Remote State Estimation Research Articles

Remote State Estimation With Enhanced Robustness in the Presence of Data Packet Dropouts

This article discusses the remote state estimation over unreliable links, where the packet dropouts occur from the sensor side to the filter, for discrete-time systems with both bounded-power disturbances and white Gaussian noises. A cascaded estimation scheme with enhanced robustness is proposed, driven by the residual signal related to the modeling mismatch. The estimation gains are characterized by two modified algebraic Riccati equations (MAREs), together with a complete and rigorous stability analysis in the mean square (MS) sense. Necessary and sufficient conditions for the MS stability of the corresponding error dynamics are then given in terms of the data arrival rate and unstable poles of the plant, i.e., the Mahler measure of the plant. Moreover, the filtering strategy is expanded into the case of distributed estimation over lossy sensor networks, where each sensor locally constructs an estimate based on its own observation and on those collected from its neighbors, and the solution is again derived by MAREs. The corresponding necessary and sufficient conditions to the MS stability in the distributed case are also characterized by relationships between data arrival rates and the Mahler measure of the plant. Finally, an example is presented to validate the current design method.

Optimal deception attacks on remote state estimators equipped with interval anomaly detectors

This paper studies the problem of optimal deception attacks on remote state estimation, where an interval χ2 detector is deployed to reveal anomalies. The information-based attack policy that can bypass the anomaly detector and cause the maximum estimation quality degradation is derived. For both attacks with strict and relaxed stealthiness, the optimal compromised measurements can be designed with three steps: obtain the minimum mean-square error estimation of the prediction error, de-correlate the estimate with historical compromised innovations, and design the compromised innovation as an optimal linear transformation. All available information for attackers is fully utilized for performance maximization while the stealthiness constraint is satisfied precisely to deceive the anomaly detector. The attack effect depends on both the amount of online information and the duration of detection interval. Contrary to well-studied innovation-based attacks using static linear combinations, the information-based deception policy is shown to be generated by a linear time-varying system, whose coefficients can be completely determined offline. The optimality of the proposed attack is verified with numerical examples and comparative studies.

An approximate minimum mean-square error estimator for linear discrete time-varying systems: Handling Try-Once-Discard protocol

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying stochastic systems under communication constraints. In order to mitigate the undesired phenomena of data collisions, a Try-Once-Discard (TOD) protocol is utilized to regulate the signal transmissions over the sensor-to-estimator communication channel, where the TOD protocol is based on the scheduling rule described by a specific switching function. Under the introduced TOD protocol, the approximate minimum mean-square error (MMSE) state estimation problem is investigated and a recursive algorithm is developed for the MMSE estimator design whose computational complexity is comparable to that of the conventional Kalman filter. Furthermore, some conditions are established for the boundedness of the estimation error covariance. A numerical example is presented to illustrate the effectiveness of the proposed MMSE estimator.

Worst-Case Stealthy Innovation-Based Linear Attacks on Remote State Estimation Under Kullback–Leibler Divergence

With the wide application of cyber-physical systems, stealthy attacks on remote state estimation have attracted increasing research attention. Recently, various stealthy innovation-based linear attack models were proposed, in which the relaxed stealthiness constraint was based on the Kullback–Leibler divergence. This article studies existing innovation-based linear attack strategies with relaxed stealthiness and concludes that all of them provided merely suboptimal solutions. The main reason is some oversight in solving the involved optimization problems: some covariance constraints were not perfectly handled. This article provides the corresponding optimal solutions for those stealthy attacks. Both one-step and holistic optimizations of stealthy attacks are studied, and the worst-case attacks with and without zero-mean constraints are derived analytically, without the necessity to numerically solve semidefinite programming problems.

Transmission schedule for jointly optimizing remote state estimation and wireless sensor network lifetime

This paper studies scheduling multiple sensors for jointly optimizing remote state estimation and wireless sensor network lifetime. Each sensor in the network sends its packet to the remote center for further estimation over wireless communication channels. Sensors in the network will inevitably run out of battery power due to transmission consumption and then reach the end of sensor network lifetime. This article focuses on the problem how to schedule multiple sensors minimizing the total remote estimation errors and simultaneously prolonging the sensor network lifetime. The problem is formulated as a special class of Markov decision process (MDP), stochastic shortest path (SSP) problem. Under this framework, we first derive a set of conditions to guarantee the optimality of structural policies in SSP. Subsequently, threshold structure of the optimal scheduling scheme is obtained by verifying the conditions. Moreover, two Value-of-Information-based multi-sensor scheduling algorithms and a linear-architecture-based learning algorithm are designed to tackle the curse of dimensionality and unknown channel statistics in large-scale sensor networks. Numerical simulations are given to validate effectiveness of our results.

