Distributed Fusion Estimator Research Articles

Stealthy Attack Against Distributed State Estimation for Cyber‐Physical Systems

ABSTRACTThis article studies a stealthy attack strategy design for distributed state estimation from the perspective of an attacker in cyber‐physical systems, where both internal and external attack scenarios are considered simultaneously. To enhance robustness against outliers and reduce communication burden, a distributed fusion estimator is developed by fusing the innovation residuals of neighboring smart sensor nodes without any attack. Based on the designed distributed fusion estimator, the evolution of the distributed estimation error covariance is analyzed, and its lower and upper bounds are obtained. Moreover, a stealthy attack framework embedding adjustable parameter is designed to weaken the estimation performance, where the constraints of the adjustable parameter are provided based on the desired attack effect. Finally, a simulation example is provided to manifest the validity of the main results.

A Recursive Non-Uniform Sampling Estimator for Asynchronous Nonlinear Systems.

In this paper, we consider the problem of asynchronous estimation in the presence of packet losses for the randomly sampling nonlinear system. Packet losses occur at the control input and at the measurement side. Firstly, the synchronization of the asynchronous sampling system is realized by weighting the state of the adjacent state update points. Secondly, the projection theorem is used to estimate the system state at the sampling time. Due to modeling errors and unmodeled dynamics, obtaining an accurate dynamic model is challenging. Therefore, observation inference based on interpolation techniques is proposed to solve the asynchronous estimation problem. Furthermore, the algorithm is extended to multi-sensor systems to obtain a distributed fusion estimator. Finally, simulation experiments are conducted to validate the effectiveness of the algorithm.

Event-Triggered Distributed Fusion Estimator for Asynchronous Markov Jump Systems with Correlated Noises and Fading Measurements.

In this study, we investigate event-triggered distributed fusion estimation for asynchronous Markov jump systems subject to correlated noises and fading measurements. The measurement noises are interrelated, and they are simultaneously coupled with the system noise. The sensor samples measurements uniformly, and the sampling rates of the sensors are different. First, the asynchronous system is synchronized at state update points; then, the local filter is obtained. Furthermore, a variance-based event-triggered strategy is introduced between the local estimator and the fusion center to decrease the energy consumption of network communication. Then, a distributed fusion estimation algorithm is proposed using a matrix-weighted fusion criterion. Finally, the effectiveness of the algorithm is verified using computer simulations.

Event-triggered fusion estimation under bounded noises

This paper is devoted to solving the event-triggered multi-sensor fusion estimation problem with bounded noises. Under the centralized fusion framework, a compensation strategy is developed to recover the non-triggered data by analyzing the intrinsic properties of Kalman filter. Then, an optimal centralized fusion estimator is derived by minimizing an upper bound of estimation error at each time. Under the distributed fusion framework, the optimal local estimator on each observable subspace is obtained by the ideas similar to those in the centralized fusion framework. Then, two different distributed fusion estimators are respectively designed by means of a spectral radius inequality and a trace inequality, where the former achieves better estimation performance and the latter is more computationally efficient. Particularly, all the proposed centralized and distributed fusion estimators are shown to be stable under moderate conditions. Finally, a classical target tracking system is utilized to verify the effectiveness and advantages of the proposed methods.

On distributed fusion estimation with stochastic scheduling over sensor networks

The paper deals with the distributed fusion estimation for linear time-varying systems over sensor networks, in which stochastic sensor scheduling and unknown exogenous inputs are taken into account. In the stochastic sensor scheduling, expensive and cheap channels are used to respectively transmit the high-precision data and the low-precision quantized data. Based on the stochastic scheduling scheme, a recursive minimum mean square error (MMSE) estimator is proposed against the unknown inputs. Then, a distributed fusion estimator is presented by combining local estimates and covariances from all sensors, relying on the covariance intersection (CI) fusion rule. Sufficient conditions are established to ensure that the proposed fusion estimator is stable with the stochastically ultimately bounded estimation error. Finally, a target tracking example is given to show the effectiveness of the proposed method.

Distributed Covariance Intersection Fusion Estimation With Delayed Measurements and Unknown Inputs

This article is concerned with the distributed covariance intersection (CI) fusion estimation for cyber-physical systems (CPSs) with delayed measurements and unknown inputs. The measurement transmission is subject to random delays described by a set of independent Bernoulli processes. Based on the provided finite-length buffers, the delayed measurements are retrieved within the corresponding buffer length. By modeling the unknown inputs with a noninformative prior distribution, a local minimum mean square error (MMSE) estimator is derived in the Bayesian framework. Then this result is extended to the multiple sensor scenario, where the sequential CI fusion approach is applied to design a recursively distributed fusion estimator. It is proved that the distributed sequential CI fusion estimator is consistent and performs better than each local estimator in state estimation. An illustrative example is provided to demonstrate the effectiveness of the proposed technique.

