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Weighted fusion robust steady-state Kalman filters for multisensor system with uncertain noise variances.

1. Introduction

The multisensor information fusion (multisource information fusion or multisensor data fusion) has been applied widely in many fields including guidance, defense, robotics, target tracking, signal processing, GPS positioning, unmanned aerial vehicle (UAV), communication, command, control, computer, and intelligent systems ([C.sup.4]I) and has attracted significant interest in recent years. Over the past two decades, many fused Kalman filtering algorithms have been developed to handle the state and signal estimation problems for the multisensor systems. The aim is how to combine the local estimators or local measurements obtained from each sensor to obtain a fused estimator, whose accuracy is higher than that of each local estimator [1]. The basic fused filtering algorithms include the centralized and distributed algorithms depending on whether the measurements' information is directly communicated to the fusion center or not [2]. For the centralized fusion algorithm, all the measurement data from local sensor are carried to the fusion center which can give the globally optimal fused state estimation, but its disadvantage is requiring a larger computation and communication burden. The distributed fusion algorithms can give the globally optimal or suboptimal state estimation by combing or weighting the local state estimators, whose advantages are that they can reduce the communication burden, and is more robust and reliable, and also has stronger fault tolerance. Under the ULMV rule, there are three optimal distributed fusion algorithms weighted by matrices, diagonal matrices, and scalars, respectively [3-5]. The optimal weighted measurement fusion algorithms can give the global optimal state estimation by weighting the local measurements to obtain a fusion measurement equation, accompanied with the state equation; based on a single Kalman filter, two optimal weighted measurements fusion algorithms were presented in [6-8].

The classical Kalman filtering is only suitable to handle the state estimation problems for the systems that the model parameters and noises variances are precisely known. However, in many application problems, the uncertainties of model parameters and noise variances are widely found. In the presence of uncertainties, the filter performance is degraded and may yield the filter divergence. In order to solve the filtering problems for uncertain systems, in recent years, several results have been derived on the design of robust Kalman filters. The so-called robust Kalman filtering is to design a filter to guarantee a minimal upper bound of the actual filtering error variances for all admissible uncertainties. There are two basic approaches to solve this problem, which are the Riccati equation approach [9-15] and the linear matrix inequality (LMI) approach [16-20]. These two methods have been applied to design the robust Kalman filter for uncertain systems with the uncertainties of model parameters [9-20], where the noise variances are assumed to be exactly known. However, the design of the robust Kalman filters with uncertain noise variances is seldom reported [21-24].

Several robust Kalman filters only consider the stochastic systems with single sensor, while the design of the multi-sensor information fusion robust Kalman filters is seldom considered and the robustness of the fused Kalman filters was not proved [25-27].

The robust Kalman filters design includes the finite-horizon (time-varying) robust Kalman filters design and the infinite-horizon (steady-state) robust Kalman filters design. The steady-state robust Kalman filters can be designed by taking the limits for the time-varying robust Kalman filters [9, 10, 14], and this is called the indirect design method. However, it is seldom reported that applying the steady-state Kalman filtering theory can directly design the steady-state robust Kalman filters, and this is called the direct design method.

For the systems with uncertain model parameters and/or noise variances, the covariance intersection (CI) fusion robust filtering method was presented in [28-31]. Its basic principle is that the robust CI fusion filter can be obtained by the convex combination of the robust local filters. Its advantage is that the cross-covariance of the local filtering errors is avoided, and it is suitable to handle the systems with unknown cross-covariance. Its disadvantage is that the local robust filters are assumed to be known, and the upper bound of the actual fused estimates has larger conservativeness because the information of cross-covariance is ignored. The geometric principle of the CI fusion is that the variance ellipse of the upper bound of actual fusion estimation error variance tightly encloses the intersection region of all variances ellipses of the upper bounds of actual local estimation error variances [32]. Recently, the ellipsoidal intersection (EI) fusion method with the cross-covariance information was presented in [33], which improves the robust accuracy of the CI fusion estimate. The comparison of the CI fusion method with several weighted fusion methods was given in [5]. The CI fusion method has been applied to many fields including remote sensing [34], simultaneous localization and mapping (SLAM) [35], rocket tracking and prediction [36], and vehicle localization [37].

Recently, the robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances were presented by our team in [24], where the Lyapunov equation method of designing robust Kalman filters, the five robust weighted fusion time-varying Kalman filters based on the minimax robust estimation principle, and the concept of robust accuracy have been presented. By taking the limits for the time-varying robust weighted fusion Kalman filters, the corresponding robust steady-state Kalman filters have been also presented. However, based on the steady-state Kalman filtering theory, the problem to design directly the robust steady-state Kalman filters is not solved, and, based on the cross-covariance information, the problem to reduce the conservativeness of the upper bound of the CI fusion estimate is not solved. In addition, only one robust weighted measurement fuser was presented in [24].

In this paper, based on the steady-state Kalman filtering theory [38, 39], for the multisensor time-invariant system with noise variances uncertainties, using the minimax robust estimation principle, for the worst-case conservative system with the conservative upper bound of noise variances, three weighted state fusion robust steady-state Kalman filters will be presented. In order to improve the robust accuracy of the CI fuser and to reduce its conservativeness, a modified robust CI fuser with the cross-covariance information will be presented. In addition, two weighted measurement fusion robust steady-state Kalman filters will be presented. The proposed robust weighted fusion steady-state Kalman filtering approach is different from that in [24]. Our approach avoids finding the time-varying robust weighted fusers and their limits.

Finally, the robustness of the local and weighting fused robust Kalman filters is proved based on the Lyapunov equation method, which is completely different from the Riccati equation method and LMI method [9-20]. Their robust accuracy relations are strictly proved. In order to verify the correctness of theoretical accuracy relations, a Monte-Carlo simulation example for a three-sensor tracking system with uncertain noise variances is given.

The remainder of the paper is organized as follows. The problem formulation is given in Section 2. The local robust steady-state Kalman filters and their robustness analysis are presented in Section 3. Six weighted fusion robust steady-state Kalman filters and their robustness analysis are proposed in Section 4. The comparison of the robust accuracies of the local and fused robust Kalman filters is given in Section 5. Section 6 gives a simulation example. The conclusions are presented in Section 7.

