Globally Optimal Distributed Fusion Filter for Descriptor Systems with Time-Correlated Measurement Noises

This paper concerns the distributed fusion filtering problem for descriptor systems with time-correlated measurement noises. The original descriptor is transformed into two reduced-order subsystems (ROSs) based on singular value decomposition. For the first ROS, a new measurement is obtained using measurement difference technology. Each sensor produces a local filter based on the fusion predictor from the fusion center and its own new measurement and then sends it to the fusion center. In the fusion center, based on local filters, a distributed fusion filter with feedback (DFFWF) in the linear minimum variance (LMV) sense is proposed by applying an innovative approach. The DFFWF for the second ROS is also obtained based on the DFFWF for the first ROS. Then, the DFFWF for the original descriptor is obtained. The proposed DFFWF can achieve the same estimation accuracy as the centralized fusion filter (CFF) under the condition that all local filter gain matrices are of full column rank. Its optimality is strictly proved. Moreover, it has robustness and reliability due to the parallel processing of local filters. Two simulation examples demonstrate the effectiveness of the developed fusion algorithm.


Introduction
In the last few decades, the problem of state estimation for descriptor (singular) systems has attracted much attention due to more widespread applications than normal systems, such as in power systems, electrical networks, chemical processes, social-economic systems, network analysis, constrained mechanical systems, time-series analysis, large-scale systems with interconnections, aerospace attitude control systems and so on [1][2][3].
Generally speaking, there are two common methods that deal with the filtering problem for descriptor systems: the full-order transformation method and the reduced-order decomposition method. Based on the full-order transformation method, the optimal linear estimators for single sensor systems [4] and distributed fusion estimators for multi-sensor systems [5] are proposed, which are directly solved based on the projection theory. However, the derivation of the estimator is complex since the descriptor system with white noise is transformed into a normal system with one-step cross-correlation colored noise. Differently from full-order methods [4,5], the original descriptor system is equivalently decomposed into two reduced-order subsystems based on the singular value decomposition, where the first reduced-order system is the normal system with white noise. Hence, the reduced-order decomposition method is more popular. Based on the reduced-order decomposition method, many estimators have been reported, including the linear minimum mean-square filter for a single sensor system with stochastic multiplicative disturbance [6], the distributed weighted state fusion optimal filter [7] and steady-state estimators [8] for systems with correlated white noise, distributed weighted state fusion filter for a system with fading measurements and stochastic nonlinearity [9], weighted measurement fusion robust estimators [10] and self-tuning estimators [11], and centralized fusion estimators for systems with different delay rates [12]. However, the above works do not consider the time-correlated measurement noises.

Problem Formulation and Preliminary Lemmas
Consider the following multi-sensor stochastic descriptor system with time-correlated measurement noises: y j (t) = D j x(t) + v j (t), j = 1, . . . , N, where x(t) ∈ n is the state and y j (t) ∈ m j is the measurement output. A, B, C, D j , U j are known constant parameter matrices with proper dimensions.. We make the following assumptions.

Assumption 1.
A is a singular square matrix, i.e., rank A) = n 1 < n .  Assumption 4. w(t) and µ j (t) are uncorrelated white noises with zero means and covariance matrices E[w(t)w T (k)] = Q w δ t,k and E[µ j (t)µ T j (k)] = Q µ j δ t,k ,j = 1, . . . , N. Assumption 5. The initial state value x(0)and measurement noise initial values v j (0), j = 1, · · · , N are mutually uncorrelated and are independent of w(t) and µ j (t), and satisfy E[ Our aim is to design the globally optimal DFFWFx d f (t|t) in the LMV sense. Besides, the global optimality of the DFFWF is proved.

Remark 1.
Descriptor systems appear in many fields, such as electrical circuit systems, largescale systems with interconnections, constrained mechanical systems. Some concrete examples of descriptor systems are presented in [1], from which readers can indeed see the existence of descriptor linear systems in our real world. In the simulation research section, an electrical circuit system is used to show the effectiveness of the proposed DFFWF algorithm.

