CN108847829A - It is a kind of that uncertain and quantization measurement non-linear filtering method occurring with random - Google Patents

It is a kind of that uncertain and quantization measurement non-linear filtering method occurring with random Download PDF

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CN108847829A
CN108847829A CN201810814464.4A CN201810814464A CN108847829A CN 108847829 A CN108847829 A CN 108847829A CN 201810814464 A CN201810814464 A CN 201810814464A CN 108847829 A CN108847829 A CN 108847829A
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filtering
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CN108847829B (en
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胡军
赵文杰
贾朝清
崔云菲
贾凤娇
高铭
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Harbin University of Science and Technology
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H21/00—Adaptive networks
    • H03H21/0012—Digital adaptive filters
    • H03H21/0016—Non linear filters
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H21/00—Adaptive networks
    • H03H21/0012—Digital adaptive filters
    • H03H21/0043—Adaptive algorithms

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Abstract

The present invention provides a kind of uncertain with random generation and quantifies the non-linear filtering method of measurement, belongs to State Estimation field.The present invention initially sets up with the random nonlinear and time-varying system dynamic model that uncertainty and quantization measurement occurs, is filtered device design to dynamic model；Then the upper bound of one-step prediction error co-variance matrix is calculated；Filtering gain matrix K is calculated by the upper bound of one-step prediction error co-variance matrixk+1；Again by filtering gain matrix Kk+1It substitutes into the filter of step 2, obtains the state estimation at k+1 momentAnd according to calculated filtering gain matrix Kk+1, calculate the upper bound Σ of filtering error covariance matrixk+1|k+1；It repeats the above steps, reaches filtering total duration until meeting.The present invention solves existing filtering technique cannot handle random generation uncertainty and quantization measurement, and then the problem for causing filtering error big simultaneously.The present invention can be used for the filtering of nonlinear and time-varying system.

Description

It is a kind of that uncertain and quantization measurement non-linear filtering method occurring with random
Technical field
Uncertain and quantization measurement non-linear filtering method occurs with random the present invention relates to a kind of, belongs to state Estimation technique field.
Background technique
Filtering be by the operation that specific band frequency filters out in signal, be selection signal and inhibit one of interference it is basic and Important measures.Filtering belong in the controls it is important study a question, in radar range finding, Target Tracking System, Image Acquisition It is widely applied in the signal estimation task in equal fields.Since the redundancy and channel width of data limit, when data pass through net When network is transferred to filter end, it will usually cause some network inducing phenomenas to occur, such as network congestion, delay, design adapt to The filtering algorithm of these network inducing phenomenas is very necessary.
Current existing method cannot handle with signal quantization simultaneously and occur at random probabilistic robust filtering side Method, reduces the accuracy of filtering algorithm, and filtering error is big；
When existing method ignores signal quantization, especially random signal quantifies, and improves the data processing difficulty of filter, Filtering performance can be reduced；
Existing method does not fully consider that Nonlinear Stochastic and random generation are uncertain, reduces the anti-of filtering algorithm Disturbance ability.
Summary of the invention
The present invention is to solve existing filtering technique to handle random generation uncertainty and quantization measurement simultaneously, and then lead The problem for causing filtering error big provides a kind of with the random non-linear filtering method that uncertainty and quantization measurement occurs.
It is of the present invention a kind of with the random non-linear filtering method that uncertainty and quantization measurement occurs, by following Technical solution is realized：
Step 1: establishing has the random nonlinear and time-varying system dynamic model that uncertainty and quantization measurement occurs；
Step 2: the dynamic model established to step 1 is filtered device design；
Step 3: calculating the upper bound Σ of one-step prediction error co-variance matrixk+1|k；
Step 4: the upper bound Σ of the one-step prediction error co-variance matrix obtained according to step 3k+1|k, calculate the k+1 moment Filtering gain matrix Kk+1；
Step 5: the filtering gain matrix K that will be obtained in step 4k+1It substitutes into the filter of step 2, when obtaining k+1 The state estimation at quarter
Judge whether k+1 reaches filtering total duration M, if k+1<M thens follow the steps six, if k+1=M, terminates；
Step 6: according to filtering gain matrix K calculated in step 4k+1, calculate filtering error covariance matrix Upper bound Σk+1|k+1；K=k+1 is enabled, enters step two, until meeting k+1=M.
