CN109787584A - A kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method - Google Patents
A kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method Download PDFInfo
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Abstract
The present invention proposes a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method, comprising: only band noise variance uncertain system will be converted into mixing uncertain system；Determine the disturbance domain of the Uncertain noise covariance of the mixing uncertain system；Calculate maximum parameter perturbation.Research of the present invention to the guaranteed cost Robust Kalman filtering of mixing uncertain system, extends the research range of guaranteed cost Robust Kalman filtering.By providing the infimum and supremum of accuracy error, the infimum or supremum for only providing accuracy error of existing Robust Kalman filtering are improved.The above both sides development and improvement, it is theoretical further to enrich guaranteed cost Robust Kalman filtering.
Description
Technical field
The present invention relates to a kind of filtering methods, and in particular to a kind of mixing uncertain system guaranteed cost Robust Kalman filtering
Device design method.
Background technique
Kalman filter is widely used in target following, signal processing, GPS positioning, robot, remote sensing, unmanned plane, satellite
The fields such as observing and controlling, basic premise are to assume that system model parameter and noise variance are accurately known.But in practical application, due to mould
The factors such as type simplification, Unmarried pregnancy and uncertain noises, so that this hypothesis is usually invalid.This will lead to system filter performance
Deterioration or even filtering divergence.Meanwhile classical Kalman filter hypothesis contains the information for being estimated state or signal, and in net
In network system, since the communication bandwidth of network is limited, sensor fault and the disturbance of various random externals lead to observation only
Include noise information.Such case, which is known as observing, loses.The case where observation noise is not white noise is also to be widely present.Mixing
Uncertain system refers to the uncertain system being made of above-mentioned uncertain mixing.
Minimax Robust Estimation principle is based on using Lyapunov equation method to noise variance uncertain system, if
Count Robust Kalman Filter.Model parameter is uncertain including determining uncertainty and stochastic uncertainty (multiplicative noise).
Determining uncertain parameter refers to that parameter is uncertain but belongs to a certain known bounded aggregate determined, such as norm-bounded is not true
Determine parameter.For band norm-bounded uncertain parameter system, robust can be designed by Riccati equation method or LMI method
Kalman filter.For multiplicative noise system, virtual noise compensation method can be used, convert original system to band and determine and join
The system of several gusts and virtual noise, and then filtering problem is solved using classical Kalman filter method.Losing observation can be used
Bernoulli Jacob distribution variables describe.For the system with colored observed noises, the method for being augmented or observation difference can be used
Method, by system converting for the system with white noise.
For the uncertainty allowed, Robust Kalman Filter can ensure the practical filtering of designed filter
Error variance has supremum.Further consider that designed Robust Kalman Filter meets the pact of preset performance indicator
Beam, this is known as guaranteed cost Robust Kalman filtering.To the permanent multisensor syste of the linear discrete with Uncertain noise covariance, answer
It is indicated with the parametrization that minimax Robust Kalman filtering method and noise variance based on Lyapunov equation method disturb
Method devises two class guaranteed cost robust fusion Kalman estimators with Lagrange Multiplier Method and linear programming method.But it is right
Lose observation, multiplicative noise and colored observed noises mixing uncertain system guaranteed cost robust fusion filtering problem also not by
It solves and the more complicated guaranteed cost robust fusion filtering problem with mixing nondeterministic network system is not solved yet
Certainly.
Summary of the invention
In view of the foregoing deficiencies of prior art, the purpose of the present invention is to provide a kind of mixing uncertain system guarantor property
It can Robust Kalman Filter design method.
In order to achieve the above objects and other related objects, the present invention provides a kind of mixing uncertain system guaranteed cost robust
Kalman filter design method, this method comprises:
Only band noise variance uncertain system will be converted into mixing uncertain system；
Determine the disturbance domain of the Uncertain noise covariance of the mixing uncertain system；
Calculate maximum parameter perturbation.
Optionally, the band mixing uncertain system includes:
The observation of band missing, sliding average colored observed noises and Uncertain noise covariance system with multiplicative noise, coloured
Observation noise and Uncertain noise covariance system.