Security Investment in Cyber-Physical Systems: Stochastic Games With Asymmetric Information and Resource-Constrained Players

This article considers remote state estimation in cyber-physical systems (CPSs) with multiple sensors. Each plant is modeled by a discrete-time stochastic linear system with measurements of each sensor transmitted to the corresponding remote estimator over a shared communication network when their securities are interdependent due to network-induced risks. A dynamic nonzero-sum game with asymmetric information is formulated in which each sensor subject to a resource budget constraint needs to decide whether to invest in security for sending data packets, taking the behaviors of other sensors into account. To overcome the difficulty in characterizing or computing the Nash equilibria (NE), the game with asymmetric information is transformed into another game with symmetric information such that the equilibrium of the original game can be obtained by solving the equilibrium of the new game. Under certain conditions, we devise a backward induction algorithm to obtain a subclass of NE of the original game, known as common information-based Markov perfect equilibria (CIBMPE). Finally, a numerical example is provided to illustrate the results obtained.

Coding for Secrecy in Remote State Estimation With an Adversary

We study the problem of remote state estimation in the presence of an eavesdropper. A sensor transmits state information over a packet-dropping link to a legitimate user. This information is randomly overheard by an eavesdropper. To reduce information leakage to the eavesdropper, previous studies have shown that by encoding the estimate with the acknowledgments (Acks), perfect secrecy can be achieved. However, this strategy greatly relies on the accuracy of the Acks and may easily fail if the Acks are compromised by cyberattacks. In this article, we tackle this issue by proposing to switch between sending an encoded state and sending the plain state to stay resilient against fake Acks. Specifically, we assume the Acks to be randomly attacked and derive recursive expressions for the minimum-mean-squared error estimates and error covariance matrices at the legitimate user and at the eavesdropper. Based upon this, we propose a transmission policy that depends on the probability of synchronization. We formulate a partially observable Markov decision process to model the evolution of the synchronization status and derive associated optimal transmission policies. Numerical examples are provided to verify the theoretical results.

Optimal linear attack for multi-sensor network against state estimation

This paper is concerned with a security problem about malicious integrity attacks in state estimation system, in which multiple smart sensors locally measure information and transmit it to a remote fusion estimator though wireless channels. A joint constraint is considered for the attacker behaviour in each channel to keep stealthiness under a residual-based detector on the remote side. In order to degrade the estimator performance, the attacker will maximize the trace of the remote state estimation error covariance which is derived based on Kalman filter theory. It is proved that the optimal linear attack strategy design problem is convex and finally turned into a semi-definite programming problem. In addition, the tendency of attack behaviour on recursive and fixed Kalman filter system is analyzed. Several examples are given to illustrate the theoretical results.

Remote State Estimation of Nonlinear Systems Over Fading Channels via Recurrent Neural Networks

In this article, we consider the remote state estimation for nonlinear dynamic systems with known linear dynamics and unknown nonlinear perturbations. The nonlinear dynamic plant is monitored by multiple distributed sensors over a random access wireless network with shared common radio channel. We focus on the communication strategy and remote state estimation algorithm design so as to achieve a remote state estimation stability subject to unknown nonlinearities in plant and various wireless impairments, such as multisensor interference, wireless fading, and additive channel noise. By exploiting the additive properties of the physical wireless channels, we propose a novel information fusion over-the-air mechanism to address the signal collision and interference among the sensors. Utilizing the partial knowledge on the linear dynamics of the plant, we also propose a novel recurrent neural network (RNN)-based remote state estimator aided by a virtual state estimation mean-square-error (MSE) process. We further propose a novel online training algorithm such that the RNN at the remote estimator can effectively learn the unknown plant nonlinearities. Using the Lyapunov drift analysis approach, we establish closed-form sufficient requirements on the communication resources needed to achieve almost sure stability of both state estimation and RNN online training in high signal-to-noise ratio (SNR) regime. As a result, our proposed scheme is asymptomatic optimal for large SNR in the sense that both the plant state and the unknown plant nonlinearities can be perfectly recovered at the remote estimator. The proposed scheme is also compared with various baselines and we show that significant performance gains can be achieved.

Stochastic detection against deception attacks in CPS: Performance evaluation and game-theoretic analysis

In this paper, we consider a security problem of remote state estimation in cyber–physical systems (CPS). The state measurements of a dynamic process, obtained by wireless sensors, may be modified deceptively by a malicious attacker, which may bypass a standard χ2 detector. On the other hand, the estimation performance with a standard χ2 detector typically is also difficult to analyze. As such, a stochastic detection mechanism with a varying triggering threshold is proposed. We provide an analytical performance evaluation for the estimation with the proposed detector, and propose a potential countermeasure to actively defend the deception attacks with the proposed detector. The interactive decision-making process between the system with the countermeasure and the attacker is then studied under a game-theoretic framework. In addition, we also extend the stochastic false data detection problem into a multi-sensor setting with a sequential data fusion process and more general attacking patterns, and analyze the estimation performance. Simulations are provided to illustrate the developed results.