2. Problem Formulation

Consider the multisensor time-invariant system with uncertain noise variances

x(t + 1) = [PHI]x(t) + [GAMMA]w(t), (1)

[y.sub.i](t) = [H.sub.i]x(t) + [v.sub.i](t), i = 1, ..., L, (2)

where t is the discrete time, x(t) [member of] [R.sup.n] is the state to be estimated, L is the number of sensors, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the measurement and measurement noise of the ith subsystem, and w(t) [member of] [R.sup.r] is the input noise. [PHI], [GAMMA], and [H.sub.i] are known constant matrices with appropriate dimensions.

Assumption 1. w(t) and [v.sub.i](t) are uncorrelated white noises with zero mean and unknown uncertain actual variances [bar.Q] and [[bar.R].sub.i], and Q and [R.sub.i] are known conservative upper bounds of [bar.Q] and [[bar.R].sub.i], respectively; that is,

[bar.Q] [less than or equal to] Q, [[bar.R].sub.i] [less than or equal to] [R.sub.i], i = 1, ..., L, (3)

in the sense that A [less than or equal to] B means that B - A [greater than or equal to] 0 is a positive semidefinite matrix.

Assumption 2. ([PHI], [GAMMA]) is a completely controllable pair and ([PHI], [H.sub.i]) is a completely observable pair.

The problem is to design the local or fused robust steady-state Kalman filter [??](t | t) such that the variances of actual filtering errors are guaranteed to have a minimal upper bound S for all admissible uncertain noise variances [bar.Q] and [[bar.R].sub.i] satisfying (3); that is, the actual filtering error variance satisfies

E [(x(t) - [??](t | t)) [(x(t) - [??](t | t)).sup.T]] [less than or equal to] S, (4)

where E is the mathematical expectation operator and the superscript T is the transpose.

Definition 3. The measurements [y.sub.i](t) generated from the systems (1) and (2) with unknown actual noise variances [bar.Q] and [[bar.R].sub.i], i = 1, ..., L, are called the actual measurements [y.sub.i](t), which are obtained via the sensors, and are available (known).

Definition 4. The measurements [y.sub.i](t) generated from the systems (1) and (2) with the conservative upper bounds Q and [R.sub.i] of noise variances are called the conservative measurement which are unavailable (unknown).

Definition 5. The Kalman filters with conservative measurements [y.sub.i](t) are called the conservative Kalman filter which is unrealizable. The Kalman filters with the actual measurements [y.sub.i](t) are called the actual Kalman filters.

3. Local Robust Steady-State Kalman Filters

According to the minimax robust optimal estimation principle [40], consider the worst-case conservative systems (1) and (2) with Assumptions 1-2 and with the conservative upper bounds Q and [R.sub.i] of noise variances; the conservative local steady-state Kalman filters are given as [38, 39]

[[??].sub.i] (t|t) = [[PSI].sub.i][[??].sub.i] (t - 1 | t - 1) + [K.sub.i][y.sub.i](t), i = 1, ..., L, (5)

[[PSI].sub.i] = [[I.sub.n] - [K.sub.i][H.sub.i]] [PHI], [K.sub.i] = [[summation].sub.i][H.sup.T.sub.i] [([H.sub.i][[summation].sub.i][H.sup.T.sub.i] + [R.sub.i]).sup.-1], (6)

where [I.sub.n] is an n x n identity matrix, [[PSI].sub.i] is a stable matrix, and [K.sub.i] is the steady-state filtering gain matrix. Here, the measurements [y.sub.i](t) are unavailable as in Definition 4.

The conservative one-step prediction error variances [[summation].sub.i] satisfy the Riccati equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The conservative local steady-state filtering error variances satisfy the Lyapunov equations

[P.sub.i] = [[PSI].sub.i][P.sub.i][[PSI].sup.T.sub.i] + [[I.sub.n] - [K.sub.i][H.sub.i]][GAMMA]Q[[GAMMA].sup.T] [[[I.sub.n] - [K.sub.i][H.sub.i]].sup.T] + [K.sub.i][R.sub.i][K.sup.T.sub.i], (8)

and the conservative local steady-state filtering error cross-covariances also satisfy the Lyapunov equations [4]

[P.sub.ij] = [[PSI].sub.i][P.sub.ij][[PSI].sup.T.sub.j] + [[I.sub.n] - [K.sub.j][H.sub.j]] [GAMMA]Q[[GAMMA].sup.T] [[[I.sub.n] - [K.sub.j][H.sub.j]].sup.T]. (9)

Notice that the conservative local Kalman filters (5) are unrealizable, because the conservative measurements [y.sub.i](t) given in Definition 4 are unavailable. Only the actual measurements [y.sub.i](t) measured via sensors are available, which are generated from systems (1) and (2) with the actual noise variances [bar.Q] and [[bar.R].sub.i], i = 1, ..., L. Therefore, replacing the conservative measurements [y.sub.i](t) with the actual measurements [y.sub.i](t) in (5), we obtain the actual local Kalman filters [[??].sub.i](t|t).

Define the actual local steady-state filtering error variance as

[[bar.P].sub.i] = E [[[??].sub.i] (t | t) [[??].sup.T.sub.i](t|t), [[??].sub.i] (t|t) = x(t) - [[??].sub.i](t | t). (10)

Substituting (1) and (5) into [[??].sub.i](t | t) = x(t) - [[??].sub.i](t | t), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Substituting the actual measurements (2) into (11) yields

[[??].sub.i](t | t) = [[PSI].sub.i][??](t - 1 | t - 1) + ([I.sub.n] - [K.sub.i][H.sub.i])[GAMMA]w (t - 1) - [K.sub.i][v.sub.i](t). (12)

Substituting (12) into (10) yields the actual steady-state filtering error variances as

[[bar.P].sub.i] = [[PSI].sub.i][[bar.P].sub.i] [[PSI].sup.T.sub.i] + [[I.sub.n] - [K.sub.i][H.sub.i]] [GAMMA][bar.Q][[GAMMA].sup.T] [[[I.sub.n] - [K.sub.i][H.sub.i]].sup.T] + [K.sub.i][[bar.R].sub.i][K.sup.T.sub.i]. (13)

Applying (12), the actual steady-state filtering error cross-covariances are obtained as

[[bar.P].sub.ij] = [[PSI].sub.i][[bar.P].sub.ij][[PSI].sup.T.sub.j] + [[I.sub.n] - [K.sub.i][H.sub.i]] [GAMMA][bar.Q][[GAMMA].sup.T] [[[I.sub.n] - [K.sub.j][H.sub.j]].sup.T], i [not equal to] j. (14)

Lemma 6 (see [38]). Consider the Lyapunov equation with U being a symmetric matrix,

P = [PSI]P[[PSI].sup.T] + U. (15)

If [PSI] is a stable matrix (all its eigenvalues are inside the unit circle) and U [greater than or equal to] 0, then P is unique and symmetric and P [greater than or equal to] 0.

