A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <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> 2) Algorithm 2. Diffusion Kalman Filtering: <s> We formulate and study distributed estimation algorithms based on diffusion protocols to implement cooperation among individual adaptive nodes. The individual nodes are equipped with local learning abilities. They derive local estimates for the parameter of interest and share information with their neighbors only, giving rise to peer-to-peer protocols. The resulting algorithm is distributed, cooperative and able to respond in real time to changes in the environment. It improves performance in terms of transient and steady-state mean-square error, as compared with traditional noncooperative schemes. Closed-form expressions that describe the network performance in terms of mean-square error quantities are derived, presenting a very good match with simulations. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <s> One of the fundamental problems in sensor networks is to estimate and track the state of targets (or dynamic processes) of interest that evolve in the sensing field. Kalman filtering has been an effective algorithm for tracking dynamic processes for over four decades. Distributed Kalman Filtering (DKF) involves design of the information processing algorithm of a network of estimator agents with a two-fold objective: 1) estimate the state of the target of interest and 2) reach a consensus with neighboring estimator agents on the state estimate. We refer to this DKF algorithm as Kalman-Consensus Filter (KCF). The main contributions of this paper are as follows: i) finding the optimal decentralized Kalman-Consensus filter and showing that its computational and communication costs are not scalable in n and ii) introducing a scalable suboptimal Kalman-Consensus Filter and providing a formal stability and performance analysis of this distributed and cooperative filtering algorithm. Kalman-Consensus Filtering algorithm is applicable to sensor networks with variable topology including mobile sensor networks and networks with packet-loss. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <s> We study the problem of distributed Kalman filtering and smoothing, where a set of nodes is required to estimate the state of a linear dynamic system from in a collaborative manner. Our focus is on diffusion strategies, where nodes communicate with their direct neighbors only, and the information is diffused across the network through a sequence of Kalman iterations and data-aggregation. We study the problems of Kalman filtering, fixed-lag smoothing and fixed-point smoothing, and propose diffusion algorithms to solve each one of these problems. We analyze the mean and mean-square performance of the proposed algorithms, provide expressions for their steady-state mean-square performance, and analyze the convergence of the diffusion Kalman filter recursions. Finally, we apply the proposed algorithms to the problem of estimating and tracking the position of a projectile. We compare our simulation results with the theoretical expressions, and note that the proposed approach outperforms existing techniques. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <s> This paper is concerned with distributed Kalman filtering for linear time-varying systems over multiagent sensor networks. We propose a diffusion Kalman filtering algorithm based on the covariance intersection method, where local estimates are fused by incorporating the covariance information of local Kalman filters. Our algorithm leads to a stable estimate for each agent regardless of whether the system is uniformly observable locally by the measurements of its neighbors which include the measurements of itself as long as the system is uniformly observable by the measurements of all the agents and the communication is sufficiently fast compared to the sampling. Simulation results validate the effectiveness of the proposed distributed Kalman filtering algorithm. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <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> BIB006 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <s> Abstract Recently the distributed estimation problem with communication constraints has been widely studied for sensor network application. Our work focus on the diffusion Kalman filter with communication constraints. To satisfy finite communication resources constraints, this paper presents a multi-channel decoupled event-triggered strategy which improves the utilization of the network communication resources. With this strategy, only some entries of sensors’ measurements are transmitted if their triggering criteria are satisfied. We apply this strategy to the step 1 of the diffusion Kalman filter and analyze its performance. The analysis shows that the multi-channel decoupled event-triggered diffusion Kalman filter is unbiased in mean sense and is convergent in mean-square sense. The theoretical steady-state mean-square deviation (MSD) and communication cost are also given in this article. Simulation results demonstrate a good match between the theory analysis and experiment. Finally this algorithm is applied to the optic-electric sensor network, and the results verify the effectiveness of the proposed strategy in terms of the communication resources utilization. <s> BIB007 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 2) Algorithm 2. Diffusion Kalman Filtering: <s> Abstract This paper deals with the distributed estimation problem for networked sensing system with event-triggered communication schedules on both sensor-to-estimator channel and estimator-to-estimator channel. Firstly, an optimal event-triggered Kalman consensus filter (KCF) is derived by minimizing the mean squared error of each estimator based on the send-on-delta triggered protocol. Then, the suboptimal event-triggered KCF is proposed in order to reduce the computational complexity in covariance propagation. Moreover, the formal stability analysis of the estimation error is provided by using the Lyapunov-based approach. Finally, simulation results are presented to demonstrate the effectiveness of the proposed filter. <s> BIB008
The diffusion Kalman filtering is developed in BIB004 by replacing consensus strategies BIB006 with diffusion strategies BIB002 after the measurement update of the Kalman filter. The purpose of diffusion stepx i,k = j ∈N i π ijφj,k (see its information version (3) and Algorithm 2 for the corresponding pseudocode) is to approximate the global filtering performance via local node interactions In comparison with Algorithm 1, this strategy is independent of the consensus iteration between two consecutive Kalman filter updates and thus improves the fusion efficiency of new measurement information BIB005 . Based on this seminal research, diffusion Kalman filtering receives great attention BIB007 and the diffusion weights π ij are further optimized through the wellknown covariance intersection approach. Especially, when the uniform observability condition is satisfied under global measurements, an improved algorithm developed in BIB005 can guarantee the boundedness of estimation error covariances of each sensor under a certain connectivity condition of time-varying topologies. 3) Algorithm 3. Kalman-Based Filtering I: Based on the consensus-based Kalman filter designed in BIB001 , a version relying on neighbors' estimate is proposed in BIB003 and the corresponding filter structure is constructed as follows: where q i,k describes the consensus error of estimated states and the consensus gain F i,k is exploited to guarantee the stability. Additionally, the formulas on K i,k , P i,k , and cross-covariance P ij,k can be obtained by using the same method in BIB003 and hence omitted in this survey paper for the convenience of presentation. For the convenience of application, the pseudocode is provided in Algorithm 3. Obviously, the accumulative error of neighbors' estimate is utilized to achieve the consensus by replacing raw measurements and covariance information. In comparison with the local Kalman filter, the introduced accumulative error leads to the coupling of filtering error dynamics, and thereby the cross-covariance needs to be handled when discussing the stability of developed filtering algorithm. Nowadays, a suboptimal algorithm ignoring cross-covariance is developed in BIB008 to deal with the filtering issue with event-triggered communication protocols.