Sequential Covariance Intersection Fusion Robust Time-Varying Kalman Filters with Uncertainties of Noise Variances for Advanced Manufacturing

This paper addresses the robust Kalman filtering problem for multisensor time-varying systems with uncertainties of noise variances. Using the minimax robust estimation principle, based on the worst-case conservative system with the conservative upper bounds of noise variances, the robust local time-varying Kalman filters are presented. Further, the batch covariance intersection (BCI) fusion and a fast sequential covariance intersection (SCI) fusion robust time-varying Kalman filters are presented. They have the robustness that the actual filtering error variances or their traces are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved based on the proposed Lyapunov equations approach. The concepts of the robust and actual accuracies are presented, and the robust accuracy relations are proved. It is also proved that the robust accuracies of the BCI and SCI fusers are higher than that of each local Kalman filter, the robust accuracy of the BCI fuser is higher than that of the SCI fuser, and the actual accuracies of each robust Kalman filter are higher than its robust accuracy for all admissible uncertainties of noise variances. The corresponding steady-state robust local and fused Kalman filters are also presented for multisensor time-invariant systems, and the convergence in a realization between the local and fused time-varying and steady-state Kalman filters is proved by the dynamic error system analysis (DESA) method and dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and the correctness of the robust accuracy relations.


Introduction
The multisensor information fusion Kalman filtering has wide applications in many high-technology fields, such as advanced manufacturing systems, mechanical industrial robots, unmanned aircraft vehicles, tracking, signal processing, remaining useful life prediction of rolling element bearings [1][2][3], improved tracking and docking of industrial mobile robots [4][5][6][7], and so on. Rolling bearings are the key components of rotating machinery, thus, the prediction of remaining useful life (RUL) is vital in condition-based maintenance (CBM). Reference 1 proposes a new method for RUL predictions of bearings based on time-varying Kalman filter, which can automatically match different degradation stages of bearings and effectively realize the prediction of RUL. Industrial mobile robots are widely used in advanced manufacturing technology systems; ref. [2] used the unscented Kalman filter to improved tracking and docking of industrial mobile robots vision-based kinematics calibration.
The basic assumption for classical Kalman filtering is that the model parameters and noise variances are exactly known, but in many practical applications, such assumption doesn't always hold. In the presence of these uncertainties, the Kalman filters may not be 1.
In Sections 2 and 3, a new methodology for designing the robust local and CI fused Kalman filters is presented for multisensor time-varying systems with uncertain noise variances, according to the minimax robust estimation rule [35,36]. Its basic principle is that based on the worst-case conservative system with the conservative upper bound of noise variances, applying the ULMV optimal estimation rule, the conservative local and CI fused Kalman filters with unavailable conservative measurements are obtained, and then replacing the conservative measurements with the actual measurements yields the robust local and CI fused Kalman filters. The classical optimal Kalman filtering methodology [22,34] is developed. The disadvantage of the original CI fusion methodology [25][26][27][28][29] is overcome where the conservative upper bounds of the local filtering error variances are assumed to be known. Hence the robust local Kalman filters are presented, which provide the conservative upper bounds of the local filtering error variances; 2.
In Section 3, the robust time-varying BCI and SCI fused Kalman filters with uncertain noise variances are presented. The steady-state optimal local, BCI and SCI fused Kalman filters [22,34] with exactly known noise variances are developed; 3.
In the process of proving Theorems 1 and 3, a Lyapunov equation method for the robustness analysis is presented by which the robustness of the local and CI fused Kalman filters is proved. Its basic principle is that the problem of proving the robustness is converted into that of deciding the positive-definiteness of the solution of a Lyapunov equation; 4. In Section 4, the concept of robust accuracy with respect to uncertainties of noise variances is presented, and the robust accuracy relations among the local, BCI and SCI fused Kalman filters with exactly known noised variances [22,34] are extended. The concept of robustness with respect to uncertain noise variances is presented, and the concept of consistency [25,26] is extended; 5.
In Section 5, for the multisensor time-invariant system with uncertain noise variances, the robust steady-state local, BCI and SCI fusion Kalman filters are also presented by replacing time-varying gains, variances and cross-covariances with their limits, respectively; 6.
Using lemma 1-3, in Theorem 7, the convergence in a realization of the local and fused time-varying and steady-state robust Kalman filters is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. To the best of our knowledge, it is presented for the first time; 7.
In Section 7, simulation 1 gives the geometric interpretation of the robust accuracy relations based on the variance ellipses and a Monte Carlo simulation example shows the correctness of the proposed robust accuracy relations and gives the sensitivity analysis of the robust SCI fuser.
The remainder of this paper is organized as follows: In Section 2, we derive the local robust time-varying Kalman filter and prove its robustness. Section 3 gives the BCI and SCI fusion robust time-varying Kalman filters and the proof of their robustness. The accuracy analysis of the local and fused Kalman filters is presented in Section 4. Section 5 gives the robust local and fused steady-state Kalman filters and their convergence. The sensitivity problem is given in Section 6. Section 7 gives a Monte Carlo simulation example. The conclusions are given in Section 8. The frequently used notations in the paper are shown in Table 1.

