This paper considers the state reconstruction problem for discrete-time cyber-physical systems when some of the sensors can be arbitrarily corrupted by malicious attacks where the attacked sensors belong to an unknown set. We first prove that the state is s-error correctable if the system under consideration is s-sparse observable where s denotes the maximum number of attacked sensors. Then, two state reconstruction methods are presented where the first method is based on searching elements with the same value in a set and the second method is developed in terms of searching element satisfying a given condition. In addition, after establishing and analyzing the conditions that the proposed state reconstruction methods are not effective, we address that it is very hard to prevent the state reconstruction when either state reconstruction method proposed in this paper is used. The correctness and effectiveness of the proposed methods are examined via an example of four-dimensional dynamic systems and a real-world example of three-inertia systems.

In this paper, we consider the distributed Kalman filtering problem over sensor networks with a finite number of sensor nodes where each node can communicate only with its neighboring nodes. We first introduce the concept of gradual information fusion estimation (GIFE) and propose an algorithm for computing the GIFE that can obtain performance improvement via reducing estimation error covariance. Also, we prove that the GIFE can be expressed as a weighted sum of local estimates. Then, based on the results of GIFE and some results proposed in this paper, two distributed Kalman filters are developed where, at time step k, each node is allowed to communicate with its neighboring nodes at most once in the first filter and each node is permitted to communicate with its neighboring nodes twice in the second filter. In addition, we prove that either of the two proposed distributed Kalman filters is unbiased and the estimation error covariance of the first distributed Kalman filter is less than or equal to that without using information fusion estimation. We prove that the estimation error covariance of the second distributed Kalman filter is less than or equal to that of any local estimate belonging to a set. An example of unmanned ground vehicle is provided to illustrate the performance of the two proposed distributed Kalman filters.

In this paper, we consider the distributed Kalman filtering problem over sensor networks with a finite number of sensor nodes where each node can communicate only with its neighboring nodes. We first introduce the concept of gradual information fusion estimation (GIFE) and propose an algorithm for computing the GIFE that can obtain performance improvement via reducing estimation error covariance. Also, we prove that the GIFE can be expressed as a weighted sum of local estimates. Then, based on the results of GIFE and some results proposed in this paper, two distributed Kalman filters are developed where, at time step k, each node is allowed to communicate with its neighboring nodes at most once in the first filter and each node is permitted to communicate with its neighboring nodes twice in the second filter. In addition, we prove that either of the two proposed distributed Kalman filters is unbiased and the estimation error covariance of the first distributed Kalman filter is less than or equal to that without using information fusion estimation. We prove that the estimation error covariance of the second distributed Kalman filter is less than or equal to that of any local estimate belonging to a set. An example of unmanned ground vehicle is provided to illustrate the performance of the two proposed distributed Kalman filters.