Scalable Distributed State Estimation for a Class of State-Saturated Systems Subject to Quantization Effects

This paper investigates the problem of a scalable distributed state estimation for a class of discrete time-variant systems with state-saturation, quantization effects, and two redundant channels over a sensor network. In transmission data from a sensor to its estimator, two phenomena are considered together. First, the data of each sensor is transmitted to its estimator through two redundant communication channels. Second, innovation data is quantized before being used by the estimator. These phenomena are beneficial in alleviating the negative effects on measurements and reducing the energy consumption and bandwidth. In the structure of proposed filter consensus is used on estimations in which consensus is first achieved on the prediction estimation, then the accuracy of computed estimation is improved by two recursive equations. The parameters of the proposed filter are obtained for each sensor node by employing an upper bound for common error covariance, therefore less computational burden is required. Eventually, the comparative simulation results are presented to show that our method has better performance compared with a rival one recently published.


I. INTRODUCTION
D ISTRIBUTED state estimation problem is a fundamental issue over wireless sensor networks in control engineering and signal processing. There is extensive research related to the study of distributed state estimation because of its extensive engineering applications in many fields, such as environment monitoring, target tracking, intelligent robotics, and battlefield surveillance [1]- [3]. One of the major challenges in distributed filtering is how to incorporate the information obtained from each sensor and its neighbours based on the given communication topology. Accordingly, several approaches have been presented for this challenge considering how to exchange data between sensors or estimators. The most popular method in the distributed state estimation is called consensus which can be classified into three main types: (1) consensus on estimation [4]- [7], in which the sum of weighted differences between the local and neighbouring estimations is added to the estimator of each node. (2) consensus on measurement [8], wherein consensus is obtained on local measurements and innovation covariance. However, it requires high communication costs and does not assure convergence. Recently, a scalable distributed extended Kalman filter with consensus on measurements is proposed in [9]. (3) consensus on information [10], in which uniform local average of information is used such that the stability of algorithm is provided. On the other hand, state saturation as a common nonlinear phenomenon appears in many practical systems because of the inherent physical restrictions of components or technological limitations; for instance, a moving vehicle with the state variables (i.e., speed and position) restrictions [11], [12]. The state saturation has a direct effect on the stability and performance of the systems. Consequently, it is essential to take the filtering process into account. So far, the problem of state estimation for state-saturated systems is investigated by a few researchers (see [13]- [19]). A time-varying filter related to the problem of estimation for a class of statesaturated systems is considered in [13] by presenting a free matrix and via a certain set of recursive nonlinear matrix inequalities. In [14], an event-based distributed state estimation problem is studied for the state-saturated systems with incomplete measurements and redundant channels. A for multi-sensor networked systems with large nodes because of using an upper bound for the cross-covariance. The remaining sections of this paper are structured as follows: The necessary background materials are recalled and the problem of a class of discrete systems with statesaturation is formulated in section 2. The structure of proposed distributed state estimation is presented in section 3. In section 4, simulation results are provided to illustrate the applicability of the suggested approach. Ultimately, the conclusion is given in section 5. Notations: The notations employed in this paper are quite standard. For a matrix B, B and tr(B) show the spectral norm and the trace, respectively. E{ξ} symbolizes the expectation of the random variable ξ. I indicates the identity matrix with appropriate dimension.

II. PROBLEM SETUP
In this paper, the sensor network is modelled as an undirected graph Ω = (ν, ϑ) where ν = {1, . . . , m} and ϑ ⊂ ν × ν are the nodes and the edges, respectively. Two nodes are connected if and only if they can exchange data between each other. The set of nodes linked to the node i is called the neighborhood of node i and is represented by Λ i . N i shows the number of neighbors of node i. It is emphasized that we consider an undirected graph in this study. Consider the following discrete time-variant system with state-saturation: where x s ∈ R n is the state vector of system that should be estimated, A s and B s are known time-variant matrices with appropriate dimensions and w s is the process noise. The saturation function σ(.) : R n → R n is denoted as follows: in which where sign denotes the signum function, ς i,max is the i-th element of vector τ max and it shows the saturation level. The measurement of the i-th sensor can be depicted as follows: z i,s ∈ R p is the output of the i-th sensor, C i,s , D 1 i,s and D 2 i,s are known time-variant matrices with appropriate dimensions and v s is the measurement noise. We assume that w s and v s are uncorrelated white noises with zero means and variances that are equal to Q w and R v . ξ 1 i,s and ξ 2 i,s are uncorrelated scalar random variables having Bernoulli distribution with meanξ j i ,ξ j i ∈ [0 1] , j = 1, 2. i.e. prob ξ j i,s = 1 =ξ j i and prob ξ j i,s = 0 = 1 −ξ j i . We assume that ξ j i,s is uncorrelated with w s and v s . Remark 1: In Eq. (4), It is assumed that the data of each sensor can be sent to its estimator through two redundant communication channels. If the transmitted state information is delivered by the first channel, other channel will be inactive, otherwise the second channel will be automatically enabled to send the state data. Redundant communication channels can help to alleviate the negative effects of packet dropouts and disconnection of partial channels in performance of networked systems [26]. It is worth noting, to avoid text clutter, we only consider two channels and adding more channels does not make a difference in the structure of the proposed method. For the sake of brevity, lets define φ i,s = (1 − ξ 1 i,s )ξ 2 i,s , therefore the measurement output in Eq. (4) can be rewritten as follows: In the networked systems, when each sensor exchanges data with its estimators/neighbors, too much energy will inevitably be consumed. To reduce the consumed energy and enhance bandwidth efficiently, the data is quantized before being transmitted. In this paper, we employ a logarithmic quantizer which is described for the i-th sensor as follows: where u j i , (j = 1, . . . , p i ) represents the j-th element of u i and the set of quantization levels for each q i,j (.) are defined as follows: in which σ i,j describes the quantization density. The logarithmic quantizer connected with the quantization level (6) is defined as the following representation: where Before ending this section, some necessary lemmas are recalled from literature. Lemma 2.1: [16], suppose that there exists a real number ε i ∈ [0, 1] for all y, x ∈ R, such that: Lemma 2.2: [27], For given matrices B, C, D and M satisfying M M T ≤ I, X is considered as a symmetric positive definite matrix, and having a scalar a > 0 such that a −1 I − DXD T > 0, then we can derive an inequality in the following form:

