CN107807906A - A kind of multi-model self calibration order filtering method - Google Patents

A kind of multi-model self calibration order filtering method Download PDF

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CN107807906A
CN107807906A CN201710853216.6A CN201710853216A CN107807906A CN 107807906 A CN107807906 A CN 107807906A CN 201710853216 A CN201710853216 A CN 201710853216A CN 107807906 A CN107807906 A CN 107807906A
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杨海峰
傅惠民
王治华
张勇波
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Beihang University
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Beihang University
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Abstract

The present invention provides a kind of multi-model self calibration order filtering method, and step is as follows：One：Establish system fundamental equation；Two：Initialization is filtered to system；Three：Time renewal is carried out to system；Four：It is iterated variable update；Five：Carry out measurement renewal；Six：It is iterated calculating；Pass through step 1 to step 6, the present invention takes full advantage of the result of calculation of two methods of order filtering and self calibration order filtering, rely on the Multiple model estimation theory based on Bayes principle, automatic distinguishing Unknown worm is zero section and non-zero section, so as to accurately select wherein most suitable result as the prior estimate of itself；The most important is that the present invention be directed to non-gaussian, nonlinear system to develop, wider compared to the other method scope of application.

Description

A kind of multi-model self calibration order filtering method
【Technical field】
The present invention provides a kind of multi-model self calibration order filtering method, belongs to non-gaussian robust filtering technical field.
【Background technology】
The problem of being influenceed for system state equation common in engineering by Unknown worm, " self calibration Kalman is filtered document Wave method [J] aviation power journal .2014,29 (06):1363-1368 " proposes a kind of self calibration kalman filter method (Self-calibration Kalman Filter, SKF), this method are being iterated the same of computing according to reset condition equation When, Unknown worm item is estimated, so that the influence of Unknown worm automatically derives compensation.On this basis, researcher Again successively developed self calibration EKF method (Self-calibration Extended Kalman Filter, SEKF) and self calibration Unscented kalman filtering method (Self-calibration Unscented Kalman Filter, SUKF), self-calibration technique has been generalized to non-linear field by them.
But due to the presence of system uncertain factor, Unknown worm also promising zero possibility.In this case, self-correcting Quasi- kalman filter method in prior estimate due to introducing the estimation to Unknown worm item, although the estimate very little, its Filtering accuracy is still not as good as the standard Kalman filtering method for not accounting for Unknown worm influence.In order to further lift self calibration card Kalman Filtering is based on Multiple model estimation theory in the filtering accuracy that Unknown worm is zero section, researcher, it is proposed that multi-model is certainly Calibrate kalman filter method (Multiple-model Self-calibration Kalman Filter, MSKF).
Especially, for nonlinear system, multi-model self calibration EKF (Multiple-model Self- Calibration Extended Kalman Filter, MSEKF) and multi-model self calibration Unscented kalman filtering method (Multiple-model Self-calibration Unscented Kalman Filter, MSUKF) is also developed, But their systems for being only applicable to Gaussian distributed, exist for zero for the nonlinear system Unknown worm of non-gaussian distribution The situation of section and non-zero section is just no longer applicable.
【The content of the invention】
It is an object of the invention to provide a kind of multi-model self calibration order filtering method (Multiple-model Self- Calibration Rank Filter, MSRF), it by Multiple model estimation theory by being incorporated into self calibration order filtering method In (Self-calibration Rank Filter, SRF), MSKF application non-gaussian, non-linear neck have been extended to Domain.It is calculated using RF and SRF simultaneously, both real-time updates weight, and then obtains state estimation.
A kind of multi-model self calibration order filtering method of the present invention, it includes following six step：
Step 1：Establish system fundamental equation
The filtering of multi-model self calibration order carries out computing using two methods of the filtering of self calibration order and order filtering, therefore system includes Two state equations, first is the state equation containing Unknown worm item, second be standard nonlinear state equation, its Expression is
