Estimation Error Covariance 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.

Joint optimal PMU and DULR placement in distribution network considering fault reconstruction and state estimation accuracy

Due to the rapid development of the distribution network, there is an urgent need for high-precision state estimation and safe and stable operation, this paper proposes a joint optimal placement method for Phasor Measurement Unit (PMU) and Dual Use Line Relay (DULR) in distribution networks considering fault reconstruction and state estimation accuracy. The proposed method aims to minimize the cost of PMU and DULR configuration. Based on the known pseudo-measurements and Supervisory Control and Data Acquisition (SCADA) measurements, we use the E-optimal standard design of the state estimation error covariance matrix to form the state estimation accuracy index and take the relay protection function of Feeder Terminal Unit (FTU) device and DULR device into consideration of the fault reconstruction. By solving the mixed integer semidefinite programming (MISDP) model, the optimal placement result is obtained, so that the network under normal condition and the network after multiple fault reconstruction can meet the constraints of state estimation accuracy and load loss rate respectively. The proposed method is tested on IEEE 33-node, 123-node systems as an example. The experimental results show that the proposed method is effective and practical.Note to Practitioners—This study is stimulated by the need for optimizing the placement of Micro-PMU (µPMU) in distribution network. Since the new µPMU device DULR has the same relay protection function as FTU, and distributed generation can supply power to the island after failure, this paper adds the load loss rate to the constraint conditions to study the influence of fault reconstruction on the placement scheme. A new two-step SE method considering mixed measurements was used in this paper to improve the computational efficiency of real-time SE of large-scale system while ensuring the accuracy of the calculation results. We have conducted tests in IEEE-33 and IEEE-123 systems. After preliminary verification, compared with the single fault scenario, we find that the placement scheme considering multiple fault scenarios can effectively improve the accuracy of state estimation and reduce the load loss rate of each fault scenario and only increase the cost a little. The increase in the number of DG and contact lines in the network will also effectively improve the feasibility of the placement scheme. The increase of the number of FTU configurations can effectively reduce the configuration of DULR by adding a small amount of µPMU, reduce the total cost as a result. In future studies, we will further consider the integrated configuration of µPMU, DULR and FTU, to meet more failure scenarios while reducing costs.

Encoding–decoding-based distributed state estimation over sensor networks with limited sensing range under DoS attacks

This paper is concerned with encoding–decoding-based distributed state estimation over sensor networks under DoS attacks. Different from most of the existing research results on distributed state estimation for sensor networks, where all sensors are assumed to have sufficiently wide sensing ranges, this paper considers sensors with limited sensing ranges. Therefore, the problem studied has more practical significance. To save the limited bandwidth resources of sensor networks, a two-channel encoding–decoding scheme (EDS) based on probability is proposed for each node to compress the transmitted data to an acceptable range, where independent DoS attacks are launched randomly on the communication channels between nodes. Then, a distributed state estimator with limited sensing ranges under DoS attacks in the presence of both the sensor-estimator channel EDS and the node–node channel EDS is constructed under the criterion of minimum mean-square error. Furthermore, considering the real-time changes of the communication topology resulting from independent DoS attacks and the uncertainty introduced by the node–node channel EDS, the upper bound of the expected estimation error covariance is derived and the boundedness of the upper bound is analyzed under given assumption conditions. Finally, a numerical example is exhibited to illustrate the effectiveness of the designed algorithm.

Recursive state estimation for delayed complex networks with random link failures and stochastic inner coupling under cyber attacks

This paper focuses on the filtering issue regarding a class of uncertain time-delayed complex networks (CNs) subject to random link failures (RLF) and stochastic inner coupling (SIC) under cyber attacks, where the attacks occur in a random way with uncertain occurrence probabilities. The phenomena of RLF and SIC are described via a Bernoulli distributed random variable and a Gaussian noise, respectively. The former reflects the presence or absence of connections between different nodes, while the latter presents the uncertainty of the inner coupling strength. In addition, denial of service attacks (DoSAs) and false data injection attacks (FDIAs) are both taken into account in the discussed cyber space, where the switching behavior of attack modes is described by two independent Bernoulli distributed random variables. A novel recursive estimator is constructed with consideration of RLF, uncertain time-delay, SIC and cyber attacks, wherein a state estimation error covariance upper bound (SEECUB) is obtained. And such upper bound is minimized via reasonable design of the estimator gain. Through theoretical deduction and analysis, it is proved that the presented SEECUB is uniformly bounded under certain conditions. Finally, the availability of the proposed filtering strategy and the gained results are verified through simulation examples.

