CN107547067A - A kind of multi-model self calibration EKF method - Google Patents

A kind of multi-model self calibration EKF method Download PDF

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CN107547067A
CN107547067A CN201710832525.5A CN201710832525A CN107547067A CN 107547067 A CN107547067 A CN 107547067A CN 201710832525 A CN201710832525 A CN 201710832525A CN 107547067 A CN107547067 A CN 107547067A
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杨海峰
傅惠民
张勇波
王治华
肖梦丽
崔轶
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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 EKF method, and step is as follows：One：Establish system fundamental equation；Two：Linearization process is carried out to forming system by formula (1), formula (2) and formula (3)；Three：Initialization is filtered to system；Four：Time renewal is carried out to system；Five：Iteration variable updates；Six：Measure renewal；Seven：It is iterated calculating；Pass through above step, the problem of non-linear system status equation frequently encountered in engineering is influenceed by Unknown worm is resolved, simultaneously as considering the situation that Unknown worm is zero, therefore self calibration EKF method is compared, the adaptability of complex environment is improved；And because the present invention is based on EKF method, it is not necessary to complete the transmission of information by the form of sampling, so the arithmetic speed of its processing nonlinear system is more quick compared with other non-linear filtering methods, be easy to practical implementation.

Description

A kind of multi-model self calibration EKF method
【Technical field】
The present invention provides a kind of multi-model self calibration EKF method, belongs to Robust Kalman Filter technology neck Domain.
【Background technology】
Kalman filtering is a kind of method estimated using system state equation and measurement equation system mode, from Nineteen sixty is widely used after being suggested in engineering field, but either linear system or nonlinear system Kalman filter method, they require that system equation is accurate.But in engineering in practice, due to the shadow of environmental factor Sound, model and parameter choose the reason such as improper, and system state equation suffers from the interference of Unknown worm, so that filtering accuracy Decline, even result in filtering divergence.For this problem, document " self calibration Kalman filter method [J] aviation power journals .2014,29(06):1363-1368 " proposes a kind of self calibration kalman filter method (Self-calibration Kalman Filter, SKF), this method is estimated Unknown worm item while computing is iterated according to reset condition equation, So that the influence of Unknown worm automatically derives compensation.
The it is proposed of self calibration kalman filter method solves system state equation and asked by what Unknown worm was influenceed well Topic, but due to the presence of system uncertain factor, Unknown worm also promising zero possibility.In this case, self calibration card Kalman Filtering method in prior estimate due to introducing the estimation to Unknown worm item, although the estimate very little, it is filtered Precision 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 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 self calibration Kalman filter method (Multiple-model Self-calibration Kalman Filter, MSKF).This method is simultaneously Be filtered using KF and SKF, according to the weight of both Bayes' theorem real-time update prior estimates, by Weighted Fusion and then Final state estimation is obtained, has given full play to two methods in the respective advantage of different phase.However, multi-model self calibration Although it is zero section that kalman filter method improves Unknown worm while ensure that Unknown worm non-zero section filtering accuracy Precision, but be only applicable to linear system, the nonlinear system of engineering generally existing in practice can not be handled by Unknown worm shadow The problem of ringing.
【The content of the invention】
It is an object of the invention to provide a kind of multi-model self calibration EKF method (Multiple-model Self-calibration Extended Kalman Filter, MSEKF), it is by the way that Multiple model estimation theory is incorporated into In SEKF, MSKF application non-linear field has been extended to.It is calculated using EKF and SEKF simultaneously, in real time more The two new weight, and then obtain state estimation.A large amount of numerical simulations show no matter non-linear system status equation is by unknown defeated Whether the influence entered is zero, and context of methods all achieves good filter result, and is calculated simply, applied widely, is easy to work Journey practical application.
A kind of multi-model self calibration EKF method of the present invention, it includes following seven steps：
Step 1：Establish system fundamental equation
Multi-model self calibration EKF is using self calibration EKF and EKF two Kind of method carries out computing, therefore system includes two state equations, and first is the state equation containing Unknown worm item, second For the nonlinear state equation of standard, 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：Linearization process is carried out to forming system by formula (1), formula (2) and formula (3)
System equation after linearization process is changed into
Zk=HkXk+Vk (9)
In formula, ΦkAnd HkRespectively state-transition matrix and measurement matrix；
Step 3：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 (12)
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；Current pin in the world Solution route to this problem is to set probability lower limit, when the probable value of a certain model is less than this lower limit, probability for system Value just keeps this lower limit no longer to change；The advantages of so handling is model is maintained while estimated result is not influenceed more Sample, but shortcoming is equally obvious, if the model that is, in probability lower limit will play a role again, when its probable value needs longer Between interative computation could recover, this real-time to Kalman filtering has a great impact.In order to overcome this shortcoming, for Context of methods only chooses two kinetic models and chooses priori estimates using maximum probability method, therefore only needs qualitative analysis two Without accurate the characteristics of calculating probable value, multi-model self calibration kalman filter method no longer makes the probability size of kind model It is iterated with the conditional probability value being calculated, but sets 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 be with The speed that guarantee probability recovers, so that the real-time of Kalman filtering is guaranteed；