Zk=hk(Xk)+Vk (3)
In formula, XkThe state vector of expression system,WithKinetic model and standard containing Unknown worm are corresponded to respectively Kinetic model, ZkRepresent system measurements vector, fk() and hk() is respectively nonlinear state recurrence equation and measurement side Journey, bkRepresent Unknown worm, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkWith Rk, and meet
In formula, Cov [] is covariance, and E [] is mathematic expectaion, δkjFor δ functions, as k=j, δkj=1, as k ≠ j When, δkj=0；
Step 2：Initialization is filtered to system
Set the initial value of state estimation and estimation error variance matrix as
Meanwhile in order to complete the fusion of two model estimated results, it is also necessary to set the probability initial value of two kinds of models
Pr(1|Z3)=Pr (2 | Z3)=0.5 (7)
And the probability initial value Pr for iterative calculationmaxAnd Prmin；Initialize PrmaxAnd PrminThe reason for it is as follows：
In multiple-model estimator calculating process, some models can be eliminated due to the gradual convergence of corresponding probability for zero, Therefore the model quantity N for participating in computing is constantly reducing, this can reduce adaptability of the system to complex environment；For context of methods Only choose two kinetic models and priori estimates chosen using maximum probability method, it is only necessary to two kinds of models of qualitative analysis it is general Without accurate the characteristics of calculating probable value, multi-model self calibration kalman filter method does not use to be calculated rate size Conditional probability value be iterated, but set two determination probability initial value PrminAnd Prmax=1-Prmin；In each step Before filtering, it is assigned to two models by the size by comparing previous step probability results respectively, and using they as initial value more The model probability at new current time；Due to PrminIt is not minimum as probability lower limit, therefore can be recovered with guarantee probability Speed so that the real-time of Kalman filtering is guaranteed；
Step 3：Time renewal is carried out to system
If the state estimation and varivance matrix at k-1 moment are respectivelyAnd Pk-1, system is carried out based on them Time updates, that is, calculates the state one-step prediction at k momentWith one-step prediction varivance matrix Pk/k-1；Filtered based on order General recurrence formula, in time renewal process firstly the need of calculate order sampling point set { χi}
As k=1,2
State one-step prediction value
Xk/k-1,i=f (χk-1,i) (9)
In formula, n is state vector XkDimension, standard normal deviator Represent Pk-1Subduplicate i-th column vector；
One-step prediction varivance matrix
In formula, ω is covariance weight coefficient, and its calculation formula is
As k ＞ 2
State one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (15)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
In above-mentioned calculating process, the one-step prediction result of formula (18) and formula (21) offer self calibration state equation, formula (20) and formula (21) provides the one-step prediction result that standard state equation calculates, and then passes through the ratio of probability size in formula (15) It is right, final one-step prediction value is completed by formula (13) and screened；The calculating of probability size relies on Bayes principle, compare be In the case of current time measuring value is known, the conditional probability of two kinds of models, as shown in formula (23) and formula (24)；
Step 4：Iteration variable updates
There are many intermediate variables to need real-time update in step 3, it is therefore necessary to obtain their recurrence formula, enter And ensure being smoothed out for whole filtering；These iteration variables include：
Each model measurement renewal
In formula
Each Model Condition probability is reset
Pr(J|Zk)=Prmax (32)
Pr[(3-J)|Zk]=Prmin(33)；
Step 5：Measure renewal
First, based on state one-step predictionWith one-step prediction varivance matrix Pk/k-1, filtered according to order general Property recurrence formula to system mode carry out resampling, obtain new order sampling point set { χk/k-1,i}
Then, according to { χk/k-1,iCalculate measurement estimate
Zk/k-1,i=h (χk/k-1,i) (35)
Calculated by formula (35) and formula (36) and measure varivance matrix PZZ
By formula (34), formula (35) and formula (36) calculation error covariance matrix PXZ
Gain matrix is calculated by formula (37) and formula (38)
Then, final state estimation and state estimation varivance matrix are obtained by measuring renewal
Step 6：Iterative calculation