A Fusion Tracking Algorithm for Electro-Optical Theodolite Based on the Three-State Transition Model.

This study presents a novel approach to address the autonomous stable tracking issue in electro-optical theodolite operating in closed-loop mode. The proposed methodology includes a multi-sensor adaptive weighted fusion algorithm and a fusion tracking algorithm based on a three-state transition model. A refined recursive formula for error covariance estimation is developed by integrating attenuation factors and least squares extrapolation. This formula is employed to formulate a multi-sensor weighted fusion algorithm that utilizes error covariance estimation. By assigning weighted coefficients to calculate the residual of the newly introduced error term and defining the sensor's unique states based on these coefficients, a fusion tracking algorithm grounded on the three-state transition model is introduced. In cases of interference or sensor failure, the algorithm either computes the weighted fusion value of the multi-sensor measurement or triggers autonomous sensor switching to ensure the autonomous and stable measurement of the theodolite. Experimental results indicate that when a specific sensor is affected by interference or the off-target amount cannot be extracted, the algorithm can swiftly switch to an alternative sensor. This capability facilitates the precise and consistent generation of data, thereby ensuring the stable operation of the tracking system. Furthermore, the algorithm demonstrates robustness across various measurement scenarios.

Unscented-Kalman-Filter-Based Remote State Estimation for Complex Networks With Quantized Measurements and Amplify-and-Forward Relays.

In this article, the remote estimation problem is addressed for a class of discrete-time complex networks under the influence of probabilistic quantization and amplify-and-forward (AF) relays. The underlying complex network model, which is inherently nonlinear and stochastic, is affected by additive process and measurement noises. Owing to the limited bandwidth of the transmission channel, the measurement outputs are quantized by a probabilistic quantizer prior to transmission. To enhance the signal quality over long-distance transmissions, the quantized measurements are sent to AF relays and subsequently forwarded to the estimator. Utilizing the unscented Kalman filter approach, a novel state estimator is designed to minimize an upper bound on the estimation error covariance. Moreover, sufficient conditions are derived to ensure that the estimation error is exponentially bounded in the mean-square sense. Lastly, the efficacy of the proposed scheme is illustrated through numerical simulations.

Event‐triggered resilient joint mobile robot localization and sensor fault estimation

SummaryThe event‐triggered joint sensor fault estimation and mobile robot (MR) localization (MRL) problem (MRLP) subject to the potential fluctuations of the estimator gain are investigated. From the external sensor to the estimator, to decrease the consumption of sensor energy and cut down network bandwidth resources, an event‐triggered scheme is considered. For the sake of characterizing the phenomenon induced by the inaccurate calculation of estimator gain, an uncertainty with the bounded second moment is employed. The purpose of this study is theoretically to find a feasible and effective approach to the addressed joint estimation problem such that the estimation error (EE) covariance (EEC) meets the given performance index. First, a minimum upper bound (UB) of the EEC is derived. Subsequently, in terms of the proposed joint estimation approach and the corresponding results obtained, an algorithm to address the resilient joint sensor fault estimation and MRLP is summarized. At the end, the effectiveness of the proposed algorithm is validated through conducting a set of comparisive experiments.

Encryption–decryption-based state estimation for nonlinear complex networks subject to coupled perturbation

This paper discusses the encryption-decryption-based state estimation (EDBSE) issue for coupled perturbation complex networks (CPCNs) in the framework of the Kalman-type filtering scheme. A uniform distributed random variable is employed to characterize the coupled perturbation among different network units. A uniform-quantization-dependent encryption-decryption (UQDED) scheme is considered here to orchestrate the transmitted data. A novel EDBSE approach is developed such that the upper bounds of prediction error (PE) covariance (PEC) and estimation error (EE) covariance (EEC) can be derived by resolving Riccati-like difference equations and the estimation parameter (EP) is determined by minimizing the trace of the upper bound of EEC. Furthermore, a uniformly bounded condition is elaborated to evaluate the algorithm performance of EDBSE. Finally, an illustrative example is conducted to verify the validity of the introduced EDBSE method.