Distributed Fusion Estimation with Sensor Gain Degradation and Markovian Delays

This paper investigates the distributed fusion estimation of a signal for a class of multi-sensor systems with random uncertainties both in the sensor outputs and during the transmission connections. The measured outputs are assumed to be affected by multiplicative noises, which degrade the signal, and delays may occur during transmission. These uncertainties are commonly described by means of independent Bernoulli random variables. In the present paper, the model is generalised in two directions: (i) at each sensor, the degradation in the measurements is modelled by sequences of random variables with arbitrary distribution over the interval [0, 1]; (ii) transmission delays are described using three-state homogeneous Markov chains (Markovian delays), thus modelling dependence at different sampling times. Assuming that the measurement noises are correlated and cross-correlated at both simultaneous and consecutive sampling times, and that the evolution of the signal process is unknown, we address the problem of signal estimation in terms of covariances, using the following distributed fusion method. First, the local filtering and fixed-point smoothing algorithms are obtained by an innovation approach. Then, the corresponding distributed fusion estimators are obtained as a matrix-weighted linear combination of the local ones, using the mean squared error as the criterion of optimality. Finally, the efficiency of the algorithms obtained, measured by estimation error covariance matrices, is shown by a numerical simulation example.

Distributed optimal linear fusion estimators

This paper is concerned with the distributed information fusion estimation problem in multi-sensor environments. A universal distributed optimal linear fusion estimation (DOLFE) algorithm, which has a Kalman-type structure with matrix gains, is presented under the linear unbiased minimum variance criterion. To reduce the computational burden, two suboptimal linear fusion estimation algorithms with diagonal-matrix gains and scalar gains are also presented. Based on the proposed DOLFE algorithm, a distributed optimal linear fusion filter (DOLFF) is presented for multi-sensor linear discrete-time stochastic systems. It has better accuracy than that based on the matrix-weighted fusion of local estimators but worse accuracy than the centralized fusion filter (CFF). The stability and steady-state property of the proposed DOLFF are analyzed. Then, the corresponding multi-step predictor and smoother are also developed based on DOLFF. To obtain the distributed fusion estimators, some estimation error cross-covariance matrices that are used to compute the gains are derived. At last, distributed optimal linear fusion estimators with feedback are also presented. Furthermore, it is strictly proved that the proposed distributed optimal linear fusion filter with feedback (DOLFFWF) has the same accuracy as the CFF. Two simulation examples show the effectiveness of the proposed algorithms.

Fusion estimation under binary sensors

Binary sensors compress the measurement information into two possible values as output and have found applications in many fields such as medical monitoring, environment monitoring and localization. This paper is concerned with the fusion estimation problem for a class of binary sensor systems with bounded noises, where the monitored physical process is described by time-varying state-space models. Since binary sensors only provide the valid information at the switching instant, a novel uncertain measurement model is proposed to characterize this feature. Then, by constructing an upper bound of fusion/estimation error at each time, convex optimization problems are established in terms of linear matrix inequalities to design stable centralized and distributed fusion estimators. An illustrative example is given to show the effectiveness of the proposed methods.

Optimal fusion state estimator for a multi-sensor system subject to multiple packet dropouts

In this note, we study the state estimation problem for a multi-sensor system subject to multiple packet dropouts. A novel optimal distributed fusion estimator is derived by applying a re-sending mechanism and a parallel information filtering structure. It is shown that the proposed distributed fusion estimator has smaller estimation error covariance and less computation complexity when compared with the centralised Kalman like estimator with multiple intermittent measurements.

