Research Journal of Applied Sciences, Engineering and Technology 6(11): 1963-1969, 2013 ISSN: 2040-7459; e-ISSN: 2040-7467 © Maxwell Scientific Organization, 2013 Submitted: December 14, 2012 Accepted: January 11, 2013 Published: July 25, 2013 Covariance Intersection Kalman Fuser for Two-sensor System with Time-delayed Measurements 1, 2 Peng Zhang and 1Zi-Li Deng Department of Automation, Heilongjiang University, Harbin 150080, China 2 Department of Computer and Information Engineering, Harbin Deqiang College of Commerce, Harbin 150025, China 1 Abstract: For a two-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the measurement transformation method, an equivalent system without measurement delays is obtained and then using the Covariance Intersection (CI) fusion method, the covariance intersection steady-state Kalman fuser is presented. It can handle the estimation fusion problem between local estimation errors for the system with unknown crosscovariances and avoid a large computed burden and computational complexity of cross-covariances. It is proved that its accuracy is higher than that of each local estimator and is lower than that of optimal Kalman fuser weighted by matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relation and indicates that its actual accuracy is close to that of the Kalman fuser weighted by matrices, hence it has good performances. Keywords: Consistency, covariance intersection fusion, information fusion kalman fuser, time-delayed measurement, unknown cross-covariance INTRODUCTION Multisensor information has been applied to many fields, such as Guidance, defense, robotics, tracking, signal processing (Hall and Llinas, 1997; Li et al., 2003; Bar-Shalom and Li, 2001). There are two kinds of optimal information fusion methods: the centralized and distributed information fusion, where the distributed weighted fusion method applies widely because of its less computational burden. Even if, to compute the optimal distributed weighted fusion Kalman estimators (Sun and Deng, 2004; Bar-Shalom and Campo, 1986), it is required to compute the crosscovariances among local Kalman estimator errors, which needs a large computational burden, because these cross-covariances are generally unknown (Deng et al., 2008) or their computation is very complex (Sun and Deng, 2009; Sun et al., 2009; Liggins et al., 2009). In order to overcome this limitation, a Covariance Intersection (CI) fusion method was presented and developed in Julier and Uhlmann (1997), Chen et al. (2002), Julier and Uhlmann (1996), Julier and Uhlmann (2009) and Arambel et al. (2001), which can solve the fused filtering problems with unknown crosscovariances and has the consistency and robustness (Julier and Uhlmann, 1997). It has wide applications (Julier and Uhlmann, 2007; Guo et al., 2010; Bolzani et al., 2007). Besides that, the systems with timedelayed measurement exist widely in control and estimation fields, which results that it is more difficult to compute the cross-covariances. Now there are augmented state method (Kailath et al., 2000) and nonaugmented state method (Sun and Deng, 2009; Zhang