CN107632959A - A kind of multi-model self calibration kalman filter method - Google Patents

A kind of multi-model self calibration kalman filter method Download PDF

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CN107632959A
CN107632959A CN201710832541.4A CN201710832541A CN107632959A CN 107632959 A CN107632959 A CN 107632959A CN 201710832541 A CN201710832541 A CN 201710832541A CN 107632959 A CN107632959 A CN 107632959A
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傅惠民
杨海峰
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 kalman filter method, and step is as follows：One：Establish system fundamental equation；Two：Initialization is filtered to the system being made up of formula (1), formula (2) and formula (3)；Three：Time renewal is carried out to system；Four：Iteration variable updates；Five：Measure renewal；Six：It is iterated calculating；Multiple model estimation theory is introduced into system state equation in by the problem of unknown input disturbances by the present invention, is filtered based on self calibration Kalman filtering and standard Kalman, has been obtained the complete procedure of multi-model self calibration kalman filter method；Both solved the problems, such as that standard Kalman was filtered in Unknown worm non-zero section filtering divergence, and be also obviously improved self calibration Kalman filtering in the filtering accuracy that Unknown worm is zero section；The invention improves the filtering accuracy that Unknown worm is zero section and non-zero section simultaneously, and the scope of application further expands, and system robustness is also further improved on the basis of self calibration kalman filter method.

