CN108228959A - Using the method for Random censorship estimating system virtual condition and using its wave filter - Google Patents
Using the method for Random censorship estimating system virtual condition and using its wave filter Download PDFInfo
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Abstract
The invention discloses a kind of method for Random censorship estimating system virtual condition and its wave filter is applied, including：Establish the state space equation of time-varying system；The state space equation includes：The measurement output of the state vector and system of system；The magnitude relationship structure exported according to the measurement of known threshold vector and the system, which deletes mistake and measures equation, obtains observable variable vector, by Random Variable Distribution Function describe described in delete the possibility for losing and measuring and occurring；The expectation of the observable variable vector is calculated according to total probability formula；It accumulates to obtain the expectation of matrix by Kronecker products and Hadamard；And obtain random matrix weighting covariance matrix；Using random matrix weighting covariance matrix as the covariance matrix of system motion process, the Kalman filtering algorithm that is expected that by using observable variable vector makes the state error covariance least estimated system virtual condition of the state space equation.And wave filter.
Description
Technical field
The present invention relates to field of signal processing, specifically a kind of method for Random censorship estimating system virtual condition
And apply its wave filter.
Background technology
In digital times, sensor measurement data is very universal.But since cell capacity is limited, data are deleted
Phenomenon is lost often to occur.Simultaneously according to pertinent literature, data delete mistake and also often appear in economy, the fields such as chemistry and information theory it
In.Therefore, it is extremely important to delete mistake phenomenon for research.Data delete mistake and necessarily information are caused to lose, then how information loss influences to filter
Wave or estimation performance are just highly studied.
One is that the data statistics of Random censorship how is made full use of under the frame of wave filter the problem of being worth challenge
Characteristic carrys out the virtual condition of preferably estimating system.
But existing filtering method has ignored this feature that Random censorship depends on measurement data.Ignore this dependence
Relationship, can influence the calculating of error co-variance matrix, and then affect the practical application of filtering algorithm.
Invention content
In view of this, the present invention provides a kind of method for Random censorship estimating system virtual condition and applies its filter
Wave device cannot effectively estimating system virtual condition and impact performance of filter with solving the problems, such as that data delete mistake.
In a first aspect, the present invention provides a kind of method for Random censorship estimating system virtual condition, including：
Establish the state space equation of time-varying system；
The state space equation, including：The measurement output of the state vector and system of system；
The magnitude relationship structure exported according to the measurement of known threshold vector and the system is deleted mistake measurement equation and is obtained
Observable variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to full probability public affairs
Formula calculates the expectation of the observable variable vector；
The coefficient matrix of the state vector of system previous step in the state vector of the system is decomposed, obtains desired value
The state vector coefficient matrix for the first previous step for being zero and the state vector coefficient matrix of the second previous step, by the system
All covariance element structure first information matrixes in first zero-mean gaussian random vector of state vector；
The random matrix of structure system；
Ask for the system random matrix and unit matrix Kronecker product after again with the first information matrix
Hadamard accumulates to obtain the expectation of matrix；
The Kronecker of unit of account column vector and unit matrix is accumulated；
It is multiplied by the unit of account column vector and unit matrix respectively in the desired both sides for obtaining matrix
The transposed matrix of Kronecker products and the matrix of the Kronecker of the unit of account column vector and unit matrix products, obtain
Random matrix weights covariance matrix；
Wherein, the element in the random matrix of the system is desired for zero, and known to the expectation of the first information matrix.
Using random matrix weighting covariance matrix as the covariance matrix of system motion process, using described considerable
Surveying the Kalman filtering algorithm that is expected that by of variable vector makes the state error covariance least estimated of the state space equation
System virtual condition.
Preferably, the state vector of the system by system previous step state vector and first zero-mean gaussian with
The combination of machine SYSTEM OF LINEAR VECTOR is formed；
The measurement output of the system is linear by the second zero-mean gaussian random vector of the state vector sum of the system
Combination is formed；
The coefficient matrix of the state vector of the system previous step and the coefficient matrix of the measurement output of system are only mutually
Vertical random parameter matrix；
The coefficient matrix of the state vector of the system of the measurement output of the system, is decomposed into first state system of vectors
Matrix number and the second state vector coefficient matrix, the desired value of the second state vector coefficient matrix is zero；
The second information matrix is built by covariance elements all in the second zero-mean gaussian random vector.
Preferably, according to the state space equation and the Kalman filtering, the state vector of first previous step
Coefficient matrix of the coefficient matrix for the state vector of system previous step, the state vector based on the system previous step, which predicts, is
Unified step predicted value.