Name Summary
A the norm of matrix A. k the discrete time E the mathematical expectation operator the trace of a matrix A s "steady-state" "i.a.r" the convergence in a realization

Local Robust Time-Varying Kalman Filters
Consider the following multisensor uncertain time-varying system with uncertainties of noise variances where x(t) ∈ R n is the state, L is the number of sensors, y i (t) ∈ R m i is the measurement of the ith subsystem, w(t) ∈ R r is the input noise and v i (t) ∈ R m i is the measurement noise of the ith sensor. φ(t), Γ(t) and H i (t) are known time-varying matrices with appropriate dimensions.
Assumption 1. w(t) and v i (t) are uncorrelated white noises with zeros mean and unknown uncertain true variances Q(t) and R i (t), respectively.
Assumption 2. Q(t) and R i (t) are known conservative upper bounds of Q(t) and R i (t), respectively, i.e., Assumption 3. The initial state x(0) is independent of w(t) and v i (t), and has mean value µ and unknown uncertain true variance P(0|0) which satisfies where P(0|0) is a known conservative upper bound of P(0|0).
Based on the ith sensor, for the worst-case conservative multisensor system (1) and (2) with the known conservative upper bounds Q(t) and R i (t) of noise variances, the conservative local optimal time-varying Kalman filters are given by [20] x From (1) and (6), the actual filtering errors are From (15), according to Assumptions 1-3, and noting that w(t) and v i (t) are uncorrelated with x i (t|t), the actual filtering error variance and cross-covariances are given by the Lyapunov equations with the initial values P ij (0|0) = P(0|0) and P ii (t|t) = P i (t|t).
Theorem 1. For multisensor uncertain system (1) and (2) with Assumptions 1-3, the actual local Kalman filters (6) is robust in the sense that for all admissible variances Q(t) and R i (t) satisfying (4) and P(0|0) ≤ P(0|0) for arbitrary time t, we have and P i (t|t) are the minimal upper bounds of P i (t|t). Hence, they are called the robust local Kalman filters.
Applying (4) yields that U i (t) ≥ 0. From (5), we have ∆P i (0|0) = P(0|0) − P(0|0) ≥ 0. Hence from (18), we have ∆P i (1|1) ≥ 0. Applying the mathematical induction method yields ∆P i (t|t) ≥ 0, for all time t, i.e., the inequalities (17) hold. If P * i (t|t) is another upper bound, then for all admissible Q(t) ≤ Q(t) and R i (t) ≤ R i (t), we have P i (t|t) ≤ P * i (t|t). Taking Q(t) = Q(t), R i (t) = R i (t), from (12) and (16), we have P i (t|t) = P i (t|t) ≤ P * i (t|t). This means that P i (t|t) is the minimal upper bounds of P i (t|t). The proof is completed. (17) is different from the consistency or non-divergent estimation [23]. The robustness means that the inequality (17) holds for all admissible uncertain Q(t) and R i (t) satisfying (4), while the consistency means that for a fixed Q(t) and R i (t), the inequality (17) holds.

The BCI Fusion Robust Time-Varying Kalman Filter
For the two-sensor uncertain systems with the Assumptions 1-3, applying the CI fused algorithm [20][21][22][23], the actual CI fusion time-varying Kalman filter with the conservative upper bounds Q(t) and R i (t) of noise variances is presented as followinĝ  1] tr ω(t)P −1 When the number of the sensors is larger than two, i.e., L ≥ 3. The actual batch covariance intersection (BCI) fusion Kalman filter is presented by the convex combination [26,35] asx wherex i (t|t) are the robust local Kalman filters, the weights ω i (t) are determined by minimizing the performance index J = trP BCI (t|t) as which can be obtained by "fimincon" function in Matlab. This needs to solve a L-dimensional nonlinear convex optimization problem, so that the larger computation burden and higher complexity are required.