Rollout approach to sensor scheduling for remote state estimation under integrity attack

We consider the sensor scheduling problem for remote state estimation under integrity attacks. We seek to optimize a trade-off between the energy consumption of communications and the state estimation error covariance when the acknowledgment (ACK) information, sent by the remote estimator to the local sensor, is compromised. The sensor scheduling problem is formulated as an infinite horizon discounted optimal control problem with infinite states. We first analyze the underlying Markov decision process (MDP) and show that the optimal scheduling without ACK attack is of the threshold type. Thus, we can simplify the problem by replacing the original state space with a finite state space. For the simplified MDP, when the ACK is under attack, the problem is modeled as a partially observable Markov decision process (POMDP). We analyze the induced MDP that uses a belief vector as its state for the POMDP. We investigate the properties of the exact optimal solution via contractive models and show that the threshold type of solution for the POMDP cannot be readily obtained. A suboptimal solution is then obtained via a rollout approach, which is a prominent class of reinforcement learning (RL) methods based on approximation in value space. We present two variants of rollout and provide performance bounds of those variants. Finally, numerical examples are used to demonstrate the effectiveness of the proposed rollout methods by comparing them with a finite history window approach that is widely used in RL for POMDP.

Remote State Estimation With Smart Sensors Over Markov Fading Channels

We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated multistate Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance in terms of the state transition matrix of the LTI system, the packet drop probabilities in different channel states, and the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach and provide new elementwise bounds of matrix powers. The stability condition is verified by numerical results and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or nonconvex depending on the transition probability matrix of the Markov channel states. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator) though the smart sensor setup achieves a better estimation performance.

Optimal innovation-based deception attacks with side information against remote state estimation in cyber-physical systems

In this paper, the problem of designing the optimal deception attacks against the remote state estimation in cyber-physical systems is investigated in the scenario where the attacker has the ability to acquire the side information about the system by employing an extra sensor. To degenerate the remote estimation performance, the attacker replaces the transmitted signals with the combination of the innovations from the system sensor and the extra sensor. Under the proposed attacks, the recursion of the error covariance of the remote estimates is derived by utilizing the orthogonality principle and the statistical properties of the related random variables. Then, the design of the attacks is transformed into a quadratic multivariate optimization problem and the optimal distribution of the modified innovations is found by taking advantage of the Lagrange multiplier method. Based on this, it is proved that the optimal policy is the solution to a semi-definite programming (SDP) problem, which results in the largest estimation errors and maintains the deceptiveness concurrently. Finally, simulation examples are given to verify the effectiveness of this work.

Energy‐efficient DoS attack against remote state estimation

In this paper, the security issue in cyber‐physical system (CPS) is investigated. In order to maximize the attack effect and save the limited energy, the energy efficiency problem is investigated from the perspective of the jammer. This problem is established as a mixed‐integer programming problem firstly, and then a necessary condition is presented to explore the optimal energy for a given attack number. Moreover, an energy efficiency optimization (EEO) algorithm is proposed to derive the optimal attack schedule. Lastly, some numerical results are shown to validate the effectiveness of the proposed algorithm.

Event-triggered remote state estimation for cyber-physical systems under malicious DoS attacks

This paper focuses on the problem of event-triggered remote state estimation for cyber-physical systems (CPSs) under malicious denial-of-service (DoS) attacks. A remote estimator with intermittent observations is derived, and a novel event-triggered communication strategy is designed by transmitting observations to the remote estimator when the predicted estimation error covariance exceeds a given threshold. By constraining the upper bound of the total attack ratio, sufficient conditions to guarantee the almost sure stability of the proposed estimator are presented. Furthermore, the event-triggering parameter is designed to balance the estimation performance and the network resource utilization. In contrast to the previous studies on remote state estimation with random packet losses, the considered DoS attack case is more challenging since the probabilistic property of the packet loss process is unavailable to describe the attacks. Finally, the efficiency of the presented method is illustrated by simulation results.

