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In this chapter, we study a remote state estimation problem in the presence of an eavesdropper. A sensor transmits local state estimates over a packet dropping link to a remote estimator, while an eavesdropper can successfully overhear each sensor transmission with a certain probability. The objective is to determine when the sensor should transmit, in order to minimize the estimation error covariance at the remote estimator, while trying to keep the eavesdropper error covariance above a certain level. Structural results on the optimal transmission policy are derived, and shown to exhibit thresholding behavior in the estimation error covariances. In the infinite horizon situation, it is shown that with unstable systems one can keep the expected estimation error covariance bounded while the expected eavesdropper error covariance becomes unbounded, for all eavesdropping probabilities strictly less than one. An alternative measure of security, constraining the amount of information revealed to the eavesdropper, is also considered, and similar structural results on the optimal transmission policy are derived. In the infinite horizon situation with unstable systems, it is now shown that for any transmission policy (within the class of stationary deterministic policies where the sensor at each time step can either transmit or not) which keeps the expected estimation error covariance bounded, the expected amount of information revealed to the eavesdropper is always lower bounded away from zero.

We consider remote state estimation in the presence of an active eavesdropper. A sensor forward local state estimates to a remote estimator over a network, which may be eavesdropped by an intelligent adversary. Aiming at improving the eavesdropping performance efficiently, the adversary may adaptively alternate between an eavesdropping and an active mode. In contrast to eavesdropping, the active attack enables the adversary to sabotage the data transfer to the estimator, and improve the data reception to itself at the same time. However, launching active attacks may increase the risk of being detected. As a result, a tradeoff between eavesdropping performance and stealthiness arises. We present a generalized framework for active eavesdropping and propose a criterion based on the packet reception rate at the estimator to evaluate the stealthiness of the eavesdropper. Moreover, the tradeoff is formulated as a constrained Markov decision process. After deriving a sufficient condition under which at least one stationary policy satisfies the stealthiness constraint and also bounds the eavesdropping performance, we develop an optimal attack policy for the eavesdropper and focus on the structural analysis of the optimal policy. Furthermore, numerical examples are provided to illustrate the developed results.

This paper considers a game-theoretic formulation of the covert communications problem with finite blocklength, where the transmitter (Alice) can randomly vary her transmit power in different blocks, while the warden (Willie) can randomly vary his detection threshold in different blocks. In this two-player game, the payoff for Alice is a combination of the coding rate to the receiver (Bob) and the detection error probability at Willie, while the payoff for Willie is the negative of his detection error probability. Nash equilibrium solutions to the game are obtained and shown to be efficiently computable using linear programming. For less covert requirements, our game-theoretic approach can achieve significantly higher coding rates than uniformly distributed transmit powers. We then consider the situation with an additional jammer, where Alice and the jammer can both vary their powers and jointly comprise one player, with Willie as the other player. The use of a jammer is shown in numerical simulations to lead to further significant performance improvements.

We consider a remote state estimation problem in the presence of an eavesdropper over packet dropping links. A smart sensor transmits its local estimates to a legitimate remote estimator, in the course of which an eavesdropper can randomly overhear the transmission. This problem has been well studied for unstable dynamical systems, but seldom for stable systems. In this article, we target at stable and marginally stable systems and aim to design an event-triggered scheduling strategy by minimizing the expected error covariance at the remote estimator and keeping that at the eavesdropper above a user-specified lower bound. To this end, we model the evolution of the error covariance as an infinite recurrent Markov chain and develop a recurrence relation to describe the stationary distribution of the state at the eavesdropper. Monotonicity and convergence properties of the expected error covariance are further investigated and employed to solve the optimization problem. Numerical examples are provided to validate the theoretical results.

This paper studies remote state estimation in the presence of an eavesdropper. A sensor transmits local state estimates over a packet dropping link to a remote estimator, while an eavesdropper can successfully overhear each sensor transmission with a certain probability. The objective is to determine when the sensor should transmit, in order to minimize the estimation error covariance at the remote estimator, while trying to keep the eavesdropper error covariance above a certain level. This is done by solving an optimization problem that minimizes a linear combination of the expected estimation error covariance and the negative of the expected eavesdropper error covariance. Structural results on the optimal transmission policy are derived, and shown to exhibit thresholding behaviour in the estimation error covariances. In the infinite horizon situation, it is shown that with unstable systems one can keep the expected estimation error covariance bounded while the expected eavesdropper error covariance becomes unbounded. An alternative measure of security, constraining the amount of information revealed to the eavesdropper, is also considered, and similar structural results on the optimal transmission policy are derived. In the infinite horizon situation with unstable systems, it is now shown that for any transmission policy which keeps the expected estimation error covariance bounded, the expected amount of information revealed to the eavesdropper is always lower bounded away from zero. An extension of our results to the transmission of measurements is also presented.