Theorem 2.
The actual BCI fusion filtering error variance is given by where P ij (t|t) are computed by (16).
Proof. From (24), we have Subtracting (27) from (23), we easily obtain the actual BCI fused filtering error which yields the formula (26). The proof is completed.
Theorem 3. For multisensor uncertain system (1) and (2) with Assumptions 1-3, the actual BCI fusion time-varying Kalman filter (23)- (25) is robust in the sense that for all admissible uncertainties of noise variances Q(t) and R i (t) satisfying (4), we have P BCI (t|t) ≤ P BCI (t|t) (29) and trP BCI (t|t) is the minimal upper bound of trP BCI (t|t). We call (23) as the robust BCI fusion Kalman filter.
Proof. In order to prove (29), we only need to prove Pre-multiplying and post-multiplying (30) by P −1 BCI , respectively, we have Substituting (24) and (26) into (31), we only need to prove From (17) for all admissible Q(t) and R i (t) satisfying (4), we have Pre-multiplying and post-multiplying (33) by P −1 i , respectively, we have From (32) and (34), we only need to prove Applying the constraint Hence, we only need to prove Exchanging the subscript symbol i with j in (37) yields Adding (37) to (38) yields which yields ∆ ≥ 0, i.e., (29) holds. Taking the trace operation for (29) yields trP BCI (t|t) ≤ trP BCI (t|t). Applying (25) yields that trP BCI (t|t) is minimal for all admissible P BCI (t|t) given in (24). The proof is completed.

Remark 2.
The proof of Theorem 3 is completely different from the proof in reference [20], where the noise variances are assumed to be exactly known, and the consistency is proved by the mathematical induction. The proof is also different from that in reference [36], where the consistency of the BCI fuser was only proved with the assumption that the local estimates are consistent, while the robustness problem was not proved.

The SCI Fusion Robust Time-Varying Kalman Filter
In order to reduce the complexity and computational burden, the sequential covariance intersection (SCI) robust time-varying Kalman fuser is presented based on the L − 1 twosensor CI fused robust Kalman filters, and it can be realized by a recursive two-sensor CI fusers [34]. Its structure is shown in Figure 1, and the comparison of the computational loads of the BCI filter and the SCI filter are shown in Table 2.
Based on the two-sensor CI fused algorithm, the actual SCI fusion time-varying Kalman filter with the conservative error variances Q(t) and R i (t) is presented as followŝ Micromachines 2022, 13, 1216 9 of 25 wherex i (t|t) are the robust local Kalman filters, and the parameters ω i (t) is determined by minimizing the performance index J as The optimization problem (44) is equivalent to the L − 1 one-dimensional optimization problems (22).

Remark 3.
When the noise variances are exactly known, the optimal steady-state SCI fuser was presented in [34]. However, for multisensor systems with uncertain noise variances, the local and SCI fusion robust time-varying Kalman filters were not presented in [34].  (1) and (2) with Assumptions 1-3, the actual SCI fused filterx SCI (t|t) and its actual error variance P SCI can be rewritten as batch representation where the weighting coefficients θ where the coefficients ω i (t) are obtained by (44).
Theorem 5. For multisensor uncertain system (1) and (2) with Assumptions 1-3, the actual SCI fusion time-varying Kalman filter (40)-(44) is robust in the sense that for all admissible uncertainties of noise variances Q(t) and R i (t) satisfying (4), we have we call (45) as the robust SCI fusion Kalman filter.
Proof. Applying Theorem 4, the SCI Kalman filter can be expressed as the equivalent BCI Kalman filter form. According to Theorem 3, the BCI time-varying fuser is robust, so that the SCI time-varying fuser is also robust. The proof is completed.
Remark 4. The proof of Theorem 5 is different from that in [34] by the consistency of the two-sensor CI fuser. We can also prove Theorem 5 based on robustness of the two-sensor CI fuser.