and Xie, 2007) to convert the system with time delays into that without time delays. In this study, for the two-sensor system with measurement delays, a measurement transformation approach is presented, by which the fused estimation problem is converted into that for systems without measurement delays (Sun and Deng, 2009; Sun et al., 2009; Liggins et al., 2009). And, to overcome the difficulty and complexity of computing crosscovariances, using the CI method (Julier and Uhlmann, 1996, 1997, 2009; Chen et al., 2002; Guo et al., 2010; Bolzani et al., 2007), a CI fusion steady-state Kalman fuser is presented, which avoids to find the crosscovariances and can reduce the computational burden. It can handle the system with unknown crosscovariances and it has higher accuracy comparing with local estimators. It is rigorously proved that the accuracy of CI fuser is higher than that of each local filter and is lower than that of optimal fuser weighted by matrices. The CI fuser is consistent and the good performance. PROBLEM FORMULATION Consider the two-sensor linear discrete timeinvariant stochastic system: Corresponding Author: Zi-Li Deng, Department of Automation, Heilongjiang University, Harbin 150080, China, Tel.: 13804583507 1963 Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 x(t += 1) Φ x(t ) + Γ w(t ) (1) zi (t= ) H i x(t − τ i ) + ξi (t ), = i 1, 2 (2) where, t is the discrete time, τ i is the measurement delay, 𝑥𝑥(𝑡𝑡)𝜖𝜖𝑅𝑅𝑛𝑛 is the state, 𝑧𝑧𝑖𝑖 (𝑡𝑡)𝜖𝜖𝑅𝑅𝑚𝑚 𝑖𝑖 is the measurement, w(t) 𝜖𝜖𝑅𝑅𝑟𝑟 and 𝜉𝜉𝑖𝑖 𝜖𝜖(𝑡𝑡)𝑅𝑅𝑚𝑚 𝑖𝑖 are uncorrelated white noises with zero mean and variances Q w and 𝑄𝑄𝜉𝜉𝜉𝜉 , respectively. The objectives are to find the local steady-state optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), (N<0, N = 0, N>0), i = 1, 2, the Kalman fusers 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁, 𝑥𝑥𝑑𝑑𝑧𝑧𝑡𝑡𝑡𝑡+𝑁𝑁, 𝑥𝑥𝑠𝑠𝑧𝑧𝑡𝑡𝑡𝑡+𝑁𝑁 weighted by matrices, diagonal matrices and scalars, the CI fusion Kalman fuser 𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) and compare their accuracy relations. 𝑥𝑥�𝐶𝐶𝐶𝐶 THE STEADY-STATE OPTIMAL WEIGHTED KALMAN FUSER Introducing the new measurement y i (t) and measurement noise v i (t): (t ) zi (t + τ i ) yi = vi = (t ) ξi (t + τ i ) (3) where, 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁). Defining the steady-state estimator error variances and cross-covariances are 𝑝𝑝𝑖𝑖𝑖𝑖𝑧𝑧 (𝑁𝑁) = 𝐸𝐸[𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + and 𝑝𝑝𝑖𝑖𝑖𝑖𝑧𝑧 (𝑁𝑁) = 𝐸𝐸[𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)𝑥𝑥�𝑖𝑖𝑧𝑧𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)] 𝑁𝑁)𝑥𝑥�𝑖𝑖𝑧𝑧𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)], respectively. So we have obtained Pijz ( N ) = Pij ( N − τ i , N − τ j ), i = 1, 2 (9) where, P ij (N-τ i , N-τ j ) = E[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁−𝜏𝜏𝑖𝑖 From (7) we can see that the problem of finding the local steady-state optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 based on the measurement (z i (t+N), z i (t+N-1), …) is converted t that of finding the optimal steady-state Kalman estimators 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 ) based on the measurement (y i (t+N-τ i ), y i (t+N-τ i -1), …), For N > τ i , N = τ i , N > τ i , the estimators are called the smoothers, filters and predictors, respectively. Lemma 1: (Sun and Deng, 2009) For the two-sensor system (1) and (5), the local steady-state Kalman predictor 𝑥𝑥�𝑖𝑖 (𝑡𝑡 + 1|𝑡𝑡) is given by: xˆˆ( = 1| t ) Ψ pi xi (t | t − 1) + K pi y= 1, 2 i t+ i (t ), i (10) (4) where, From (2), we have the new measurement equation without measurement delays. yi (t ) = H i x(t ) + vi (t ), i = 1, 2 = L( yi (t + N − τ i ), yi (t + N − τ i − 1),) (7) where 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 ) is the linear minimum variance filters of x(t) based on (y i (t+N-τ i ), y i (t+N-1-τ i ), …) and 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) is the linear minimum variance filters based on L(z i (t+N), z i (t+N-1), …). So the relation between the estimation errors 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) and 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 is: xiz (t | t + N= ) xi (t | t + N − τ i ),= i 1, 2 (12) The prediction error variance matrix Σ i satisfies the steady-state Riccati equation: Σ i =Φ [Σ i − Σ i H iΤ ( H i Σ i H iΤ + Qvi ) −1 H i Σ i ]Φ Τ + Γ Qw Γ Τ and the cross-covariance matrix (13) ∑𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡 + 1𝑡𝑡𝑥𝑥𝑖𝑖𝑇𝑇𝑡𝑡+1𝑡𝑡 between local prediction errors satisfies the Lyapunov equation: Σ ij = Ψ pi Σ ijΨ pjΤ + Γ Qw Γ Τ , i, j = 1, 2, i ≠ j (6) So we have the relation as: xˆˆiz (t | t + N= ) xi (t | t + N − τ i ) (11) (5)= K pi ΦΣ i H iΤ ( H i Σ i H iΤ + Qvi ) −1 where, the new measurement noise v i (t) is white noise with zero mean and variance Q vi = Q ζi . Denoting the linear space spanned by the stochastic variables (z i (t+N), z i (t+N-1), …) as L (z i (t+N), z i (t+N1), …) and the linear space spanned by the stochastic variables ( yi (t + N − τ i ), yi (t + N − τ i − 1),) as L( yi (t + N − τ i ), yi (t + N − τ i − 1),) , we have the relation between the two linear measurement spaces as: L( zi (t + N ), zi (t + N − 1),) Ψ pi= Φ − K pi H i (14) The -k i steps steady-state Kalman predictor 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) is given by: | t + ki ) Φ − ki −1 xi (t + ki + 1| t + ki ), ki ≤ −2 xˆˆ( i t= (15) The -k i steps Kalman prediction error variance 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and crosscovariance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E�𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 �𝑡𝑡�𝑡𝑡�𝑡𝑡 + 𝑘𝑘𝑗𝑗 �� are given by: (8) 1964 Pi (ki , ki ) = Φ − ki −1Σ iΦ ( − ki −1) Τ − ki + ∑Φ − ki − j Γ Qw Γ ΤΦ ( − ki − j ) Τ , ki ≤ −2 j =2 (16) Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 = Pij (ki , k j ) Φ − ki −1Σ ijΦ ( − ki −1) Τ + − max( ki , k j ) − 2 ∑ j =0 (17) Φ j Γ Qw Γ ΤΦ jΤ , ki , k j ≤ −2 The steady-state Kalman filter 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡) is as following: xˆˆ( | t ) Ψ fi xi (t − 1| t − 1) + K fi yi (t= ), i 1, 2 i t= (18) (19) = K fi Σ i H iΤ ( H i Σ i H iΤ + Qvi ) −1 (29) Pi (ki , ki ) = Σ i , ( ki = −1) (30) Pij (ki , k j ) = Σij , (ki = −1, k j = −1) (31) − ki − 2 ∑Φ j j =0 Γ Qw Γ ΤΦ jΤ , k −k j Pij (ki , k j ) = Φ − ki −1Ψ pii −k j −2 ∑ + Σ ijΦ ( − k −1) Τ i (33) Φ − k −1Ψ pik + r +1Γ Qw Γ ΤΦ r Τ + i i r= − ki −1 − ki − 2 ∑Φ (21) Γ Qw Γ ΤΦ r Τ , (k j ≤ ki ≤ −2) r r =0 ( k − ki ) Τ Pij (ki , k j ) = Φ − ki −1Σ ijΨ pi j Pij Ψ fi PijΨ fjΤ + ( I n − K fi H i )Γ Qw Γ Τ ( I n − K fj H j )Τ = (22) − ki − 2 ∑ + Φ ( − k j −1) Τ k j + r +1 ( −1− k j ) Τ k j + r +1 pj ( −1− k j ) Τ Φ r Γ Qw Γ ΤΨ pj Φ (34) r= − k j −1 The steady-state Kalman smoother 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ), (k i >0) is given by: ki xˆˆ( = xi (t | t − 1) + ∑ K i (r )ε i (t + r ) i t |t+k i) i = 1, 2 (32) (ki ≤ −2) The corresponding steady-state Kalman filtering error variance 𝑃𝑃𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡)𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡)] and crosscovariance 𝑃𝑃𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡)𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡)] are given by: i, j = 1, 2 , i ≠ j Pij (ki , k j ) = Pij = Pi (ki , ki ) Φ − ki −1Σ iΦ ( − ki −1) Τ + (20) P = ( I n − K fi H i )Σ i ii (28) Case 2: when k i <0, k j <0 where, Ψ= ( I n − K fi H i )Φ fi Pi (ki , ki ) = Pii − ki − 2 ∑ + Φ Γ Qw Γ Ψ Τ r r= − k j −1 −k j −2 ∑Φ (23) r r =0 Φ + Γ Qw Γ ΤΦ rΤ , (ki ≤ k j ≤ −2) r =0 − k j −1 Pij (ki , k j ) = Ψ pi where, + = K i (r ) Σ iΨ piΤr H iΤ ( H i Σ i H iΤ + Qvi ) −1 ε i (t ) = yi (t ) − H i xˆi (t | t − 1) −k j −2 r =0 (24) (25) ∑Ψ ki Pi (ki , ki ) = Σ i − ∑ K i (r )( H Σ i H Τ + Qvi ) K iΤ (r ), ki > 0 ( − k j −1) Τ (35) Γ Qw Γ ΤΦ r Τ , (ki = −1, k j ≤ −2) = Pij (ki , k j ) Φ − ki −1Σ ijΨ pi( − ki −1) Τ + − ki − 2 ∑Φ r r =0 The Kalman smoothing error variance 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and cross-covariance 𝑃𝑃𝑖𝑖𝑖𝑖 �𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 � = 𝐸𝐸�𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 �𝑡𝑡�𝑡𝑡 + 𝑘𝑘𝑗𝑗 �� are given by: r pi Σ ijΦ (36) Γ Qw Γ ΤΨ pjrΤ , (ki ≤ −2, k j = −1) Case 3: when k i >0, k j >0 ki Pii (ki , ki ) = Σ i − ∑ K i (r )[ H i Σ i H iΤ + Qvi ]K iΤ (r ), (ki > 0) (37) r =0 (26) kj ki Pij (ki , k j ) = ∑ [ I nδ r 0 − Ki (r ) H iΨ pir ]Σ ij ∑ [ I nδ s 0 − K j (s) HjΨ pjs ]Τ r =0 (38) = r 0=s 0 min( ki , k j ) Pij (ki , k j ) = Σ ij − ∑ K i (r ) H Σ ij H Τ K Τj (r ), ki , k j > 0 + (27) min( ki , k j ) r −1 ∑ ∑ K (r ) H Ψ = r 0= u 0 i i uΤ pi Γ Qw Γ ΤΨ pjuΤ H Τj K Τj (r ), (i ≠ j ) r =0 Case 4: when k i <0, k j <0≥ Lemma 2: (Sun and Deng, 2009) For the two-sensor system (1) and (5), the steady-state Kalman filtering error variance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = E [𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and cross-covariance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E�𝑥𝑥�𝑖𝑖 �𝑡𝑡�𝑡𝑡 + 𝑘𝑘𝑗𝑗𝑥𝑥𝑖𝑖𝑇𝑇𝑡𝑡𝑡𝑡+𝑘𝑘𝑖𝑖, (k i = N-τ i , k j = N- τ i ) are given by: = Pij (ki , k j ) Φ − ki −1Σ ijΨ pj( −1− ki ) Τ − Case 1: when k i = 0, k j = 0 1965 kj ∑Φ − ki −1 s =0 − ki − 2 ∑Φ r r =0 Σ ijΨ pj( s − k −1) Τ H Τj K Τj ( s ) + i Γ Qw Γ ΤΨ pjrΤ + k j − ki − 2 ∑ ∑Φ r 0 =s 0= r Γ Qw Γ ΤΨ pj( s + r ) Τ H Τj K Τj ( s ) (39) Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 Case 5: when k i ≥0, k