Preferably, according to the Kalman filtering algorithm, based on random matrix weighting covariance matrix, described first
The coefficient matrix of the covariance matrix of zero-mean gaussian random vector and the first zero-mean gaussian random vector construction system
The covariance matrix of system motion process；
Wherein, the random matrix of the system in the random matrix weighting covariance matrix is the mistake of system previous step
Poor covariance matrix.
Preferably, kalman gain is determined；
System present status is predicted using the state vector of the system previous step and the kalman gain is multiplied by institute
The difference of the expectation of observable variable vector and observable variable vector is stated, obtains the current state vector of system, simultaneously
To current vector covariance matrix.
Preferably, the Random Variable Distribution Function uses Bernoulli random variable distribution function.
Preferably, according to the Kalman filtering algorithm, covariance matrix and described first is weighted by the random matrix
The covariance of zero-mean gaussian random vector forms the covariance matrix of the system motion process.
Preferably, the coefficient matrix of the state vector of the system first exported the measurement of the system, is decomposed into the
Then one state vector coefficient matrix and the second state vector coefficient matrix are deleted mistake measurement equation according to known threshold value structure and are obtained
To observable variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to full probability
Formula and the expectation for calculating the observable variable vector.
Preferably, the kalman gain, including：First gain coefficient and the second gain coefficient；
The kalman gain is multiplied by second gain coefficient for first gain coefficient；
The probability of the magnitude relationship of corresponding element in the measurement output of the known threshold vector and the system is calculated,
Probability matrix is obtained according to the probability；
First gain coefficient is the second state vector coefficient matrix premultiplication predicting covariance matrix, and the right side multiplies
The probability matrix.
Second aspect, the present invention provide a kind of wave filter for Random censorship estimating system virtual condition, including：
Memory and processor and storage on a memory and the computer program that can run on a processor, the calculating
Machine program is such as a kind of above-mentioned method for Random censorship estimating system virtual condition, when the processor performs described program
Realize following steps：
Establish the state space equation of time-varying system；
It is described including state space equation：The measurement output of the state vector and system of system；
The magnitude relationship structure exported according to the measurement of known threshold vector and the system is deleted mistake measurement equation and is obtained
Observable variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to full probability public affairs
Formula calculates the expectation of the observable variable vector；
The coefficient matrix of the state vector of system previous step in the state vector of the system is decomposed, obtains desired value
The state vector coefficient matrix for the first previous step for being zero and the state vector coefficient matrix of the second previous step, by the system
All covariance element structure first information matrixes in first zero-mean gaussian random vector of state vector；
The random matrix of structure system；
Ask for the system random matrix and unit matrix Kronecker product after again with the first information matrix
Hadamard accumulates to obtain the expectation of matrix；
The Kronecker of unit of account column vector and unit matrix is accumulated；
It is multiplied by the unit of account column vector and unit matrix respectively in the desired both sides for obtaining matrix
The transposed matrix of Kronecker products and the matrix of the Kronecker of the unit of account column vector and unit matrix products, obtain
Random matrix weights covariance matrix；
Wherein, the element in the random matrix of the system is desired for zero, and known to the expectation of the first information matrix.
Using random matrix weighting covariance matrix as the covariance matrix of system motion process, using described considerable
Surveying the Kalman filtering algorithm that is expected that by of variable vector makes the state error covariance least estimated of the state space equation
System virtual condition.
The present invention has the advantages that：
Invention provides a kind of method for Random censorship estimating system virtual condition, and mistake boundary is deleted for different, can
Obtain the difference of measured value.Simultaneously for the time-varying system that random parameter of becoming estranged is deleted with data, the present invention can effectively estimate
Dbjective state is counted out, and becoming smaller for boundary is lost, and it is smaller to filter evaluated error with deleting.The filtering for the algorithm that the present invention obtains
Error, which is much smaller than, has obtained method in document [3], this illustrates that the obtained algorithm of the present invention has advantage.Namely
It says, filter design method of the invention considers Random censorship and is present in Discrete Time-Varying Systems to system with random parameter simultaneously
It is inclined dependent on the calculating that this feature of measurement data is caused to solve Random censorship using conditional expectation for the influence of output performance
Poor problem.