Accuracy Analysis
From (53), we can see that P SCI (t|t) is the upper bound of the unknown actual fused variances P SCI (t|t) for all possible P i (t|t) and all admissible unknown P ij (t|t) satisfying (16), so that P SCI (t|t) can be viewed as the global accuracy of the SCI fuser. From (46), we see that P SCI (t|t) is independent of actual variances P i (t|t) and cross-covariances P ij (t|t). So that the global accuracy of the SCI fuser has the robustness with respect to uncertain P i (t|t) and P ij (t|t). From (16), we see that the uncertainties of P i (t|t) and P ij (t|t) are yielded by the uncertainties of Q(t) and R i (t) satisfying (4).

Definition 1.
The robustness with respect to uncertainties of noise variances of a Kalman filter is defined as its actual filtering error variances or their traces yielded by all admissible uncertainties of noise variances, which are guaranteed to have a minimal or less-conservative upper bound and this upper bound is independent of uncertainties of noise variances. The Kalman filter with robustness is called to be robust.

Remark 5.
The accuracy relations (54) and (55) mean that for all admissible uncertainties of variances satisfying (4) and (5), the actual accuracies trP θ (t|t), θ = 1, · · · , L, BCI, SCI of the local or fused time-varying Kalman filter are globally controlled by trP θ (t|t), therefore the robust accuracy trP θ (t|t) is also called the global accuracy of a robust Kalman filter. The robustness of the local and fused filters means that the robust accuracy trP θ (t|t) is independent of arbitrarily variances satisfying (4) and (5).

Remark 6.
From the definition 2, the smaller trP θ (t|t) (or trP θ (t|t)) means the higher robust (or actual) accuracy. From (54)-(58), we conclude that the robust accuracy of the robust SCI fuser is higher than that of each local robust Kalman filter, and the robust accuracy of the BCI fuser is higher than that of the SCI fuser. The actual accuracies of a robust Kalman filter are higher than its robust accuracy for all admissible uncertainties.
Remark 7. Theorem 1 shows that P i (t|t) is the minimal upper bound of P i (t|t) in the matrix inequality sense. Theorem 3 shows that trP BCI (t|t) is the minimal upper bound of trP BCI (t|t) in the trace inequality sense. From (55), (57) and (58) yields that trP SCI (t|t) ≤ trP SCI (t|t) ≤ trP i (t|t), i = 1, · · · , L so that trP SCI (t|t) is a less-conservative upper bound of trP SCI (t|t).

Robust Local and Fused Steady-State Kalman Filters
Now we investigate the asymptotic properties of the local and fused robust timevarying Kalman filters, we shall present the corresponding steady-state robust Kalman filters. We shall also rigorously prove the convergence in a realization between the robust time-varying and steady-state Kalman filters, by the DESA method and DVESA method [37,38].
Lemma 1 [39]. Consider the following Lyapunov equation with F being a symmetric matrix where P, Ψ and F are the n × n matrices, Ψ is a stable matrix (i.e., all its eigenvalues are inside the unit circle). If F ≥ 0, then P is symmetric and unique, and P ≥ 0.
Lemma 2 [38]. Consider the time-varying Lyapunov equation where t ≥ 0, the output P(t) and the input U(t) are the n × n matrices, and the n × n matrices F 1 (t) and F 2 (t) are uniformly asymptotically stable, i.e., there exist constants 0 < ρ j < 1 and If U(t) is bounded, then P(t) is bounded. If U(t) → 0 , then P(t) → 0 , as t → ∞ . Notice that U(t) is called to be bounded, if U(t) ≤ c (constant), for arbitrary t ≥ 0.

Lemma 3 [37]. Consider a dynamic error system
where δ(t) ∈ R n , u(t) ∈ R n , and F(t) is uniformly asymptotically stable. If u(t) is bounded, then δ(t) is bounded. If u(t) → 0 , then δ(t) → 0 , as t → ∞ . (1) and (2) with Assumptions 1-2, where φ(t) = φ, Γ(t) = Γ, H i (t) = H i , Q(t) = Q, R i (t) = R i , Q(t) = Q and R i (t) = R i are all the constant matrices. If each subsystem with conservative noise variances Q and R i is completely observable and completely controllable, then the actual local steady-state Kalman filters are given asx