According to the state estimation at k momentWith varivance matrix Pk, repeat step three, step 4 and step 5, enter And the state estimation and varivance matrix at k+1 moment are obtained, reciprocal iteration, until filtering terminates；
By step 1 to step 6, the present invention takes full advantage of two methods of order filtering and the calculating of self calibration order filtering As a result, the Multiple model estimation theory based on Bayes principle is relied on, automatic distinguishing Unknown worm is zero section and non-zero section, so as to Accurately to select wherein most suitable result to be used as the prior estimate of itself.The most important is, the present invention be directed to Non-gaussian, nonlinear system are developed, wider compared to the other method scope of application.
Beneficial effects of the present invention include
(1) Multiple model estimation theory is incorporated into self calibration order filtering method, has been derived by multi-model self calibration order The complete procedure of filtering method.
(2) it can play order filtering method respectively using automatic identification Unknown worm as zero section and non-zero section and filtered with self calibration order The respective advantage of wave method.
(3) filtering accuracy when system is influenceed by Unknown worm is further increased, while improves the steady of filtering It is qualitative, enhance the robustness of system.
(4) nonlinear system for disobeying Gaussian Profile can be handled, the scope of application is extended, and meeting engineer applied needs Ask.
【Brief description of the drawings】
Fig. 1 is the inventive method schematic flow sheet.
Fig. 2 is the step 3 time to update schematic flow sheet.
Sequence number, symbol, code name are described as follows in invention：
Xk：The state vector of system
The state vector of kinetic model containing Unknown worm
The state vector of Standard kinetic model
Zk：System measurements vector
f(·)、h(·)：Non-gaussian, Nonlinear Vector function
bk：Unknown worm
Wk：System noise vector
Vk：Measure noise vector
Qk：System noise vector variance matrix
Rk：Measure noise vector variance matrix
Cov[·]：Covariance
E[·]：Mathematic expectaion
δkj：δ functions
{χi}：Order sampling point set
Standard normal deviator
Pk-1Subduplicate i-th column vector
Pr(1|Zk)、Pr(2|Zk)：The probability initial value of iterative calculation
Prmax、Prmin：Probability initial value for iterative calculation
Pk/k-1：One-step prediction varivance matrix
ω：Covariance weight coefficient
argmax[f(x)]：Function maxima return function
Measure estimate
PZZ:Error in measurement variance matrix
PXZ:Error co-variance matrix
State estimation
【Embodiment】
The present invention is elaborated below in conjunction with the accompanying drawings.
The present invention proposes a kind of multi-model self calibration order filtering method, and its flow chart is as shown in figure 1, time more new technological process Figure is as shown in Fig. 2 it includes following six step：
Step 1：Establish system fundamental equation
In formula, XkThe state vector of expression system,WithKinetic model and standard containing Unknown worm are corresponded to respectively Kinetic model, ZkRepresent system measurements vector, fk() and hk() is respectively nonlinear state recurrence equation and measurement side Journey, bkRepresent Unknown worm, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkWith Rk, and meet
In formula, Cov [] is covariance, and E [] is mathematic expectaion, δkjFor δ functions, as k=j, δkj=1, as k ≠ j When, δkj=0.
Step 2：Filtering initialization
Set state estimation and the initial value of estimation error variance matrix
Set the probability initial value of two kinds of models
Pr(1|Z3)=Pr (2 | Z3)=0.5 (47)
And the probability initial value Pr for iterative calculationmaxAnd Prmin。
Step 3：Time updates
As k=1,2
State one-step prediction value
Xk/k-1,i=f (χk-1,i) (49)
In formula, n is state vector XkDimension, standard normal deviator Represent Pk-1Subduplicate i-th column vector.
One-step prediction varivance matrix
In formula, ω is covariance weight coefficient, and its calculation formula is
As k ＞ 2
State one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (55)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
Its time renewal schematic flow sheet is as shown in Figure 2；
Step 4：Iteration variable updates
Each model measurement renewal
In formula
Each Model Condition probability is reset
Pr(J|Zk)=Prmax (72)
Pr[(3-J)|Zk]=Prmin (73)
Step 5：Measure renewal
Resampling is carried out, obtains new order sampling point set { χk/k-1,i}
Then, according to { χk/k-1,iCalculate measurement estimate
Zk/k-1,i=h (χk/k-1,i) (75)
Calculate and measure varivance matrix PZZ
Calculation error covariance matrix PXZ
Calculate gain matrix
State estimation
Step 6：Iterative calculation
K=k+1 (83)
Repeat step three, step 4 and step 5, until filtering terminates.

