CN107783944A - A kind of multi-model self calibration Unscented kalman filtering method - Google Patents

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

Info

Publication number
CN107783944A
CN107783944A CN201710853217.0A CN201710853217A CN107783944A CN 107783944 A CN107783944 A CN 107783944A CN 201710853217 A CN201710853217 A CN 201710853217A CN 107783944 A CN107783944 A CN 107783944A
Authority
CN
China
Prior art keywords
mrow
msub
msubsup
mover
mtd
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN201710853217.0A
Other languages
Chinese (zh)
Inventor
杨海峰
傅惠民
张勇波
王治华
肖梦丽
崔轶
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Beihang University
Original Assignee
Beihang University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Beihang University filed Critical Beihang University
Priority to CN201710853217.0A priority Critical patent/CN107783944A/en
Publication of CN107783944A publication Critical patent/CN107783944A/en
Pending legal-status Critical Current

Links

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/18—Complex mathematical operations for evaluating statistical data, e.g. average values, frequency distributions, probability functions, regression analysis

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Mathematical Physics (AREA)
  • Data Mining & Analysis (AREA)
  • Mathematical Analysis (AREA)
  • Theoretical Computer Science (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Computational Mathematics (AREA)
  • Algebra (AREA)
  • Databases & Information Systems (AREA)
  • Software Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Operations Research (AREA)
  • Computing Systems (AREA)
  • Life Sciences & Earth Sciences (AREA)
  • Bioinformatics & Cheminformatics (AREA)
  • Bioinformatics & Computational Biology (AREA)
  • Evolutionary Biology (AREA)
  • Probability & Statistics with Applications (AREA)
  • Medicines Containing Antibodies Or Antigens For Use As Internal Diagnostic Agents (AREA)

Abstract

The present invention provides a kind of multi-model self calibration Unscented kalman 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 Unscented kalman filtering and self calibration Unscented kalman 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 strongly non-linear system exploitation, face is more extensive for benefiting from practical implementation, has very positive application value.