Step 4：Time renewal is carried out to system
Work as k=1, when 2, state one-step prediction value
One-step prediction varivance matrix
As k ＞ 2, state one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (19)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
S1=P1 (27)
In above-mentioned calculating process, formula (15) provides the one-step prediction result of self calibration state equation, and formula (16) provides mark The one-step prediction result that quasi- state equation calculates, then by the comparison of probability size in formula (12), completed by formula (14) final One-step prediction value screening.The calculating of probability size relies on Bayes principle, and what is compared is known to current time measuring value In the case of, the conditional probability of two kinds of models, as shown in formula (19) and formula (20), other calculating process then come from multi-model and estimated Meter is theoretical；
Step 5：Iteration variable updates
There are many intermediate variables to need real-time update in step (3), it is therefore necessary to their recurrence formula is obtained, And then ensure being smoothed out for whole filtering.These iteration variables include：
Each model measurement renewal
Each Model Condition probability is reset
Pr(J|Zk)=Prmax (33)
Pr[(3-J)|Zk]=Prmin (34)
Step 6：Measure renewal
Filtering gain matrix
State estimation
The calculation formula of this step is related to Multiple model estimation theory due to the basic assumption for not being related to self-calibration technique Content, so calculating process and the Posterior estimator formula of standard Kalman filtering method are consistent；It should be noted simultaneously that Because multi-model self calibration kalman filter method is obtained by the fusion calculation of two model results, therefore calculating In journey and transmission of the estimation error variance matrix between adjacent moment is not needed, so not providing P in calculation procedurekMeter Calculate formula；
Step 7：Iterative calculation
The state estimation respectively obtained according to the two of the k moment modelsWith varivance matrix Pk, repeat step (4), (five) and (six), and then the state estimation at k+1 moment is obtained, reciprocal iteration, until filtering terminates；
By above step, the problem of non-linear system status equation frequently encountered in engineering is influenceed by Unknown worm It is resolved, simultaneously as considering the situation that Unknown worm is zero, therefore compares self calibration EKF side Method, the adaptability of complex environment is improved.And because the present invention is based on EKF method, it is not necessary to logical The form of over-sampling completes the transmission of information, so it handles the arithmetic speed of nonlinear system compared with other non-linear filtering methods It is more quick, it is easy to practical implementation.
Advantages of the present invention is with good effect：
(1) Multiple model estimation theory is introduced into non-linear system status equation in by the problem of unknown input disturbances, be based on Self calibration extends Kalman Filtering and EKF, has been derived by the filter of multi-model self calibration spreading kalman in theory The complete procedure of wave method.
(2) the advantages of invention has merged two methods of EKF and SEKF each, both solved EKF not The problem of knowing input non-zero section filtering divergence, it is zero section to be also obviously improved self calibration EKF in Unknown worm Filtering accuracy.
(3) invention improves the filtering accuracy that Unknown worm is zero section and non-zero section simultaneously, and is directed to nonlinear system It is proposed.The scope of application has obtained great extension, and simultaneity factor robustness is also in the base of self calibration EKF method It is further improved on plinth.
【Brief description of the drawings】
Fig. 1 is the inventive method schematic flow sheet.
Fig. 2 is the step (4) 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
f(·)、h(·)：Nonlinear Vector function
Zk：System measurements vector
Φk：State-transition matrix
Hk：Measurement matrix
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
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
argmax[f(x)]：Function maxima return function
rk：Residual error
Tk：Measure estimation error variance matrix
Kk：Filtering gain 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 EKF method, and its flow chart is as shown in figure 1, the time Flow chart is updated as shown in Fig. 2 it includes following seven steps：
Step 1：Establish system fundamental equation
Zk=hk(Xk)+Vk (39)
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：System linearization
Zk=HkXk+Vk (43)
In formula, ΦkAnd HkRespectively state-transition matrix and measurement matrix；
Step 3：Filtering initialization
Set the initial value of state estimation and estimation error variance matrix as
Meanwhile set the probability initial value of two kinds of models
Pr(1|Z3)=Pr (2 | Z3)=0.5 (46)
And the probability initial value Pr for iterative calculationmaxAnd Prmin；
Step 4：Time updates
Work as k=1, when 2, state one-step prediction value
One-step prediction varivance matrix
As k ＞ 2, state one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (53)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
S1=P1 (61)
Its time updates schematic flow sheet, as shown in Figure 2；
Step 5：Iteration variable updates
Each model measurement renewal
Each Model Condition probability is reset
Pr(J|Zk)=Prmax (67)
Pr[(3-J)|Zk]=Prmin(68)；
Step 6：Measure renewal
Filtering gain matrix
State estimation
Step 7：Iterative calculation
K=k+1 (73)
Repeat step four, step 5 and step 6, until filtering terminates.

Claims (1)

  1. A kind of 1. multi-model self calibration EKF method, it is characterised in that：It includes following seven steps：
    Step 1：Establish system fundamental equation
    Multi-model self calibration EKF is using self calibration EKF and two kinds of sides of EKF Method carries out computing, therefore system includes two state equations, and first is the state equation containing Unknown worm item, and second is mark Accurate nonlinear state equation, its expression are
    <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：Linearization process is carried out to forming system by formula (1), formula (2) and formula (3)
    <mrow> <msub> <mi>&amp;Phi;</mi> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>f</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>X</mi> <mi>k</mi> </msub> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>h</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>X</mi> <mi>k</mi> </msub> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    System equation after linearization process is changed into
    <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mn>1</mn> </msubsup> <mo>=</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <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>7</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>X</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>X</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>W</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Zk=HkXk+Vk·············(9)
    In formula, ΦkAnd HkRespectively state-transition matrix and measurement matrix；
    Step 3：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>10</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>11</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 (12)
    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 be joined Model quantity N with computing is constantly reducing, and this can reduce adaptability of the system to complex environment；Only chosen for the present invention Two kinetic models and use maximum probability method selection priori estimates, therefore only need the probability of two kinds of models of qualitative analysis big It is small without accurate the characteristics of calculating probable value, multi-model self calibration kalman filter method does not use the bar being calculated Part probable value is iterated, but sets the probability initial value Pr of two determinationsminAnd Prmax=1-Prmin；Filtered in each step Before, it is assigned to two models respectively by comparing the size of previous step probability results, and worked as using them as initial value renewal The model probability at preceding moment；Due to PrminIt is not minimum as probability lower limit, therefore can guarantee that the speed that probability recovers, So that the real-time of Kalman filtering is guaranteed；
    Step 4：Time renewal is carried out to system
    Work as k=1, when 2, state one-step prediction value
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mn>1</mn> <mo>/</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mn>2</mn> <mo>/</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
    One-step prediction varivance matrix
    <mrow> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;Phi;</mi> <mn>0</mn> </msub> <msub> <mi>P</mi> <mn>0</mn> </msub> <msubsup> <mi>&amp;Phi;</mi> <mn>0</mn> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>Q</mi> <mn>0</mn> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
    P2/1=Φ1P1Φ1 T+Q1············(16)
    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>17</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>18</mn> <mo>)</mo> </mrow> </mrow>
    In formula
    J=argmaxjPr(j|Zk)···········(19)
    <mrow> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <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> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <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>f</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</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> <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> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <msubsup> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mi>T</mi> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>S</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;times;</mo> <msubsup> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mi>T</mi> </msubsup> <mo>-</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <msubsup> <mi>S</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <msub> <mi>H</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <msub> <mi>H</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>&amp;times;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</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> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <msup> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>23</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>k</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>24</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> <msup> <mrow> <mo>|</mo> <msub> <mi>T</mi> <mi>k</mi> </msub> <mo>|</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mn>...</mn> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>S</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <msub> <mi>H</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>3</mn> </mrow> </msub> <msubsup> <mi>S</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mi>T</mi> </msubsup> <mo>-</mo> <msub> <mi>Q</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <msub> <mi>H</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>&amp;rsqb;</mo> <mn>....</mn> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
    S1=P1··············(27)
    <mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <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>28</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>T</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <msubsup> <mi>H</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>
    In above-mentioned calculating process, formula (15) provides the one-step prediction result of self calibration state equation, and formula (16) provides standard shape The one-step prediction result that state equation calculates, then by the comparison of probability size in formula (12), final one is completed by formula (14) Walk predicted value screening；The conditional probability of two kinds of models, as shown in formula (19) and formula (20)；
    Step 5：Iteration variable updates
    There are many intermediate variables to need real-time update in step (3), it is therefore necessary to their recurrence formula is obtained, and then Ensure 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>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <msubsup> <mi>H</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <msubsup> <mi>H</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mn>...</mn> <mrow> <mo>(</mo> <mn>30</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> <msub> <mi>H</mi> <mi>k</mi> </msub> <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> <mn>...</mn> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msubsup> <mi>P</mi> <mi>k</mi> <mi>j</mi> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msubsup> <mi>K</mi> <mi>k</mi> <mi>j</mi> </msubsup> <msub> <mi>R</mi> <mi>k</mi> </msub> <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>32</mn> <mo>)</mo> </mrow> </mrow>
    Each Model Condition probability is reset
    Pr(J|Zk)=Prmax············(33)
    Pr[(3-J)Zk]=Prmin···········(34)；
    Step 6：Measure renewal
    Filtering gain matrix
    <mrow> <msub> <mi>K</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>H</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>H</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>R</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mn>...</mn> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>
    State estimation
    <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> <mo>&amp;lsqb;</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>h</mi> <mi>k</mi> </msub> <mrow> <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> <mo>&amp;rsqb;</mo> <mn>...</mn> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>
    The calculation formula of this step due to not being related to the basic assumption of self-calibration technique and the related content of Multiple model estimation theory, So calculating process and the Posterior estimator formula of standard Kalman filtering method are consistent；
    Step 7：Iterative calculation
    The state estimation respectively obtained according to the two of the k moment modelsWith varivance matrix Pk, repeat step (four), (5) and (six), and then the state estimation at k+1 moment is obtained, reciprocal iteration, until filtering terminates.
CN201710832525.5A 2017-09-15 2017-09-15 A kind of multi-model self calibration EKF method Pending CN107547067A (en)