Present invention feature the most prominent and significant beneficial effect are：
It is according to the present invention a kind of with the random non-linear filtering method that uncertainty and quantization measurement occurs, simultaneously The random influence that uncertainty and quantization measurement occurs to filtering performance is considered, is obtained using the method for Extended Kalman filter The concrete form of filtering algorithm, compared with the filtering method of existing nonlinear time_varying system, the present invention can be handled simultaneously Nonlinear disturbance, it is random uncertain and quantization measurement occurs, be satisfied the filtering method of recursive form, reached resist it is non- Linear perturbation and probabilistic purpose occurs at random, the present invention is suitable for the robust filtering problem of nonlinear and time-varying system.
For having the random nonlinear and time-varying system that uncertainty and quantization measurement occurs, the filter invented is set Meter method can effectively estimate dbjective state and the mark in the filtering error covariance upper bound can be maintained at corresponding square Above error.Compared to existing method, filters relative error and reduce about 25%.
Detailed description of the invention
Fig. 1 is the method for the present invention flow chart；
Fig. 2 is the virtual condition path x of system1,kPath is filtered with itComparison diagram, solid line is the reality of system in figure Border Phase Pathway x1,k, dotted line is corresponding filtering path
Fig. 3 is the virtual condition path x of system2,kPath is filtered with itComparison diagram, solid line is the reality of system in figure Border Phase Pathway x2,k, dotted line is corresponding filtering path
Fig. 4 is the relational graph of the mark in log (MSE1) and the corresponding filtering error covariance upper bound, and solid line is x in figure1,k? The logarithm log (MSE1) of square error, dotted line are mark (the MSE1 expression state x in the corresponding filtering error covariance upper bound1,kEstimate Count mean square error)；
Fig. 5 is the relational graph of the mark in log (MSE2) and the corresponding filtering error covariance upper bound, and solid line is x in figure2,k? The logarithm log (MSE2) of square error, dotted line are mark (the MSE2 expression state x in the corresponding filtering error covariance upper bound2,kEstimate Count mean square error).
Specific embodiment
Specific embodiment one：Present embodiment is illustrated in conjunction with Fig. 1, one kind that present embodiment provides have with Uncertain and quantization measurement non-linear filtering method occurs for machine, specifically includes following steps：
Step 1: establishing has the random nonlinear and time-varying system dynamic model that uncertainty and quantization measurement occurs；
Step 2: the dynamic model established to step 1 is filtered device design；
Step 3: calculating the upper of one-step prediction error co-variance matrix according to one-step prediction error and mathematical processing methods Boundary Σk+1|k；
Step 4: the upper bound Σ of the one-step prediction error co-variance matrix obtained according to step 3k+1|k, calculate the k+1 moment Filtering gain matrix Kk+1；
Step 5: the filtering gain matrix K that will be obtained in step 4k+1It substitutes into the filter of step 2, when obtaining k+1 The state estimation at quarter
Judge whether k+1 reaches filtering total duration M, if k+1<M thens follow the steps six, if k+1=M, terminates；
Step 6: according to filtering gain matrix K calculated in step 4k+1, calculate filtering error covariance matrix Upper bound Σk+1|k+1；K=k+1 is enabled, enters step two, until meeting k+1=M.
Specific embodiment two：The present embodiment is different from the first embodiment in that described in step 1 have with State space form that is uncertain and quantifying the nonlinear and time-varying system dynamic model measured occurs for machine：
xk+1=(Ak+αk△Ak)xk+f(xk,ξk)+Bkωk (1)
yk=Ckxk+νk (2)
Wherein, xkIt is the state vector in k moment nonlinear and time-varying system dynamic model, xk+1For the k+1 moment it is non-linear when Become the state vector in system dynamic model；ykThe measurement output vector of etching system when for k；AkIt is k moment nonlinear and time-varying system Sytem matrix, BkIt is noise profile matrix, the C of k moment nonlinear and time-varying systemkIt is the measurement of k moment nonlinear and time-varying system Matrix, Ak、BkAnd CkIt is known matrix；ξkIt is the Gaussian sequence of zero-mean；ωkIt is to be desired for zero, variance Qk's Process noise, Qk>0；νkIt is to be desired for zero, variance RkMeasurement noise, Rk>0；△AkFor Bounded uncertainties matrix, Meet △ Ak=HkFkMk, wherein HkFor △ AkLeft end metric matrix, MkFor △ AkRight end metric matrix, FkMeet conditionαkIt is the stochastic variable for obeying Bernoulli Jacob's distribution, meets following statistical property：
Wherein Prob { } indicates the probability of happening of event, For αkMathematic expectaion；f(xk,ξk) it is random generation nonlinear function, meet Wherein fT(xj,ξj) Indicate f (xk,ξk) transposition,Indicate mathematic expectaion, ΠiAnd ΓiIt is known matrix, ΠiFor i-th of nonlinear function Weight matrix, ΓiFor the quadratic form matrix of i-th of state, i ΠiAnd ΓiSerial number, i ∈ { 1 ..., s }, s are non-linear letters The sum of number weight matrix.