Hybrid-attack-resistant distributed state estimation for nonlinear complex networks with random coupling strength and sensor delays

In this paper, a recursive distributed hybrid-attack-resistant state estimation (SE) scheme is proposed for a class of time-varying nonlinear complex networks (NCNs) subject to random coupling strength (RCS) and random sensor delays (RSDs) under hybrid attacks. A hybrid-attack model is considered to characterize the random occurrence of denial-of-service (DoS) attacks and deception attacks. The objective of the problem to be solved is to develop a recursive distributed estimation method such that, in the presence of RCS, RSDs and hybrid attacks, a locally optimized upper bound (UB) on the estimation error covariance (EEC) is ensured. By employing the mathematical induction method, a UB is firstly derived on the EEC. Subsequently, the obtained UB is minimized by appropriately designing the estimator gain (EG). Furthermore, a sufficient criterion guaranteeing the exponential boundedness (EB) of SE error is elaborated in the mean square sense (MSS). Finally, simulation experiments with localization applications of multiple mobile indoor robots are conducted to illustrate the applicability of the proposed SE scheme.

Security Analysis of Distributed Consensus Filtering Under Replay Attacks.

This work studies the security of consensus-based distributed filtering under the replay attack, which can freely select a part of sensors and modify their measurements into previously recorded ones. We analyze the performance degradation of distributed estimation caused by the replay attack, and utilize the Kullback-Leibler (K-L) divergence to quantify the attack stealthiness. Specifically, for a stable system, we prove that under any replay attack, the estimation error is not only bounded, but also can re-enter the steady state. In that case, we prove that the replay attack is ϵ -stealthy, where ϵ can be calculated based on two Lyapunov equations. On the other hand, for an unstable system, we prove that the trace of estimation error covariance is lower bounded by an exponential function, which indicates that the estimation error may diverge due to the attack. In view of this, we provide a sufficient condition to ensure that any replay attack is detectable. Furthermore, we analyze the case that the adversary starts to attack only if the current measurement is close to a previously recorded one. Finally, we verify the theoretical results via several numerical simulations.

Outlier‐resistant state estimation for complex networks with random false data injection attacks under encoding–decoding mechanism

SummaryThis article focuses on the outlier‐resistant state estimation problem for discrete time‐varying complex networks (TVCNs) affected by random false data injection attacks (FDIAs) under an encoding–decoding mechanism (EDM). From the perspective of information security, a uniform‐quantization‐based EDM is employed to encrypt the transmitted data. During the data transmission process, a set of independent random variables governed by Bernoulli distribution is introduced to characterize the occurrence of random FDIAs. For the purpose of alleviating the passive impact of potential measurement outliers, a saturation structure is adopted during the estimator design. The gain matrix is given by minimizing the upper bound of estimation error covariance. According to the stochastic analysis method, it is shown that the state estimation error is bounded exponentially in mean‐square sense by providing new sufficient condition. It should be noted that we make the first attempt to develop new outlier‐resistant state estimation method with performance evolution criterion in the time‐varying perspective for TVCNs with random FDIAs under EDM. Finally, a simulation example with comparative experiment is presented to illustrate the effectiveness of the newly presented outlier‐resistant estimation algorithm.

A distributed state and fault estimation scheme for state-saturated systems with quantized measurements over sensor networks

This article explores a new framework of distributed state and fault estimation (DSFE) for the state-saturated systems over sensor networks. To this aim, the upper bound on estimation error covariance (EEC) is ensured and the explicit expression of the corresponding estimator gains is given with both quantization effects and state saturations. Further, a feasible upper bound is located on EEC and minimized by parameterizing the estimator gain. The matrix simplification technique is adopted to deal with the sensor network topology’s sparseness problem. Additionally, the estimation performance is first analyzed and then ensured by conducting a sufficient condition. At last, experiments are carried out to verify the feasibility of the developed DSFE method.