Description of the drawings
By below with reference to description of the attached drawing to the embodiment of the present invention, the above and other purposes of the present invention, feature and
Advantage is apparent, in the accompanying drawings：
Fig. 1 is a kind of flow signal of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Figure；
Fig. 2 is surveyed when being a kind of Γ=I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
The curve of the one-component of amount；
Fig. 3 is surveyed when being a kind of Γ=- I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
The curve of the one-component of amount；
Fig. 4 is filtered when being a kind of Γ=I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Wave error one-componentCurve；
Fig. 5 is filtered when being a kind of Γ=I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Second component of wave errorCurve；
Fig. 6 is filtered when being a kind of Γ=- I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Wave error one-componentCurve；
Fig. 7 is filtered when being a kind of Γ=- I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Wave error one-componentCurve；
Fig. 8 is filtered when being a kind of Γ=- I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Wave error one-componentCurve, utilize the method for document [3]；
When Fig. 9 is a kind of Γ=- 1 of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Filtering error one-componentCurve, utilize the method for document [3].
Specific embodiment
Below based on embodiment, present invention is described, but what deserves to be explained is, the present invention is not limited to these realities
Apply example.Below to the present invention datail description in, it is detailed to describe some specific detail sections.However, for not detailed
The part described to the greatest extent, those skilled in the art can also understand the present invention completely.
In addition, it should be understood by one skilled in the art that the attached drawing provided simply to illustrate that the purpose of the present invention,
Feature and advantage, attached drawing are not to be actually drawn to scale.
Meanwhile unless the context clearly requires otherwise, otherwise throughout the specification and claims " comprising ", "comprising" etc.
Similar word should be construed to the meaning included rather than exclusive or exhaustive meaning；That is, it is " including but not limited to "
Meaning.
Symbol description：In the present invention, RnRepresent Euclidean n-space, Rn×mRepresent the collection of all n × m ranks real matrixes
It closes.MTThe transposition of representing matrix M.I represents the unit matrix for being suitble to dimension.Represent Kronecker product operation.diag{N1,
N2..., NmRepresent that diagonal blocks are matrix Ns1, N2..., NmBlock diagonal matrix.Prob { E } represents the possibility that event E occurs.Represent the mathematic expectaion and variance of random matrix R respectively with Var (R).The covariance of Cov (x, y) representation vector x, y.If
Somewhere does not have clear and definite specified matrix dimension in text, then assumes that its dimension is suitble to the algebraic operation of matrix.
Specifically, the present invention realizes the technology of the present invention by following phases, and illustrate beneficial effects of the present invention.
Stage one：According to Practical Project background, establish with Random censorship and the time-varying system mould under random parameter situation
Type.
Stage two：It obtains information matrix and deletes the calculation formula for losing the model measured and its conditional expectation.
Stage three：The calculation formula of the information matrix of covariance information between reflection random matrix all elements is obtained, and is transported
It is accumulated to obtain the calculation formula of random matrix weighting covariance matrix with Hadamard products and Kronecker.
Stage four：According to the time-varying system model that the stage one establishes, to minimize state error covariance, operational phase
Two and the calculation formula in stage three, and pass through " prediction " and " update " two stages, construct system Tobit Kalman filter.
Stage five：The Tobit Kalman filter of (i.e. without random parameter) is provided under degenerate case.
Stage six：By the obtained Tobit kalman filter methods based on conditional expectation of the invention with it is existing
Tobit Kalman filtering design methods are compared, and illustrate its superiority.
The filter design method of the present invention considers Random censorship and is present in Discrete Time-Varying Systems with random parameter simultaneously
Influence to system output performance solves what Random censorship was caused dependent on this feature of measurement data using conditional expectation
Offset issue is calculated, compared with existing Tobit Design on Kalman Filter method, the present invention is based on the Tobit of conditional expectation
Kalman filter method can not only handle Random censorship and random parameter phenomenon simultaneously, but also algorithm for design can be clearly seen
Influence of the Random censorship to filtering performance.
When with state space method analysis system, the dynamic characteristic of system is with the first differential side being made of state variable
Journey group describes.It can reflect the variation of whole independent variables of system, so as to determine the totality fortune of system simultaneously
Dynamic state, but also can easily handle primary condition.In this way, in design system, it is no longer confined to input quantity, output
Amount, the margin of error provide strong tool to improve system performance.In addition analysis design and in real time control are carried out using computer
System, thus can be applied to nonlinear system, time-varying system, multiinput-multioutput system and random process etc..Specific configuration
The method of time-varying system state space can refer to《Signal and system》The content of middle one chapter of state space description.