Other steps and parameter are same as the specific embodiment one.
Specific embodiment three：Present embodiment is unlike specific embodiment two, filter described in step 2 The detailed process of design includes：
Firstly, exporting y to measurementkQuantified to obtain q (yk)：
Wherein q () is indicated to quantifying, qjIndicate q's J-th of component,For vector ykJ-th of component,For j-th of logarithmic quantization device；J=1,2 ..., m；
After handling by quantizer, the measured value that filter termination receives is：
Wherein, Λk:=diag { λk,1,λk,2,…,λk,m, λk,jIt is the stochastic variable for obeying Bernoulli Jacob's distribution, meetsWithWhereinIt is λk,jMathematic expectaion, Prob () indicate thing The probability of happening of part；
Construct following filter formula：
Wherein,For xkIn the state estimation vector at k moment,For the state estimation at k+1 moment,For xk In the one-step prediction at k moment,It is the measured value at the k+1 moment after quantification treatment,It is Λk+1Mathematic expectaion, Kk+1For the filtering gain matrix at k+1 moment.
Other steps and parameter are identical with embodiment two.
Specific embodiment four：Present embodiment is unlike specific embodiment three, logarithmic quantization device's Specifically calculating process is：
For eachThere is following quantization level：
Wherein, χ(j)Quantization resolution is represented,For the quantization level initial value of corresponding j-th of element,For quantization level Value；Quantization level embodies the degree of quantization.
Then, logarithmic quantization deviceForm is：
AndIt follows that
Wherein,Enable γ=diag { δ1,δ2,…δm,WithIt is to meet conditionMatrix, wherein γ-1For the inverse matrix of γ,It is matrix Transposition, diag { δ1,δ2,…δmIt is diagonal matrix, the element on diagonal line is respectively δ1,δ2,…δm,It is a diagonal matrix, diagonal entry is followed successively by WithActually be not in, therefore do not consider.
Other steps and parameter are identical with embodiment two.
Specific embodiment five：Unlike specific embodiment three or four, step 3 is specially present embodiment：
According to formulaObtain the one-step prediction error at k momentThen it utilizesCalculate the one-step prediction error co-variance matrix P at k momentk+1|k, but due in one-step prediction error There are indeterminate in covariance matrix, its exact value is hardly resulted in, therefore utilizes inequality processing method, by Pk+1|kIn it is unknown Item is processed into the form of known matrix, to find the upper bound of one-step prediction error co-variance matrix, i.e. Pk+1|k≤Σk+1|k。
Calculate one-step prediction error co-variance matrix Pk+1|kUpper bound Σk+1|k：
Wherein, Σk|kFor filtering error covariance matrix Pk|kThe upper bound,WithRespectively Ak、Bk、 MkAnd HkTransposition,ε1、ε2It is weight coefficient, is Known normal amount,WithRespectively indicate ε1And ε2Inverse,ForTransposition,Representing matrixMark.
Other steps and parameter are identical as specific embodiment three or four.
Specific embodiment six：Present embodiment is unlike specific embodiment five, the k+1 moment described in step 4 Filtering gain matrix Kk+1Specific calculating process be：
Wherein, AndyK+1,1It is to meet inequalityStochastic variable, ε3、ε4、ε5、ε6It is weight coefficient, is known constant；Respectively γk+1,1、ε3、ε4、 ε5、ε6Inverse；I indicates that unit matrix, ο are Hadamard (Hadamard) products,For Ck+1Transposition,It is random Matrix Λk+1Mathematic expectaion, the mark of tr () representing matrix, diag { } indicate diagonal matrix.