A robust state estimation method for power systems using generalized correntropy loss function

The accurate estimation of power system states is crucial for effective monitoring and control. However, the performance of conventional state estimators, which assume Gaussian measurement noise and do not account for denial-of-service attacks, can deteriorate significantly in real power systems. To address these issues, this paper proposes a novel robust state estimation method based on the quadratic function (QF) and the generalized correntropy loss function (GCL). The proposed QF-GCL state estimation method can effectively deal with non-Gaussian measurement noise and denial-of-service attacks. To enhance the computational efficiency, an influence function based solving method is developed. To determine the optimal parameters for the proposed QF-GCL state estimation method, a new state estimation error covariance equation is further derived. Simulations are performed on the IEEE 30-bus, 118-bus and 300-bus systems, to demonstrate the accurate and robust performance of the proposed QF-GCL robust state estimation method.

Distributed Energy-Based Estimation Over Harvesting-Constrained Sensor Networks.

This article investigates the distributed joint state and fault estimation issue for a class of nonlinear time-varying systems over sensor networks constrained by energy harvesting. It is assumed that data transmission between sensors requires energy consumption, and each sensor can harvest energy from the external environment. A Poisson process models the energy harvested by each sensor, and the sensor's transmission decision depends on its current energy level. One can obtain the sensor transmission probability through a recursive calculation of the probability distribution of the energy level. Under such energy harvesting constraints, the proposed estimator only uses local and neighbor data to simultaneously estimate the system state and the fault, thereby establishing a distributed estimation framework. Moreover, the estimation error covariance is determined to possess an upper bound, which is minimized by devising energy-based filtering parameters. The convergence performance of the proposed estimator is analyzed. Finally, a practical example is presented to verify the usefulness of the main results.

A new robust dynamic state estimation approach for power systems with non-Gaussian noise

The Gaussian noise distribution is typically used in dynamic state estimation (DSE) but it is not always true in practice because of abnormal system inputs, impulsive noise and measurement outliers. In this paper, a new robust DSE approach based on a new robust Lp norm based estimator and the cubature Kalman filter (CKF) is developed for power systems with non-Gaussian noise statistics. The Lp norm based estimator is derived from the Lp norm formula and the quadratic formula in order to alleviate the impacts from bad data and outliers. The proposed Lp-CKF DSE approach exhibits good accuracy because a new estimation error covariance is obtained by using the influence function. The robustness of the proposed Lp-CKF DSE approach is verified by performing simulations on a generator in the IEEE 39-bus system.

Application of the adaptive method to determine the process noise in the extended Kalman filter to estimate the parameters of the two dimensional inverse heat transfer problem

To deal with inverse nonlinear heat transfer problems in a two-dimensional heat conduction problem, this paper presents a hybrid approach combining fuzzy logic and the extended Kalman filter (EKF). The proposed algorithm has been applied to the real-time reconstruction of the temperature field, time-varying convective heat transfer coefficient, and time-varying boundary heat flux using temperature data from a set of sensors. The effects of the covariance of the initial estimation error, the time step of data acquisition by the sensors, and the number of installed sensors on the precision and stability of the estimation results has been investigated by numerical experiments. To compare and validate the results, all these parameters were investigated using the EKF method. The results show that the adaptive fuzzy extended Kalman filter (FEKF) method for estimating temperature, convection coefficient, and heat flux at the boundaries is precise and stable. The comparison shows that the performance of the FEKF method in parameter estimation is better than the EKF method, and the highest increase in precision and stability of the results is related to the convective heat transfer coefficient.

A Joint State and Fault Estimation Scheme for State-Saturated System with Energy Harvesting Sensors.

