A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> Consensus algorithms for networked dynamic systems provide scalable algorithms for sensor fusion in sensor networks. This paper introduces a distributed filter that allows the nodes of a sensor network to track the average of n sensor measurements using an average consensus based distributed filter called consensus filter. This consensus filter plays a crucial role in solving a data fusion problem that allows implementation of a scheme for distributed Kalman filtering in sensor networks. The analysis of the convergence, noise propagation reduction, and ability to track fast signals are provided for consensus filters. As a byproduct, a novel critical phenomenon is found that relates the size of a sensor network to its tracking and sensor fusion capabilities. We characterize this performance limitation as a tracking uncertainty principle. This answers a fundamental question regarding how large a sensor network must be for effective sensor fusion. Moreover, regular networks emerge as efficient topologies for distributed fusion of noisy information. Though, arbitrary overlay networks can be used. Simulation results are provided that demonstrate the effectiveness of consensus filters for distributed sensor fusion. <s> BIB001 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> This note addresses Distributed State Estimation (DSE) over sensor networks. Two existing consensus approaches for DSE, i.e., consensus on information (CI) and consensus on measurements (CM), are combined to provide a novel class of hybrid consensus filters (named Hybrid CMCI) which enjoy the complementary benefits of CM and CI. Novel theoretical results, limitedly to linear systems, on the guaranteed stability of the Hybrid CMCI filters under collective observability and network connectivity are proved. Finally, the effectiveness of the proposed class of consensus filters is evaluated on a target tracking case study with both linear and nonlinear sensors. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> This letter addresses the consensus-based nonlinear state estimation in distributed sensor networks with unknown measurement noise statistics. The existence of naive nodes and the communication constraint in sensor networks requires a hybrid consensus filtering method. In the frame of consensus filtering, a novel consensus nonlinear filtering approach named variational Bayesian consensus cubature Kalman filter (VB-CCKF) is proposed, in which the CKF is employed to handle the nonlinear state estimation and the VB approximation is adopted to iteratively estimate the sufficient statistics of the measurement noise covariance on each step. Simulations are performed in order to demonstrate the effectiveness of the proposed approach. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> In this brief, we study the detectability issues in the context of distributed state estimation problems for a class of locally undetectable sensor networks. First, we introduce a novel detectability condition, i.e., weightedly uniform detectability (WUD), which is a sufficient condition to prove that the error covariances of the consensus filtering are uniformly bounded even though the local sensor nodes are undetectable. Different from the existing detectability (or observability) conditions, our condition includes the interacting weights which could further optimize the lower detectability Gramian bound. Hence, a new weights selection method is derived in term of the criterion of WUD. This new rule of selecting weights provides a new framework for distributed state estimation. The advantages of this approach lead to a better performance in estimation without extra computational burden to the filtering process. Finally, an example shows the effectiveness of the proposed method. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> In this paper, the high-dimensional distributed state estimation problem is investigated for a class of sensor networks within the cubature Kalman filtering (CKF) framework. The network consists of two types of nodes, i.e., communication ones and sensor ones. First, a hybrid consensus-based cubature Kalman filtering (HCCKF) is developed by blending the two existing approaches, namely, consensus on measurements (CM) and consensus on information (CI). As a result, the proposed filtering algorithm has complementary features of CM and CI, which turns out to be a better solution to the distributed state estimation problem. Secondly, estimation errors in HCCKF are proved to be exponentially bounded in mean square. Finally, a target tracking case-study in an example sensor network is given to demonstrate the effectiveness of the proposed HCCKF. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> Algorithm 1 1: Prediction updates Calculatex <s> This paper is concerned with the distributed state estimation problem over wireless sensor networks. The communication links are unreliable that are subject to random link failures modeled as a set of independent Bernoulli processes. To estimate the plant state collaboratively, a Kalman-consensus filtering approach is developed where the sensors spread the local information obtained from the Kalman filtering algorithm by performing a consensus of the inverse covariance matrices at each time instant. Sufficient conditions for the stochastic boundedness of the Kalman-consensus filter are established. It is shown that the filtering performance is directly influenced by the network connectivity and the collective observability. A numerical example is illustrated to verify the proposed results. <s> BIB006
Initialize Ω i,k (0), q i,k (0) and the step length L; 4: Repeat each step until l = L − 1: 5: Communications Iteration updates the number of neighbors. In addition, the induced weighted matrix of this graph is denoted as Additionally, the shorthand diag i,n {M i } used in this paper denotes a block diagonal matrix with diagonal blocks being matrices With the help of adopted consensus mechanisms and filter structures, Kalman-based distributed filtering algorithms can be roughly divided into the following six cases. 1) Algorithm 1. Kalman-Consensus Filtering: Inspired by the result in BIB001 , a two-stage Kalman-consensus filtering algorithm is developed in BIB002 . This kind of algorithm described by (2) is composed of a classical local Kalman filter (1) and a consensus update of information matrices and vectors with , where π ij is the consensus weight and L is the selected step length of consensus calculation. Its pseudocode is provided in Algorithm 1. Note that, even if the local sensor was undetectable, a sufficient condition is proposed in BIB004 to expose the uniform boundedness under the assumption of weightedly uniform detectability, which includes uniform detectability as a special case. Such an algorithm is further extended to two more general cases: the one is under the unscented Kalman filtering framework; and the other is under the cubature Kalman filtering (CKF) framework. For the first case, the boundedness condition of a consensusbased algorithm is derived in  with the help of a series of assumptions on system parameters. For the second one, an improved version, named as variational Bayesian consensus CKF, is proposed in BIB003 , where the variational Bayesian approximation is utilized to estimate the covariance of measurement noises. Furthermore, a hybrid Kalman algorithm via CKF is designed in BIB005 by blending the consensus strategies on both measurements and information matrices. Finally, when the random link failure  is a concern, in BIB006 , a renewed weighted matrix is introduced in consensus steps to implement distributed filtering and the effect on boundedness from the statistics information of random link failures is thoroughly investigated.