Description

A kind of multi-model self calibration Unscented kalman filtering method
【Technical field】
The present invention provides a kind of multi-model self calibration Unscented kalman filtering method, belongs to Robust Kalman Filter technology neck Domain.
【Background technology】
The problem of system state equation is influenceed by Unknown worm is the problem of generally existing in engineering.Traditional Kalman filtering Method either linear system or nonlinear system, it is accurate to require system equation, therefore can not be solved above-mentioned It is difficult.Document " self calibration Kalman filter method [J] aviation power journal .2014,29 (06):1363-1368 " proposes one Kind self calibration kalman filter method (Self-calibration Kalman Filter, SKF), this method is according to original shape While state equation is iterated computing, Unknown worm item is estimated, so that the influence of Unknown worm automatically derives benefit Repay.On this basis, researcher has successively developed self calibration EKF method (Self-calibration again Extended Kalman Filter, SEKF) and self calibration Unscented kalman filtering method (Self-calibration Unscented Kalman Filter, SUKF), self-calibration technique has been promoted the use of non-linear field by them.But due to The presence of system uncertain factor, Unknown worm also promising zero possibility.In this case, self calibration kalman filter method Due to introducing the estimation to Unknown worm item in prior estimate, although the estimate very little, its filtering accuracy is still too late not to be had There is the standard Kalman filtering method for considering that Unknown worm influences.
It is based on to further lift self calibration Kalman filtering in the filtering accuracy that Unknown worm is zero section, researcher Multiple model estimation theory, it is proposed that multi-model self calibration kalman filter method (Multiple-model Self- Calibration Kalman Filter, MSKF).This method is filtered using KF and SKF simultaneously, according to Bayes' theorem The weight of both real-time updates prior estimate, final state estimation is obtained by Weighted Fusion, has given full play to two Kind method is in the respective advantage of different phase.In addition, by the way that nonlinear system is linearized, a kind of multi-model self calibration expansion card Kalman Filtering method (Multiple-model Self-calibration Extended Kalman Filter, MSEKF) quilt Developing, it can solve state estimation problem of the weakly non-linear system under the influence of Unknown worm, but in engineering The relevant issues run into common strongly non-linear system, this method do not apply to simultaneously.
【The content of the invention】
It is an object of the invention to provide a kind of multi-model self calibration Unscented kalman filtering method (Multiple-model Self-calibration Unscented Kalman Filter, MSUKF), it is by the way that Multiple model estimation theory is incorporated into In SUKF, MSKF application strong nonlinearity field has been extended to.It is calculated using UKF and SUKF simultaneously, in real time The two weight of renewal, and then obtain state estimation.
A kind of multi-model self calibration Unscented kalman filtering method of the present invention, it includes following six step：
Step 1：Establish system fundamental equation
Multi-model self calibration Unscented kalman filtering is using self calibration Unscented kalman filtering and Unscented kalman filtering 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：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 the present invention only Choose two kinetic models and priori estimates are chosen using maximum probability method, it is only necessary to the probability of two kinds of models of qualitative analysis For size without accurate the characteristics of calculating probable value, multi-model self calibration kalman filter method does not use what is be calculated Conditional probability value is iterated, but sets the probability initial value Pr of two determinationsminAnd Prmax=1-Prmin；Filtered in each step Before ripple, it is assigned to two models by the size by comparing previous step probability results respectively, and is updated using them as initial value The model probability at 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；Based on without mark card The general recurrence formula of Kalman Filtering, firstly the need of calculating Sigma point sets { χ in time renewal processi}
As k=1,2
State one-step prediction value
Xk/k-1,i=f (χk-1,i) (9)
In formula, λ=α2(n+ κ)-n is the scale parameter of Sigma points, and n is state vector XkDimension, κ is adjustment parameter, It is general take 0 or 3-n, α be another adjustment parameter；
One-step prediction varivance matrix
In formula,For covariance weight coefficient, 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 Unscented kalman The general recurrence formula of ripple carries out resampling to system mode, obtains new Sigma point sets { 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 Unscented kalman filtering and the filter of self calibration Unscented kalman Two methods of the result of calculation of ripple, relies on the Multiple model estimation theory based on Bayes principle, and 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 Be, the present invention be directed to strongly non-linear system exploitation, in practical implementation to benefit from face more extensive, have very Positive application value.
In summary, beneficial effects of the present invention include
(1) Multiple model estimation theory is incorporated into self calibration Unscented kalman filtering method, has been derived by multi-model The complete procedure of self calibration Unscented kalman filtering method.
(2) Unscented kalman filtering method and oneself can be played respectively using automatic identification Unknown worm as zero section and non-zero section Calibrate the respective advantage of Unscented kalman filtering method.
(3) filtering accuracy when nonlinear system is influenceed by Unknown worm is further increased, while improves and filtered The stability of journey, enhance the robustness of system.
【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(·)：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}：Sigma point sets
λ：The scale parameter of Sigma points
n：The dimension of state vector
κ、α：Adjustment parameter
Average weight coefficient
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
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 Unscented kalman filtering 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 six step：
Step 1：Establish system fundamental equation
Zk=hk(Xk)+Vk (43)
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 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 (47)
And the probability initial value Pr for iterative calculationmaxAnd Prmin。
Step 3：Time renewal is carried out to system
As k=1,2
State one-step prediction value
Xk/k-1,i=f (χk-1,i) (49)
In formula, λ=α2(n+ κ)-n is the scale parameter of Sigma points, and n is state vector XkDimension, κ is adjustment parameter, It is general take 0 or 3-n, α be another adjustment parameter.
One-step prediction varivance matrix
In formula,For covariance weight coefficient, 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 to system mode, obtains new Sigma point sets { χk/k-1,i}
Calculate and measure 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
Calculate 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 Unscented kalman filtering method, it is characterised in that：It includes following six step：
    Step 1：Establish system fundamental equation
    Multi-model self calibration Unscented kalman filtering is using self calibration Unscented kalman filtering and two kinds of sides of Unscented kalman filtering 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> <mo>...</mo> <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> <mo>...</mo> <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> <mo>...</mo> <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 (2Z3)=0.5 (7)
    And the probability initial value Pr for iterative calculationmaxAnd Prmin；Two kinetic models are only chosen for the present invention and are adopted Priori estimates are chosen with maximum probability method, it is only necessary to which the probability size of two kinds of models of qualitative analysis calculates generally without accurate The characteristics of rate value, multi-model self calibration kalman filter method do not use the conditional probability value being calculated and are iterated, and It is the probability initial value Pr for setting two determinationsminAnd Prmax=1-Prmin；Before the filtering of each step, by comparing previous step It is assigned to two models by the size of probability results respectively, and the model probability at current time is updated using them as initial value；By In PrminIt is not minimum as probability lower limit, therefore can guarantee that the speed that probability recovers, so that Kalman filtering Real-time 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；Based on Unscented kalman The general recurrence formula of filtering, firstly the need of calculating Sigma point sets { χ in time renewal processi}
    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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>m</mi> <mi>i</mi> </msubsup> <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> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    In formula, λ=α2(n+ κ)-n is the scale parameter of Sigma points, and n is state vector XkDimension, κ is adjustment parameter, typically Take 0 or 3-n, α be another adjustment parameter；
    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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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,For covariance weight coefficient, its calculation formula is
    <mrow> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>i</mi> <mo>&amp;NotEqual;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mi>&amp;lambda;</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>m</mi> <mi>i</mi> </msubsup> <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> </mrow> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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> <mi>1</mi> </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> <mo>...</mo> <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> <mi>/</mi> <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> <mi>/</mi> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mo>...</mo> <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> <mo>...</mo> <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> <mo>...</mo> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>m</mi> <mi>i</mi> </msubsup> <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> </mrow> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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> <mo>,</mo> <mi>i</mi> </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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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>j</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, according to Unscented kalman filtering General recurrence formula carries out resampling to system mode, obtains new Sigma point sets { 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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </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> <mrow> <mo>(</mo> <msqrt> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>)</mo> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>/</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>m</mi> <mi>i</mi> </msubsup> <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> </mrow> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <msubsup> <mi>w</mi> <mi>c</mi> <mi>i</mi> </msubsup> <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> <mo>...</mo> <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> <mo>...</mo> <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.
CN201710853217.0A 2017-09-20 2017-09-20 A kind of multi-model self calibration Unscented kalman filtering method Pending CN107783944A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201710853217.0A CN107783944A (en) 2017-09-20 2017-09-20 A kind of multi-model self calibration Unscented kalman filtering method

