Resilient State Estimation Research Articles

Mean-square exponential convergence for Byzantine-resilient distributed state estimation

This paper studies the problem of distributed resilient state estimation (RSE) for linear measurement models in the presence of locally-bounded Byzantine nodes that may arbitrarily deviate from the prescribed update rule. Necessary and sufficient conditions for the existence of a distributed algorithm to solve the RSE problem in the almost sure sense are characterized in term of the topology-associated robustly collective observability. Under these conditions, a distributed projected stochastic resilient filtering algorithm is proposed. Compared with the existing results where asymptotic or probabilistic finite-time analysis is established, the exponential convergence (in sense of mean square) of the proposed algorithm is proved. To further improve computational performance of the algorithm, an adaptive event-triggered mechanism is constructed without compromising its correctness of the estimate.

Zonotopic resilient state estimation for unmanned surface vehicles subject to integrity attacks

This work concerns the problem of resilient state estimation for unmanned surface vehicle systems subject to integrity attacks under peak-bounded perturbations. A novel resilient remote state estimation method is proposed based on zonotopic reachability analysis. The compromised sensors are identified based on the so-called predicted output set and zonotope support strip. Then, a resilient state reachable set is obtained by using the sensor data from the remaining benign measurements. For enhancing the performance of estimation, segment minimization is introduced to optimize the obtained state reachable set. Simulation results are provided to testify the effectiveness of the proposed method.

Distributed Resilient State Estimation for Cyber-Physical Systems Against Bit Errors: A Zonotopic Set-Membership Approach

This paper addresses the distributed resilient state estimation issue for a class of large-scale cyber-physical systems (CPSs) against bit errors. Owing to the reliability of the binary data, a novel binary encoding-decoding scheme is developed to enable digital communication of the CPSs. We consider the situation where the transmitted binary bits may be flipped over noisy communication channels. The primary purpose of this paper is to design a distributed resilient estimation scheme under a bit-flipped detection mechanism to achieve a reliable estimation of the CPSs. First, by means of the Minkowski sum of sets, a novel distributed zonotopic resilient estimation scheme is proposed where the optimal estimator gain is obtained in the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F$</tex-math></inline-formula> -radius sense. Furthermore, a two-stage bit-flipped detection scheme is developed by taking advantage of 1) the prediction output zonotope and 2) the intersection between the prediction zonotope and the estimation zonotope. Furthermore, in order to improve the algorithm efficiency, a reduction operator is introduced to reduce the order of the zonotope. Finally, the IEEE 30-bus systems are applied to verify theoretical results of the developed distributed resilient estimation algorithm.

Finite-time distributed resilient state estimation subject to hybrid cyber-attacks: a new dynamic event-triggered case

This paper is concerned with the issues of finite-time distributed resilient state estimation subject to hybrid cyber-attacks. The information exchanges among estimators are governed by an improved dynamic event-triggered mechanism, in which the time-varying threshold with predetermined upper and lower bounds is updated by artificial internal dynamics. With the help of the Lyapunov stability theory combined with the S-procedure, a sufficient condition is developed such that the augmented error dynamics are stochastic finite-time bounded. Furthermore, the desired estimator gains are acquired in terms of the solution to certain matrix inequalities which involve the information of communication topology, cyber-attack probabilities as well as the uncertainty of gain matrices. Finally, the effectiveness of the designed distributed state estimator is illustrated by a numerical example.

Observability Decomposition-Based Decentralized Kalman Filter and Its Application to Resilient State Estimation under Sensor Attacks.

This paper considers a discrete-time linear time invariant system in the presence of Gaussian disturbances/noises and sparse sensor attacks. First, we propose an optimal decentralized multi-sensor information fusion Kalman filter based on the observability decomposition when there is no sensor attack. The proposed decentralized Kalman filter deploys a bank of local observers who utilize their own single sensor information and generate the state estimate for the observable subspace. In the absence of an attack, the state estimate achieves the minimum variance, and the computational process does not suffer from the divergent error covariance matrix. Second, the decentralized Kalman filter method is applied in the presence of sparse sensor attacks as well as Gaussian disturbances/noises. Based on the redundant observability, an attack detection scheme by the test and a resilient state estimation algorithm by the maximum likelihood decision rule among multiple hypotheses, are presented. The secure state estimation algorithm finally produces a state estimate that is most likely to have minimum variance with an unbiased mean. Simulation results on a motor controlled multiple torsion system are provided to validate the effectiveness of the proposed algorithm.