Theorem 7. Formultisensor uncertain systems (1) and (2) with Assumptions 1-2, the actual local Kalman filters (5) with conservative upper bounds Q and [R.sub.i] of noise variances are robust in the sense that, for all admissible actual variances [bar.Q] and [[bar.R].sub.i] satisfying (3), one has

[[bar.P].sub.i] [less than or equal to] [P.sub.i], i = 1, ..., L, (16)

and they are called the local robust steady-state Kalman filters, and [P.sub.i] is the minimal upper bound of [[bar.P].sub.i].

Proof. Defining [DELTA][P.sub.i] = [P.sub.i] - [[bar.P].sub.i], subtracting (13) from (8) yields the Lyapunov equation

[DELTA][P.sub.i] = [[PSI].sub.i][DELTA][P.sub.i][[PSI].sup.T.sub.i] + [U.sub.i], (17)

where

[U.sub.i] = [[I.sub.n] - [K.sub.i][H.sub.i]][GAMMA](Q - [bar.Q])[[GAMMA].sup.T][[[I.sub.n] - [K.sub.i][H.sub.i]].sup.T] + [K.sub.i] ([R.sub.i] - [[bar.R].sub.i]) [K.sup.T.sub.i]. (18)

Applying (3) and (18) yields [U.sub.i] [greater than or equal to] 0, and noting that [[PSI].sub.i] is a stable matrix and applying Lemma 6, we have [DELTA][P.sub.i] [greater than or equal to] 0; that is [[bar.P].sub.i] [less than or equal to] [P.sub.i] holds. Taking [bar.Q] = Q, [[bar.R].sub.i] = [R.sub.i], then the constraints (3) are satisfied, and [U.sub.i] = 0. Applying Lemma 6 to (17) yields [DELTA][P.sub.i] = 0; that is, [P.sub.i] = [[bar.P].sub.i]. For arbitrary other upper bound [P.sup.*.sub.i], we have [P.sub.i] = [[bar.P].sub.i] [less than or equal to] [P.sup.*.sub.i], which yields that [P.sub.i] is the minimal upper bound of [[bar.P].sub.i]. The proof is completed.

4. Weighted Fusion Robust Steady-State Kalman Filters

4.1. Four Robust Weighted State Fusion Steady-State Kalman Filters. For the worst-case conservative multisensor systems (1) and (2) with Assumptions 1-2, and with conservative upper bounds Q and [R.sub.i], under the ULMV fusion rule, the four conservative steady-state optimal weighted fusion Kalman filters are given by [3-5]

[[??].sub.[theta]] (t | t) = [L.summation over (i=1)] [[OMEGA].sup.[theta].sub.i][[??].sub.i] (t | t), [theta] = m, s, d, CI (19)

with the constraint of unbiasedness

[L.summation over (i=1)][[OMEGA].sup.[theta].sub.i] = [I.sub.n], (20)

where [theta] = m, s, d and CI denote the fusers weighted by matrices, scalars, diagonal matrices, and the CI fuser, respectively.

The optimal weighted matrices are computed as [3-5]

[[[OMEGA].sup.m.sub.1] ... [[OMEGA].sup.m.sub.L]] = [([e.sup.T][P.sup.-1]e).sup.-1][e.sup.T][P.sup.-1], e = [[[I.sub.n] ... [I.sub.n]].sup.T], (21)

P = [([P.sub.ij]).sub.nLxnL] (22)

with the definition [P.sub.ii] = [P.sub.i].

The conservative fused filtering error variance is given as

[P.sub.m] = [([e.sup.T][P.sup.-1]e).sup.-1]. (23)

The optimal scalars weights are computed as

[[[omega].sub.1], ..., [[omega].sub.L]] = [([e.sup.T][P.sup.-1.sub.tr]e).sup.-1] [e.sup.T] [P.sup.-1.sub.tr],

[[OMEGA].sup.s.sub.i] = [[omega].sub.i][I.sub.n], i = 1, ..., L, (25)

where e = [[1 ... 1].sup.T] and the L x L matrix [P.sub.tr] (t | t) is defined as

[P.sub.tr] = [(tr [P.sub.ij]).sub.LxL], (26)

where tr [P.sub.ij] denotes the trace of [P.sub.ij]. The conservative fused error variance is given as

[P.sub.s] = [L.summation over (i=1)][L.summation over (j=1)] [[omega].sub.i][[omega].sub.j][P.sub.ij]. (27)

The optimal diagonal matrix weights are computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

where [P.sup.ii.sub.sk] is the (i, i)th diagonal element of [P.sub.sk], s, k = 1, ..., L.

The conservative fused error variance is given as

[P.sub.d] = [L.summation over (i=1)][L.summation over (j=1)][[OMEGA].sup.d.sub.i][P.sub.ij][[OMEGA].sup.dT.sub.j]. (29)

The CI fusion weights are computed as [5, 31, 32]

[[OMEGA].sub.CI] = [[[OMEGA].sup.CI.sub.1], ..., [[OMEGA].sup.CI.sub.L]], [[OMEGA].sup.CI.sub.i] = [[omega].sub.i][P.sup.*.sub.CI][P.sup.-1.sub.i], i = 1, ..., L, (30)

[P.sup.*.sub.CI] = [[[L.summation over (i=1)][[omega].sub.i][P.sup.-1.sub.i]].sup.-1]. (31)

The optimal weighting coefficients [[omega].sub.i] are obtained by minimizing the performance index

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

This needs to solve L-dimension nonlinear convex optimization problem, which can be solved by "fmincon" function in MATLAB toolbox.