j <0 − k j −1 = Pij (ki , k j ) Ψ pj ki Σ ijΦ ( − k −1) Τ − ∑ Ki (s) H iΨ pi ( s − k j −1) s =0 + −k j −2 ∑Ψ =r 0 where the optimal weighted diagonal matrices A j is given as: i Σ ijΦ (40) ( − k j −1) Τ ki − k j − 2 r pi Γ Qw Γ ΤΦ rΤ + ∑ ∑ K ( s) H Ψ i =s 0=r 0 j s+r pi a j1 Aj = Γ Qw Γ ΤΦ rΤ Lemma 3: (Sun and Deng, 2004) For the two-sensor system (1) and (5), the optimal Kalman fuser weighted by matrices is given by: z xˆˆˆ( N ) Ω1 x1z (t | t + N ) + Ω2 x2z (t | t + N ) m t | t += = Ω1 xˆˆ( 1 t | t + N − τ 1 ) + Ω2 x2 (t | t + N − τ 2 ) xˆˆzj1 z z ˆˆ x j = , xd = xˆˆzjn −1 Τ −1 = P(ki , k j ) (= Pij (ki , k j )) NL× NL , (i, j 1, 2) (42) (43) where, e = [I n , I n ] and the fused error variance matrix P m is given as: −1 −1 Pm = [e P (ki , k j )e] The optimal Kalman fuser by scalars is given by: (44) 𝑥𝑥�𝑠𝑠𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) weighted z xˆˆˆ( N ) a1 x1z (t | t + N ) + a2 x2z (t | t + N ) s t | t += = a1 xˆˆ( 1 t | t + N − τ 1 ) + a2 x2 (t | t + N − τ 2 ) xdz1 xdnz (52) (45) 2 ∑a j =1 jl x zjl (t | t= + N ), l 1, , n esΤ Ptr−1 esΤ Ptr−1es = al a1l a2l ] [= edΤ ( P ll ) −1 edΤ ( P ll ) −1 ed where 𝑒𝑒𝑑𝑑𝑇𝑇 = [1 … 1] and defining: P (ll ) P ll = 1(ll ) P21 P12(ll ) P2(ll ) trP (k , k ) trP12 (k1 , k2 ) Ptr = 1 1 1 trP21 (k2 , k1 ) trP2 (k2 , k2 ) (48) The corresponding optimal fused estimation error variance P s is given by: 2 From the proposed interpretation, it is clear that the computation of getting the cross-covariances between local filtering is very complex and the computed burden is very large. So a CI fusion method was presented (Julier and Uhlmann, 1997; Chen et al., 2002). Consider a two-sensor system (1) (2) and (5), when P 1 and P 2 are known, but the cross-covariance P 12 is unknown, the CI Kalman fuser without P 12 is given by: xˆˆCIz= (t | t + N ) PCI [ω P1−1 x1z (t | t + N ) + (49) (1 − ω ) P2−1 xˆ2z (t | t + N )] =i 1 =j 1 The optimal Kalman fuser 𝑥𝑥�𝑑𝑑𝑧𝑧 (𝑡𝑡|𝑡𝑡|𝑡𝑡 + 𝑁𝑁) weighted by diagonal matrices is given by: z xˆˆˆ( N ) A1 x1z (t | t + N ) + A2 x2z (t | t + N ) d t | t += ˆˆ( = A1 x1 t | t + N − τ 1 ) + A2 x2 (t | t + N − τ 2 ) (56) THE STEADY-STATE CI KALMAN FUSER (47) 2 (55) =i 1 =j 1 (46) e = [1 1] Ps = ∑∑ ai a j Pij (ki , k j ) (54) where 𝑃𝑃𝑖𝑖𝑖𝑖𝑙𝑙𝑙𝑙 is the (l, l)th diagonal component of P ij (k i , k j ) and the optimal component fused error variance P dl and the optimal fused error variance P d are given by: Pd = ∑∑ Ai Pij AΤj where, Τ s (53) where the optimal weighted coefficient vector is: Τ ll −1 −1 where the optimal weighted coefficients a = [a 1 , a 2 ] is= Pdl [e= 1, , n d ( P ) ed ] , l given as: 2 2 a= (51) So the equation (50) is equivalent with: T Τ j = 1, 2 , (41) xˆˆdlz (t = | t + N) [Ω1 , Ω2 ] = (e P (k1 , k2 )e) e P (k1 , k2 ) −1 a jn The component expression of 𝑥𝑥�𝑗𝑗𝑧𝑧 and 𝑥𝑥�𝑑𝑑𝑧𝑧 are given by: where the weighted matrix is given by: Τ a j2 (57) where the CI fusion error variance matrix P CI is given by: (50) 1966 P= [ω P1−1 + (1 − ω ) P2−1 ]−1 CI (58) Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 where, Pi Pi (ki , ki ), i = 1, 2 , P12 P12 (k1 , k2 ) , P21 P21 (k2 , k1 ) and the weighted coefficient 