Resilient Event-Triggered Distributed State Estimation for Nonlinear Systems Against DoS Attacks.

This article investigates the resilient event-triggered (ET) distributed state estimation problem for nonlinear systems under denial-of-service (DoS) attacks. Different from the existing results mainly considering linear or specified nonlinear systems, more general nonlinear systems are considered in this study. Moreover, the considered DoS attacks are able to compromise different communication links among estimators independently. In this context, by resorting to the techniques of incremental homogeneity, a nonlinear ET distributed estimation scheme is designed to estimate the states and regulate the data transmission. Under this scheme, the resilient state estimation is achieved by employing a multimode switching estimator, and the problem of efficiency loss of the ET mechanism caused by DoS attacks is solved by designing a dynamic trigger threshold with switched update laws. Then, based on the decay rates of the Lyapunov function corresponding to different communication modes, sufficient conditions are given to guarantee the stability of the estimation error system under DoS attacks. Finally, simulation results are provided to verify the effectiveness of the proposed method.

Dynamic event-triggered resilient state estimation for time-varying complex networks with Markovian switching topologies

This paper addresses a resilient state estimation problem for an array of nonlinear complex networks with switching topologies under the dynamic event-triggered mechanism (ETM). To reduce the unnecessary data delivery, the dynamic ETM is introduced to schedule the data delivery from sensors to estimators. The model of the switched complex networks is established by adopting a Markov chain which is better to reflect the characteristics of practical complex networks. A set of novel estimators is obtained by using the properties of Kronecker product combining with the Lyapunov–Krasovskii method, and some easy-to-check conditions are derived such that the dynamics of state estimation error satisfies the prescribed H∞ performance index. In addition, the parameters of the designed resilient state estimators can be acquired by solving a series of convex optimization problems. In the end, a simulation example is given to demonstrate the validity of the proposed theoretical results in this paper.

Encoding–decoding strategy based resilient state estimation for bias-corrupted stochastic nonlinear systems

This paper is concerned with the resilient state estimation problem for a type of stochastic nonlinear systems, in which the possible dynamical bias is considered that is depicted by a dynamical equation. In pursuit of enhancing the robustness of the propagated data, a binary encoding strategy (BES) is exploited in the binary symmetric channel (BSC). While the random bit errors caused by the channel noise may take place during the propagation of the binary bit string via the memoryless BSC. To characterize the occurrence of the bit errors, a series of Bernoulli distributed random variables is adopted. More specifically, in order to deal with the possible gain fluctuation of the estimator in the execution process, a resilient state estimator is employed. This paper intends to put forward a novel resilient estimation scheme under the BES, which can assure that the estimation error dynamics is exponentially ultimately bounded in mean square. A sufficient criterion is first acquired for the existence of the expected resilient estimator and the estimator parameter is achieved by solving a convex optimization problem. Finally, an illustrative simulation example is provided to verify the validity of the theoretical results.

A Robust and Resilient State Estimation for Linear Systems

This article is concerned with the state estimation of linear dynamic systems when some sensors are corrupted by attackers. This problem is known as resilient state estimation (RSE), and aims to achieve, under some conditions, the estimation of the true state despite the malicious attacks on sensors. The state-of-art RSE methods provides a bound on estimation errors when external disturbance exists. However, it is shown in this article that the effect of the disturbance on estimation error may be larger than that for conventional observers, or even worse, resiliency may be lost for the disturbance that exceeds the bound. To resolve this issue, unknown input observer (UIO) mechanism is adopted in RSE for the purpose of estimating true plant state under both sensor attacks and external disturbance. Also achieved in this work is the method of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">partial</i> state UIO synthesis, which relaxes the design requirement for full state UIO. In relation to resiliency, it is shown that 2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> redundant detectability is a necessary condition for robust and resilient state estimator in order to tolerate up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> sensor attacks. Numerical examples are given to validate the effectiveness of the proposed method.