Description

A kind of multi-model self calibration kalman filter method
【Technical field】
The present invention provides a kind of multi-model self calibration kalman filter method, belongs to Robust Kalman Filter technical field.
【Background technology】
Kalman filtering is a kind of method estimated using system state equation and measurement equation system mode, from Nineteen sixty has been widely used in the fields such as signal transacting, navigation, survey of deep space and fault diagnosis after being suggested.The karr of standard Graceful filtering method is just for linear system, and on this basis, researcher successively proposes EKF (Extended Kalman Filter, EKF), Unscented kalman filtering (Unscented Kalman Filter, UKF) and particle filter Serial of methods such as (Particle Filter, PF), they extend to the scope of application of kalman filter method non-linear System, and further increase filtering accuracy.
But above-mentioned filtering method either linear system or nonlinear system, it is accurate to require system equation 's.But in engineering in practice, due to the influence of environmental factor, model and parameter, to choose reason, the system state equation such as improper past Toward being disturbed by Unknown worm, so that filtering accuracy declines, filtering divergence is even resulted in.For this problem, document " self-correcting Quasi- Kalman filter method [J] aviation power journal .2014,29 (06):1363-1368 " proposes a kind of self calibration Kalman Filtering method (Self-calibration Kalman Filter, SKF), this method is iterated according to reset condition equation While computing, Unknown worm item is estimated, so that the influence of Unknown worm automatically derives compensation.
But due to the uncertainty of system, the Unknown worm of system state equation may also be zero.In this case, Self calibration kalman filter method is due to introducing Unknown inputs item, although this can gradually be received with the progress of filtering Hold back to zero, but the presence of the fluctuation in view of convergence process, time delay and error, its filtering accuracy are just filtered not as good as standard Kalman Wave method.Therefore, while Unknown worm non-zero section filtering accuracy is kept, the precision that lifting Unknown worm is zero section is self-correcting The further perfect direction of quasi- kalman filter method.
Multiple model estimation theory (Multiple-Model Estimation, MME) has provided for the solution of this problem Effect approach.Its general principle is：Because artificial modeling can not possibly be completely accurate, then multiple power are established simultaneously for a system Model to be learned to be described, all models both participate in computing and obtain multiple one-step prediction values and one-step prediction varivance matrix, The conditional probability corresponding to each kinetic model under measuring value known case is iterated to calculate out further according to Bayes' theorem, by adding Power merges and then obtains final state estimation.
【The content of the invention】
It is an object of the invention to provide a kind of multi-model self calibration kalman filter method (Multiple-model Self- Calibration Kalman Filter, MSKF), it is entered simultaneously using self calibration Kalman filtering and standard Kalman filtering Row iteration computing, and weight corresponding to two methods filter result is obtained at each moment, by Weighted Fusion play they The respective advantage of different phase.A large amount of simulation results show no matter whether system is influenceed by Unknown worm, and the present invention can Enough ensure good filter result.
A kind of multi-model self calibration kalman filter method of the present invention, it includes following six step：
Step 1：Establish system fundamental equation
Multi-model self calibration Kalman filtering is entered using two methods of self calibration Kalman filtering and standard Kalman filtering Row computing, therefore system includes two state equations, first is self calibration state equation, second be standard state equation, Its expression is
Zk=HkXk+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, ΦkAnd HkRespectively state-transition matrix and measurement matrix, bkRepresent unknown Input, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkAnd 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 the system being made up of formula (1), formula (2) and formula (3)
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；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 3：Time renewal is carried out to system
Work as k=1, when 2, state one-step prediction value
One-step prediction varivance matrix
P1/0=Φ0P0Φ0 T+Q0 (10)
P2/1=Φ1P1Φ1 T+Q1 (11)
As k ＞ 2, state one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (14)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
S1=P1 (22)
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；Its time renewal schematic flow sheet is as shown in Figure 2；
Step 4：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 (28)
Pr[(3-J)|Zk]=Prmin (29)
；
Step 5：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 6：Iterative calculation
The state estimation respectively obtained according to the two of the k moment modelsWith varivance matrix Pk, repeat step (3), (four) and (five), and then the state estimation at k+1 moment is obtained, reciprocal iteration, until filtering terminates.
In summary：Step (1) establishes two kinetic models (state equation) needed for Multiple model estimation theory, Prerequisite is provided for follow-up estimated result fusion；Step (2) ensure that the normal startup of filtering method；Step (3) As the core procedure of the present invention, the fusion of two model estimated results is completed, makes the present invention be able to adapt to different complexity Environment；Step (4) is then the smooth important guarantee of step (3), passes through its renewal to intermediate variable, step (3) can normal use at each moment；Step (5) is by introducing measurement information, there is provided after state Estimation is tested, filtering accuracy is further lifted；Step (6) has ensured being normally carried out for filtering, it is run to actual need The end time wanted；That is, Multiple model estimation theory is introduced system state equation by unknown input disturbances by the present invention In problem, filtered based on self calibration Kalman filtering and standard Kalman, be derived by multi-model self calibration card in theory The complete procedure of Kalman Filtering method；The invention had both solved standard Kalman filtering in Unknown worm non-zero section filtering divergence Problem, self calibration Kalman filtering has also been obviously improved in the filtering accuracy that Unknown worm is zero section；The invention improves simultaneously Unknown worm is the filtering accuracy of zero section and non-zero section, and the scope of application further expands, and system robustness is also in self calibration karr It is further improved on the basis of graceful filtering method.
Advantages of the present invention is with good effect：
(1) Multiple model estimation theory is introduced into system state equation in by the problem of unknown input disturbances, based on self calibration Kalman filtering and standard Kalman filtering, have been derived by the complete of multi-model self calibration kalman filter method in theory Process.
(2) the advantages of various methods have been merged in the invention, both solved standard Kalman filtering in Unknown worm non-zero section The problem of filtering divergence, self calibration Kalman filtering has also been obviously improved in the filtering accuracy that Unknown worm is zero section.
(3) invention improves the filtering accuracy that Unknown worm is zero section and non-zero section simultaneously, and the scope of application further expands Exhibition, system robustness are also further improved on the basis of self calibration kalman filter method.
【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
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 kalman filter method, and its flow chart is as shown in figure 1, time renewal Flow chart is as shown in Fig. 2 it includes following six step：
Step 1：Establish system fundamental equation
Zk=HkXk+Vk (34)
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, ΦkAnd HkRespectively state-transition matrix and measurement matrix, bkRepresent unknown Input, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkAnd 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 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 (38)
And the probability initial value Pr for iterative calculationmaxAnd Prmin。
Step 3：Time updates
Work as k=1, when 2, state one-step prediction value
One-step prediction varivance matrix
P1/0=Φ0P0Φ0 T+Q0 (41)
P2/1=Φ1P1Φ1 T+Q1 (42)
As k ＞ 2, state one-step prediction value
One-step prediction varivance matrix
In formula
J=argmaxjPr(j|Zk) (45)
Wherein, function argmax [f (x)] returns to the value of the x when f (x) is maximum, and
S1=P1 (53)
Its time renewal schematic flow sheet is as shown in Figure 2；
Step 4：Iteration variable updates
Each model measurement renewal
Each Model Condition probability is reset
Pr(J|Zk)=Prmax (59)
Pr[(3-J)|Zk]=Prmin (60)
；
Step 5：Measure renewal
Filtering gain matrix
State estimation
；
Step 6：Iterative calculation
K=k+1 (65)
Repeat step three, step 4 and step 5, until filtering terminates.