It optionally, will be system converting for only with missing observation, sliding average colored observed noises and Uncertain noise covariance
Band noise variance uncertain system, specifically includes:
Determine the equivalent state spatial model of sliding average colored observed noises；
By the state equation in the observation of band missing, sliding average colored observed noises and Uncertain noise covariance system
It is augmented with the equivalent state spatial model of the sliding average colored observed noises；
To the observational equation in the observation of band missing, sliding average colored observed noises and Uncertain noise covariance system
It is converted.
Optionally, using the method that is augmented to the observation of band missing, sliding average colored observed noises and uncertain noise
The equivalent state spatial model of state equation and the sliding average colored observed noises in variance system is augmented.
Optionally, it to the observation of band missing, sliding average colored observed noises and is not known using virtual noise technology
Observational equation in noise variance system is converted.
It optionally, will be system converting for only band noise side with multiplicative noise, colored observed noises and Uncertain noise covariance
Poor uncertain system, specifically includes:
To in multiplicative noise, colored observed noises and Uncertain noise covariance system state equation and colored observation make an uproar
Sound equation is augmented；
It is converted to the observational equation in multiplicative noise, colored observed noises and Uncertain noise covariance system.
Optionally, using the method that is augmented to the band multiplicative noise, colored observed noises and Uncertain noise covariance system
In state equation and color observation noise equation be augmented.
Optionally, using virtual noise technology to band multiplicative noise, colored observed noises and Uncertain noise covariance system
In observational equation converted.
As described above, a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method of the invention, tool
Have it is following the utility model has the advantages that
Research of the present invention to the guaranteed cost Robust Kalman filtering of mixing uncertain system, extends guaranteed cost robust
The research range of Kalman filter.By providing the infimum and supremum of accuracy error, existing robust is improved
The infimum or supremum for only providing accuracy error of Kalman filter.The above both sides development and improvement, further
It is theoretical to enrich guaranteed cost Robust Kalman filtering.
Specific embodiment
Illustrate embodiments of the present invention below by way of specific specific example, those skilled in the art can be by this specification
Other advantages and efficacy of the present invention can be easily understood for disclosed content.The present invention can also pass through in addition different specific realities
The mode of applying is embodied or practiced, the various details in this specification can also based on different viewpoints and application, without departing from
Various modifications or alterations are carried out under spirit of the invention.It should be noted that in the absence of conflict, following embodiment and implementation
Feature in example can be combined with each other.
Robust Kalman Filter is designed to uncertain system, requires consideration for how the problem of measuring Robust Estimation precision.
The mark of filtering error variance matrix is defined as precision index, the value of mark is smaller to mean that precision is higher.Define practical valuation error
The mark of variance is available accuracy, and the mark for defining practical valuation error variance supremum is robust precision.Define robust precision and
The deviation of available accuracy is accuracy error.
Symbol description: RnIndicate that n ties up the space Euclidean；The mark of trP expression matrix P.
Guaranteed cost Robust Kalman filtering is designed to band mixing uncertain system, it first can be by being augmented method and virtually making an uproar
Audio technology carries out system conversion, and grandfather tape is mixed uncertain system and is converted into the only band uncertain system of noise variance.Then
Based on minimax Robust Estimation principle, using Uncertain noise covariance parametric method, Lagrange Multiplier Method and
Lyapunov equation method designs two class guaranteed cost Robust Kalman Filters.Two kinds of following discussion mix uncertain system
Conversion process, as follows:
(1) observation of band missing, sliding average colored observed noises and Uncertain noise covariance system
X (t+1)=Φ x (t)+Γ w (t) (1)
yi(t)=γi(t)Hix(t)+η(t)+ei(t), i=1 ..., L (2)
Wherein, x (t) ∈ RnFor system mode,For the observation of i-th of subsystem, w (t) ∈ RrIt makes an uproar for process
Sound,For the observation noise of i-th of subsystem, η (t) is the autocorrelative sliding average colored observed noises of limited step,
Random parameter ci(t) it is
WhereinFor normal parameter, ξi(t) for zero-mean, with known variance RξiWhite noise.q-1For unit lag operator, q-1
α (t)=α (t-1), Ct(q-1) it is to introduce q-1Polynomial matrix, ncFor the order of sliding average colored observed noises.α (t) is
Zero-mean white noise.