Finite-horizon resilient state estimation for complex networks with integral measurements from partial nodes

Finite-horizon resilient state estimation for complex networks with integral measurements from partial nodes

Distributed resilient state estimation over sensor networks with random nonlinearities, fading measurements, and stochastic gain variations

AbstractThe distributed resilient state estimation problem of nonlinear discrete systems in sensor networks is investigated in this article. The system model under consideration involves three phenomena of incomplete information: randomly occurring nonlinearities, fading measurements, and random gain variations. The probabilistic characteristics of the above phenomena are depicted by three sets of independent random variables subject to more general random distribution. Based on the above model, by applying Lyapunov functional approach and random distribution solution method, the asymptotic stability in the mean square sense of the estimation error system with a given attenuation level is proved. Further, the estimator parameters are solved by introducing a novel linearization method. Finally, a numerical simulation is given to illustrate the validity of the theoretical results.

Distributed resilient state estimation for nonlinear systems with randomly occurring communication delays and missing measurements

SummaryThis article is concerned with the distributed H∞ resilient state estimation problem for a class of nonlinear systems with randomly occurring communication delays and missing measurements in sensor networks. A novel sensor model is proposed, in which two Bernoulli distributed white sequences are introduced to describe the random communication delay and missing measurements in a unified framework. Meanwhile, the estimator gain is allowed to fluctuate within a certain range. Based on the developed model, a novel Lyapunov–Krasovskii functional with multiple delay information terms is constructed, then the stochastic analysis technique and the extended integral inequality are used to calculate the functional derivative. Consequently, the existence conditions for the required distributed estimator are established to ensure that the estimation error system is asymptotically mean‐square stable and satisfies the prescribed H∞ performance constraint, and the desired gain of distributed resilient estimator is also solved by linearizing the nonlinear terms. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Resilient set-membership state estimation for nonlinear complex networks with time-delay and incomplete measurements

Taking the incomplete measurements and the weighted try-once-discard (WTOD) protocol into account, this paper develops a novel resilient set-membership state estimation (RSMSE) method for time-varying nonlinear complex networks with time-invariant delay. A classic interval matrix technique is utilized to describe incomplete measurements. The Taylor series expansion is applied to dispose the nonlinearities, where the high-order terms of the linearization errors are described by norm-bounded uncertainties. To mitigate the communication burden, the WTOD protocol is introduced, where only one node can send updated data through a shared communication network at each certain transmission step. Using the recursive linear matrix inequalities (RLMIs), a series of ellipsoidal sets including the state vector can be determined. The desirable estimator gain and a smallest possible estimation ellipsoid can be calculated via solving the convex optimization problem. Lastly, we use an illustrative example to show the feasibility of the introduced RSMSE technique.

Fully Distributed Resilient State Estimation Based on Distributed Median Solver

In this paper, we present a scheme of fully distributed resilient state estimation for linear dynamical systems under sensor attacks. The proposed state observer consists of a network of local observers, where each of them utilizes local measurements and information transmitted from the neighbors. As a fully distributed scheme, it does not necessarily collect a majority of sensing data for the sake of attack identification, while the compromised sensors are eventually identified by the distributed network and excluded from the observers. For this, the overall network (not the individual local observer) is assumed to have redundant sensors and assumed to be connected. The proposed scheme is based on a novel design of a distributed median solver, which approximately recovers the median value of local estimates.

Resilient state estimation for time‐varying uncertain dynamical networks with data packet dropouts and switching topology: an event‐triggered method

In this study, the recursive state estimation method based on the event-triggered protocol is given for time-varying uncertain complex networks with switching topology (ST), data packet dropouts (DPDs), and randomly occurring non-linearities (RONs). The event-triggered strategy is given to regulate the communications and then save the limited network resources when the measurement information is transmitted. The phenomena of RONs, ST, and DPDs are depicted by some random variables obeying the Bernoulli distribution. Besides, the estimator gain perturbations are dealt with. The major aim of this study is to present a new state estimation algorithm for addressed time-varying uncertain complex networks such that, for all STs, DPDs, and RONs, a minimal upper bound of state estimation error covariance matrix is derived by designing an optimal estimator gain matrix. Finally, a numerical simulation is presented to show the effectiveness of the provided time-varying state estimation method.