Claims (1)

  1. A kind of 1. multi-model self calibration kalman filter method, it is characterised in that：It includes following six step：
    Step 1：Establish system fundamental equation
    Multi-model self calibration Kalman filtering is transported using two methods of self calibration Kalman filtering and standard Kalman filtering Calculate, therefore system includes two state equations, first is self calibration state equation, and second state equation for standard, it has Body expression formula is
    <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>1</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>2</mn> <mo>)</mo> </mrow> </mrow>
    Zk=HkXk+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, ΦkAnd HkRespectively state-transition matrix and measurement matrix, bkRepresent unknown defeated Enter, WkWith VkRespectively system noise vector sum measures noise vector, and its variance matrix is respectively QkAnd Rk, 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 the system being made up of formula (1), formula (2) and formula (3)
    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；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 the speed that guarantee probability recovers, from And the real-time of Kalman filtering is set to be guaranteed；
    Step 3：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>&amp;Phi;</mi> <mn>0</mn> </msub> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>8</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>&amp;Phi;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mn>...</mn> <mrow> <mo>(</mo> <mn>9</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>10</mn> <mo>)</mo> </mrow> </mrow>
    P2/1=Φ1P1Φ1 T+Q1············(11)
    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>12</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>13</mn> <mo>)</mo> </mrow> </mrow>
    In formula
    J=argmaxjPr(j|Zk)···········(14)
    <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> <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> <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>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>15</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>&amp;Phi;</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mn>...</mn> <mrow> <mo>(</mo> <mn>16</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>17</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>18</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>19</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>20</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>21</mn> <mo>)</mo> </mrow> </mrow>
    S1=P1··············(22)
    <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>23</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>24</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 4：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>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> <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>26</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>27</mn> <mo>)</mo> </mrow> </mrow>
    Each Model Condition probability is reset
    Pr(J|Zk)=Prmax············(28)
    Pr[(3-J)|Zk]=Prmin···········(29)
    ；
    Step 5：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>30</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> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <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> <mn>...</mn> <mrow> <mo>(</mo> <mn>31</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；It should be noted simultaneously that due to Multi-model self calibration kalman filter method is obtained by the fusion calculation of two model results, therefore in calculating process And transmission of the estimation error variance matrix between adjacent moment is not needed, so not providing P in calculation procedurekCalculating it is public Formula；
    Step 6：Iterative calculation
    The state estimation respectively obtained according to the two of the k moment modelsWith varivance matrix Pk, repeat step (three), (4) and (five), and then the state estimation at k+1 moment is obtained, reciprocal iteration, until filtering terminates.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108537360A (en) * 2018-03-01 2018-09-14 长安大学 One kind taking polyfactorial adaptable Kalman filter slide prediction method into account
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
CN117909850A (en) * 2024-03-18 2024-04-19 中铁电气化局集团有限公司 Carrier cable supporting device vibration signal processing method based on fusion algorithm

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108537360A (en) * 2018-03-01 2018-09-14 长安大学 One kind taking polyfactorial adaptable Kalman filter slide prediction method into account
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
CN117909850A (en) * 2024-03-18 2024-04-19 中铁电气化局集团有限公司 Carrier cable supporting device vibration signal processing method based on fusion algorithm
CN117909850B (en) * 2024-03-18 2024-06-04 中铁电气化局集团有限公司 Carrier cable supporting device vibration signal processing method based on fusion algorithm

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Application publication date: 20180126