Covariance Intersection Fusion Kalman Estimators for Multi-sensor System with Colored Measurement Noises

For multi-sensor system with colored measurement noises, using the observation transformation, the system can be converted into an equivalent system with correlated measurement noises. Based on this method, using the classical Kalman filtering, this study proposed a Covariance Intersection (CI) fusion Kalman estimator, which can handle the fused filtering, prediction and smoothing problems. The advantage of the proposed method is that it can avoid the computation of the cross-covariances among the local filtering errors and can reduce the computational burden significantly, as well as the CI fusion algorithm can be used in the uncertain system with unknown cross-covariances. Based on classical Kalman filtering theory, the centralized fusion and three weighted fusion (weighted by matrices, scalars and diagonal) estimators are also presented respectively. Their accuracy comparisons are given. The geometric interpretations based on covariance ellipses are also given. The experiment results show that the accuracy of the CI fuser is higher than that of the each local smoothers and is lower that that of the centralized fusion Kalman smoother or the optimal fuser weighted by matrix. The MSE curves show that the accuracy of the CI fuser is close to the optimal fuser weighted by matrix in most instances, which means that our proposed method has higher accuracy and good performance.


INTRODUCTION
Multisensor information fusion filtering has been widely applied to many fields, including guidance, navigation, GPS positioning and so on.Now the commonly used method of information fusion is centralized and distributed fusion Kalman methods.The centralized fusion Kalman filters can give the globally optimal state estimation by directly combing all the local measurement equations, but the computation burden is larger.Distributed fusion Kalman filters are given by three weighted fusion algorithms with the matrix weights, scalar weights or diagonal matrix weights (Deng et al., 2005;Sun et al., 2010;Deng et al., 2012).Compared with the centralized fuser, the weighted fuser can reduce the calculation burden, but they are globally suboptimal.
The above weighting fusion filters require to calculate the cross-covariances of local filtering errors.However, in many theoretical and application problems, the cross-covariance is unknown, or the computation of the cross-covariances is very complicated (Sun et al., 2010).
In order to overcome the above drawback and limitation, the covariance intersection fusion Kalman method is presented in Julier andUhlman (1997, 2009), which can avoid computing local cross-covariance and can solve the fused filtering problems for multi-sensor systems with unknown cross-covariance.The accuracy comparison in Deng et al. (2012) is given only for systems with uncorrelated white measurement noises.In this study, a Covariance Intersection (CI) fusion Kalman estimator is presented for multi-sensor system with colored measurement noises, whose accuracy is higher than that of each local Kalman estimator and is lower than that of the centralized fusion estimator or the optimal Kalman fuser weighted by matrices and the accuracy comparison is given.