𝜔𝜔𝜔𝜔[0, 1] and minimizes the performance index: J = min trPCI = min tr{[ω P1−1 + (1 − ω ) P2−1 ]−1} ω (59) ω∈[0,1] where the notation tr denotes the trace of matrix. The optimal weighting coefficient ω can be solved by the gold section method or Fibonacci method (Julier and Uhlmann, 1996). � 𝐶𝐶𝐶𝐶 ≤ 𝑃𝑃 � 𝐶𝐶𝐶𝐶 and which yields (63). It has been proven 𝑃𝑃 trP m ≤trP d ≤trP s ≤trP i , = 1, 2 in the reference (Sun and Deng, 2004; Julier and Uhlmann, 1997; Chen et al., 2002), so we have (64) and (65) hold. In the performance index (59), taking ω = 0, we have J = trP 2 and taking ω = 1, we have J = trP 1 . Hence the optimal weighting coefficient 𝜔𝜔𝜔𝜔[0, 1] yields trP CI ≤trP 1 ≤trP CI ≤trP 2 , i.e. trP CI ≤ trP i , i = 1, 2. Taking trace operation for (63) and (64), we have (66) holds. The proof is completed. SIMULATION EXAMPLE THE CONSISTENCY AND ACCURACY OF CI KALMAN FUSER Consider the tracking system with two-sensor and with time-delayed measurements: Theorem1: The actual CI fusion error variance matrix 𝑃𝑃�𝐶𝐶𝐶𝐶 is given by: = PCI PCI [ω 2 P1−1 + ω (1 − ω ) P1−1 P12 P2−1 + 0.5T02 1 T0 = x(t + 1) x(t ) + 0 1 T0 ) H i x(t − τ i ) + ξi (t ), = zi (t= i 1, 2 (60) ω (1 − ω ) P2−1 P21 P1−1 + (1 − ω ) 2 P2−1 ]PCI Proof: From (58), we can obtain which yields 𝑃𝑃𝐶𝐶𝐶𝐶 [𝜔𝜔𝑃𝑃1−1 = x(t ) PCI [ω P1−1 x(t ) + (1 − ω ) P2−1 x(t )] + 𝑃𝑃2−1 ] = 𝐼𝐼, (61) Subtracting (57) from (61), we get the actual CI fuser error: xCI= (t | t + N ) PCI [ω P1−1 x1 (t | t + N ) + H1 = The actual CI fusion error variance 𝑃𝑃�𝐶𝐶𝐶𝐶 = 𝑇𝑇 (𝑡𝑡|𝑡𝑡 𝐸𝐸[𝑥𝑥�𝐶𝐶𝐶𝐶 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)𝑥𝑥�𝐶𝐶𝐶𝐶 + 𝑁𝑁)], where Ε denotes the mathematical expectation, Τ denotes the transpose. Substituting (62) into 𝑃𝑃�𝐶𝐶𝐶𝐶 yields that (60) holds. The proof is completed. Theorem 2: For local and fused Kalman filters, we have the accuracy relations: Pm ≤ Pd , Pm ≤ Ps , Pm ≤ PCI , Pm ≤ Pi , i = 1, 2 (63) PCI ≤ PCI (64) trPm ≤ trPd ≤ trPs ≤ trPi , i = 1, 2 (65) trPm ≤ trPCI ≤ trPCI ≤ trPi , i = 1, 2 (66) Proof: from (41), 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) is the Unbiased Linear Minimum Variance (ULMV) estimate of x(t ) based on the linear space 𝐿𝐿𝑚𝑚 = 𝐿𝐿𝑚𝑚 �𝑥𝑥�1𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)� spanned by �𝑥𝑥�1𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)�. From (45), (50), (57), we have 𝑥𝑥�𝜃𝜃𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝜖𝜖𝐿𝐿𝑚𝑚 , 𝜃𝜃 = 𝑠𝑠, 𝑑𝑑, 𝐶𝐶𝐶𝐶, 𝑖𝑖, 1 0 0] , = H2 ,τ 1 1,= τ2 2 = 0 1 where, 𝑥𝑥(𝑡𝑡)𝜖𝜖𝑅𝑅 2 is the state, z i (t) is the measurement with delays for sensor i, w(t) and ζ i (t) are white noise with zero mean and variances Q w and Q ζi , respectively. According to the corresponding transformation, we obtain the measurement y i (t) without time delays: yi (t ) = zi (t + τ i ) = H i x(t ) + vi (t ) (62) (1 − ω ) P2−1 x2 (t | t + N )] [1 where v i (t) = ζ i (t+τ i ) is the transformed measurement white noise with zero mean and variance Q vi = Q ζi . Thus the problem of finding the local steady-state optimal Kalman smoother 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), (𝑖𝑖 = 1, 2) based on the measurement z i (t) is converted into the problem