In this article, the issue of joint state and fault estimation is ironed out for delayed state-saturated systems subject to energy harvesting sensors. Under the effect of energy harvesting, the sensors can harvest energy from the external environment and consume an amount of energy when transmitting measurements to the estimator. The occurrence probability of measurement loss is computed at each instant according to the probability distribution of the energy harvesting mechanism. The main objective of the addressed problem is to construct a joint state and fault estimator where the estimation error covariance is ensured in some certain sense and the estimator gain is determined to accommodate energy harvesting sensors, state saturation, as well as time delays. By virtue of a set of matrix difference equations, the derived upper bound is minimized by parameterizing the estimator gain. In addition, the performance evaluation of the designed joint estimator is conducted by analyzing the boundedness of the estimation error in the mean-squared sense. Finally, two experimental examples are employed to illustrate the feasibility of the proposed estimation scheme.

Distributed cyber attack detection and physical fault diagnosis for a class of interconnected large-scale systems

In this paper we focus on the distributed cyber attack detection and physical fault diagnosis problem for a class of interconnected large-scale systems (ILSSs). In the proposed scheme, apart from node measurement, edge measurement is also used to construct distributed Kalman filter to estimate the state of each subsystem. The gain matrices of Kalman filter are determined by minimizing the covariance of estimation error in the attack-free and fault-free case, which reduces the false alarm rate of cyber attack detection and physical fault diagnosis. Based on this filter, a bank of adjacent residual generators is constructed to characterize the influence of cyber attack on the edge measurement, and the Chi-square test is used to detect whether the received edge measurements are attacked. At the same time, a local residual generator is constructed for each subsystem to characterize the influence of physical faults on it, and the residual signal is evaluated by variance and directional residual, so as to make distributed fault detection and isolation of each subsystem. It is worth noting that at each step, each subsystem first performs attack detection on the received edge measurements, and then estimates its own state using the attack-free edge measurements and node measurement, which further improves the accuracy of fault detection and isolation. In addition, a sufficient condition that ensuring the mean square exponential boundedness of the estimation error is given. Finally, the proposed scheme is verified by an illustrative example.

Greedy selection of sensors with measurements under correlated noise

We address the sensor selection problem where linear measurements under correlated noise are gathered at the selected nodes to estimate the unknown parameter. Since finding the best subset of sensor nodes that minimizes the estimation error requires a prohibitive computational cost especially for a large number of nodes, we propose a greedy selection algorithm that uses the log-determinant of the inverse estimation error covariance matrix as the metric to be maximized. We further manipulate the metric by employing the QR and LU factorizations to derive a simple analytic rule which enables an efficient selection of one node at each iteration in a greedy manner. We also make a complexity analysis of the proposed algorithm and compare with different selection methods, leading to a competitive complexity of the proposed algorithm. For performance evaluation, we conduct numerical experiments using randomly generated measurements under correlated noise and demonstrate that the proposed algorithm achieves a good estimation accuracy with a reasonable selection complexity as compared with the previous novel selection methods.

Optimal design and performance analysis of secure estimator for cyber‐physical systems against deception attacks

AbstractCyber‐physical systems (CPSs) are increasingly vulnerable to the threat from malicious adversaries owing to the deep integration of networks and facilities; therefore, CPSs' security against cyber attacks has attracted extensive attention. The deception attack is one of the typical cyber attacks targeting CPSs, and a deception attack mode is established to describe the simultaneous occurrence of actuator and sensor attacks in this paper. Compared with single‐target deception attacks with constant value, the designed attack mode that can describe deception attacks against multiple components and embedded random characteristics has higher applicability in practical engineering processes. To counter such attack behavior, a secure estimator is proposed based on the ideology of augmented state to achieve the optimal design of secure state estimation and deception attack identification. Taking the energy shortage problem of CPSs into account, a novel event‐triggered communication mechanism is introduced on the basis of measurement residuals to improve resource efficiency by reducing low‐value information transmission. The communication scheme and the augmented state estimator are co‐designed by adopting the implicit information of the event‐triggered mechanism. Then, the performance of the co‐designed estimator is analyzed by the upper bound of the estimation error covariance matrix. Besides, the reconciliation between estimation performance and communication energy consumption is achieved by the derivation of Pareto equilibrium. Finally, applications of our approaches to a 3D target tracking system are provided to illustrate the effectiveness of the proposed method.