Fusion estimation for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises

PurposeThis paper aims to give the centralized and distributed fusion estimator for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises and give the corresponding square-root cubature Kalman filter.Design/methodology/approachBased on the Gaussian approximation recursive filter framework, the authors derive the centralized fusion filter and using the projection theorem, the authors derive the centralized fusion smoother. Then, based on the fast batch covariance intersection fusion algorithm, the authors give the corresponding results for distributed fusion estimators.FindingsDesigning the fusion estimators for nonlinear multi-sensor networked systems with packet loss compensation and correlated noises is necessary. It is useful for general nonlinear systems.Originality/valueThroughout the whole study, the main highlights of this paper are as follows: packet loss compensation and correlated noises are considered in nonlinear multi-sensor networked systems. There are no relevant conclusions in the existing literature; centralized and distributed fusion estimators are derived based on the above system; for the posterior covariance with compensation factor and correlated noises, a new square-root factor of the error covariance is derived; and the new square-root factor of the error covariance is used to replace the numerical implementation of the covariance in cubature Kalman filter (CKF), which simplified the problem in calculating the posterior covariance in CKF.

Fusion Estimation for Discrete Time-Varying Systems With Bounded Nonlinearities

This paper studies the distributed fusion estimation problem for a class of time-varying systems with bounded nonlinearities, where nonlinear terms are considered under two different cases. By using matrix analysis approach, each local estimator with time-varying gain is designed by minimizing an upper bound of the square error of the estimator. Based on the obtained local estimators, the distributed fusion estimators are designed by establishing a class of convex optimization problems. Notice that the designed local estimators and fusion estimators are stable, and the established convex optimization problems can be easily solved by standard software packages. Localization of a mobile robot is given to demonstrate the effectiveness of the proposed methods.

Distributed Fusion Estimator for Multisensor Multirate Systems With Correlated Noises

A distributed fusion estimation algorithm is studied for multisensor multirate systems with correlated noises, where the state update rate is positive integer multiples of the measurement sampling rates and different sensors sample uniformly with different sampling rates. The measurement noises from different sensors are correlated with each other and are also correlated with the process noise. First, the state space model is established at the measurement sampling points (MSPs). Then, the optimal local filter at the MSPs and optimal local estimators (LEs) at the state update points are presented by an innovation analysis approach, respectively. Moreover, the cross-covariance matrices of estimation errors between any two LEs are derived, which involves three jointly recursive difference equations. At last, a distributed fusion estimator is proposed by applying the matrix-weighted fusion estimation algorithm in the linear minimum variance sense. The stability of the proposed algorithms is analyzed. Simulation results illustrate the effectiveness of the algorithms.

Distributed Asynchronous Fusion Estimator for Stochastic Uncertain Systems With Multiple Sensors of Different Fading Measurement Rates

This paper is concerned with the distributed fusion estimation problem for time-varying stochastic uncertain systems with multiple asynchronous sampling sensors of different fading measurement rates. Stochastic uncertainties of multiplicative noises exist in the state and measurement equations. The phenomena of fading measurements of different sensors are depicted by a group of stochastic variables with known statistics. Different sensors have different sampling rates. Sampling period of each sensor is uniform and an integer multiple of the state update period. By transforming the multiplicative noises into the additive noises, local estimators (LEs) and estimation error cross-covariance matrices between any two LEs are derived at the state update points. A real-time distributed fusion estimator (DFE) is obtained by using the matrix-weighted fusion estimation algorithm in the linear unbiased minimum variance sense. The asymptotic stability and steady-state property are analyzed. A new conception of $\boldsymbol{{period}}$ steady state is presented and the $\boldsymbol{{period}}$ steady-state properties of the LEs, cross-covariance matrices, and DFE are proven. Though DFE has lower accuracy than the centralized fusion estimator, it has better robustness and flexibility since it has a parallel structure. To avoid the calculation of cross-covariance matrices, a sequential covariance intersection fusion estimator is given by using two-sensor covariance intersection fusion algorithm. It has lower accuracy but smaller computational cost than DFE, and better accuracy than LEs. Two examples are given to show the effectiveness of the proposed algorithms.

Fusion estimation for multi-sensor networked systems with packet loss compensation

This paper is concerned with information fusion estimation problems for multi-sensor networked systems with random packet losses. Based on a recent developed compensation strategy of packet losses that the predictor of lost observation is used as the observation when a packet is lost, centralized fusion estimators (CFEs), including the filter, predictor and smoother, in the linear unbiased minimum variance (LUMV) sense are first designed by completing square method. Then, local optimal estimators are designed for each sensor subsystem. Estimation error cross-covariance matrices between any two local estimators are derived. Based on local estimators and cross-covariance matrices, distributed fusion estimators (DFEs) are presented by using the matrix-weighted fusion estimation algorithm in the LUMV sense. Compared with the existing results with zero-input and hold-input compensations, the proposed algorithms with prediction compensations can obviously improve the estimation accuracy. Two simulation examples show their effectiveness.