System Transformation
Under Assumption 3, there exist the non-singular matrices M and N [7,8], which satisfy where A 1 ∈ n 1 ×n 1 and B 3 ∈ (n−n 1 )×(n−n 1 ) are both non-singular lower triangular matrices, B 1 ∈ n 1 ×n 1 is the quasi-lower triangular matrix and A 2 , j , D (2) j , j = 1, . . . , N are matrices with appropriate dimensions. By introducing x(t) = N (x (1) T T , the original descriptor system can be transformed into the following two ROSs: x (2) It is clear that the first ROS (5) is a normal system with time-correlated measurement noise v j (t), and the second ROS (6) is a linear combination of x (1) (t) and w(t).
First, we adopt the measurement difference method used in ref. [20] to remove the time-correlated noise v j (t). Using the measurement difference, the new measurement can be expressed as: In the above derivation, we use the fact that x (1) according to the state update equation in Equation (5). This is acceptable since the state transition matrix must be invertible [20,35].
Then, the first ROS can be expressed as: where (8) that the new measurement noise η j (t) is a one-step autocorrelation and cross-correlation with process noise w(t), which brings a challenge to obtaining the globally optimal linear filter.

Remark 2. It is clear from Equation
For the sake of convenience in discussion, we introduce the augmented vectors: Then, the augmented system can be written as: Further, we determined the following noise statistic information using Assumptions 4 and 5, which play an important role in the design of DFFWF.
Before ending this section, we recall the following CFF for the considered descriptor system, which serves in the subsequent sections.

Lemma 1.
For the first ROS (9) under Assumptions 1-5, the CFF is computed by: The centralized fusion predictor is computed bŷ P (1) The new measurement noise one-step predictor is computed bŷ The process noise filter is computed bŷ with P z c (t) = DP where the initial values arex 0 is the first n 1 components of N −1 x 0 , and P (1) 0 is the first n 1 × n 1 sub-block of N −1 P 0 N − T .
Proof. The proof is similar to the case for normal systems with one-step auto-and crosscorrelated measurement noises under the data receiving rate α = 1 [36].

Lemma 2.
For the second ROS (6) under Assumptions 1-5, the CFF is provided by: The cross-covariance matrix between the two subsystems is computed by The fusion state filter and its filtering error covariance matrix of the original descriptor (1)-(3) are provided bŷ Proof. The proof is straightforward from ref. [24].

Main Results
In this section, we design the DFFWFx d f (t|t) in Figure 1 based on local filter inputŝ x (1) j (t|t) and j = 1, · · · , N. We first design the globally optimal DFFWFx (1) d f (t|t) for the first ROS (8) using an innovation analysis approach. Then, the DFFWFx (2) d f (t|t) for the second ROS can be obtained based onx (1) d f (t|t) and the process noise filterŵ(t|t).
The fusion state filter and its filtering error covariance matrix of the original descriptor (1)-(3) are provided by (1) T Proof. The proof is straightforward from ref. [24]. □

Main Results
In this section, we design the DFFWF ˆ( | ) df x t t in Figure 1 based on local filter inputs (1) ( | ) j x t t and 1, , j N =  . We first design the globally optimal DFFWF (1) ( | ) df x t t for the first ROS (8) using an innovation analysis approach. Then, the DFFWF (2) ( | ) df x t t for the second ROS can be obtained based on (1) x t t and the process noise filter ˆ( | ) w t t .

Local Filter with Feedback
In this subsection, we will derive the local filter (1) ( | ) j x t t based on the feedback in- are computed by Theorem 2, and ˆ( | 1) are computed by Theorem 3. In view of the definitions above, we know that ˆ( | 1)

Local Filter with Feedback
In this subsection, we will derive the local filterx (1) j (t|t) based on the feedback informationx and P xη f (t|t − 1) are computed by Theorem 3. In view of the definitions above, we know thatη j f (t|t − 1) is the jth row block ofη f (t|t − 1), and P (8) under Assumptions 1-5, the local state filter with feedback is given by:

Theorem 1. For ROS
The gain matrix iscomputed by and innovation and its variance matrix are computedby where fusion predictorsx and P η j f (t|t − 1) and the cross-covariance matrix P xη j f (t|t − 1) are the feedback information from the fusion center to the local filter. The initial values arex Proof. Along the same line as the proof of CFF, the local filter can be obtained. The difference is thatx and P xη j f (t|t − 1) are the feedback information, not the local information.