Φ, Γ and HiFor the matrix of appropriate dimension.Bernoulli scalar white noise γi(t), i=1 ..., L indicates missing
Observation, the probability that value is 1 or 0 are respectively
Prob{γi(t)=1 }=πi,Prob{γi(t)=0 }=1- πi
Wherein 0≤πi≤ 1, i=1 ..., L, and γi(t) with w (t), ei(t) and α (t) is incoherent.
Mixing uncertain system (1)-(3) are converted, steps are as follows:
1) sliding average colored observed noises η (t) has state-space model of equal value
xη(t+1)=Φηxη(t)+Γηα(t) (5)
It is augmented matrix ΦηAnd ΓηRespectively
WhereinFor (nc-1)×(nc- 1) unit matrix is tieed up.
It brings formula (4) into formula (6), has
Wherein
U-th of element be 1, other elements 0.
2) state equation (1) and (5) are augmented using the method that is augmented.
Method is augmented to formula (1) and formula (5) use below, is augmented state and matrix
Wherein,To be augmented state, na=n+nc, 0 indicates the null matrix of appropriate dimension.Then there is the state being augmented
Equation
xa(t+1)=Φaxa(t)+Γawa(t) (9)
3) it applies virtual noise technology and is augmented method and observational equation is converted
Define virtual white noise Υ0i(t)=γi(t)-πi, then the observational equation y of formula (2)i(t) it can be changed and turn to
Wherein
vai(t)=ei(t)+α (t), i=1 ..., L
It has conservative and actual noise variance RaijWithRespectively
Wherein, ReiWithRespectively white noise ei(t) conservative and actual noise variance.RαWithRespectively white noise α
(t) conservative and actual noise variance.δijFor Kronecker function, δii=1, δij=0 (i ≠ j).
Definition is augmented matrixWith
Wherein HiIt is appropriate dimension null matrix for systematic observation matrix, 0.Then observational equation (10) can be converted into
It re-definesMatrix
Then formula (10) is converted into
Because of Υ0i(t) and ξu(t) it is incoherent white noise, then can introduces virtual observation noise, there is final observation side
Journey
yi(t)=Haixa(t)+vβi(t) (15)
Wherein vβi(t) the virtual observation white noise to introduce
System (9) and (15) after conversion are system only with Uncertain noise covariance, guaranteed cost Robust Kalman filtering
Algorithm can be used for the design of the guaranteed cost robust filter of the system.
(2) band multiplicative noise, colored observed noises and Uncertain noise covariance system
xc(t+1)=Φ xc(t)+Γw(t) (17)
ηi(t+1)=Aiηi(t)+αi(t) (19)
Wherein xc(t)∈RnFor common condition,For the observation of i-th of subsystem, w (t) ∈ RrFor process noise,WithFor white noise.For colored observed noises.ξiμ(t)∈R1For scalar white noise sequence, and
With w (t), ei(t) and αi(t) uncorrelated.Φ,Γ,Hi0,HiμAnd AiFor the matrix of appropriate dimension.
Mixing uncertain system (17)-(19) are converted, steps are as follows:
1) (17) and (19) are augmented using the method that is augmented
The state being augmented and matrix are introduced for system (17) and (19), is had
Wherein,For mi×miUnit matrix is tieed up, 0 indicates the null matrix of appropriate dimension.Then there is the property carried after being augmented to make an uproar
Multi-sensor and multi-model (different partial models) system of sound
xi(t+1)=Φaixi(t)+Γiwi(t) (20)
Whereinni=n+mi, it is the state of partial model, referred to as local state.