Fig. 1 is a kind of flow signal of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Figure, as shown in Figure 1, a kind of method for Random censorship estimating system virtual condition, including：Step 101 establishes time-varying system
State space equation；It is described including state space equation：The measurement output of the state vector and system of system；Step 102 piece
The magnitude relationship structure exported according to the measurement of known threshold vector and the system deletes mistake measurement equation and obtains observable variable
Vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；Institute is calculated according to total probability formula
State the expectation of observable variable vector；Step 103 is by the state vector of the system previous step in the state vector of the system
Coefficient matrix decomposes, obtain the state vector coefficient matrix of the first previous step that desired value is zero and the state of the second previous step to
Coefficient of discharge matrix, all covariance elements structures the in the first zero-mean gaussian random vector by the state vector of the system
One information matrix；Step 104 builds the random matrix of system；Ask for the random matrix and unit matrix of the system
After Kronecker products the expectation of matrix is obtained with the first information Hadamard matrix nature again；Unit of account column vector with
The Kronecker products of unit matrix；It is multiplied by the unit of account column vector and list respectively in the desired both sides for obtaining matrix
What the transposed matrix and the Kronecker of the unit of account column vector and unit matrix of the Kronecker products of bit matrix accumulated
Matrix obtains random matrix weighting covariance matrix；The random matrix is weighted into covariance matrix as system with step 105
The covariance matrix of motion process, the Kalman filtering algorithm that is expected that by using observable variable vector make the state
The state error covariance least estimated system virtual condition of space equation.
At step 104, the element in the random matrix of the system is desired for zero, and the phase of the first information matrix
Known to prestige.
Stage one：According to Practical Project background, establish with Random censorship and the time-varying system mould under random parameter situation
Type, i.e. step 101 establish the state space equation of time-varying system；It is described including state space equation：The state vector x of systemk
Measurement with system exports
The state vector x of the system previous stepk-1Coefficient matrices Ak-1Measurement with system exportsCoefficient matrix Ck
For random parameter matrix independent mutually.
Such as：Time-varying system state space equation is：
In formula,The state vector of expression system；The measurement output of expression system；wk-1And vkRespectively
First zero-mean gaussian random vector and the second zero-mean gaussian random vector, wk-1And vkCovariance is respectivelyWithIt is denoted as the covariance matrix and the second zero-mean of the first zero-mean gaussian random vector respectively
The covariance matrix of Gaussian random vector, AndIt is only mutually
Vertical random parameter matrix；For the known matrix for being suitably for that there is suitable dimension.
Stage two is the calculation formula for obtaining information matrix and deleting the model for losing measurement and its conditional expectation.
Specifically, step 103 is by the state vector x of the systemkIn system previous step state vector xk-1's
Coefficient matrices Ak-1It decomposes, obtains the state vector coefficient matrix of the first previous step that desired value is zeroWith the second previous step
State vector coefficient matrixBy the first zero-mean gaussian random vector w of the state vector of the systemk-1In all association sides
Poor element builds first information matrix.The state vector x of the systemkBy the state vector x of system previous stepk-1With described
One zero-mean gaussian random vector wk-1Linear combination is formed, the first zero-mean gaussian random vector wk-1Coefficient be Bk-1；It is described
The measurement output of systemBy the state vector x of the systemkSecond zero-mean gaussian random vector v of sumkLinear combination structure
Into the second zero-mean gaussian random vector vkCoefficient be 1.
The measurement output of the systemThe system state vector xkCoefficient matrix Ck, it is decomposed into first state
Vectorial coefficient matrixWith the second state vector coefficient matrixThe second state vector coefficient matrixDesired value be
Zero.
The coefficient matrix of the state vector of the system of the measurement output of the system, is decomposed into first state system of vectors
Matrix number and the second state vector coefficient matrix.
By the second zero-mean gaussian random vector vkIn all covariance elements build the second information matrix.
Such as：Without loss of generality, it is assumed that random matrix AkAnd CkConstruction is as follows：
Their statistical property is as follows：
Wherein,WithRespectivelyWithS rows, t row element.
For its simplicity, definition includesAndIn the information matrixs of all covariance elements be respectively
With
Wherein,AndRespectivelyWithS rows, t row block matrix；AndRespectivelyAnd
T rows, m row element.
Note：The covariance information of [1] and document [2] description random matrix is often by matrix element in previous literature
Form describes one by one, very cumbersome, and the matrix after being unfavorable for calculates.
[1] Y.Luo, Y.Zhu, D.Luo, J.Zhou, E.Song, and D.Wang, Globally optimal
multisensor distributed random parameter matrices Kalman filtering fusionwith
Applications, Sensors, vol, 8, no.12, pp.8086-8103,2008.
[2] D.Ding, Z.Wang, H.Dong, and H.Shu, Distributed H ∞ state estimation
with stochastic parameters and nonlinearities through sensor networks：the
Finite-horizon case, Automatica, vol.48, no.8, pp.1575-1585,2012.
