1.Introduction

Kalman filtering is an important state estimation algorithm.It can solve the filtering problems for the time-varying system with unstable or multidimensional signals.In this perspective,the Kalman filtering algorithm is better than the Wiener filtering algorithm and the modern time series analysis algorithm[1].So far,the Kalman filtering algorithm has been widely applied to many fields such as signal processing and navigation[2,3].However,its disadvantage is that the model parameters and noise statistic of the systems must be exactly known[4].In the practical application process,the influence of the surrounding environment,the error of measurement equipment or improper selection for the model and the parameter always lead to observation system errors[5,6].However,they are difficult to be eliminated by applying the traditional Kalman filtering algorithm[7].Now there have been a series of incremental filtering algorithms.For the linear discrete system under poor observation condition,an incremental Kalman filtering algorithm was presented in[8].Those for the nonlinear systems were given by[9]and[10].The extended incremental Kalman filtering problem and the incremental particle filtering problem were solved respectively.However,the incremental smoother has not been presented.

The multi-sensor estimation problems are more common in practice.Many different information fusion Kalman filtering algorithms and results have been presented under different information fusion criteria and methods[11].Compared with the centralized fusion algorithm,the weighted measurement fusion algorithm can yield a globally optimal fuser with lower computation burden[12,13].

In this paper,the detail derivation of an incremental Kalman smoother is given.Furthermore,a weighted measurement fusion incremental Kalman smoother based on the global optimalweighted measurement fusion algorithm is given.The simulation results show its effectiveness and feasibility.

2.Problem formulation

Consider the multi-sensor target tracking systems,with the state equation for target motion denoted by

where xk∈ Rnis the state at time k,Φk ∈ Rn×nis the state transformation matrix,Γk∈ Rn×ris the system noise distribution matrix,and wk∈Rris the system noise vector.

Assumption 1 wkis Gauss white noise with zero mean and variance Qk.x0is the initial state for the target,and Cov[wk,wj]=Qkδkj,Cov[x0,wk]=0.

Suppose the target is observed by N sensors,and the observation equation can be denoted by

where zik∈Rmis the observation vector for the ith sensor at time k,∈ Rm×nis the observation matrix of the ith sensor at time k,and ∈Rmis the observation noise vector of the ith sensor at time k.is a Gauss white noise with zero mean and variance .is the observation system error of the ith sensor at time k.

Assumption 2 Suppose the observation noises are independent of each other at the same time k,as well as those

at different times,i.

e,Cov[]= δkl,Cov[wl,]=0 and Cov[x0,]=0.

The objectives are to find the local and weighted fusion linear minimum variance incremental smoothersandfor the state xk,based on the measurementat time k.

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3.Single sensor incremental Kalman smoother

Based on (2), the incremental observation equation is given as follows:

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where is the observation vector increment for the ith sensor at time k,andSuppose that the stochastic vectorsand are Gauss white noises uncorrelated with each other,thenis also a Gauss white noise with zero mean and variance 2

Remark 1 In the practical Engineering application,the observation system errors ofandare usually close to each other.Thus the system error of Δis comparatively small and can be omitted.

Furthermore,applying(1)and(3)yields

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In(4),the unknown observation error is eliminated.Equations(1)and(4)contribute to the multi-sensor incremental filtering fusion equations.The minimum performance index for the local incremental Kalman filteris

Based on(5),the problem is equivalent to computing the following projection

Using the iterative projection formulation,we have the following iterative relationship

Iterating(1),we find that xkis a linear combination of wk-1,...,w0,x0,i.e.,

where is the Kalman filtering gain.

Computing the projection for(1)on Lit follows that

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From(4),we have the following containment relationship:

Applyingyieldsand from(22)we have the innovation matrix

Furthermore,applying assumption 1 and assumption 2 yields

Applying the projection formula and Ewk=0 yields

then(9)turns into

Thus the weighted measurement fusion equation is given by

From(11),it is obtained that

From(22),it is obtained that

The innovation expression is given as

Suppose that the filtering error and its variance matrix for state xkare

withand as the filtering error variance matrix and the one step prediction error variance matrix,i.e.,

Substituting(4)into(18)yields the innovation expression

Subtracting(14)from(1),we have the error relationship

Applying(7),it follows that

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Substituting(22)into(24)yields

Applying the following relationship:

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it follows thatand from(23)we obtain

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From(25),it is obtained that

Based on(8),it is required to computein order to computeApplying the projection orthogonality,and substituting(1),(22)andintowe haveNote thatit is obtained that

From(8),(28)and(30),it follows that

In summary,we have the incremental Kalman filter as follows.

Applying(23)and(25),we have

In practice,the Kalman smoother has important applications,such as the initial velocity estimation problem in launching a guided missile,the initial concentration estimation problem in the chemical reaction process,and the initial velocity estimation problem in launching the artificial satellite into the orbit.

Theorem 2 For the systems(1)and(4)with assumption 1 and assumption 2,the iterative Kalman smoother is given by

Proof Based on the iterative projection formula,we have the following iterative relationship:

where the smoothing gain is

so=0,and(15)turns into

From(28),it follows that

In a word,the smoothing gain is summed up as computing

Theorem 1 For the systems(1)and(4)with assumption 1 and assumption2,the iterative Kalman filter is given as

Iterating(44)for N times yields the relationship as follows:

where we define

with Inas an n×n unit matrix.Applying(42)yields

Note the following relationships:

and substituting(46)into(48),it follows that

Using(47),it is obtained that

Substituting(53)and(43)into(41),we have

From(40)it is obtained that

这里,,,, 考虑ε≪1, 连续性方程和N-S方程将以围绕基本流场线性化, 得到三维线性稳定性方程如下:

Based on the orthogonality of the projection

and Lwe have

then from(56),it is obtained that

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Substituting(43)into(59),we have

The proof is completed.