Fusion Filter with Feedback
In the preceding subsection, we obtained the local filter based on the fusion state and measurement noise predictors. In this subsection, we will propose the fusion filterx (1) d f (t|t) based on the local filterx (1) j (t|t) and its gain K j (t), j = 1, · · · , N from individual sensors. In the fusion center, we regard local filtersx ). In the following text, we will derive the fusion filterx Theorem 2. For the first ROS (8) under Assumptions 1-5, in the fusion center, the DFFWF and its covariance matrix are provided by: and the gain matrix is computed by where The fusion predictor and its covariance matrix are computed by: where G = [I n , · · · , I n ] T .ŵ(t|t),P w (t|t), P xw (t|t), P η f (t|t − 1) and P xη f (t|t − 1) are addressed in Theorem 3. The initial values arex Proof. From the recursive projection formula [37], we obtain where the innovation x(t) and filtering gain matrix L(t) are defined as From the local filter (28), the input to the fusion centerx(t|t) can be expressed aŝ Substituting the second equation of (8) and (31) into (42) and T ,x(t|t) can be further rewritten aŝ which together with (39) and (40) yield (33). Using (43), the innovation associated witĥ x(t|t) can be rewritten as Taking projection on both sides of the state update equation of (8) onto L(x(0|0), · · · , x(t|t)), (37) follows directly.
Noting that x and substituting (46) and (47) (34) and (38) The process noise filter is computed bŷ where P x (t) and L(t) are computed by Theorem 2. The initial value isx

Remark 4.
It is worth noting thatŵ(t|t), K w (t|t), P w (t|t)andP xw (t|t)computed in the fusion center are used to produce the fusion one-step predictor (37) and do not need to be sent to the local filter.
The cross-covariance matrix between the two subsystems is computed by The fusion state filter and its filtering error covariance matrix of the original descriptor (1)-(3) are provided bŷ d f (t|t) (P (12) d f (t|t)) Proof. The proof is straightforward from ref. [24].
To describe the implementation of the proposed DFFWF algorithm clearly and intuitively, the following Algorithm 1 environment is used:

Initialization:
Set the initial valuesx and P xη j f (0| − 1) = 0 in each individual sensor and the initial valuesx and P xη f (0| − 1) = 0 in the fusion center. for t:=1 to N do (if there are N samples) Step 1: Compute local filterx (1) j (t|t) and gains K j (t) based on Theorem 1 in each individual sensor.
Step 3: Read all local filtersx (1) j (t|t) and filter gains K j (t) to produce the augmented T in the fusion center.
Step 10: Compute DFFWFx d f (t|t) and its covariance matrix P d f (t|t) for the original descriptor by (61)-(62) in Corollary 1.
Output the DFFWFx d f (t|t) and P d f (t|t).
Step 11: if t==N break else set t = t + 1, return to step 1. end

Estimation Performance of the DFFWF
In the proceeding subsection, we obtained the DFFWFx d f (t|t) that has better reliability, flexibility and robustness since the used measurements in the fusion center are not raw measurements but the local filtersx (1) j (t|t), j = 1, · · · , N, that have been received from individual sensors. Subsequently, let us analyze the global optimality of the proposed DFFWF. According to Lemma 2 and Corollary 1, it is clear that the global optimality of

Lemma 3. Let A(t) be a full column rank matrix and R(t) be a non-singular matrix. Then, we have
Theorem 4. For the first ROS (8) under Assumptions 1-5, if K j (t), j = 1, · · · , N are of full column rank, the DFFWF is equivalent to the CFF, i.e., under the same initial valueŝ The following results hold: Proof. Substituting (43) into (33), the fusion filter becomeŝ Substituting (36) into (34), the fusion filtering error equation becomes It follows from (35), (49) and (53) that If K j (t), j = 1, · · · , N are of full column rank, K(t) can be guaranteed to be of full column rank. By applying Lemma 3 and (24), we obtain Substituting (68) into (65)-(67) and comparing with (13), (17) and (53), we obtain which shows that L(t)K(t) and K η (t + 1|t)K(t) are the centralized optimal estimation gain matrices K c (t) for state and K η c (t + 1|t) for measurement noise, respectively. Further we obtain K c (t)P z c (t)K T c (t) = L(t)P x (t)L T (t), which shows that P (1) Substituting (31) into (28) and noting the definitionx(t|t) = [(x (1) T ,x(t|t) can be expressed aŝ Substituting (70) into (33) and comparing with (11), we obtainx The proof is completed.

Remark 5.
In Theorem 4, the global optimality of DFFWF algorithm is analyzed. Now, we compare the computational cost with distributed fusion filter weighted by matrices (DFFWM). Here, we give the computational cost by calculating the times of multiplication and division. For ease of comparisons, without loss of generality, we only give the computational cost of the first ROS. In the fusion center, DFFWF and DFFWM have the same computational cost, the computational order of magnitude is O (Nn 1 ) 3 . Hence, the proposed DFFWF algorithm is superior to the DFFWM in accuracy, which will be shown in the simulation research.