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CN108871325A (en) * 2018-06-26 2018-11-23 北京航空航天大学 A kind of WiFi/MEMS combination indoor orientation method based on two layers of Extended Kalman filter
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
CN110729982A (en) * 2019-09-30 2020-01-24 中国船舶重工集团公司第七0七研究所 Kalman filtering algorithm optimization method based on matrix sparsity
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method
CN117216482A (en) * 2023-11-07 2023-12-12 南开大学 Method for enhancing wild value interference resistance of Kalman filter by means of iterative strategy

Cited By (9)

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Publication number Priority date Publication date Assignee Title
CN108871325A (en) * 2018-06-26 2018-11-23 北京航空航天大学 A kind of WiFi/MEMS combination indoor orientation method based on two layers of Extended Kalman filter
CN108871325B (en) * 2018-06-26 2019-06-07 北京航空航天大学 A kind of WiFi/MEMS combination indoor orientation method based on two layers of Extended Kalman filter
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
CN110729982A (en) * 2019-09-30 2020-01-24 中国船舶重工集团公司第七0七研究所 Kalman filtering algorithm optimization method based on matrix sparsity
CN110729982B (en) * 2019-09-30 2023-03-10 中国船舶重工集团公司第七0七研究所 Kalman filtering algorithm optimization method based on matrix sparsity
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method
CN117216482A (en) * 2023-11-07 2023-12-12 南开大学 Method for enhancing wild value interference resistance of Kalman filter by means of iterative strategy
CN117216482B (en) * 2023-11-07 2024-03-01 南开大学 Method for enhancing wild value interference resistance of Kalman filter by means of iterative strategy

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