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201710853217.0A CN107783944A (en) 2017-09-20 2017-09-20 A kind of multi-model self calibration Unscented kalman filtering method

Publications (1)

Publication Number Publication Date
CN107783944A true CN107783944A (en) 2018-03-09

Family

ID=61437920

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201710853217.0A Pending CN107783944A (en) 2017-09-20 2017-09-20 A kind of multi-model self calibration Unscented kalman filtering method

Country Status (1)

Country Link
CN (1) CN107783944A (en)

Cited By (13)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108599737A (en) * 2018-04-10 2018-09-28 西北工业大学 A kind of design method of the non-linear Kalman filtering device of variation Bayes
CN108680162A (en) * 2018-04-20 2018-10-19 浙江工业大学 A kind of human body target tracking method based on progressive Unscented kalman filtering
CN109100649A (en) * 2018-06-25 2018-12-28 南京南瑞继保电气有限公司 Parameter estimation method for generator excitation system and speed regulation system based on phasor measurement
CN109240289A (en) * 2018-09-01 2019-01-18 哈尔滨工程大学 Wave glider yawing information self-adapting filtering method
CN109269497A (en) * 2018-07-31 2019-01-25 哈尔滨工程大学 Based on AUV cutting method to the multiple dimensioned Unscented kalman filtering estimation method of rate pattern
CN110109049A (en) * 2019-03-27 2019-08-09 北京邮电大学 Unscented kalman filtering method and device for the estimation of extensive aerial angle
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
CN110580661A (en) * 2019-08-07 2019-12-17 云南电网有限责任公司昆明供电局 32-path vector Fourier and Kalman correction algorithm for power protection
CN111336246A (en) * 2020-04-03 2020-06-26 中国北方车辆研究所 State observation method for electro-hydraulic control system of vehicle transmission device
CN111551785A (en) * 2020-04-29 2020-08-18 南京理工大学 Frequency and harmonic detection method based on unscented Kalman filter
CN112989260A (en) * 2021-02-07 2021-06-18 湖北理工学院 Interactive multi-model filtering method of automatic feeding weighing system
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method
CN114647935A (en) * 2022-03-11 2022-06-21 北京航空航天大学 Composite material structure reliability real-time evaluation method considering time-varying failure threshold