Define

[[OMEGA].sub.[theta]] = [[[OMEGA].sup.[theta].sub.1], ..., [[OMEGA].sup.[theta].sub.L]], [theta] = m,s,d, CI. (33)

From (20), we have

x(t) = [L.summation over (i=1)] [[OMEGA].sup.[theta].sub.i]x(t), [theta] = m,s,d, CI. (34)

Subtracting (19) from (34) yields the conservative fused filtering errors as

[[??].sub.[theta]] (t | t) = [L.summation over (i=1)] [[OMEGA].sup.[theta].sub.i] [[??].sub.i] (t | t), [theta] = m,s,d, CI. (35)

Applying (34) and (35) yields the conservative fused filtering error variances having a unified form

[P.sub.[theta]] = [[OMEGA].sub.[theta]]P[[OMEGA].sup.T.sub.[theta]], [theta] = m,s,d, CI (36)

with P defined in (22). Replacing the conservative local Kalman filters [[??].sub.i](t | t) in (19) by the actual local Kalman filters [[??].sub.i](t | t), we obtain the actual weighted fusion Kalman filters.

Define the actual weighted fusion filtering error variance [[bar.P].sub.[theta]] (t | t) as

[[bar.P].sub.[theta]] (t | t) = E[[[??].sub.[theta]] (t | t) [[??].sup.T.sub.[theta]] (t | t)], [theta] = m, s, d, CI, (37)

where [[??].sub.[theta]] (t | t) = x(t) - [[??].sub.[theta]] (t | t) and [[??].sub.[theta]] (t | t) is the actual fused filters (19) with [[??].sub.i] (t | t) (i = 1, ..., L) being the actual local Kalman filters. From (35) and (37), we obtain the actual fused filtering error variances as

[[bar.P].sub.[theta]] = [[OMEGA].sub.[theta]] [bar.P] [[OMEGA].sup.T.sub.[theta]], [theta] = m, s, d, CI, (38)

with the definition

[bar.P] = [([[bar.P].sub.ij]).sub.nLxnL]. (39)

In particular, from (30), (36), and (38), taking [theta] = CI, the CI fuser has the conservative and actual steady-state fused error variances as

[P.sub.CI] = [P.sup.*.sub.CI] [[L.summation over (i=1)] [L.summation over (j=1)] [[omega].sub.i] [P.sup.-1.sub.i][P.sub.ij][P.sup.-1.sub.j][[omega].sub.j]] [P.sup.*.sub.CI], (40)

[[bar.P].sub.CI] = [P.sup.*.sub.CI] [[L.summation over (i=1)][L.summation over (j=1)] [[omega].sub.i][P.sup.-1.sub.i] [[bar.P].sub.ij][P.sup.-1.sub.j][[omega].sub.j]] [P.sup.*.sub.CI]. (41)

Notice that [P.sub.CI] is defined with the conservative cross-covariance [P.sub.ij].

Lemma 8 (see [24]). Let [LAMBDA] be the r x r positive semidefinite matrix; that is, [LAMBDA] [greater than or equal to] 0; then the following rL x rL matrix [[LAMBDA].sub.[delta]] is also positive semidefinite; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

Lemma 9 (see [24]). Let [R.sub.i] be the [m.sub.i] x [m.sub.i] positive semidefinite matrix; that is, [R.sub.i] [greater than or equal to] 0; the following m x m block-diagonal matrix [R.sub.[delta]] is also positive semidefinite; that is,

[R.sub.[delta]] = diag ([R.sub.1], ..., [R.sub.L]) [greater than or equal to] 0 (43)

with m = [m.sub.1] + ... + [m.sub.L].

Theorem 10. For multisensor uncertain systems (1) and (2) with Assumptions 1-2 and with conservative upper bounds Q and [R.sub.i] of noise variances, the actual four steady-state weighted Kalman fusers are robust in the sense that, for all admissible actual variances [bar.Q] and [[bar.R].sub.i] satisfying (3), one has

[[bar.P].sub.[theta]] [less than or equal to] [P.sub.[theta]], [theta] = m, s, d, CI, (44)

and they are called the robust weighted fusion steady-state Kalman filters, and [P.sub.[theta]] is the minimal upper bound of [[bar.P].sub.[theta]].

Proof. Defining [DELTA][P.sub.[theta]] = [P.sub.[theta]] - [[bar.P].sub.[theta]], subtracting (36) from (38) yields

[DELTA][P.sub.[theta]] = [[OMEGA].sub.[theta]] (P - [bar.P]) [[OMEGA].sup.T.sub.[theta]]. (45)

In order to prove the robustness [DELTA][P.sub.[theta]] = [P.sub.[theta]] - [[bar.P].sub.[theta]] [greater than or equal to] 0, we only need to prove that the inequality P - [bar.P] [greater than or equal to] 0 holds.

Applying (8) and (9) yields the following Lyapunov equation

P = [PSI]P[[PSI].sup.T] + U[Q.sub.a][U.sup.T] + KR[K.sup.T], (46)

where we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

Similarly, applying (13) and (14), [bar.P] can be expressed as

[bar.P] = [PSI][bar.P][[PSI].sup.T] + U[[bar.Q].sub.a][U.sup.T] + K[bar.R][K.sup.T]. (48)

with the definitions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

Since [[PSI].sub.i] is a stable matrix, then the eigenvalues of the matrix [[PSI].sub.i] are all within the unit circle and are determined from its characteristic equation det([lambda][I.sub.n] - [[PSI].sub.i]) = 0. The eigenvalues of the matrix [PSI] are determined from the characteristic equation

det ([lambda][I.sub.nL] - [PSI]) = det ([lambda][I.sub.n] - [[PSI].sub.1]) ... det ([lambda][I.sub.n] - [[PSI].sub.L]) = 0, (50)

which yields that [PSI] is also a stable matrix, because the eigenvalues of [[PSI].sub.i] are also the eigenvalues of [PSI].