of finding the local steady-state optimal Kalman filter 𝑥𝑥�1 (𝑡𝑡|𝑡𝑡) and Kalman predictor 𝑥𝑥�2 (𝑡𝑡|𝑡𝑡 − 1) based on the new measurement y i (t), respectively. Further, applying the Kalman filtering method yields the steady-state optimal fusion Kalman smoothers 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡 + 1, 𝑥𝑥𝑑𝑑𝑧𝑧𝑡𝑡𝑡𝑡+1, 𝑥𝑥𝑠𝑠𝑧𝑧𝑡𝑡𝑡𝑡+1 weighted by matrices, Fig. 1: The accuracy comparison of P 1 , P 2 , P 0 , P s , P CI and 𝑃𝑃�𝐶𝐶𝐶𝐶 1967 Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 0.78 0.68 0.58 0.48 0.38 0.28 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 MSE1 MSEd MSE2 MSECI MSE0 trP1 MSEs trP2 Fig. 2: The curves of MSEi(t) diagonal matrices and scalars and CI fused Kalman 𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 1). In simulation, we take smoother 𝑥𝑥�𝐶𝐶𝐶𝐶 0 4 = T0 0.4,= Qw 1,= Qξ 1 0.36,= Qξ 2 0 0.16 In order to give a powerful geometric interpretation with respect to accuracy relations of local and fusers, the covariance ellipse for a covariance matrix P is defined as the locus of points {x: xTP-1x = c} where c is a constant. In the sequel, c = 1 will be assumed without loss of generality. It was proved (Julier and Uhlmann, 1996) that P a ≤ P b is equivalent to that the ellipse for P a is enclosed in the ellipse for P b . The simulation results are shown in Fig. 1. From Fig. 1, it is clearly shown that the covariance ellipse for P m lies within the intersection of ellipses for P 1 and P 2 , the ellipse for CI fused theoretical covariance P CI encloses the intersection region of ellipses for P 1 and P 2 (Chen et al., 2002) and passes through the four points of intersection of ellipses for P 1 and P 2 . The ellipse for CI fused actual covariance 𝑃𝑃�𝐶𝐶𝐶𝐶 lies within the ellipse for theoretical covariance 𝑃𝑃�𝐶𝐶𝐶𝐶 . In order to verify the above theoretical results for accuracy relation, 200 Monte-Carlo runs are performed for t = 1, …, 300 the results are shown in Fig. 2. where the straight lines denote trP i , i = 0, 1, 2, CI and trPCI , the curves denote the corresponding mean square errors MSE i (t). We see the CI fuser has good performance, whose accuracy is approximates to the accuracy of the optimal smoother, because MSE CI (t) is approximates to MSE m (t). CONCLUSION matrices, a CI fusion steady-state Kalman fuser has been presented, which has the advantage that the computation of cross-covariances can be avoided. It is proved that its accuracy is higher than that of each local estimator and is lower than that of optimal fuser weighted by matrices. A two-sensor system MonteCarlo simulation example shows that its accuracy is approximates to the accuracy of the optimal fuser and is higher than those of the fusers weighted by diagonal matrices and scalars. ACKNOWLEDGMENT This study is supported by the Natural Science Foundation of China under grant NSFC-60874063, the 2012 Innovation and Scientific Research Foundation of graduate student of Heilongjiang Province under grant YJSCX2012-263HLJ and the Support Program for Young Professionals in Regular Higher Education Institutions of Heilongjiang Province under grant 1251G012. REFERENCES