Second-Order Fault Tolerant Extended Kalman Filter for Discrete Time Nonlinear Systems

As missing sensor data may severely degrade the overall system performance and stability, reliable state estimation is of great importance in modern data-intensive control, computing, and power systems applications. Aiming at providing a more robust and resilient state estimation technique, this paper presents a novel second-order fault-tolerant extended Kalman filter estimation framework for discrete-time stochastic nonlinear systems under sensor failures, bounded observer-gain perturbation, extraneous noise, and external disturbances condition. The failure mechanism of multiple sensors is assumed to be independent of each other with various malfunction rates. The proposed approach is a locally unbiased, minimum estimation error covariance based nonlinear observer designed for dynamic state estimation under these conditions. It has been successfully applied to a benchmark target-trajectory tracking application. Computer simulation studies have demonstrated that the proposed second-order fault-tolerant extended Kalman filter provides more accurate estimation results, in comparison with traditional first- and second-order extended Kalman filter. Experimental results have demonstrated that the proposed second-order fault-tolerant extended Kalman filter can serve as a powerful alternative to the existing nonlinear estimation approaches.

Detection of Sensor Attack and Resilient State Estimation for Uniformly Observable Nonlinear Systems having Redundant Sensors

This paper presents a detection algorithm for sensor attacks and a resilient state estimation scheme for a class of uniformly observable nonlinear systems. An adversary is supposed to corrupt a subset of sensors with the possibly unbounded signals, while the system has sensor redundancy. We design an individual high-gain observer for each measurement output so that only the observable portion of the system state is obtained. Then, a nonlinear error correcting problem is solved by collecting all the information from those partial observers and exploiting redundancy. A computationally efficient, on-line monitoring scheme is presented for attack detection. Based on the attack detection scheme, an algorithm for resilient state estimation is provided. The simulation results demonstrate the effectiveness of the proposed algorithm.

Resilient state estimation for nonlinear complex networks with time-delay under stochastic communication protocol

In this paper, the resilient state estimation problem is discussed for a class of discrete nonlinear complex networks with time-delay subject to the stochastic communication protocol (SCP). The SCP is employed to determine which node has the priority of accessing to the communication networks at each transmission instant. A switch-based approach is utilized to establish the updating rule of the measurement output under the SCP. The aim of the paper is to design a resilient state estimator for the addressed complex networks with time-delay and SCP, where sufficient conditions are given to ensure that the estimation error is exponentially ultimate bounded in the mean-square sense and the asymptotic upper bound of the estimation error is given. Moreover, the expression forms of the desired estimator gains are obtained via the solutions to a set of linear matrix inequalities. Finally, a numerical example is provided to illustrate the validity of the proposed resilient state estimation method.

Towards Resilient UAV: Escape Time in GPS Denied Environment with Sensor Drift

Abstract This paper considers a resilient state estimation framework for unmanned aerial vehicles (UAVs) that integrates a Kalman filter-like state estimator and an attack detector. When an attack is detected, the state estimator uses only IMU signals as the GPS signals do not contain legitimate information. This limited sensor availability induces a sensor drift problem questioning the reliability of the sensor estimates. We propose a new resilience measure, escape time, as the safe time within which the estimation errors remain in a tolerable region with high probability. This paper analyzes the stability of the proposed resilient estimation framework and quantifies a lower bound for the escape time. Moreover, simulations of the UAV model demonstrate the performance of the proposed framework and provide analytical results.

Extended dissipative resilient estimator design for discrete-time switched neural networks with unreliable links

This paper addresses the problem of extended dissipativity-based resilient state estimation for discrete-time switched neural networks in the presence of unreliable links. To overcome the difficulty of random fluctuation and communication constraints, a novel and highly flexible form of estimator is designed by considering random uncertainty, signal quantization and data packet dropout phenomena simultaneously. The considered mode-dependent average dwell time (MDADT) switching law is shown to be more general than the traditional ADT for permitting each subsystem to has its own average dwell time. By constructing the MD Lyapunov–Krasovaskii functional (LKF), sufficient conditions are given to ensure the exponential mean-square stability and extended stochastic dissipativity of the augmented system and an explicit expression of the desired estimator is presented. Finally, a example with two cases is provide to illustrate the feasibility and effectiveness of the developed theoretical results.