Denoting [DELTA]P = P - [bar.P], subtracting (48) from 46) yields the Lyapunov equation

[DELTA]P = [PSI][DELTA]P[[PSI].sup.T] + U([Q.sub.a] - [[bar.Q].sub.a]) [U.sup.T] + K(R - [bar.R]) [K.sup.T]. (51)

Noting that [bar.Q] [less than or equal to] Q, [[bar.R].sub.i] [less than or equal to] [R.sub.i], applying Lemmas 8 and 9 yields [[bar.Q].sub.a] [less than or equal to] [Q.sub.a], [bar.R] [less than or equal to] R; therefore U([Q.sub.a] - [[bar.Q].sub.a])[U.sup.T] + K(R - [bar.R]) [K.sup.T] [greater than or equal to] 0. Noting that [PSI] is a stable matrix and applying Lemma 6 to (51), we have [DELTA]P [greater than or equal to] 0; that is,

[bar.P] [less than or equal to] P. (52)

From (45) and (52), we have [DELTA][P.sub.[theta]] [greater than or equal to] 0, so (44) holds. If [P.sup.*.sub.[theta]] is other upper bound of [[bar.P].sub.[theta]], taking [bar.Q] = Q, [[bar.R].sub.i] = [R.sub.i], we have [[bar.Q].sub.a] = [Q.sub.a], [bar.R] = R, so applying Lemma 6 to (51) yields [DELTA]P = 0; that is, [bar.P] = P. Hence, applying (45) yields [DELTA][P.sub.[theta]] = 0, so [P.sub.[theta]] = [[bar.P].sub.[theta]] [less than or equal to] [P.sup.*.sub.[theta]], which yields that [P.sub.[theta]] is the minimal upper bound of [[bar.P].sub.[theta]]. The proof is completed.

Remark 11. From (19) and (30), the CI fusion Kalman filter can be rewritten as

[[??].sub.CI] (t | t) = [P.sup.*.sub.CI] [L.summation over (i=1)] [[omega].sub.i][P.sup.-1.sub.i][[??].sub.i](t | t), (53)

which can be reviewed as a special fuser weighted by matrices with weights [[OMEGA].sup.CI.sub.i] = [[omega].sub.i][P.sup.*.sub.CI][P.sup.-1.sub.i].

From (31) and (53), the CI fuser has the convex combination form as

[([P.sup.*.sub.CI]).sup.-1] [[??].sub.CI] (t | t) = [L.summation over (i=1)] [[omega].sub.i] [P.sup.-1.sub.i][[??].sub.i](t | t), (54)

[([P.sup.*.sub.CI]).sup.-1] = [L.summation over (i=1)][[omega].sub.i][P.sup.-1.sub.i], (55)

and it is proved [5] that [P.sup.*.sub.CI] is a conservative upper bound of [[bar.P].sub.CI],

[[bar.P].sub.CI] [less than or equal to] [P.sup.*.sub.CI]. (56)

From (31) or (55), the upper bound [P.sup.*.sub.CI] is defined without the cross-covariance information and is only determined by the conservative local variances [P.sub.i], so that [P.sup.*.sub.CI] has certain conservativeness; that is, [P.sup.*.sub.CI] is not a minimal upper bound of [[bar.P].sub.CI] for all admissible uncertainties of noise variances. From Theorem 10, we have the robustness

[[bar.P].sub.CI] [less than or equal to] [P.sub.CI] (57)

and [P.sub.CI] defined by (40) with the conservative cross-covariance [P.sub.ij], is the minimal upper bound of [[bar.P].sub.CI]. Hence, we have

[P.sub.CI] [less than or equal to] [P.sup.*.sub.CI]; (58)

that is, the upper bound [P.sub.CI] has less conservativeness than [P.sup.*.sub.CI].

4.2. Two Robust Weighted Measurement Fusion Steady-State Kalman Filters. For the worst-case conservative systems (1) and (2) with Assumptions 1-2, and with the conservative upper bounds Q and [R.sub.i] of noise variances, if [H.sub.i] have the common m x n right factor H, that is,

[H.sub.i] = [M.sub.i]H, i = 1, ..., L, (59)

where [M.sub.i] is [m.sub.i] x n matrix, and the matrix [M.sup.(0)T][R.sup.(0)-1][M.sup.(0)] or [H.sup.(0)T][R.sup.(0)-1][H.sup.(0)] is assumed to be invertible, with the definition

[M.sup.(0)] = [[[M.sup.T.sub.1], ..., [M.sup.T.sub.L]].sup.T], [H.sup.(0)] = [[[H.sup.T.sub.1], ..., [H.sup.T.sub.L]].sup.T], (60)

we have the conservative centralized fusion measurement equation

[y.sup.(0)](t) = [H.sup.(0)]x(t) + [v.sup.(0)](t), (61)

[y.sup.(0)](t) = [[[y.sup.T.sub.1](t), ..., [y.sup.T.sub.L](t)].sup.T], [v.sup.(0)](t) = [[[v.sup.T.sub.1](t), ..., [v.sup.T.sub.L](t)].sup.T], (62)

Where, according to Assumption 1, the fused noise [v.sup.(0)](t) has the conservative variance matrix

[R.sup.(0)] = diag ([R.sub.1], ..., [R.sub.L]). (63)

If (59) holds, then (61) becomes

[y.sup.(0)](t) = [M.sup.(0)]Hx(t) + [v.sup.(0)](t). (64)

if [M.sup.(0)T][R.sup.(0)-1][M.sup.(0)] is invertible, applying the weighted least squares (WLS) method [34], Hx(t) has the WLS estimate

[y.sup.(1)](t) = [[[M.sup.(0)T][R.sup.(0)-1][M.sup.(0)]].sup.-1] [M.sup.(0)T][R.sup.(0)-1][y.sup.(0)](t). (65)

Substituting (64) into (65) yields the first conservative weighted measurement fusion equation

[y.sup.(1)](t) = [H.sup.(1)]x(t) + [v.sup.(1)](t), [H.sup.(1)] = H, (66)

[v.sup.(1)](t) = [[[M.sup.(0)T][R.sup.(0)-1][M.sup.(0)]].sup.-1][M.sup.(0)T][R.sup.(0)-1][v.sup.(0)](t), (67)