A Solution to Estimation Fusion for Multirate Measurements with Delays

This paper presents the distributed estimation fusion algorithms for a class of multirate multisensor systems with measurement delays. Different sensors uniformly sample measurements with different sampling rates and time delays. First, a new state-space model at the measurement sampling points is constructed and the original time delayed system is transformed to a delay-free one with correlated noises in limited time intervals to obtain the real-time estimation. Based on the new state-space model, the optimal local filter at the measurement sampling points is obtained for each single-sensor subsystem, and then the local estimator at the state update points is derived by using the predictor based on the filter at the measurement sampling points. Then, the estimation error cross-covariance matrices between any two local estimators are derived. Finally, the real-time distributed fusion estimator at the state update points is obtained based on the optimal fusion criterion weighed by matrices in the linear unbiased minimum variance sense. Besides, to avoid the correlation between system and measurement noises in the developed state-space model, a simple alternative but nonreal-time estimation fusion algorithm is also presented by employing dummy measurements. A numerical example is given to illustrate the effectiveness of the proposed algorithms.

Distributed Covariance Intersection Fusion Estimation for Cyber-Physical Systems With Communication Constraints

This technical note addresses the distributed covariance intersection fusion estimation problem for a class of cyber-physical systems (CPSs), where the raw measurements are preprocessed in each sink node to obtain the local state estimates of a CPS. When each local estimate is transmitted to a remote fusion center (FC), the communication network between the sink node and the FC is subject to finite bandwidth, varying delays and packet dropouts. Based on a new communication constraint model covering all these network phenomena, a recursively distributed fusion estimator (DFE) is designed for the addressed CPSs. Since the estimation performance directly impacts the stability of control operation in CPSs, stability condition is derived such that the mean square error of the designed DFE is bounded, and it is proved that the DFE is independent of the initial values under certain conditions. The design procedure is illustrated using a numerical example.

Distributed fusion estimator for multi‐sensor asynchronous sampling systems with missing measurements

The fusion estimation problem for a class of multi-sensor asynchronous sampling linear stochastic systems with missing measurements is considered, where the state is updated uniformly and each sensor non-uniformly samples one measurement at most within a state update period. Based on the sampled measurement data of each sensor, the optimal local state estimators are designed at the state and measurement points by using the projection theory. The cross-covariance matrices between estimation errors of any two local estimators are derived. Based on the obtained local estimators and cross-covariance matrices, the distributed optimal fusion estimator is given by using matrix-weighted fusion estimation algorithm in the linear minimum variance sense. A simulation example verifies the effectiveness of the algorithms.

Distributed fusion filters from uncertain measured outputs in sensor networks with random packet losses

This paper addresses the distributed fusion filtering problem for discrete-time random signals from measured outputs perturbed by random parameter matrices and correlated additive noises. These measurements are obtained by a sensor network with a given topology, where random packet dropouts occur during the data transmission through the different network communication channels. The distributed fusion estimation is accomplished in two phases. Firstly, by an innovation approach and using the last observation that successfully arrived if a packet is lost, a preliminary distributed least-squares estimator is designed at each sensor node using its own measurements and those from its neighbors. Secondly, every sensor collects the preliminary filters that are successfully received from its neighbors and fuses this information with its own one to generate the least-squares linear matrix-weighted distributed fusion estimator. The accuracy of the proposed estimators, which is measured by the estimation error covariances, is examined by a numerical simulation example.

Fusion estimation using measured outputs with random parameter matrices subject to random delays and packet dropouts

This paper investigates the centralized and distributed fusion estimation problems for discrete-time random signals from multi-sensor noisy measurements, perturbed by random parameter matrices, which are transmitted to local processors through different communication channel links. It is assumed that both one-step delays and packet dropouts can randomly occur during the data transmission, and different white sequences of Bernoulli random variables with known probabilities are introduced to depict the transmission delays and losses at each sensor. Using only covariance information, without requiring the evolution model of the signal process, a recursive algorithm for the centralized least-squares linear prediction and filtering estimators is derived by an innovation approach. Also, local least-squares linear estimators based on the measurements received by the processor of each sensor are obtained, and the distributed fusion method is then used to generate fusion predictors and filters by a matrix-weighted linear combination of the local estimators, using the mean squared error as optimality criterion. In order to compare the performance of the centralized and distributed fusion estimators, recursive formulas for the estimation error covariance matrices are also derived. A numerical example illustrates how some usual network-induced uncertainties can be dealt with the current observation model with random matrices.