Step 102 is exported according to the measurement of known threshold vector Γ and the systemMagnitude relationship structure delete mistake survey
Amount equation obtain observable variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；Root
The expectation of the observable variable vector is calculated according to total probability formula.Specifically, the measurement due to system exportsIt is one
A latent variable can not be directly measured or observe, therefore we introduce observable variable (that is, observable variable is vectorial) ykIt goes
Expression is deleted mistake and is measured, definition
Wherein, observable variable vector is：
τiFor aboutKnown threshold.
By introducing Bernoulli random variableTo describe to delete the possibility for losing and measuring and occurring, random change
Distribution function is measured using Bernoulli random variable distribution function, obtains deleting that lose measurement model as follows：
Observable variable vector ykConditional expectation and observable variable vector ykExpectation it is as follows：
Wherein：The state vector x of system previous stepk|k-1For the one-step prediction value at k-1 moment,
α=[α1 α2 … αm]T,
σ=diag { σ1, σ2..., σm,
Wherein, what deserves to be explained is step 102 and step 103 have no precedence relationship.
Stage three is the calculation formula for obtaining including the information matrix of covariance information between random matrix all elements, and is transported
It is accumulated to obtain the calculation formula of random matrix weighting covariance matrix with Hadamard products and Kronecker.That is step 104 structure system
The random matrix of system；Ask for the system random matrix and unit matrix Kronecker product after again with the first information
Hadamard matrix nature obtains the expectation of matrix；The Kronecker of unit of account column vector and unit matrix is accumulated；It is described to obtain
The desired both sides of matrix be multiplied by respectively the transposed matrix of the Kronecker products of the unit of account column vector and unit matrix with
And the matrix of the Kronecker of the unit of account column vector and unit matrix products, random matrix weighting covariance matrix is obtained,
Specific method is as follows：
1. provide a special diagonal random matrixWeight covariance matrix.
For diagonal random matrix pkWith a Weighted random matrix Gk=[gIj, k]m×m＞ 0, if GkDependent on pk, that
Wherein：
Operation is defined for two diagonal matrix, is defined as follows：
2. define random diagonal matrix pkInformation matrix.
By random matrix pkIt represents as follows：
So its information matrix is represented by：
The random matrix weighting covariance matrix calculation formula under 3. general scenario is given below：
For Q=[qj]n×m, W=[wj]m×m, R=[rj]m×n, wherein wj, rj, qjFor stochastic variable, enableAndWithIt is known that with
It is as follows that Hadamard products and Kronecker products can obtain random matrix weighting covariance matrix calculation formula：
Wherein, W be positive definite random matrix, [R]st=QsRt, QsAnd RtThe s row of respectively Q and R, the random square of line n
Battle array,
It 4. can be according to information matrixAnd formula (5) obtains
Random matrix weights covariance matrixAnd it is able to verify that
It is of equal value with the formula in situation 1.
Stage four is specially：According to the time-varying system model that the stage one establishes, to minimize state error covariance, fortune
With the calculation formula in stage two and stage three, and pass through " prediction " and " update " two stages, construction system Tobit Kalman's filters
Wave device.
In general, using Kalman filtering algorithm, the process model using system is first had to, to predict NextState
System.Assuming that present system mode is k, according to the model of system, it can predict and appear in based on the laststate of system
State：X (k | k-1)=A X (k-1 | k-1)+B U (k), X (k | k-1) it is the one-step prediction value of filtering or one-step prediction value.
In formula, the one-step prediction value X of filtering (k | k-1) is the prediction result using previous step state vector, and X (k-1 | k-
1) it is that laststate is optimal as a result, U (k) is the controlled quentity controlled variable of present status, if without controlled quentity controlled variable, it can be 0.
Till now, our system results have had updated, and can correspond to the covariance of X (k | k-1) also not more
Newly, covariance is represented with P：P (k | k-1)=A P (k-1 | k-1) AT+Q.In formula, P (k | k-1) is the corresponding associations of X (k | k-1)
Variance (that is, filtering one-step prediction error covariance matrix or one-step prediction error covariance matrix), and P (k-1 | k-1) it is X (k-
1 | k-1) corresponding covariance, ATRepresent the transposed matrix of A, Q is the covariance of systematic procedure.
If having there is the prediction result of present status, the measured value of present status is then regathered.With reference to predicted value and
Measured value can obtain optimization estimated value (that is, filter value) X (k | k) of present status k：X (k | k)=X (k | k-1)+Kg
(k)(Z(k)-HX(k|k-1))；Wherein, Kg is kalman gain, and Kg (k)=P (k | k-1) HT/(H P(k|k-1)HT+R)。
Till now, have been obtained for inscribing optimal estimated value (that is, filter value) X (k | k) during k.But it is intended to
Kalman filter is enabled constantly to run down until systematic procedure terminates, we will also update the association side of X under k states (k | k)
Difference：P (k | k)=(I-Kg (k) H) P (k | k-1)；Wherein, I is 1 matrix, single model list is measured, I=1.When system into
When entering k+1 states, P (k | k) is exactly the P (k-1 | k-1) of formula.In this way, algorithm can autoregressive operation go down.