Where, from (67), the fused noise [v.sup.(1)](t) has the variance matrix

[R.sup.(1)] = [[[M.sup.(0)T][R.sup.(0)-1][M.sup.(0)]].sup.-1]. (68)

If [H.sup.(0)T][R.sup.(0)-1][H.sup.(0)] is invertible, from (61), x(t) has the WLS estimate

[y.sup.(2)](t) = [[[H.sup.(0)T][R.sup.(0)-1][H.sup.(0)]].sup.-1] [H.sup.(0)T] [R.sup.(0)-1] [y.sup.(0)] (t). (69)

Substituting (61) into (69) yields the second conservative measurement fusion equation

[y.sup.(2)](t) = [H.sup.(2)]x(t) + [v.sup.(2)](t), [H.sup.(2)] = [I.sub.n], (70)

[v.sup.(2)](t) = [[[H.sup.(0)T][R.sup.(0)-1][H.sup.(0)]].sup.-1] [H.sup.(0)T] [R.sup.(0)-1] [v.sup.(0)] (t), (71)

where from (71) the fused noise [v.sup.(2)](t) has the variance matrix

[R.sup.(2)] = [[[H.sup.(0)T][R.sup.(0)-1][H.sup.(0)]].sup.-1]. (72)

Hence, the centralized fusion system and two measurement fusion systems have a unified form as

x(t + 1) = [PHI]x(t) + [GAMMA]w(t), y(j) (t) = [H.sup.(j)]x(t) + [v.sup.(j)](t), j = 0,1,2. (73)

Theorem 12. For the worst-case multisensor uncertain system with Assumptions 1-2, and with the conservative upper bounds Q and [R.sub.i] of noise variances, the robust centralized fusion Kalman filter and two robust weighted measurement fusion Kalman filters have a unified form as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (74)

and the conservative and actual fused error variances are, respectively, given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (75)

From (62), (67), and (71), applying Assumption 1 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (76)

We have three equivalent robust Kalman filters

[[??].sup.(0)] (t|t) = [[??].sup.(1)] (t | t) = [[??].sup.(2)] (t | t) (77)

with the robustness

[[bar.P].sup.(J)] [less than or equal to] [P.sup.(j)], j = 0,1,2, (78)

and we have the accuracy relations

[P.sup.(0)] = [P.sup.(1)] = [P.sup.(2)], [[bar.P].sup.(0)] = [[bar.P].sup.(1)] = [[bar.P].sup.(2)], (79)

tr [P.sup.(0)] = tr [P.sup.(1)] = tr [P.sup.(2)], tr [[bar.P].sup.(0)] = tr [[bar.P].sup.(1)] = tr [[bar.P].sup.(2)]. (80)

Proof. The robust Kalman filters (74) have the equivalent information filter form [38]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (81)

From (81), we see that, in order to prove (77) and (79), we only need to prove

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (82)

Applying (59), (60), (63), (65), (68), (69), and (72), we easily verify that (82) hold. In order to prove (78), from (79), we only need to prove [[bar.P].sup.(0)] [less than or equal to] [P.sup.(0)]; that is, the centralized fusion Kalman filter is robust. In fact, applying (3) and Lemma 9 yields [[bar.R].sup.(0)] [less than or equal to] [R.sup.(0)], so applying the derivation similar to Theorem 7 yields [[bar.P].sup.(0)] [less than or equal to] [P.sub.(0)] The proof is completed.

5. Robust Accuracy Analysis

Theorem 13. For multisensor uncertain systems (1) and (2) with Assumptions 1-2, the local and fused Kalman filters have the following accuracy relations:

[[bar.P].sub.[theta]] [less than or equal to] [P.sub.[theta]], [theta] = 1, ..., L,m,s,d,CI, (83)

[[bar.P].sup.(j)] [less than or equal to] [P.sup.(j)], j = 0, 1, 2, (84)

[P.sup.(0)] = [P.sup.(1)] = [P.sup.(2)], [[bar.P].sup.(0)] = [[bar.P].sup.(1)] = [[bar.P].sup.(2)], (85)

[P.sup.(0)] [less than or equal to] [P.sub.m], [P.sub.m] [less than or equal to] [P.sub.[theta]], [theta] = 1, ..., L,m,s,d, (86)

[P.sub.CI] [less than or equal to] [P.sup.*.sub.CI], [[bar.P].sub.CI] [less than or equal to] [P.sup.*.sub.CI], (87)

tr[[bar.P].sub.[theta]] [less than or equal to] tr [P.sub.[theta]], [theta] = 1, ..., L, m, s, d, CI, (88)

tr[[bar.P].sup.(j)] [less than or equal to] tr[P.sup.(j)], j = 0, 1, 2, (89)

tr[P.sup.(0)] = tr[P.sup.(1)] = tr[P.sup.(2)], tr [[bar.P].sup.(0)] = tr [[bar.P].sup.(1)] = tr [[bar.P].sup.(2)], (90)

tr[P.sup.(0)] [less than or equal to] tr[P.sub.m], tr[P.sub.m] [less than or equal to] tr[P.sub.[theta]], [theta] = 1, ..., L, m, s, d, (91)

tr [P.sub.CI] [less than or equal to] tr [P.sup.*.sub.CI], tr [[bar.P].sub.CI] [less than or equal to] tr [P.sup.*.sub.CI], (92)

tr [P.sub.m] [less than or equal to] tr [P.sub.d] [less than or equal to] tr [P.sub.s] [less than or equal to] tr [P.sub.i], i = 1, ..., L. (93)

Proof. The relations (83)-(85) were proved in Theorems 7-12. The relation 86) was proved in 5]. The relation (87) was proved in Remark 11. Taking the trace operations for (83)-(87) yields (88)-(92). The relation (93) was proved in 5]. The proof is completed.