Other steps and parameter are identical as specific embodiment five.
Specific embodiment seven：Present embodiment is filtered described in step 6 and is missed unlike specific embodiment six The specific calculating process in the upper bound of poor covariance matrix is：
According to formulaObtain the filtering error at k+1 momentThen it utilizes Calculate the filtering error covariance matrix P at k+1 momentk+1|k+1, but due to existing in filtering error covariance matrix not really Determine item, hardly result in its exact value, therefore utilize inequality processing method, by Pk+1|k+1In the unknown be processed into known matrix Form, to find the upper bound of filtering error covariance matrix, i.e. Pk+1|k+1≤Σk+1|k+1。
Wherein, Σk+1|k+1For filtering error covariance matrix Pk+1|k+1The upper bound,WithRespectively ForAnd Kk+1Transposition.
Other steps and parameter are identical as specific embodiment six.
Embodiment
Beneficial effects of the present invention are verified using following embodiment：
System parameter：
F=sin (3k), Hk=[0.02 0.03]T,Mk=[0.02 0.04],
In addition, xk=[x1,k x2,k]T,
Qk=0.15, Rk=0.2,χ(1)=0.4, ε1= 0.01, ε2=0.1, ε3=1, ε4=1, ε5=0.1, ε6=1, γk+1,1=0.42,And system Initial valueΣ0|0=2I2(I2It is the unit matrix that dimension is 2).
Filtering method of the invention is carried out using above-mentioned parameter, filter effect is as shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5：
Fig. 2 is the virtual condition path x of system1,kPath is filtered with itComparison diagram；Fig. 3 is the practical shape of system State path x2,kPath is filtered with itComparison diagram；Fig. 4 is log (MSE1) and the corresponding filtering error covariance upper bound The relational graph of mark；Fig. 5 is the relational graph of the mark in log (MSE2) and the corresponding filtering error covariance upper bound.
By Fig. 2 to Fig. 5 as it can be seen that uncertain and quantization measurement nonlinear and time-varying system occurs for random, The filter design method invented can effectively estimate dbjective state and the mark in the filtering error covariance upper bound can It is maintained above corresponding mean square error.Compared to existing method, filtering error reduces about 25%.
The present invention can also have other various embodiments, without deviating from the spirit and substance of the present invention, this field Technical staff makes various corresponding changes and modifications in accordance with the present invention, but these corresponding changes and modifications all should belong to The protection scope of the appended claims of the present invention.

Claims (7)

1. a kind of occur uncertain and quantization measurement non-linear filtering method with random, which is characterized in that the method Specifically include following steps：
Step 1: establishing has the random nonlinear and time-varying system dynamic model that uncertainty and quantization measurement occurs；
Step 2: the dynamic model established to step 1 is filtered device design；
Step 3: calculating the upper bound Σ of one-step prediction error co-variance matrixk+1|k；
Step 4: the upper bound Σ of the one-step prediction error co-variance matrix obtained according to step 3k+1|k, calculate the k+1 moment filter Gain matrix Kk+1；
Step 5: the filtering gain matrix K that will be obtained in step 4k+1It substitutes into the filter of step 2, obtains the k+1 moment State estimation
Judge whether k+1 reaches filtering total duration M, if k+1<M thens follow the steps six, if k+1=M, terminates；
Step 6: according to filtering gain matrix K calculated in step 4k+1, calculate the upper bound of filtering error covariance matrix Σk+1|k+1；K=k+1 is enabled, enters step two, until meeting k+1=M.