According to the time-varying system model that the stage one establishes, to minimize state error covariance, operational phase two and rank
The calculation formula of section three, and pass through " prediction " and " update " two stages, construct system Tobit Kalman filter, wave filter
Model is as follows：
Filter model is as follows：
The one-step prediction value of filtering is：
Filtering one-step prediction error covariance matrix is：
Filter value is：
Predicting covariance matrix (that is, filtering error covariance matrix) is：
Wherein,
Stage five is specially：Tobit Kalman filter when (that is, without random parameter), mould are provided under degenerate case
Type is as follows：
Wherein：
Stage six is specially：By the obtained Tobit kalman filter methods based on conditional expectation of the invention with it is existing
Tobit Kalman filtering design methods compare, illustrate its superiority：
With existing Tobit kalman filter methods, in the stage fiveIt calculates as follows：
With the further perfect Tobit kalman filter methods of the present invention, in the stage fiveIt calculates as follows：
By the comparative analysis of (7) formula and (8) formula, the present invention obtains the Tobit Kalman filtering sides based on conditional expectation
Method can not only handle Random censorship and random parameter phenomenon simultaneously, but also algorithm for design can be clearly seen Random censorship to filter
The influence of wave performance.
Meanwhile the present invention provides a kind of wave filter for Random censorship estimating system virtual condition, including：Memory and
Processor and storage are on a memory and the computer program that can run on a processor, the computer program are above-mentioned one kind
For the method for Random censorship estimating system virtual condition, the processor realizes following steps when performing described program, specifically
It can refer to the content of a kind of above-mentioned method for Random censorship estimating system virtual condition.
Establish the state space equation of time-varying system.
It is described including state space equation：The measurement output of the state vector and system of system；
The magnitude relationship structure exported according to the measurement of known threshold vector and the system is deleted mistake measurement equation and is obtained
Observable variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to full probability public affairs
Formula calculates the expectation of the observable variable vector.
The coefficient matrix of the state vector of system previous step in the state vector of the system is decomposed, obtains desired value
The state vector coefficient matrix for the first previous step for being zero and the state vector coefficient matrix of the second previous step, by the system
All covariance element structure first information matrixes in first zero-mean gaussian random vector of state vector.
The random matrix of structure system, specifically, random matrix is random diagonal matrix.
Ask for the system random matrix and unit matrix Kronecker product after again with the first information matrix
Hadamard accumulates to obtain the expectation of matrix.
The Kronecker of unit of account column vector and unit matrix is accumulated.
It is multiplied by the unit of account column vector and unit matrix respectively in the desired both sides for obtaining matrix
The transposed matrix of Kronecker products and the matrix of the Kronecker of the unit of account column vector and unit matrix products, obtain
Random matrix weights covariance matrix.
Wherein, the element in the random matrix of the system is desired for zero, and known to the expectation of the first information matrix.
Using random matrix weighting covariance matrix as the covariance matrix of system motion process, using described considerable
Surveying the Kalman filtering algorithm that is expected that by of variable vector makes the state error covariance least estimated of the state space equation
System virtual condition.
[3] B.Allik, C.Miller, M.J.Piovoso, and R.Zurakowski, The Tobit Kalman
filter：An estimator for censored measurements, IEEE Transactions on Control
Systems Technology, vol.24, no.1, pp.365-371,2016.
Fig. 2 is surveyed when being a kind of Γ=I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
The curve of the one-component of amount.Fig. 3 is a kind of side for Random censorship estimating system virtual condition of the embodiment of the present invention
The curve of one-component measured during Γ=- I of method.The comparison of Fig. 2 and Fig. 3 can be seen that deletes mistake boundary for different,
The difference of measured value can be obtained.
Fig. 4 is filtered when being a kind of Γ=I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Wave error one-componentCurve.Fig. 5 is that one kind of the embodiment of the present invention is real for Random censorship estimating system
Second component of filtering error during Γ=I of the method for border stateCurve.Fig. 6 is the one of the embodiment of the present invention
Filtering error one-component when kind is directed to Γ=- I of the method for Random censorship estimating system virtual conditionSong
Line.It filters and misses when Fig. 7 is a kind of Γ=- I of method for Random censorship estimating system virtual condition of the embodiment of the present invention
Poor one-componentCurve.Fig. 8 is that one kind of the embodiment of the present invention is directed to the practical shape of Random censorship estimating system
Filtering error one-component during Γ=- I of the method for stateCurve, utilize the method for document [3].Fig. 9 is this
The first point of filtering error during a kind of Γ=- I of method for Random censorship estimating system virtual condition of inventive embodiments
AmountCurve, utilize the method for document [3].