Remark 14. The trace of the error variance matrix is used to represent the filtering accuracy which is equal to the sum of the filtering error variances for the components of state. The smaller trace means the higher accuracy and the larger trace means the lower accuracy. The accuracy relations (88) and (89) mean that, for any admissible [bar.Q] and [[bar.R].sub.i] satisfying (3), the actual accuracy tr [[bar.P].sub.[theta]] or tr[[bar.P].sup.(j)] of the local or fused filters [[??].sub.[theta]] (t | t), [theta] = 1, ..., L, m, d, s, CI, j = 0,1,2, is globally controlled by tr [P.sub.[theta]] or tr [P.sup.(j)], and tr [P.sub.[theta]] or tr [P.sup.(j)] is independent of all admissible [bar.Q] and [[bar.R].sub.i] satisfying (3). Therefore, tr [P.sub.[theta]] or tr [P.sup.(j)] is called robust accuracy of [[??].sub.[theta]] (t | t) and is also called global accuracy [24], and tr [[bar.P].sub.[theta]] or tr [[bar.P].sup.(j)] is called its actual accuracy. Notice that, for different [bar.Q] and [[bar.R].sub.i], the corresponding actual accuracies are also different; that is, tr [[bar.P].sub.[theta]] or tr [[bar.P].sup.(j)] is related to admissible [bar.Q] and [[bar.R].sub.i]. From (88) and (89), we see that the robust accuracy is the admissible lowest actual accuracy; that is, it is the lowest bound of the actual accuracies. The smaller tr [P.sub.[theta]] or tr [P.sup.(j)] means the higher robust accuracy, and the larger tr [P.sub.[theta]] or tr [P.sup.(j)] means the lower robust accuracy.

Remark 15. From Theorem 10, we see that [P.sub.CI] with the cross-covariance information is a minimal upper bound of [[bar.P].sub.CI], and, from (57) and (58), we have the robust accuracy relation

tr [[bar.P].sub.CI] [less than or equal to] tr [P.sub.CI] [less than or equal to] tr [P.sup.*.sub.CI]. (94)

This means that the robust accuracy of the CI fuser is tr [P.sub.CI] rather than tr [P.sup.*.sub.CI], so that the modified robust accuracy tr [P.sub.CI] is higher than the original robust accuracy tr [P.sup.*.sub.CI] in [24], and it also develops and extends the ellipsoidal intersection (EI) fuser with the cross-covariance information [33].

6. Simulation Example

Consider a three-sensor tracking system with uncertain noise variances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (95)

where [T.sub.0] = 0.25 is the sampled period, x(t) = [[[x.sub.1](t), [x.sub.2](t)].sup.T] is the state, [x.sub.1](t) and [x.sub.2](t) are the position and velocity of target at time t[T.sub.0] x w(t), and [v.sub.i](t) are independent Gaussian white noises with zero mean and unknown actual variances [bar.Q] and [[bar.R].sub.i], respectively. Taking the conservative noise variances Q and [R.sub.i] satisfies [bar.Q] [less than or equal to] Q and [[bar.R].sub.i] [less than or equal to] [R.sub.i]. In the simulation, we take Q = 1, [R.sub.1] = 0.8, [R.sub.2] = diag(8, 0.36), [R.sub.3] = 0.64, [bar.Q] = 0.8, [[bar.R].sub.1] = 0.65, [[bar.R].sub.2] = diag (6, 0.25), and [[bar.R].sub.3] = 0.54.

Applying the local and fused robust steady-state Kalman filters, the actual and conservative filtering error variances are obtained in Table 1.

Table 1 verifies the accuracy relations (83)-(87), and it is easy to be verified that [P.sub.CI] [less than or equal to] [P.sup.*.sub.CI] is satisfied. The traces of the error variances of the local and fused Kalman filters are compared in Table 2, which verify the accuracy relations (88)-(93).

In order to give a geometric interpretation of the matrix accuracy relations, the covariance ellipse is defined as the locus of points [x: [x.sup.T][P.sup.-1]x = c}, where P is the variance matrix and c is a constant. Generally, we select c = 1. It has been proved [32] that [P.sub.1] [less than or equal to] [P.sub.2] is equivalent to the covariance ellipse form [P.sub.1] by contains that form by [P.sub.2].

From Figures 1 and 2, we see that the ellipse of [P.sup.(j)] (j = 0,1,2) is enclosed in these of [P.sub.m], [P.sub.d], [P.sub.s], and [P.sub.i] (i = 1,2,3) which verifies the matrix accuracy relations (85)-(87). From Figure 3, we see that the ellipses of [[bar.P].sup.(j)] (j = 0, 1, 2) or [[bar.P].sub.[theta]] ([theta] = 1, 2, 3, m, s, d, CI) are enclosed in these of the conservative upper bound [P.sup.(j)] or [P.sub.[theta]], respectively, which verify the robustness (83)-(87).

In order to verify the above theoretical accuracy relations, taking [rho] = 200 Monte-Carlo simulation runs, the mean square error (MSE) value at time t of local or fused Kalman filters is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (96)

where [x.sup.(k)](t) or [[??].sup.(k).sub.[theta]](t | t) denotes the kth realization of x(t) or [[??].sub.[theta]] (t | t) and [[??].sup.(k)(0)](t | t) denotes the kth realization of the centralized fuser. t = 1, ..., [t.sub.f] are the recursive steps, the final step is [t.sub.f] = 500, and the MSE curves of the local and weighted fusion robust Kalman filters are shown in Figure 4, where the straight lines denote the traces of the theoretical error variances, respectively, the curves denote the values of the [MSE.sub.[theta]](t) or [MSE.sub.j](t), and tr[P.sup.(0)] = tr[P.sup.(1)] = tr[P.sup.(2)], tr[[bar.P].sup.(0)] = tr [[bar.P].sup.(1)] = tr [[bar.P].sup.(2)].

According to the consistency of the sampled variance, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (97)

From Figure 4, we see that when t is sufficiently large, the values of [MSE.sub.i](t) (i = [theta], j) are close to the corresponding theoretical values tr [[bar.P].sub.i], which verifies the consistency (97). We also see that [MSE.sub.i](t) [less than or equal to] tr [P.sub.i], which verifies the accuracy relations (88) and (89).

7. Conclusions

For the multisensor time-invariant uncertain systems with uncertainties of noise variances, according to the minimax robust estimation principle, based on the worst-case conservative system with conservative upper bound of noise variances, using the ULMV optimal estimation rule and the steady-state Kalman filtering theory, the six robust weighted fusion steady-state Kalman filters have been presented. Their robustness was proved by using the Lyapunov equation method and their robust accuracy relations were proved. Compared with the method and results in [24], the main new contributions are as follows.