Theorem 7. For multisensor uncertain time-invariant system
where y i (t) are the actual measurements, and the initial valuex s i (0|0) can arbitrarily be selected. Σ i satisfies the steady-state Riccati equations and the conservative cross-covariances P ij and the actual cross-covariances P ij satisfy the steady-state Lyapunov equations with the definition P i = P ii , P i = P ii , and we have P ij (t|t) → P ij , as t → ∞ , i, j = 1, · · · , L (78) The actual local steady-state Kalman filters (72) are robust in the sense that for all admissible uncertainties of Q and R i satisfying Q ≤ Q, R i ≤ R i , then and P i is the minimal upper bound of P i . They are called the robust local steady-state Kalman filters.
Proof. According to the complete observability and complete controllability of each subsystem, we have [40] Then from (7), (8) and (11), we have where Ψ i are stable matrices [40], and Ψ i (t) are uniformly asymptotically stable [40]. When t → ∞ , taking the limit operations for (6)-(11), (12) and (16), we obtain (72)-(77). From K i (t) → K i , the gains K i (t) are bounded, which yields the boundedness of the input of the Lyapunov Equation (12). Hence, applying Lemma 2 to (12) yields that P ij (t|t) are bounded. Setting Ψ i (t) = Ψ i + ∆Ψ i (t) with ∆Ψ i (t) → 0 , and subtracting (76) from (12) with Applying K i (t) → K i , the boundedness of P ij (t|t), and ∆Ψ i (t) → 0 yields that U ij (t) → 0 . Applying Lemma 2 to (83) yields ∆ ij (t) → 0 , as t → ∞ , i.e., (78) holds. Similarly, we can prove (79). Taking the limit operation for (17), as t → ∞ , and applying (78) and (79) yields (80). Taking Q = Q, R i = R i , subtracting (77) from (76), and applying Lemma 1 yields P i = P i , if P * i is arbitrary other upper bound of P i for all admissible Q and R i satisfying Q ≤ Q, R i ≤ R i , then we have P i = P i ≤ P * i , which yields that P i is the minimal. The proof is completed. (1) and (2) with Assumptions 1-2, if each subsystem with conservative noise variances Q and R i is completely observable and completely controllable, then the actual steady-state BCI fusion Kalman filter is given aŝ

Theorem 8. For multisensor uncertain time-invariant system
wherex s i (t|t) are given in Theorem 7, and the optimal weighting coefficients ω i are obtained by minimizing the performance index J = trP BCI as It has the robustness in the sense that for all admissible uncertainties of Q and R i satisfying where the actual fused steady-state filtering error covariance is given as and trP BCI is the minimal upper bound of trP BCI . It is called the robust steady-state BCI fusion Kalman filter.

Theorem 9. For multisensor uncertain time-invariant system
where the weighting coefficients θ (r) i can be computed recursively by and it is robust in the sense that for all admissible uncertainties Q and R i satisfying Q ≤ Q, R i ≤ R i , we have P SCI ≤ P SCI (96) It is called the robust steady-state SCI fusion Kalman filter.

Theorem 12.
Under the conditions of Theorem 10, the robust accuracy comparison of the local and the fused robust steady-state Kalman filters is given by Proof. Applying (78), (79), (103) and (104) yields that P i (t|t) → P i , P BCI (t|t) → P BCI , P SCI (t|t) → P SCI . As t → ∞ , taking the limit operations for (54)-(58) yields Theorem 12.
The proof is completed.

Sensitivity Problem
For the SCI fusion robust Kalman filter, the fused schemes are different with respect to different orders of sensors. For example, in the case where there are three fused structures as shown in Figure 2, the problem is that whether the SCI fused robust accuracy is sensitive with respect to the fused orders of sensors. The following two sensor simulation examples will show that the robust accuracy of the SCI fuser is not very sensitive with respect to the orders of the sensors.