Simulation Research
In this section, we use a numerical example and a circuit system to illustrate the estimation performance of the proposed fusion filtering algorithm. Example 1. Consider a numerical example described in ref. [7]: where w(t) and µ j (t),j = 1, 2, 3 are mutually uncorrelated zero mean white noises with variances Q w and Q µ j . We know from (71) that M = N = I 4 since the original descriptor is already the canonical form. In the simulation, we set D 1 = 1 0.5 1 0 0 1 0 1 , D 2 = 1 0 0 1 0 0.8 1 0 ,    Figure 3 shows the filtering error variances of DFFWF, CFF, DFFWF and all local filters with feedback (LFWF). In Figures 2 and 3, each curve is drawn at each 2-step. The true values and filters are given in Table 1 at time 0 and 50. From Figures 2 and 3 and Table 1, it is conclude that the designed DFFWF is numerically equivalent to the CFF for the same initial values. That is to say the designed DFFWF also has global optimality. To show the superiority of the proposed DFFWF, DFFWM is also computed and shown in Figure 3. It shows that estimation accuracy of the proposed DFFWF is higher than that of any LFWF and DFFWM. Moreover, for the first and second components, the estimation accuracy of DFFWM is lower than that of LFWF measured by sensor 1. But for the third and fourth components, the result is just the opposite. The reason is that DFFWM is obtained by weighting all the local filters without feedback. On the other hand, DFFWF requires the feedback communication from the fusion center to individual sensors. Figure 4 shows the filtering error variances of LFWF with and without feedback for sensor 1. It is clear that the estimation accuracy of the proposed local filter with feedback is higher than that of the local filter without feedback, which demonstrates that feedback does improve the local estimation accuracy.
The measurement equation is the same as in example 1. Taking the sample period 0 0.05 1 T =  from Euler's approximation, the corresponding discrete-time model can be obtained as:  Figure 5, where the voltage sourceu e is the control input. It is effected by white noise w(t) due to the equipment installation, circuitry interference and voltage fluctuation. For R, L 0 and C i , i = 1, 2 denote the resistor, inductor and the ith capacity, respectively. Selecting the state x(t) = [u e1 (t), u e2 (t), i 1 (t), i 2 (t)] T , u e1 (t) and i 1 (t) are the voltage and currents of C 1 , and u e2 (t) and i 2 (t) are the voltage and current of C 2 . According to Kirchoff's second law, we can establish the following state equation [1,11]:

Example 2. Consider the circuit system measured by three sensors shown in
The measurement equation is the same as in example 1. Taking the sample period T 0 = 0.05 1 from Euler's approximation, the corresponding discrete-time model can be obtained as: In the simulation, we set C 1 = 1, and B 3 = 1. The filtering performance is provided in Figure 6. It shows the expected tracking results.
The measurement equation is the same as in example 1. Taking the sample period 0 0.05 1 T =  from Euler's approximation, the corresponding discrete-time model can be obtaine as:

Conclusions
This paper investigated the problem of distributed fusion filters was investigated for multi-sensor descriptor systems with time-correlated measurement noise. Using singular value decomposition, the original descriptor system was transformed into two reducedorder non-descriptor subsystems. First, an equivalent new system with a new measure-

Conclusions
This paper investigated the problem of distributed fusion filters was investigated for multi-sensor descriptor systems with time-correlated measurement noise. Using singular value decomposition, the original descriptor system was transformed into two reducedorder non-descriptor subsystems. First, an equivalent new system with a new measurement noise was established using a different approach to remove the time-correlated measurement noises. The new measurement noise was one-step auto-and cross-correlated. Based on the local measurement and fusion predictor from the fusion center, the local filters were obtained in the LMV sense. Then, the local filters and filtering gains were sent to the fusion center and used as the measurement inputs to produce the fusion filters. Under the condition that all local filtering gains were of full column rank, the presented DFFWF has global optimality. Furthermore, the obtained feedback can also improve the estimation of each local filter. In the future, we will try to deal with the state estimation problem for descriptor systems with time-correlated noises and some network-induced phenomena such as random transmission delays [12,38], losses [36,39] and deception attacks [40].

Conflicts of Interest:
The authors declare no conflict of interest.