(1) Reference [24] presented an indirect design method for obtaining the robust weighted fusion steady-state Kalman filters which are obtained by taking the limit operations for the proposed time-varying robust Kalman fusers. This paper presented a simple direct design method based on the steady-state Kalman filtering theory, which can directly obtain the same steady-state results in [24], and avoided finding the time-varying robust Kalman fusers.

(2) A modified CI fusion method has been presented. A minimal upper bound of actual CI fusion error variances was presented based on the cross-covariance information. It reduces the conservativeness of the original upper bound without the cross-covariance information, and it improves and increases the robust accuracy of the CI fuser as shown in Remark 15. The ellipsoidal intersection (EI) fuser [33] with the cross-covariance information was developed and extended.

(3) Two robust weighted measurement fusion algorithms and the robust centralized fusion algorithm have been presented in a unified framework, their equivalence was proved, and they have the highest robust accuracy than the above other fusers. Only one weighted measurement fusion algorithm was presented in [24].

This paper is limited to the multisensor systems with uncertain noise variances; the extension of the proposed results to the multisensor systems with both the uncertain model parameters and noise variances is in the investigation.

http://dx.doi.org/10.1155/2014/369252

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China under Grant NSFC-60874063 and Innovation and Scientific Research Foundation of Graduate Student of Heilongjiang Province under Grant YJSCX2012-263HLJ. The authors thank the reviewers and editors for their helpful and constructive comments, which are very valuable for improving quality of the paper.

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Wen-Juan Qi, Peng Zhang, and Zi-Li Deng

Department of Automation, Heilongjiang University, Harbin 150080, China

Correspondence should be addressed to Zi-Li Deng; dzl@hlju.edu.cn

Received 18 November 2013; Accepted 7 March 2014; Published 30 April 2014

Academic Editor: Jong Hae Kim

TABLE 1: The actual and conservative filtering error
variances of the local and fused robust Kalman filters.

[P.sub.1]            [P.sub.2]          [P.sub.3]

[MATHEMATICAL      [MATHEMATICAL      [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT    EXPRESSION NOT
REPRODUCIBLE IN   REPRODUCIBLE IN    REPRODUCIBLE IN
ASCII]                 ASCII]            ASCII]

[[bar.P].sub.1]   [[bar.P].sub.2]    [[bar.P].sub.3]

[MATHEMATICAL      [MATHEMATICAL      [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT    EXPRESSION NOT
REPRODUCIBLE IN   REPRODUCIBLE IN    REPRODUCIBLE IN
ASCII]                 ASCII]            ASCII]

[P.sub.s]         [P.sup.*.sub.CI]     [P.sub.CI]

[MATHEMATICAL      [MATHEMATICAL      [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT    EXPRESSION NOT
REPRODUCIBLE IN   REPRODUCIBLE IN    REPRODUCIBLE IN
ASCII]                 ASCII]            ASCII]

[[bar.P].sub.s]   [[bar.P].sub.CI]

[MATHEMATICAL      [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT
REPRODUCIBLE IN   REPRODUCIBLE IN
ASCII]                 ASCII]

[P.sub.1]             [P.sub.4]            [P.sub.5]

[MATHEMATICAL       [MATHEMATICAL        [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT       EXPRESSION NOT
REPRODUCIBLE IN    REPRODUCIBLE IN      REPRODUCIBLE IN
ASCII]                 ASCII]               ASCII]

[[bar.P].sub.1]    [[bar.P].sub.m]      [[bar.P].sub.d]

[MATHEMATICAL       [MATHEMATICAL        [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT       EXPRESSION NOT
REPRODUCIBLE IN    REPRODUCIBLE IN      REPRODUCIBLE IN
ASCII]                 ASCII]               ASCII]

[P.sub.s]            [P.sup.(0)]         [P.sup.(1)] =
                                          [P.sup.(2)]

[MATHEMATICAL       [MATHEMATICAL        [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT       EXPRESSION NOT
REPRODUCIBLE IN    REPRODUCIBLE IN      REPRODUCIBLE IN
ASCII]                 ASCII]               ASCII]

[[bar.P].sub.s]   [[bar.P].sub.(0)]   [[bar.P].sub.(1)] =
                                       [[bar.P].sub.(2)]

[MATHEMATICAL       [MATHEMATICAL        [MATHEMATICAL
EXPRESSION NOT     EXPRESSION NOT       EXPRESSION NOT
REPRODUCIBLE IN    REPRODUCIBLE IN      REPRODUCIBLE IN
ASCII]                 ASCII]               ASCII]

TABLE 2: The accuracy comparison of local and fused robust
Kalman filters.

tr[P.sub.1]   tr[P.sub.2]    tr[P.sub.3]    tr[P.sub.m]

0.5538          0.5245         0.4952         0.1942

tr[[bar.P]    tr[[bar.P]     tr[[bar.P]     tr[[bar.P]
.sub.1]         .sub.2]        .sub.3]        .sub.m]

0.4465          0.3815         0.4069         0.1485

tr[P.sup      tr[[bar.P]    tr[P.sup.(1)]   tr[[bar.P]
.(0)]          .sup.(0)]        = tr        .sup.(1)] =
                             [P.sup.(2)]    tr[[bar.P]
                                             .sup.(2)]

0.1795          0.1367         0.1795         0.1367

tr[P.sub.1]   tr[P.sub.d]   tr[P.sub.s]     tr[P.sup.*
                                             .sub.CI]
                                           tr[P.sub.CI]

0.5538          0.2212        0.2725      0.4022, 0.2360

tr[[bar.P]    tr[[bar.P]    tr[[bar.P]      tr[[bar.P]
.sub.1]         .sub.d]       .sub.s]        .sub.CI]

0.4465          0.1711        0.2131          0.1817

tr[P.sup
.(0)]

0.1795
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Article Details
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Title Annotation:Research Article
Author:Qi, Wen-Juan; Zhang, Peng; Deng, Zi-Li
Publication:Journal of Applied Mathematics
Article Type:Report
Date:Jan 1, 2014
Words:8925
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