Arambel, P.O., C. Rago and R.K. Mehra, 2001. Covariance intersection algorithm for distributed spacecraft state estimation. Proceedings of the American Control Conference. Arington, VA, pp: 4398-4402. Bar-Shalom, Y. and L. Campo, 1986. The effect of the common process noise on the two-sensor fusedtrack covariance. IEEE T. Aero. Elec. Syst., 22: 803-805. Bar-Shalom, Y. and X.R. Li, 2001. Estimation with Applications to Tracking and Navigation. John Wiley and Sons Inc., Thiagalingam Kirubarajan. For the two-sensor system with time-delayed measurements and with unknown cross-covariance 1968 Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013 Bolzani, J.C., C. Ferreira and J. Waldmann, 2007. Covariance intersection-based sensor fusion for sounding rocket tracking and impact area prediction. Control Eng. Pract., 15: 389-409. Chen, L.J., P.O. Arambel and R.K. Mehra, 2002. Estimation under unknown correlation: Covariance intersection revisited. IEEE T. Automat. Contr., 47: 1879-1882. Deng, Z.L., Y. Gao, C.B. Li and G. Hao, 2008. Selftuning decoupled information fusion Wiener state component filters and their convergence. Automatica, 44: 685-695. Guo, Q., S.Y. Chen, H.R. Leung and S.T. Liu, 2010. Covariance intersection based image fusion technique with application to pan sharpening in remote sensing. Inform. Sci., 180: 3434-3443. Hall, D.L. and J. Llinas, 1997. An introduction to multisensor data fusion. Proc. IEEE, 85: 6-23. Julier, S.J. and J.K. Uhlmann, 1996. General decentralized data fusion with unknown cross covariances. Proceedings of the SPIE Aerosence Conference, SPIE, 2775: 536-547 Julier, S.J. and J.K. Uhlmann, 1997. Non-divergent estimation algorithm in the presence of unknown correlations. Proceedings of the IEEE American Control Conference. Albuquerque, NM, USA, pp: 2369-2373. Julier, S.J. and J.K. Uhlmann, 2007. Using covariance intersection for SLAM. Robot. Auton. Syst., 55: 3-20. Julier, S.J. and J.K. Uhlmann, 2009. General Decentralized Data Fusion with Covariance Intersection. In: Liggins, M.E., D.L. Hall and J. Llinas (Eds.), Handbook of Multisensor Data Fusion, Theory and Practice. 2nd Edn., CRC Press, pp: 319-342. Kailath, T., A.H. Sayed and B. Hassibi, 2000. Linear Estimation. Upper Saddle River, Prentice-Hall, New Jersey. Li, X.R., Y.M. Zhu, J. Wang and C.J. Han, 2003. Unified optimal linear estimation fusion, Part I: Unified fusion rules. IEEE T. Inform. Theory, 49: 2192-2208. Liggins, M.E., D.L. Hall and J. Llinas, 2009. Handbook of Multisensor Data Fusion: Theory and Practice. 2nd Edn., CRC Press, Boca Raton, FL. Sun, S.L. and Z.L. Deng, 2004. Multi-sensor optimal information Kalman filter. Automatica, 40: 1017-1023. Sun, X.J. and Z.L. Deng, 2009. Information fusion Wiener filter for the multisensor multichannel ARMA signals with time-delayed measurements. IET Signal Process., 3(5): 403-415. Sun, X.J., Y. Gao, Z.L. Deng, C. Li and J.W. Wang, 2009. Multi-model information fusion kalman filtering and white noise deconvolution. Inform. Fusion, 11: 163-173. Zhang, H.S. and L.H. Xie, 2007. Control and Estimation of Systems with Input/output Delays. Springer, Berlin. 1969