By the comparison of Fig. 4 and Fig. 6 or Fig. 5 and Fig. 7 as it can be seen that the time-varying system for deleting random parameter of becoming estranged with data
System, the present invention can effectively estimate dbjective state, and lose becoming smaller for boundary, and it is smaller to filter evaluated error with deleting.
The filtering error of algorithm that the present invention obtains it can be seen from the comparison of Fig. 6 and Fig. 8 or Fig. 7 and Fig. 9 is much smaller than
Has obtained method in document [3], this illustrates that the obtained algorithm of the present invention has advantage.
The present invention obtains a kind of Tobit Design on Kalman Filter method, to be a kind of for Random censorship and random ginseng
Several kalman filter methods is related to Kalman filter when unilateral Random censorship and stochastic parameter variation occur for time-varying system
Design.The present invention obtains including for the first time the information matrix of covariance information between random matrix all elements.According to information matrix simultaneously
It is accumulated using Kronecker product and Hadamard, obtains the calculation formula of random matrix weighting covariance matrix for the first time.Pass through condition
It is expected the feature that Random censorship depends on measurement data is effectively treated.Include Random censorship in the filtering algorithm of final design
Boundary, it is aobvious to be contained in predicting covariance matrix, it can be seen that its influence to predicting covariance matrix.With it is existing
Tobit Kalman filtering design methods compare, the Tobit kalman filter methods the present invention is based on conditional expectation not only can be with
Random censorship and random parameter phenomenon are handled, and algorithm for design can be clearly seen shadow of the Random censorship to filtering performance simultaneously
It rings.
Obviously, those skilled in the art should be understood that each unit of the above-mentioned present invention or each step can be with general
Computing device realize that they can concentrate on single computing device or be distributed in multiple computing devices compositions
On network, optionally, they can be realized with the program code that computing device can perform, it is thus possible to be stored in
They are either fabricated to each integrated circuit unit respectively or will be in them by computing device to perform in storage device
Multiple units or step be fabricated to single integrated circuit unit to realize.In this way, the present invention is not limited to any specific hard
Part and software combine.
Embodiment described above only expresses embodiments of the present invention, and description is more specific and detailed, but can not
Therefore it is interpreted as the limitation to the scope of the claims of the present invention.It should be pointed out that those skilled in the art,
Under the premise of not departing from present inventive concept, several deformations, equal replacement can also be made, improved etc., these belong to the present invention
Protection domain.Therefore, the protection domain of patent of the present invention should be determined by the appended claims.
Claims (10)
- A kind of 1. method for Random censorship estimating system virtual condition, which is characterized in that including：Establish the state space equation of time-varying system；The state space equation, including：The measurement output of the state vector and system of system；The magnitude relationship structure exported according to the measurement of known threshold vector and the system delete mistake measure equation obtain it is considerable Survey variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to total probability formula meter Calculate the expectation of the observable variable vector；The coefficient matrix of the state vector of system previous step in the state vector of the system is decomposed, it is zero to obtain desired value The state vector coefficient matrix of the first previous step and the state vector coefficient matrix of the second previous step, by the state of the system All covariance element structure first information matrixes in first zero-mean gaussian random vector of vector；The random matrix of structure system；Ask for the system random matrix and unit matrix Kronecker product after again with the first information matrix Hadamard accumulates to obtain the expectation of matrix；The Kronecker of unit of account column vector and unit matrix is accumulated；The unit of account column vector is multiplied by the desired both sides for obtaining matrix respectively and the Kronecker of unit matrix is accumulated Transposed matrix and the Kronecker of the unit of account column vector and unit matrix product matrix, obtain random matrix and add Weigh covariance matrix；Using random matrix weighting covariance matrix as the covariance matrix of system motion process, become using the Observable The Kalman filtering algorithm that is expected that by of amount vector makes the state error covariance least estimated system of the state space equation Virtual condition；Wherein, the element in the random matrix of the system is desired for zero, and the expectation of the first information matrix is Know.