4.Multi-sensor weighted measurement fusion incremental Kalman smoother

Applying the augmented method, a centralized fusion measurement equation is contributed as

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whereis the combined measurement vectoris the combined measurement matrix,andis the combined measurement noise with the variance matrix as.

Applying the incremental Kalman filtering algorithm similar to that in Theorem 1,we can obtain the centralized fusion incremental Kalman filterand predictorand their error variance matrices andAnd applying the incremental Kalman smoothing algorithm similar to that in Theorem 2,we can obtain the centralized fusion incremental Kalman smootherand its error variance matrix

The centralized fusion measurement fusion equation(61)can be considered as the measurement model for the state xk.Applying the weighted least squares(WLS)method,we can obtain the WLS estimator as follows:

Taking projection for(4),it follows that

where the measurement errorhas the minimum variance matrix

Multiplying(67)by

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Substituting(62),(63)and(65)into the above equation and introducing the average measurement fusion yield the following weighted estimator:

Introduce the average inverse fusion variance matrix as follows:

and define the average fusion measurement noise as

Multiplying(70)by 1/N,it follows that and the fusion measurement noise has the variance matrix as

Remark 2 For the same measurement matrices ora simplified incremental measurement fusion equation is given by

Theorem 3 For the weighted measurement fusion incremental systems(1)and(74)with assumption 1 and assumption 2,the weighted measurement fusion incremental Kalman filter is given as

Proof It is similar to that given by Theorem 1. □

Theorem 4 For the systems(1)and(74)with assumption 1 and assumption 2,the weighted measurement fusion incremental Kalman smoother is given by

Proof It is similar to that given by Theorem 2.

5.Global optimality of the weighted fusion incremental Kalman smoother

Theorem 5 The weighted measurement fusion incremental Kalman filerand predictorfor the multi-sensor weighted fusion incremental systems(1)and(74)has the global optimality,i.e.,it is numerically identical to the centralized fusion incremental Kalman predictorand predictoras

with the same error variance matrices

It is only required to take the same initial values

Proof See that in[1].

Theorem 6 The weighted measurement fusion incremental Kalman smootherfor the multi-sensor weighted fusion incremental systems(1)and(74)has the global optimality,i.e.,it is numerically identical to the centralized fusion incremental Kalman smootheras

with the same error variance matrix

Proof See that in[1].

6.Simulation model and result analysis

Many simulation experiments show that the system errors cannot be eliminated by the traditional Kalman filtering and self-adaptive Kalman filtering when the observation data has unknown errors due to the environments.In this paper,the presented local and weighted fusion incremental Kalman smoothers can eliminate the system error and greatly improve the filtering accuracy.

Example 1 Consider the one dimension discrete system

where wkand vkiare the system noise and observation noise at time k,respectively.They both are independent Gauss white noises.The mean and variance for wkare 0 and Qk=0.1.The mean and variance forareThey are all known.(i=1,2,3)are unknown observation errors.Suppose=3(i=1,2,3).

The simulation results are shown in Fig.1 and Table 1.A hundred Monte Carlo runs are carried out and the mean square error(MSE)curves are shown in Fig.1,and the numerical comparison at time k=50 is given in Table 1.They show the accuracy of the proposed weighted fusion incremental Kalman smoothing algorithm,which is effective and feasible.

Fig.1 MSE curves for the local and weighted fusion incremental smoothers in Example 1

Table 1 Numerical comparison of MSEs for the local and weighted fusion smoothers at time k=50 in Example 1

Remark 3For Φk=0.9,Γ k=1,=1,=3,Qk=0.1,=1(i=1,2,3),the numerical comparison between the proposed fuser and that in[14]is given by Table 2.It shows the advantages of the proposed fuser at complexity and accuracy.

Sensor 1 Sensor 2 Sensor 3 Fuser 0.946 6 1.002 0 0.834 4 0.570 9

Table 2 Numerical comparison between the proposed estimator and that given in[14]

Estimator Running time/s Root MSE Sequential fusion incremental KF 0.034 5 0.702 0 Weighted fusion smoother 0.030 0 0.536 8

Example 2 Consider the two dimensions discrete system

where wkand are the system noise and observation noise at time k,respectively.They are independent Gauss white noises.The mean and variance for wkare 0 and Qk= 10.The mean and variance forare 0 and=0.005,=0.006 5,=0.009 5(i=1,2,3).They are all known.(i=1,2,3)are unknown observation errors.Suppose=3(i=1,2,3).

The simulation results are shown in Fig.2 and Table 3.A hundred Monte Carlo runs are carried out and the MSE curves are shown by Fig.2,and the numerical comparison at time k=50 is given by Table 3.They show the accuracy of the proposed weighted fusion incremental Kalman smoothing algorithm,which is effective and feasible.

Fig.2 MSE curves for the local and weighted fusion incremental smoothers in Example 2

Table 3 Numerical comparison of MSEs for the local and weighted fusion smoothers at time k=50 in Example 2

Sensor 1 Sensor 2 Sensor 3 Fuser 0.282 5 0.280 2 0.279 5 0.277 9

7.Conclusions

In this paper, the incremental measurement equation is presented,which can effectively eliminate the unknown system errors.Based on the incremental observation equation,an incremental Kalman smoother is given,which solves the state estimation problem for the system with unknown systems errors.Moreover,a globally optimal weighed measurement fusion incremental Kalman smoother is also given based on a globally optimal weighted measurement fusion algorithm.The simulation results show its effectiveness and feasibility.

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