It can be obtained from formula (17) and formula (20), xc(t) be all local states common condition, i.e.,
xc(t)=Cxxi(t),Cx=[In 0] (22)
2) observational equation is converted using virtual noise technology
Virtual observation white noise is introduced for (21), then system (20) and (21) after being augmented can be exchanged into
xi(t+1)=Φaixi(t)+Γiwi(t) (23)
Wherein virtual observation white noiseFor
System (23) and (24) after conversion are system only with Uncertain noise covariance, guaranteed cost Robust Kalman filtering
Algorithm can be used for the design of the guaranteed cost robust filter of the system.
2, to, only with the uncertain system of Uncertain noise covariance, following methods can be used to design phase after above-mentioned conversion
The guaranteed cost Robust Kalman Filter answered.Uncertain system is mixed to above-mentioned two band first and does following hypothesis:
Assuming that 1w (t), ei(t) and αiIt (t) is band zero-mean, not knowing realized variance is Q, ReiAnd RαiUncorrelated white noise
Sound,WithThe respectively conservative upper bound of actual noise variance, meets relationship
Therefore have
Note 1. illustrates that the conservative and realized variance of white noise α (t) is respectively in the model of the first uncertain system
RαWithIt can be respectively seen as RαiWithSpecial case, therefore it is subsequent calculating only refer to RαiWith
Assuming that 2 Φ and AiIt is stable matrix respectively.
Assuming that 3 variances disturb Δ Q, Δ Rei, Δ RαiParameterisable is
Wherein, εk≥0,WithIndicate uncertain parameter disturbance.Disturb orientation battle array Qk≥0,WithIt is known positive semidefinite symmetrical matrix.
To two kinds after conversion only with the system of Uncertain noise covariance, using minimax Robust Estimation principle, to band
The system of conservative upper bound variance situation can obtain conservative and actual Kalman filter accordingly and accordingly conservative (robust)
Error variance P and actual error varianceDefinitionFor the deviation of Robust Variance and realized variance, Robust Variance is defined
Mark with realized variance is respectively robust precision trP and available accuracyThe deviation of robust precision and available accuracyFor
Accuracy error
The step of design guaranteed cost Robust Kalman Filter is given below, specific as follows:
(1) accuracy error tr Δ P is parameterized, and determines that uncertain variance disturbs domain
The deviation of error varianceMeet Lyapunov equation, and by assuming existence and unique solution known to 2.Pass through vacation
If 3, and to Δ P track taking operation, it is as follows that tr Δ P parameterized form can be obtained:
Wherein bk, cij, dijIt is to be got in calculating process to obtained matrix track taking operation.
Note 2. is because of the b that in calculating process, obtainsk, cij, dijThough specific representation it is different, subsequent theory deduction without
Their concrete form is closed, therefore unified representation here.
By assuming the 1 disturbance domain Ω that can obtain Uncertain noise covariancemFor
Maximum noise variance disturbs
Next maximum parameter perturbation is foundWithAnd obtain maximum noise variance disturbance domain Δ
Qm,WithThen there is the maximum interference domain Ω of uncertain variancem, to all disturbances (the Δ Q, Δ R in this domainei,ΔRαi)
∈Ωm, accuracy error tr Δ P is guaranteed the r > 0 in preset indication range.
(2) maximum parameter perturbation is calculated
By formula (33)-formula (38) it is found that finding maximum perturbation domain ΩmIt is equivalent to find maximum parameter perturbation domain
Problem is converted into the volume of maximization hypercube formula (39)
Therefore, it is equivalent to the optimization problem of the maximization J under following constraint condition again by formula (31) problem.
LnJ is the monotonically increasing function of J, therefore lnJ and J have identical maximum point.So problem is equivalent to again about
It is asked under beam
Using Lagrange Multiplier Method, multiplier λ is introduced, then problem is converted into without maximizing under constraining
Partial derivative is taken to formula (43) both sides, and is enabledIt can obtain
(3) first kind guaranteed cost robust Klaman filter
Above two mixing uncertain system (1)-(3) and (17)-(19) are assuming that under 1-3, to preset accuracy error
Index r > 0, there are the maximum perturbation domain Ω of corresponding Uncertain noise covariancem, to all in this domain allow disturbance (Δ Q,
ΔRei,ΔRαi)∈Ωm, the accuracy error of corresponding practical Kalman filterAlways within the specified range,
I.e.