Claims (1)

  1. A kind of 1. multi-model self calibration order filtering method, it is characterised in that：It includes following six step：
    Step 1：Establish system fundamental equation
    The filtering of multi-model self calibration order carries out computing using two methods of the filtering of self calibration order and order filtering, therefore system includes two State equation, first is the state equation containing Unknown worm item, and second nonlinear state equation for standard, its is specific Expression formula is
    <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mn>1</mn> </msubsup> <mo>=</mo> <msub> <mi>f</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>f</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>W</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    Zk=hk(Xk)+Vk·············(3)
    In formula, XkThe state vector of expression system,WithCorresponding kinetic model and standard containing Unknown worm is dynamic respectively Mechanical model, ZkRepresent system measurements vector, fk() and hk() is respectively nonlinear state recurrence equation and measurement equation, bkRepresent Unknown worm, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkAnd Rk, and And meet
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mi>o</mi> <mi>v</mi> <mo>&amp;lsqb;</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>W</mi> <mi>j</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <msubsup> <mi>W</mi> <mi>j</mi> <mi>T</mi> </msubsup> <mo>&amp;rsqb;</mo> <mo>=</mo> <msub> <mi>Q</mi> <mi>k</mi> </msub> <msub> <mi>&amp;delta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mi>o</mi> <mi>v</mi> <mo>&amp;lsqb;</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>V</mi> <mi>j</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> <msubsup> <mi>V</mi> <mi>j</mi> <mi>T</mi> </msubsup> <mo>&amp;rsqb;</mo> <mo>=</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <msub> <mi>&amp;delta;</mi> <mrow> <mi>k</mi> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mi>o</mi> <mi>v</mi> <mo>&amp;lsqb;</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>V</mi> <mi>j</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <msubsup> <mi>V</mi> <mi>j</mi> <mi>T</mi> </msubsup> <mo>&amp;rsqb;</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>...</mn> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    In formula, Cov [] is covariance, and E [] is mathematic expectaion, δkjFor δ functions, as k=j, δkj=1, as k ≠ j, δkj=0；
    Step 2：Initialization is filtered to system
    Set the initial value of state estimation and estimation error variance matrix as
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>&amp;rsqb;</mo> <mn>...</mn> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>E</mi> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>&amp;rsqb;</mo> <mn>...</mn> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    Meanwhile in order to complete the fusion of two model estimated results, it is also necessary to set the probability initial value of two kinds of models
    Pr(1|Z3)=Pr (2 | Z3)=0.5 (7)
    And the probability initial value Pr for iterative calculationmaxAnd Prmin；
    Two kinetic models are only chosen for the present invention and priori estimates are chosen using maximum probability method, it is only necessary to qualitative point The probability size of two kinds of models is analysed without accurate the characteristics of calculating probable value, multi-model self calibration kalman filter method is not Reuse the conditional probability value being calculated to be iterated, but set the probability initial value Pr of two determinationsminAnd Prmax=1- Prmin；Before the filtering of each step, it is assigned to by the size for comparing previous step probability results by two models respectively, and with it For initial value update current time model probability；Due to PrminIt is not minimum as probability lower limit, therefore can protects The speed that probability recovers is demonstrate,proved, so that the real-time of Kalman filtering is guaranteed；
    Step 3：Time renewal is carried out to system
    If the state estimation and varivance matrix at k-1 moment are respectivelyAnd Pk-1, based on them is carried out to system the time Renewal, that is, calculate the state one-step prediction at k momentWith one-step prediction varivance matrix Pk/k-1；One based on order filtering As property recurrence formula, in time renewal process firstly the need of calculate order sampling point set { χi}
    As k=1,2
    State one-step prediction value
    <mrow> <msub> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>3</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>4</mn> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>......</mn> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Xk/k-1,i=f (χk-1,i)············(9)
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    In formula, n is state vector XkDimension, standard normal deviatorRepresent Pk-1Subduplicate i-th column vector；
    One-step prediction varivance matrix
    <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>....</mn> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
    In formula, ω is covariance weight coefficient, and its calculation formula is
    <mrow> <mi>&amp;omega;</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <msubsup> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mn>3.0105...</mn> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
    As k ＞ 2
    State one-step prediction value
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>J</mi> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
    One-step prediction varivance matrix
    <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>J</mi> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
    In formula
    J=argmaxjPr(j|Zk)···········(15)
    <mrow> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>3</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>4</mn> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>.....</mn> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>3</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>4</mn> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>......</mn> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>-</mo> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>3</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>4</mn> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>......</mn> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