Power Allocation for Remote Estimation Over Known and Unknown Gilbert-Elliott Channels

In this paper, we consider the problem of power scheduling of a sensor that transmits over a (possibly) unknown Gilbert-Elliott (GE) channel for remote state estimation. The sensor supports two power modes, namely low power, and high power. The scheduling policy determines when to use low power or high power for data transmission over a fading channel with temporal correlation while satisfying the energy constraints. Although error-free acknowledgement/negative-acknowledgement (ACK/NACK) signals are provided by the remote estimator, they only provide meaningful information about the underlying channel state when low power is utilized. This leads to a partially observable Markov decision process (POMDP) problem and we derive conditions that preserve the optimality of a stationary schedule derived for its fully observable counterpart. However, implementing this schedule requires knowledge of the parameters of the GE model which are not available in practice. To address this, we adopt a Bayesian framework to learn these parameters online and propose an algorithm that is shown to satisfy the energy constraint while achieving near-optimal performance via simulation.

Communication Scheduling by Deep Reinforcement Learning for Remote Traffic State Estimation With Bayesian Inference

Traffic awareness is the prerequisite of autonomous driving. Given the limitation of on-board sensors (e.g., precision and price), remote measurement from either infrastructure or other vehicles can improve traffic safety. However, the wireless communication carrying the measurement result undergoes fading, noise and interference and has a certain probability of outage. When the communication fails, the vehicle state can only be predicted by Bayesian filtering with a low precision. Higher communication resource utilization (e.g., transmission power) reduces the outage probability and hence results in an improved estimation precision. The power control subject to an estimate variance constraint is a difficult problem due to the complicated mapping from transmit power to vehicle-state estimate variance. In this paper, we develop an estimator consisting of several Kalman filters (KFs) or extended Kalman filters (EKFs) and an interacting multiple model (IMM) to estimate and predict the vehicle state. We propose to apply deep reinforcement learning (DRL) for the transmit power optimization. In particular, we consider an intersection and a lane-changing scenario and apply proximal policy optimization (PPO) and soft actor-critic (SAC) to train the DRL model. Testing results show satisfactory power control strategies confining estimate variances below given threshold. SAC achieves higher performance compared to PPO.

Jamming attack against remote state estimation over multiple wireless channels: A reinforcement learning based game theoretical approach

In this paper, we design a potential game to investigate the interplay between sensors and attackers in wireless remote estimation systems (RESs), where we consider both closed-loop and open-loop scenarios that depend on whether the sensor’s actions are known to the attacker or not. As a typical application of Cyber–Physical Systems (CPSs), wireless RESs have been showing vulnerable to channel jamming attacks. As a countermeasure, the wireless sensor in the estimation system can switch to another independent wireless channel to transmit sensor data. However, a smart attacker can also switch the target channel to launch a jamming attack, resulting in a game between the sensor and attacker. Moreover, the reinforcement learning based methods are proposed to obtain solutions of the designed game models in two scenarios. We also provide extensive simulation results to validate the effectiveness of the proposed methods.

DoS Attack Allocation Against Remote State Estimation Over Packet-Dropping Networks

We consider the security problem of remote state estimation in cyber-physical systems. The sensor sends data under round-robin protocol to the remote side while denial-of-service (DoS) attacker could interfere this process. Due to the energy-constrained DoS attack, it needs to decide whether to launch attacks to the wireless channel. In this work, two different cases for multi-sensor running Kalman filter are studied, i.e., sensors have limited or sufficient computational abilities. Furthermore, to capture the unreliability of the communication network, the occurrence of packet loss in the absence of attack is considered. Based on the evolution process of estimation error covariance, we first propose the sufficient condition to guarantee the optimality of the attack allocation strategy which maximally degrades the system performance. Then, the optimal attack sequence is given in a closed-form and shown to be monotonic with respect to time. Finally, a simulation example is given to illustrate the effectiveness of our proposed attack strategy.

Securing Remote State Estimation against Sequential Logic Attack of Sensor Data

The SCADA system, which is widely used in the continuous monitoring and control of the physical process of modern critical infrastructure, relies on the feedback control loop. The remote state estimation system triggers the control algorithm or control condition of the controller according to the monitoring data returned by the sensor. The controller sends the control command to the actuator, and the actuator executes the command to control the physical process. Since SCADA system monitoring and control data are usually transmitted through unprotected wireless communication networks, attackers can use false sensor data to trigger control algorithms to make wrong decisions, disrupt the physical processing of the SCADA system, and cause huge economic losses, even casualties. We found an attack strategy based on the sequential logic of sensor data. This kind of attack changes the time logic or sequence logic of the response data, so that the false data detector can be successfully deceived. This would cause the remote state estimation system to trigger wrong control algorithms or control conditions, and eventually disrupt or destroy the physical process. This paper proposes a sequential signature scheme based on the one-time signature to secure the sequential logic and transmission of sensor data. The security analysis proves that the proposed scheme can effectively resist counterfeiting, forgery, denial, replay attacks, and selective forwarding attacks.