And the infimum of accuracy error is zero, supremum r.Then corresponding practical Kalman filter is referred to as first
Class guaranteed cost Robust Kalman Filter, and formula (48) is referred to as its guaranteed cost robustness.
(4) second class guaranteed cost Robust Kalman Filters
The disturbance domain Ω of given Uncertain noise covariancemFor formula (32)-(35), problem is converted into searching accuracy errorInfimum and supremum.This is the inverse problem of first kind guaranteed cost Robust Kalman Filter design.
By the disturbance domain Ω for the Uncertain noise covariance that formula (32)-(35) givem, it is equivalent to given Uncertain noise covariance
Parameter perturbation domain
According to formula (31) it is found that parameter perturbation domain to given Uncertain noise covarianceGuaranteed cost index will take tr Δ P
Maximum value,
Above two uncertain system (1)-(3) and (17)-(19) give Uncertain noise covariance assuming that under 1-3
Parameter perturbation domainAll in this domain are allowed to disturb, the accuracy error of corresponding practical Kalman filterWith infimum zero, supremum rm, i.e.,
Wherein supremum rm is given by
Then corresponding practical Kalman filter is referred to as the second class guaranteed cost Robust Kalman Filter, and formula (51) is referred to as
Its guaranteed cost robustness.
The above-described embodiments merely illustrate the principles and effects of the present invention, and is not intended to limit the present invention.It is any ripe
The personage for knowing this technology all without departing from the spirit and scope of the present invention, carries out modifications and changes to above-described embodiment.Cause
This, institute is complete without departing from the spirit and technical ideas disclosed in the present invention by those of ordinary skill in the art such as
At all equivalent modifications or change, should be covered by the claims of the present invention.
Claims (8)
1. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method, which is characterized in that this method comprises:
Only band noise variance uncertain system will be converted into mixing uncertain system；
Determine the disturbance domain of the Uncertain noise covariance of the mixing uncertain system；
Calculate maximum parameter perturbation.
2. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 1,
It is characterized in that, the band mixing uncertain system includes:
Band lacks observation, sliding average colored observed noises and Uncertain noise covariance system and with multiplicative noise, colored observation
Noise and Uncertain noise covariance system.
3. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 2,
It is characterized in that, it will be system converting for only band noise with missing observation, sliding average colored observed noises and Uncertain noise covariance
Variance uncertain system, specifically includes:
Determine the equivalent state spatial model of sliding average colored observed noises；
The band is lacked to state equation and the institute in observation, sliding average colored observed noises and Uncertain noise covariance system
The equivalent state spatial model for stating sliding average colored observed noises is augmented；
Observational equation in the observation of band missing, sliding average colored observed noises and Uncertain noise covariance system is carried out
Conversion.
4. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 3,
It is characterized in that, using the method that is augmented to the observation of band missing, sliding average colored observed noises and Uncertain noise covariance system
The equivalent state spatial model of state equation and the sliding average colored observed noises in system is augmented.
5. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 3,
It is characterized in that, using virtual noise technology to the observation of band missing, sliding average colored observed noises and uncertain noise side
Observational equation in poor system is converted.
6. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 1,
Be characterized in that, will with multiplicative noise, colored observed noises and Uncertain noise covariance it is system converting for only band noise variance it is not true
Determine system, specifically include:
To in multiplicative noise, colored observed noises and Uncertain noise covariance system state equation and colored observed noises side
Cheng Jinhang is augmented；
It is converted to the observational equation in multiplicative noise, colored observed noises and Uncertain noise covariance system.
7. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 6,
It is characterized in that, using the method that is augmented to the shape in multiplicative noise, colored observed noises and Uncertain noise covariance system
State equation and colored observed noises equation are augmented.
8. a kind of mixing uncertain system guaranteed cost Robust Kalman Filter design method according to claim 6,
It is characterized in that, using virtual noise technology to the sight in multiplicative noise, colored observed noises and Uncertain noise covariance system
Equation is surveyed to be converted.
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