    <mrow> <mi>Pr</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>|</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>p</mi> <mi>d</mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <mi>j</mi> <mo>)</mo> </mrow> <mi>Pr</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>|</mo> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mi>p</mi> <mi>d</mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>Pr</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>|</mo> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mn>...</mn> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mi>p</mi> <mi>d</mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <mfrac> <mrow> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>r</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msubsup> <mi>T</mi> <mi>k</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>)</mo> </mrow> <mrow> <mi>q</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>|</mo> <msub> <mi>T</mi> <mi>k</mi> </msub> <msup> <mo>|</mo> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mn>...</mn> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
    In above-mentioned calculating process, formula (18) and formula (21) provide the one-step prediction result of self calibration state equation, formula (20) and Formula (21) provides the one-step prediction result that standard state equation calculates, then by the comparison of probability size in formula (15), by formula (13) final one-step prediction value screening is completed；The calculating of probability size relies on Bayes principle, and what is compared is when current In the case of carving known to measuring value, the conditional probability of two kinds of models, as shown in formula (23) and formula (24)；
    Step 4：Iteration variable updates
    There are many intermediate variables to need real-time update in step 3, it is therefore necessary to obtain their recurrence formula, Jin Erbao Demonstrate,prove being smoothed out for whole filtering；These iteration variables include：
    Each model measurement renewal
    <mrow> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mo>=</mo> <msubsup> <mi>P</mi> <mrow> <mi>X</mi> <mi>Z</mi> </mrow> <mi>j</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mi>Z</mi> <mi>Z</mi> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mn>...</mn> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>j</mi> </msubsup> <mo>=</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>+</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>P</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mo>=</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <msubsup> <mi>P</mi> <mrow> <mi>Z</mi> <mi>Z</mi> </mrow> <mi>j</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mn>...</mn> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
    In formula
    <mrow> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mi>h</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>P</mi> <mrow> <mi>Z</mi> <mi>Z</mi> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <mn>....</mn> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>P</mi> <mrow> <mi>X</mi> <mi>Z</mi> </mrow> <mi>j</mi> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>j</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mn>.....</mn> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>
    Each Model Condition probability is reset
    Pr(J|Zk)=Prmax············(32)
    Pr[(3-J)|Zk]=Prmin···········(33)；
    Step 5：Measure renewal
    First, based on state one-step predictionWith one-step prediction varivance matrix Pk/k-1, passed according to the generality that order filters Apply-official formula carries out resampling to system mode, obtains new order sampling point set { χk/k-1,i}
    <mrow> <msub> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>3</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </msub> <msub> <mrow> <mo>(</mo> <msqrt> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>4</mn> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mn>.....</mn> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>
    Then, according to { χk/k-1,iCalculate measurement estimate
    Zk/k-1,i=h (χk/k-1,i)···········(35)
    <mrow> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>
    Calculated by formula (35) and formula (36) and measure varivance matrix PZZ
    <mrow> <msub> <mi>P</mi> <mrow> <mi>Z</mi> <mi>Z</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mo>+</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <mn>.....</mn> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>
    By formula (34), formula (35) and formula (36) calculation error covariance matrix PXZ
    <mrow> <msub> <mi>P</mi> <mrow> <mi>X</mi> <mi>Z</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&amp;omega;</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </munderover> <mo>{</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;chi;</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>}</mo> <mn>.....</mn> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>
    Gain matrix is calculated by formula (37) and formula (38)
    <mrow> <msub> <mi>K</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>X</mi> <mi>Z</mi> </mrow> </msub> <msubsup> <mi>P</mi> <mrow> <mi>Z</mi> <mi>Z</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mrow>
    Then, final state estimation and state estimation varivance matrix are obtained by measuring renewal
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 6：Iterative calculation
    According to the state estimation at k momentWith varivance matrix Pk, repeat step three, step 4 and step 5, and then To the state estimation and varivance matrix at k+1 moment, reciprocal iteration, until filtering terminates.
CN201710853216.6A 2017-09-20 2017-09-20 A kind of multi-model self calibration order filtering method Pending CN107807906A (en)

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CN110110711A (en) * 2019-06-06 2019-08-09 郑州轻工业学院 A kind of iterative learning control systems input signal estimation method under noisy communication channel
CN111158382A (en) * 2020-01-19 2020-05-15 郑州轻工业大学 Unmanned vehicle positioning model construction method and system based on wireless ultra-wideband network
CN111189454A (en) * 2020-01-09 2020-05-22 郑州轻工业大学 Unmanned vehicle SLAM navigation method based on rank Kalman filtering
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110110711A (en) * 2019-06-06 2019-08-09 郑州轻工业学院 A kind of iterative learning control systems input signal estimation method under noisy communication channel
CN110110711B (en) * 2019-06-06 2021-06-04 郑州轻工业学院 Iterative learning control system input signal estimation method under noise channel
CN111189454A (en) * 2020-01-09 2020-05-22 郑州轻工业大学 Unmanned vehicle SLAM navigation method based on rank Kalman filtering
CN111158382A (en) * 2020-01-19 2020-05-15 郑州轻工业大学 Unmanned vehicle positioning model construction method and system based on wireless ultra-wideband network
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method

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