III. MAIN RESULTS
In this section, first a novel scalable distributed state estimation is presented for a linear system with state-saturated, then we obtain upper bounds for common, prediction and filter error covariances and finally the parameters of proposed filter are calculated such that the upper bounds of the estimation error covariances are minimized. The proposed distributed state estimation equations are defined as follows: xi,s|s is the common estimation ,x i,s|s andx i,s+1|s are the current estimation of state, and its one-step ahead prediction at i-th sensor, respectively.
denotes the innovation sequence before being quantized, G i,s , L i,s and K i,s are the time-varying parameters of proposed filter which will be determined later by minimizing the error covariance matrices.

Remark 2:
In the proposed method, differently from existing consensus on estimation algorithms, the consensus is first done on the prediction estimations based on the given topology called common estimation in Eq. (8a). Then the common estimation is used with the prediction estimation in Eq. (8b) and the current estimation in Eq. (8c) to improve the accuracy of computed estimation. As can be seen similarly to [28]- [31], differently from traditional Kalman filter, term L i,s q(z i,s ) is added to the prediction estimation equation to obtain more accurate estimation. Based on Eqs. (6)- (7) and (8a), the quantized signal q(z i,s ) is described as follows: By merging Eqs. (1)-(9), the common error e − i,s|s = x s − x − i,s|s , the prediction error e i,s+1|s = x s −x i,s+1|s and the filter error e i,s|s = x s −x i,s|s are defined as follows: The covariances of the common, prediction and filtering errors are defined as follows: based on Eqs. (10)-(15), the common, prediction and filter error covariances are obtained as follows: in which According to Eqs. (16)- (18), due to the existence of uncertainty matrix Θ i,s and unknown terms‫א‬ 1 , ℵ l , l = 1, .., 3 the common, prediction and filter error covariances cannot be obtained in an explicit way. Therefore, we will obtain upper bounds for them in Theorem 1. Before proceeding further, we introduce a useful lemma that will be used for computing the parameters of proposed filter. Lemma 3.1: The state covariance matrix q i,s = E x s x T s for Eq. (1) can be calculated as follows: where d = n i=1 ς i 2 and Σ − i,s|s is an upper bound for the common error covariance that will be introduced later. Proof. based on Eq. (1) the state covariance matrix is obtained as follows: using elementary inequality xy T + yx T ≤ xx T + yy T and selecting the third and fourth terms on the right-hand side of Eq. (21), one can conclude that: substituting Eqs. (22) into (21), we have: merging Eqs. (20)-(23), we can obtain Eq. (19). Theorem 3.2: If there exists a positive scalar α i,s such that The parameters of proposed filter in Eqs. (8a)-(8c) are obtained as follows: Proof. using elementary inequality xy T +yx T ≤ xx T +yy T and selecting the second and third terms on the right-hand side of Eq. (16), we have: Now, substituting Eqs. For computing Σ i,s+1|s , selecting the first term on the righthand side of Eq. (17) and using lemma 2.1, we can conclude that: using the aforementioned inequality for unknown terms ℵ j , j = 1, ..., 3, we have: Moreover, an upper bound for the third term on the right-hand side of Eq. (17) is obtained as follows: And finally, using lemma 2.2 in the fourth and fifth terms on the right-hand side of Eq. (17), we have: Eq. (25) can be attained by substituting Eqs. (32)-(37) into Eq. (17). Similarly, for computing Σ i,s|s , based on Eq. (21) and utilizing the abovementioned inequality for‫א‬ 1 , we have: using lemma 2.2, the upper bounds are calculated for other terms on the right-hand side of Eq. (18) as follows: (41) Therefore, substituting Eqs. (38)-(41) into Eq. (18), the upper bound of filter error covariance is obtained as follows: The prediction and filter gains are calculated by setting the first variation of Eqs. (24) and (42) to zero as follows: The main issue of this paper is also examined in [25]. In [25], a distributed filter with consensus on measurements is presented and an augment vector is used to calculate the parameters of proposed filter. In the mentioned method the cross-covariance matrices between coupling nodes are calculated recursively and used to obtain the gain matrices of nodes simultaneously, but in our method an upper bound is implemented for cross-covariance to compute gain matrix. Therefore, computational burden of our method is remarkably less than [25]. Moreover, other challenge in the method proposed in [25] is that the gain matrix can be non-singular in the case of a link failure or the loss of information from the neighbours, but our method overcomes this challenge. Also, the augmented approach in [25] cannot be applied for a graph with a lot of nodes.
Remark 4: For computing the parameters of the proposed filter, the common, prediction and filter error covariances are computed in Eqs. (16)- (18). According to Eqs. (16)- (18), due to the existence of the uncertainty matrix Θ i,s and unknown terms‫א‬ 1 , ℵ l , l = 1, ..., 3 the common, prediction and filter error covariances cannot be obtained in an explicit way. To overcome this challenge, in Theorem 1 we obtained upper bounds for the common, prediction and filter error covariances. On the other hand, the cross-covariance matrices appear in the second and the third terms on the right-hand side of Eq. (16). If the cross-covariance matrices are directly computed, then the proposed distributed filter is not scalable. Therefore, we attain upper bounds for them in Eqs. (21)- (22) using the inequality xy T +yx T ≤ xx T +yy T . Therefore, the parameters of the proposed filter are obtained by minimizing Eqs. (24)- (25) and (42).

