true initial state of target is x₀=[349.14,6500.4 ,-6.7967 ,-1.8093,0.6932]^T, P_x0=diag [10^-6, 10^-6, 10^-6, 10^-6, 0].State-noise covariance for driving target truthful data to produce is P_v= diag[2.4064×10^-5, 2.4064 × 10^-5, 0], and the state-noise covariance for driving wave filter to run is P_v= diag[2.4064×10^-5, 2.4064 × 10^-5, 10^-6].Wave filter original state is x_0|0=[349.14,6500.4 ,- 6.7967 ,-1.8093,0]^T, initial covariance is P_x0|0=diag [10^-6, 10^-6, 10^-6, 10^-6, 1].Use traditional respectively EKF, common UKF and we the UKF4 of offer is provided system mode is carried out Recursive Filtering estimation, carry out the Monte of 25 times Carlo emulates, and result is as shown in Figure 5 and Figure 6.Fig. 5 is RMSE pair that three kinds of algorithms are estimated about each component of target state According to figure, Fig. 6 is the target root-mean-square position and the comparison figure of velocity error estimated.Being clear to, the estimated accuracy of UKF4 is best, secondly It is UKF, is finally EKF；Error occurs mainly in target and produces the moment of huge negativeacceleration because of aerodynamic drag.

The content not being described in detail in description of the invention belongs to prior art known to professional and technical personnel in the field.

Claims (3)

1. the tasteless conversion of any dimension high-order and a Unscented Kalman Filter method, described high-order is more than or equal to 4 rank, mainly Under general tasteless conversion and Unscented Kalman Filter method frame, be extended to any dimension and also can provide to The accurate coupling of machine amount High Order Moment, it is characterised in that provide and a kind of go for any dimension and can accurately mate High Order Moment The method for designing of σ point set, this σ point set is by the key of the tasteless conversion of any dimension high-order and the tasteless conversion of Unscented Kalman Filter Basis, specifically comprises the following steps that

(1) set up the dynamic system model of Nonlinear Filtering Problem, including state equation and observational equation, wherein system mode with The dimension of machine amount x is N_x；

(2) the σ point set corresponding with said system state random quantity x in tasteless conversion is made to beWherein σ point Sum is p+1, and i-th σ point is x⁽ⁱ⁾, corresponding weights are w⁽ⁱ⁾, lower same；Design provides dimension equal to N_xStandard normal distribution The σ point set of random quantityThe method for designing of this σ point set should apply to any dimension and can realize right The accurate coupling of random quantity High Order Moment (>=4 rank)；

(3) initiation parameter of unbalanced input wave filter, or the average about previous moment k-1 system mode random quantityAnd covarianceAnd the system quantities measured value y of current time k_k；

(4) based onWithThe σ point set Γ that conversion is obtained by step (2)_I, generate and be applicable to any dimension and can accurate Join the σ point set on High Order Moment (>=4 rank)

(5) the application tasteless conversion of any dimension high-order respectivelyEstimate to draw about currently 5 statistics of moment k one-step prediction: the average of system mode random quantityAnd covarianceSystem measurements random quantity AverageAnd covarianceAnd the cross covariance of system mode and measurement random quantity

(6) line in application linear minimum mean-squared error (Linear Minimum Mean Square Error, LMMSE) meaning Property optimal estimation estimate the output average of system mode random quantity about current time kAnd covariance

(7) if there being new measurement to arrive, returning step (3), otherwise terminating.

Any dimension high-order the most according to claim 1 (>=4 rank) tasteless conversion and Unscented Kalman Filter method, its feature It is: in described step (2), design is given and is applicable to any dimension N and can realize standard normal distribution random quantity high-order The σ point set Γ of square (>=4 rank) accurately coupling_I, concrete grammar is to be processed respectively according to following three types by σ point therein:

First kind σ point one, for initial point beWeights are w₀；

Equations of The Second Kind σ point 2N altogether_xIndividual, it is positioned on the coordinate axes of each dimension, is all s apart from initial point₁, i.e.Weights are

3rd class σ point is altogetherIndividual, it is positioned on the bisector of All Quardrants, its coordinate figure in each dimension is+s₂Or-s₂, and take All over institute's likely value, and weights are

Above-mentioned parameter w₀、w₁、w₂、s₁、s₂Determination method as follows:

Work as N_xIt is that the solution of following Simultaneous Equations is time=1 (the most one-dimensional):

$w_0+2N_x{w}_1+2^{N_x}{w}_2-1=0---\left(1\right)$

$2w_1{s}_1^2+2^{N_x}{w}_2{s}_2^2-1=0---\left(2\right)$

$2w_1{s}_1^4+2^{N_x}{w}_2{s}_2^4-3=0---\left(3\right)$

$2w_1{s}_1^6+2^{N_x}{w}_2{s}_2^6-15=0---\left(4\right)$

$2w_1{s}_1^8+2^{N_x}{w}_2{s}_2^8-105=0---\left(5\right)$

Work as N_xIt is the solution of following Simultaneous Equations time >=2 (i.e. multidimensional):

$w_0+2N_x{w}_1+2^{N_x}{w}_2-1=0---\left(6\right)$

$2w_1{s}_1^2+2^{N_x}{w}_2{s}_2^2-1=0---\left(7\right)$

$2w_1{s}_1^4+2^{N_x}{w}_2{s}_2^4-3=0---\left(8\right)$

$2^{N_x}{w}_2{s}_2^4-1=0---\left(9\right)$

$2w_1{s}_1^6+2^{N_x}{w}_2{s}_2^6-15=0---\left(10\right).$

Any dimension high-order the most according to claim 1 (>=4 rank) tasteless conversion and Unscented Kalman Filter method, its feature It is: the tasteless conversion of any dimension high-order of application in described step (5)It is crucial Improvement is the design applying any dimension high-order (>=4 rank) the σ point set jointly provided by step (2) and step (4).