Simulation Examples
Example 1. Consider a 3-sensor tracking system with uncertain noise variances T is the state, x 1 (t) and x 2 (t) are the position and velocity of target at time tT 0 . y i (t) is the measurement, w(t) and v i (t) are independent Gaussion white noises with zero mean and unknown variances Q and R i , respectively, Q and R i are conservative upper bounds of Q and R i satisfying Q ≤ Q, R i ≤ R i . In the simulation, we take Q = 1, R 1 = 0.8, R 2 = diag(8, 0.36), R 3 = 0.5,Q = 0.8, R 1 = 0.65, R 2 = diag(6, 0.25),R 3 = 0.45.
The traces of the conservative and actual local robust filtering error variances are compared in Figure 3. For Figure 3, we see that the traces of the local and fused robust time-varying Kalman filters quickly converge to these of the corresponding steady-state Kalman filters, which verify the robust accuracy relations (54)-(58), and their steady-state robust and actual accuracy relations (107)-(110).
The robust and actual accuracy comparisons are shown in Tables 3 and 4. From  Tables 3 and 4, we see that the SCI fused robust accuracy trP SCI123 , trP SCI132 and trP SCI321 are close or equal to the BCI fused robust accuracy trP BCI , and the accuracy of the SCI fuser is not very sensitive with respect to the orders of sensor. We also see that the actual accuracy of the SCI fuser, and trP SCI123 , trP SCI132 and trP SCI321 are close to or equal to the actual accuracy of the BCI fuser trP BCI ; they are all higher than the robust accuracy of each local filter, which verify the accuracy relations (54)-(58) and their steady-state robust and actual accuracy relations (107)-(110). The traces of the conservative and actual local robust filtering error variances are compared in Figure 3. For Figure 3, we see that the traces of the local and fused robust time-varying Kalman filters quickly converge to these of the corresponding steady-state Kalman filters, which verify the robust accuracy relations (54)-(58), and their steady-state robust and actual accuracy relations (107)-(110).    In order to give a geometric interpretation of the accuracy relations, The covariance ellipses of the robust time-varying Kalman filters at time t = 10 and robust steady-state Kalman filters are shown in Figures 4-9.
From Figures 4-9, we see that the ellipses of the actual variances P i (i = 1, 2, 3) are all enclosed in that of the conservative variances P i , respectively, which verify the robustness (17). The ellipses of actual BCI and SCI fused variances P BCI and P SCIijk (ijk = 123, 132, 231) are respectively enclosed in those of P BCI and P SCIijk , which verifies the robustness (29) and (53). Moreover, we see that the ellipse of P BCI is close to or equal to that of P SCIijk , the ellipse of P BCI is close to or equal to that of P SCIijk , which means that the robust accuracies of the SCI fusers with different orders of sensors are close to those of the BCI fusers, and the robust and actual accuracies of the SCI fusers are not very sensitive to the orders of sensors.
In order to verify the above theoretical accuracy relations, taking N = 200 runs, the mean square error (MSE) value at time t of the local and fused robust Kalman filters are shown in Figure 10. From Figure 10, we see that when t is sufficiently large, we have the accuracy relations MSE θ (t) ≤ trP θ , θ = 1, 2, 3, BCI, SCI and the curves of MSE θ (t) are close to the straight lines corresponding to trP θ , which verify the robust accuracy relations (107) and the robust accuracy relations in Table 3.
In order to give a geometric interpretation of the accuracy relations, The covariance ellipses of the robust time-varying Kalman filters at time           From Figures 4-9, we see that the ellipses of the actual variances i P     Table 5 shows the sensitivity of the actual and robust accuracies for the SCI fuser with respect to the orders of sensors  9). . Figure 3, for the sensor number L = 4, there are 12 fused orders as follows:

Similar to
SCI1234, SCI1243, SCI1324, SCI1342, SCI1423, SCI1432, SCI2314, SCI2341, SCI2413, SCI2431, SCI3412, SCI3421 Table 5 shows the sensitivity of the actual and robust accuracies for the SCI fuser with respect to the orders of sensors From Table 5, we see that all values of trP SCIijkr or trP SCIijkr are close to these of trP BCI or trP BCI , respectively. This means that the robust or actual accuracies of the SCI fusers are not very sensitive to the orders of sensors.

Conclusions
Sequential covariance intersection fusion robust time-varying Kalman filters are presented for the multi-sensor systems with uncertainties of noise variances, the main contributions of this paper are as follows: A minimax robust estimation approach of designing the robust local, BCI and SCI fused Kalman filters has been presented for the multisensor system with uncertain noise variances. For the multisensor time-invariant systems with uncertain noise variances, the convergence problem of the robust local and fused time-varying Kalman filters has been solved. The robust local, BCI and SCI fused steady-state Kalman filters have been presented by replacing the time-varying gains, variances and cross-covariances with their limits, respectively. The convergence in a realization of the local and fused time-varying and steady-state Kalman filters was proved by the dynamic error system analysis (DESA) method [39] and the dynamic variance error system analysis (DVESA) method [40].
The proposed results can be applied to some simulation application research, including target tracking systems, uninterruptible power supply systems, mass spring random vibration systems, and so on. The proposed results are limited to multisensor systems with uncertainties of noise variances. The extensions of the proposed results to multisensor systems with uncertainties of both model parameters and noise variances are under investigation.