IV. SIMULATION RESULTS
In this section, numerical simulation results are employed to indicate the merits of proposed method in Eqs. (8a)-(8c). The uncorrelated noises w s and v s have zero-mean and unity covariances. The initial values of the states and the estimators are considered as follows: The saturation levels are ς 1,max = 10 and ς 2,max = 2.    As seen, the proposed filter has acceptable performance for four different sensor nodes. In the following to demonstrate the scalability of our method, we consider a new sensor network topology with 50 nodes. As shown in Figure. 2, all 50 sensors are organized in a network with 273 edges. Figure. 3 illustrates the averaged root mean square errors (RMSE) of estimations obtained by our method and a rival method [25]. It is worth noting that our method is implemented by two different topologies while the algorithm proposed in [25] is not scalable for the network graph of 50 nodes. The averaged root mean square error (RMSE) is calculated by: where x 1 s and x 2 s denote the first and second states, and (x 1 j i,s|s ,x 2 j i,s|s ) denotes the first and second estimates of the i-th sensor at the j-th Monte Carlo run. The mean square errors (MSEs) for different sensor nodes are reported in Table   VOLUME   1 to demonstrate estimation accuracy for each sensor node in the given topology. As it is obvious, each sensor node has a different accuracy depending on the number of its neighbours and its local information such as measurement of matrix and noise. It should be noted that if we consider a topology with more nodes, the MSEs of sensor nodes become more accurate. Example 2: Consider the wheeled mobile robot (WMR) in Figure. 4 which consists of two rear driving wheels (A 1 , A 2 ) [32]. The parameters of the WMR are reported as follows: F and P are the projection of the mass center and the center of two front wheels of the robot, respectively. Also, (x P , y P ) and (x F , y F ) symbolize the coordinates of P and the coordinates of F , respectively. l F stands for the distance between the points F and P . θ P denotes the heading angle of the robot and v P is the transitional speed at the point P . ω shows the angular speed of WMR . If the state variables of the system are taken as x = θ p ,ẋ f ,ẏ f ,θ f T , then, the dynamical equations of a typical WMR can be obtained. Discretization of the WMR model with sampling interval T = 0.1 yields to:  Figure. 5 and Figure. 6 illustrate the trajectories for the actual states and the corresponding estimates for three randomly selected nodes from 500 Monte Carlo simulation.

V. CONCLUSIONS
In this paper, a novel scalable distributed state estimation has been presented for a class of state-saturated systems with quantization effects and two redundant channels. Both statesaturation and quantization phenomena have been considered in the proposed estimator. In our approach, the consensus is first achieved on the prediction estimate which is subsequently improved by two successive relations. Finally, by solving three Riccati-like difference equations, the parameters of filter have been determined at each sensor node by utilizing an upper bound for the common error covariance.  Comparative simulation results have been reported to show that the averaged root mean square errors (RMSE) of estimation gets decreased by our method compared to a rival one in literature. The main results of this paper can be developed to the following issues in order to improve the proposed method in the future works: (1) consideration of the roundrobin protocol for reducing the data exchange between nodes [19], [33], (2) attachment of the gain variations that usually occur in the practical system because of computational or implementation uncertainties in the hardware [34], [35], (3) consideration of the issue that when data is exchanged among the sensors, due to the vulnerability of communication networks, the information can be overheard and modified by the adversary [36], [37], (4) consideration of the Fault Estimation [38] . Thus, based on the view of the author, considering the above topics can improve the performance of the proposed method in practical applications.