- 2. a kind of method for Random censorship estimating system virtual condition according to claim 1, it is characterised in that：The state vector of the system is linear by the state vector of system previous step and the first zero-mean gaussian random vector Combination is formed；The measurement of the system is exported by the second zero-mean gaussian random vector linear combination of the state vector sum of the system It forms；The coefficient matrix of the state vector of the system previous step and the coefficient matrix of the measurement output of system are independent mutually Random parameter matrix；The coefficient matrix of the state vector of the system of the measurement output of the system, is decomposed into first state vector coefficient square Battle array and the second state vector coefficient matrix, the desired value of the second state vector coefficient matrix is zero；The second information matrix is built by covariance elements all in the second zero-mean gaussian random vector.
- 3. a kind of method for Random censorship estimating system virtual condition according to claim 1, it is characterised in that：According to the state space equation and the Kalman filtering, the state vector coefficient matrix of first previous step is is The coefficient matrix of the state vector for previous step of uniting, the state vector based on the system previous step predict system one-step prediction Value.
- 4. a kind of method for Random censorship estimating system virtual condition according to claim 1, it is characterised in that：According to the Kalman filtering algorithm, based on random matrix weighting covariance matrix, first zero-mean gaussian The coefficient matrix of the covariance matrix of random vector and the first zero-mean gaussian random vector constructs system motion process Covariance matrix；Wherein, the random matrix of the system in the random matrix weighting covariance matrix is assisted for the error of system previous step Variance matrix.
- 5. a kind of method for Random censorship estimating system virtual condition according to claim 1, it is characterised in that：Determine kalman gain；Predict system present status using the state vector of the system previous step and the kalman gain be multiplied by it is described can The difference of the expectation of observational variable vector and observable variable vector obtains the current state vector of system, while is worked as Preceding vector covariance matrix.
- 6. a kind of method for Random censorship estimating system virtual condition according to claim 1, it is characterised in that：The Random Variable Distribution Function uses Bernoulli random variable distribution function.
- 7. a kind of method for Random censorship estimating system virtual condition according to claim 2, it is characterised in that：According to the Kalman filtering algorithm, by the random matrix weight covariance matrix and first zero-mean gaussian with The covariance of machine vector forms the covariance matrix of the system motion process.
- 8. a kind of method for Random censorship estimating system virtual condition according to claim 2, it is characterised in that：The coefficient matrix of the state vector of the system that first measurement of the system is exported, is decomposed into first state system of vectors Then matrix number and the second state vector coefficient matrix delete mistake measurement equation according to known threshold value structure and obtain observable variable Vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to total probability formula and calculate The expectation of the observable variable vector.
- 9. a kind of method for Random censorship estimating system virtual condition according to claim 6, it is characterised in that：The kalman gain, including：First gain coefficient and the second gain coefficient；The kalman gain is multiplied by second gain coefficient for first gain coefficient；The probability of the magnitude relationship of corresponding element in the measurement output of the known threshold vector and the system is calculated, according to The probability obtains probability matrix；First gain coefficient is the second state vector coefficient matrix premultiplication predicting covariance matrix, and the right side multiplies described Probability matrix.
- 10. a kind of wave filter for Random censorship estimating system virtual condition, which is characterized in that including：Memory and processor and storage on a memory and the computer program that can run on a processor, the computer journey Sequence is a kind of method for Random censorship estimating system virtual condition, the processor as described in any one of claim 1~9 Following steps are realized when performing described program：Establish the state space equation of time-varying system；It is described including state space equation：The measurement output of the state vector and system of system；The magnitude relationship structure exported according to the measurement of known threshold vector and the system delete mistake measure equation obtain it is considerable Survey variable vector, by Random Variable Distribution Function describe described in delete lose measure occur possibility；According to total probability formula meter Calculate the expectation of the observable variable vector；The coefficient matrix of the state vector of system previous step in the state vector of the system is decomposed, it is zero to obtain desired value The state vector coefficient matrix of the first previous step and the state vector coefficient matrix of the second previous step, by the state of the system All covariance element structure first information matrixes in first zero-mean gaussian random vector of vector；The random matrix of structure system；Ask for the system random matrix and unit matrix Kronecker product after again with the first information matrix Hadamard accumulates to obtain the expectation of matrix；The Kronecker of unit of account column vector and unit matrix is accumulated；The unit of account column vector is multiplied by the desired both sides for obtaining matrix respectively and the Kronecker of unit matrix is accumulated Transposed matrix and the Kronecker of the unit of account column vector and unit matrix product matrix, obtain random matrix and add Weigh covariance matrix；Wherein, the element in the random matrix of the system is desired for zero, and known to the expectation of the first information matrix；Using random matrix weighting covariance matrix as the covariance matrix of system motion process, become using the Observable The Kalman filtering algorithm that is expected that by of amount vector makes the state error covariance least estimated system of the state space equation Virtual condition.
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