Cited By (21)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108599737A (en) * 2018-04-10 2018-09-28 西北工业大学 A kind of design method of the non-linear Kalman filtering device of variation Bayes
CN108599737B (en) * 2018-04-10 2021-11-23 西北工业大学 Design method of nonlinear Kalman filter of variational Bayes
CN108680162A (en) * 2018-04-20 2018-10-19 浙江工业大学 A kind of human body target tracking method based on progressive Unscented kalman filtering
CN109100649B (en) * 2018-06-25 2020-10-16 南京南瑞继保电气有限公司 Parameter estimation method for generator excitation system and speed regulation system based on phasor measurement
CN109100649A (en) * 2018-06-25 2018-12-28 南京南瑞继保电气有限公司 Parameter estimation method for generator excitation system and speed regulation system based on phasor measurement
CN109269497A (en) * 2018-07-31 2019-01-25 哈尔滨工程大学 Based on AUV cutting method to the multiple dimensioned Unscented kalman filtering estimation method of rate pattern
CN109269497B (en) * 2018-07-31 2022-04-12 哈尔滨工程大学 Multi-scale unscented Kalman filtering estimation method based on AUV tangential velocity model
CN109240289A (en) * 2018-09-01 2019-01-18 哈尔滨工程大学 Wave glider yawing information self-adapting filtering method
CN110109049A (en) * 2019-03-27 2019-08-09 北京邮电大学 Unscented kalman filtering method and device for the estimation of extensive aerial angle
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
CN110580661A (en) * 2019-08-07 2019-12-17 云南电网有限责任公司昆明供电局 32-path vector Fourier and Kalman correction algorithm for power protection
CN110580661B (en) * 2019-08-07 2023-04-25 云南电网有限责任公司昆明供电局 32-path vector Fourier-Kalman correction algorithm for power protection
CN111336246A (en) * 2020-04-03 2020-06-26 中国北方车辆研究所 State observation method for electro-hydraulic control system of vehicle transmission device
CN111336246B (en) * 2020-04-03 2021-07-02 中国北方车辆研究所 State observation method for electro-hydraulic control system of vehicle transmission device
CN111551785B (en) * 2020-04-29 2022-08-12 南京理工大学 Frequency and harmonic detection method based on unscented Kalman filter
CN111551785A (en) * 2020-04-29 2020-08-18 南京理工大学 Frequency and harmonic detection method based on unscented Kalman filter
CN112989260A (en) * 2021-02-07 2021-06-18 湖北理工学院 Interactive multi-model filtering method of automatic feeding weighing system
CN112989260B (en) * 2021-02-07 2022-11-01 湖北理工学院 Interactive multi-model filtering method of automatic feeding weighing system
CN113091748A (en) * 2021-04-12 2021-07-09 北京航空航天大学 Indoor self-calibration navigation positioning method
CN114647935A (en) * 2022-03-11 2022-06-21 北京航空航天大学 Composite material structure reliability real-time evaluation method considering time-varying failure threshold

Similar Documents

Publication Publication Date Title
CN107783944A (en) A kind of multi-model self calibration Unscented kalman filtering method
CN106487358B (en) A kind of maneuvering target turning tracking
CN107547067A (en) A kind of multi-model self calibration EKF method
CN106767780B (en) The extension ellipsoid set-membership filtering method approached based on Chebyshev polynomial interopolations
CN104713560B (en) Multi-source distance measuring sensor spatial registration method based on expectation maximization
CN107402381A (en) A kind of multiple maneuver target tracking methods of iteration self-adapting
CN107590317A (en) A kind of generator method for dynamic estimation of meter and model parameter uncertainty
CN103063212B (en) A kind of Combinated navigation method based on nonlinear mapping ADAPTIVE MIXED Kalman/H ∞ wave filter
CN107807906A (en) A kind of multi-model self calibration order filtering method
CN110501696B (en) Radar target tracking method based on Doppler measurement adaptive processing
CN109186601A (en) A kind of laser SLAM algorithm based on adaptive Unscented kalman filtering
CN104021289B (en) Non-Gaussian unsteady-state noise modeling method
CN107565931A (en) A kind of self calibration Unscented kalman filtering method
CN108763167A (en) A kind of adaptive filter method of variation Bayes
CN103973263B (en) Approximation filter method
CN104166989B (en) Rapid ICP method for two-dimensional laser radar point cloud matching
CN103684350B (en) A kind of particle filter method
CN103237320B (en) Wireless sensor network quantizes based on mixing the method for tracking target that Kalman is merged
CN105929340A (en) Method for estimating battery SOC based on ARIMA
CN107797106A (en) A kind of PHD multiple target tracking smooth filtering methods of the unknown clutter estimations of acceleration EM
CN105701292B (en) A kind of parsing discrimination method of maneuvering target turning rate
CN107632959A (en) A kind of multi-model self calibration kalman filter method
CN106707234B (en) A kind of sensor network target localization method for combining delay inequality and angle measurement
CN106054167A (en) Intensity filter-based multi-extended target tracking method
CN102005050B (en) Gaussian log model single-side curvature threshold fitting method used for change detection

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
WD01 Invention patent application deemed withdrawn after publication

Application publication date: 20180309

WD01 Invention patent application deemed withdrawn after publication