2. a kind of according to claim 1 occur uncertain and quantization measurement non-linear filtering method with random, It is characterized in that there is the random nonlinear and time-varying system dynamic model that uncertainty and quantization measurement occurs described in step 1 State space form is：
xk+1=(Ak+αk△Ak)xk+f(xk,ξk)+Bkωk (1)
yk=Ckxk+νk (2)
Wherein, xkIt is the state vector in k moment nonlinear and time-varying system dynamic model, xk+1For k+1 moment nonlinear time-varying system State vector in system dynamic model；ykThe measurement output vector of etching system when for k；AkIt is that k moment nonlinear and time-varying system is System matrix, BkIt is noise profile matrix, the C of k moment nonlinear and time-varying systemkIt is the measurement square of k moment nonlinear and time-varying system Battle array；ξkIt is the Gaussian sequence of zero-mean；ωkIt is to be desired for zero, variance QkProcess noise, Qk>0；νkIt is to be desired for Zero, variance RkMeasurement noise, Rk>0；△AkFor Bounded uncertainties matrix, meet △ Ak=HkFkMk, wherein HkFor △AkLeft end metric matrix, MkFor △ AkRight end metric matrix, FkMeet conditionαkIt is to obey Bernoulli Jacob's distribution Stochastic variable, meet following statistical property：
Wherein Prob { } indicates the probability of happening of event,For αk Mathematic expectaion；f(xk,ξk) it is random generation nonlinear function, meet Wherein fT(xj,ξj) Indicate f (xk,ξk) transposition,Indicate mathematic expectaion, ΠiFor i-th of nonlinear function weight matrix, ΓiFor i-th of shape The quadratic form matrix of state, i ΠiAnd ΓiSerial number, i ∈ { 1 ..., s }, s are the sums of nonlinear function weight matrix.
3. a kind of according to claim 2 occur uncertain and quantization measurement non-linear filtering method with random, It is characterized in that, the detailed process of the design of filter described in step 2 includes：
Firstly, exporting y to measurementkQuantified to obtain q (yk)：
For logarithmic quantization device；J=1,2 ..., m；
After handling by quantizer, the measured value that filter termination receives is：
Wherein, Λk:=diag { λk,1,λk,2,…,λk,m, λk,jIt is the stochastic variable for obeying Bernoulli Jacob's distribution, meetsWithWhereinIt is λk,jMathematic expectaion；
Construct following filter formula：
Wherein,For xkIn the state estimation vector at k moment,For the state estimation at k+1 moment,For xkIn k The one-step prediction at quarter,It is the measured value at the k+1 moment after quantification treatment,It is Λk+1Mathematic expectaion, Kk+1For k The filtering gain matrix at+1 moment.
4. a kind of according to claim 3 occur uncertain and quantization measurement non-linear filtering method with random, It is characterized in that, logarithmic quantization deviceSpecific calculating process be：
Quantization levelFor：
Wherein, χ(j)Quantization resolution is represented,For the quantization level initial value of corresponding j-th of element,For taking for quantization level Value；
Then：
AndIt follows that
Wherein,Enable Υ=diag { δ1,δ2,…δm,With It is to meet conditionMatrix, wherein Υ-1For the inverse matrix of Υ,It is matrixTransposition, diag {δ1,δ2,…δmIt is diagonal matrix, the element on diagonal line is respectively δ1,δ2,…δm,It is one Diagonal matrix, diagonal entry are followed successively by。
5. according to a kind of non-linear filtering method with random generation uncertainty and quantization measurement of claim 3 or 4, It is characterized in that, step 3 is specially：
Calculate one-step prediction error co-variance matrix Pk+1|kUpper bound Σk+1|k：
Wherein, Σk|kFor filtering error covariance matrix Pk|kThe upper bound,WithRespectively Ak、Bk、MkAnd Hk Transposition,ε1、ε2It is weight coefficient,With Respectively indicate ε1And ε2Inverse,ForTransposition,Representing matrixMark.
6. a kind of according to claim 5 occur uncertain and quantization measurement non-linear filtering method with random, It is characterized in that, the filtering gain matrix of k+1 moment described in step 4 Kk+1Specific calculating process be：
Wherein, Andγk+1,1It is to meet inequalityStochastic variable, ε3、 ε4、ε5、ε6It is weight coefficient；Respectively γk+1,1、ε3、ε4、ε5、ε6Inverse；I indicates unit Battle array, o are Hadamard products,For Ck+1Transposition, For random matrix Λk+1Mathematic expectaion, the mark of tr () representing matrix, diag { } indicate diagonal matrix.
7. a kind of according to claim 6 occur uncertain and quantization measurement non-linear filtering method with random, It is characterized in that, the specific calculating process in the upper bound of filtering error covariance matrix described in step 6 is：
Wherein, Σk+1|k+1For filtering error covariance matrix Pk+1|k+1The upper bound,WithRespectivelyAnd Kk+1Transposition.
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