PROBLEM FORMULATION
Consider a multi-sensor tracking system with colored measurement noises: ( ) ( ) ( ) we have the new measurement equation: (5) From ( 7) we easily obtain w(t) and v i (t) are correlated noise with zeros mean and variance: where, E is the mathematical expectation operator and the superscript T denotes the transpose.
The local steady-state kalman filters: Assume that the system (1) and ( 5) is completely observable and completely stable, then the local steady-state Kalman filter of the th i sensor is given as: where, ∑ i is the one-step predicting error variance and satisfies the steady-state Riccati equation: The local filtering error variances are given as: and the local filtering error cross-covariance satisfies the Lyapunov equation: The local steady-state kalman predictors: For multisensor system (1) and ( 5), the local steady-state Kalman one-step predictor of the th i sensor is given as: The local one-step predictor error variances are computed by Eq. ( 14) The local one-step predictor error cross-covariance satisfies the Lyapunov equation: and the N-step predictor of the th i sensor is: The N-step predictor error variances and crosscovariance are given as the following formula: where the ∑ i and ∑ ij are the one-step error variances and cross-covariance which are obtained by the Eq. ( 14) and ( 21).
The local steady-state kalman smoothers: For multisensor system (1) and ( 5), the local steady-state Kalman N-step smoother of the th i sensor is given as: where, we define that Ψ ் = ൫Ψ ் ൯݇, ∑ i is the one-step predicting error variance and ‫ݔ‬ ො ‫ݐ|ݐ(‬ − 1 ) is the local steady-state one-step predictor.N -step error variance matrix and error cross- covariance of smother are obtained by: where, ∑ ij is the one-step predicting error crosscovariance and when min (r, s)>0, we have: when min (r, s) = 0: The centralized fusion steady-state kalman estimators: Introducing the augmented measurement equation: with the definitions: For the system (1) and ( 34), the centralized fusion steady-state Kalman filter is given as: where, ∑ c satisfies the steady-state Riccati equation: ) The centralized fusion error variance is given by: For the system (1) and ( 34), the centralized fusion steady-state Kalman N-step predictor is given as: For the system (1) and ( 34), the centralized fusion steady-state Kalman smoother is given as: where, ‫ݔ‬ ො ‫ݐ|ݐ(‬ − 1 ) is the certralized fusion steady-state one step predictor: The fused error variance is given by: The three weighted fusion steady-state kalman estimators: When the local smoothing error variance Pi(N) (i = 1, …, L) and cross-covariances P ij (N) (i≠j) are exactly known, the three weighted fusion steadystate Kalman estimators are given as follows: The optimal fusion estimator weighted by matrices is: with the constraints ∑ Ω ୀଵ = ‫ܫ‬ .
The optimal matrix weights are given by: ( ) where, e T = [I n , …, I n ],P = (P ij (N)) nL×Ln the optimal fused error variance matrix is: The optimal fusion smoother weighted by scalars is: with the constraints ∑ ߱ = 1 ୀଵ .The optimal scalar weights are given by: [ ] ( ) where, the symbol tr denotes the trace of matrix and e T = [1, …, 1], P tr = (trP ij (N)) L×L .
The optimal fused error variance is given as: ( ) ( ) The optimal fusion smoother weighted by diagonal matrix is: with the constraints: The optimal diagonal matrix weights are given by: ( ) ( ) P and the fused error variance is given by: The Covariance Intersection (CI) fusion steady-state kalman smoother: When the local smoothing error variance ( ) are exactly known, but the cross-covariances ( ) , ij P i j ≠ are unknown, using the CI fusion algorithm ,the CI fusion Kalman smoother without cross-covariances is presented as follows: with the constraints ∑ ߱ = 1, ୀଵ ߱ ≥ 0, where, the optimal weighted coefficients ߱ (1, … , ‫)ܮ‬ are determined by minimizing the performance index such that: This is a nonlinear optimization problem with constraints in Euclidean space L R , it can be solved by "fmincon" function in MATLAB toolbox.
The cross-covariances can be obtained by the local steady-state Kalman smoothing formula (29), so the actual fused error variance ( ) CI P N is given by: Form Eq. ( 66) the P CI (N) only depend on the local error variance P i (i = 1, …, L), but is independent on cross-covariances P ij , (i ≠ j).Form (68) the actual error variance ܲ ത ூ (ܰ) not only depend on the local error variance P i (i = 1, …, L) but also dependent on crosscovariances P ij , (i ≠ j).So P CI (N) is the common upper bound of the actual fused error variance.

The accuracy comparison of local and fused kalman smoothers:
For the multi-sensor system (1)and ( 5)with exactly known local error variances P i (i = 1, …, L) and cross-covariance P ij , (i ≠ j), the accuracy relations based on error covariance matrix and their trace are given as: , ( ) ( ) where, the matrix inequality A≤B means that B−A≥0 is positive semi-definite.
Remark1: the accuracy of the local and fused smoothers is defined as the trace of their error variances, the smaller trace means higher accuracy and the larger trace means lower accuracy.

SIMULATION RESULTS
Consider the 3-sensor tracking system with colored measurement noises: ( ) ( ) ( ) In the simulation, we take: The accuracy comparison of the local, the centralized fuser, three weighted fusers and the CI fuser are compared in Table 1.
Form Table 1, we see that the accuracy of the centralized fuser trP c (N) is the highest, the actual accuracy ‫ܲݎݐ‬ ത ூ (ܰ) is higher than the local smoothers, but lower than the optimal fusion smoother weighted by matrix trP m (N).The accuracy weighted by scalars trP s (N) is close to that weighted by diagonal matrix trP d (N) and both of them are lower than the accuracy  weighted by matrix.Table 1 verifies the accuracy relations ( 69) and ( 70).
In order to give a geometric interpretation of the accuracy relations, the variance ellipse is defined as the locus of points {x: x T P −1 x = c}, where P is the variance matrix and c is a constant.Generally, we select c = 1.It has been proved in Deng et al. (2012) that P 1 ≤P 2 is equivalent to that the variance ellipse of 1 P is enclosed in that of 2 P .theaccuracy comparison of the variance ellipses is shown in Fig. 1-3.
Form Fig. 1-3, we see that the covariance ellipse of P m (N) is enclosed in the ellipse of ܲ ത ூ (ܰ) and the ellipse of ( ) CI P N is enclosed in that of P CI (N).The ellipse of P s (N) and P d (N) encloses the ellipse of P m (N). the ellipse of P c (N) is enclosed in all ellipses for P i (N), i

Fig. 6 :
Fig. 6: The MSE curves of local and fused smoothers

Table 1 :
The accuracy comparison of the local and fused Kalman