We consider the communication channel pricing and selection problem in a networked control system. To encompass the sequentialized nature of the decision-making process, we use game theory and formulate a Stackelberg game framework, where the server first determines the channel pricing strategy, and the clients then make channel selection decisions. Both single-server-single-client (SSSC) scenario and single-server-multi-client (SSMC) scenario are discussed. The existence of an optimal stationary and deterministic policy for the clients is proved. We show that for the SSSC scenario, the server's optimal pricing strategy in terms of maximizing revenue is to ensure that the client uses the good channel all the time. For the SSMC scenario, it is assumed that the channel price remains invariant. As a consequence, each client has an optimal policy with threshold structure. Some properties of the optimal policy pair for both scenarios are obtained. Simulation results confirm the structure and properties of both the server and clients' optimal strategies.

Remote state estimation problems in the presence of eavesdroppers have recently been investigated in the literature. For unstable systems, it has been shown that it is possible to keep the expected estimation error covariance bounded, while the expected eavesdropper error covariance becomes unbounded in the infinite horizon. In this note, we consider an alternative notion of security based on the amount of information revealed to the eavesdropper. Upper and lower bounds on the information revealed are derived. In particular, in the infinite horizon, it is shown that with unstable systems, any transmission policy (within the class of stationary deterministic policies where the sensor at each time step can either transmit its local state estimate or not) which keeps the expected estimation error covariance bounded must always reveal a non-zero expected amount of information to the eavesdropper.

This paper studies transmission scheduling for remote state estimation in the presence of an eavesdropper. A sensor transmits local state estimates over a packet dropping link to a remote estimator, while an eavesdropper can successfully overhear each sensor transmission with a certain probability. The objective is to determine when the sensor should transmit, in order to minimize the estimation error covariance at the remote estimator, while trying to keep the eavesdropper error covariance above a certain level. This is done by solving an optimization problem that minimizes a linear combination of the expected estimation error covariance and the negative of the expected eavesdropper error covariance. Structural results on the optimal transmission policy are derived, and shown to exhibit thresholding behavior in the estimation error covariances. In the infinite horizon situation, it is shown that with unstable systems one can keep the expected estimation error covariance bounded while the expected eavesdropper error covariance becomes unbounded, for all eavesdropping probabilities strictly less than one.

This paper studies the optimal output-feedback control of a linear time-invariant system where a stochastic event-based scheduler triggers the communication between the sensor and the controller. The primary goal of the use of this type of scheduling strategy is to provide significant reductions in the usage of the sensor-to-controller communication and, in turn, improve energy expenditure in the network. In this paper, we aim to design an admissible control policy, which is a function of the observed output, to minimize a quadratic cost function while employing a stochastic event-triggered scheduler that preserves the Gaussian property of the plant state and the estimation error. For the infinite horizon case, we present analytical expressions that quantify the tradeoff between the communication cost and control performance of such event-triggered control systems. This tradeoff is confirmed quantitatively via numerical examples. Besides, numerical simulations justify that the event-triggered control provides better quadratic control performance than the (traditional) periodic time-triggered control at the same average sampling rate.

We consider pricing and selection with fading channels in a Stackelberg game framework. A channel server decides the channel prices and a client chooses which channel to use based on the remote estimation quality. We prove the existence of an optimal deterministic and Markovian policy for the client, and show that the optimal policies of both the server and the client have threshold structures when the time horizon is finite. Value iteration algorithm is applied to obtain the optimal solutions for both the server and client, and numerical simulations and examples are given to demonstrate the developed result.

The problem of remote state estimation in the presence of eavesdroppers has recently been investigated in the literature. For unstable systems it has been shown that one can keep the expected estimation error covariance bounded, while the expected eavesdropper error covariance becomes unbounded in the infinite horizon, using schemes based on transmission scheduling. In this paper we consider an alternative approach to achieve security, namely injecting noise into sensor transmissions, similar to the artificial noise technique used in physical layer security for wireless communications. Numerical results demonstrate significant performance improvements using this approach, with respect to the trade-off between the expected estimation error covariance and expected eavesdropper covariance.

This brief introduces wireless communications ideas and techniques into the study of networked control systems. It focuses on state estimation problems in which sensor measurements (or related quantities) are transmitted over wireless links to a central observer. Wireless communications techniques are used for energy resource management in order to improve the performance of the estimator when transmission occurs over packet dropping links, taking energy use into account explicitly in Kalman filtering and control. The brief allows a reduction in the conservatism of control designs by taking advantage of the assumed. The brief shows how energy-harvesting-based rechargeable batteries or storage devices can offer significant advantages in the deployment of large-scale wireless sensor and actuator networks by avoiding the cost-prohibitive task of battery replacement and allowing self-sustaining sensor to be operation. In contrast with research on energy harvesting largely focused on resource allocation for wireless communication systems design, this brief optimizes estimation objectives such as minimizing the expected estimation error covariance. The resulting power control problems are often stochastic control problems which take into account both system and channel dynamics. The authors show how to pose and solve such design problems using dynamic programming techniques. Researchers and graduate students studying networked control systems will find this brief a helpful source of new ideas and research approaches.

This paper studies a remote state estimation problem where a sensor, equipped with energy harvesting capabilities, observes a dynamical process and transmits local state estimates over a packet dropping channel to a remote estimator. The objective is to decide, at every discrete time instant, whether the sensor should transmit or not, in order to minimize the expected estimation error covariance at the remote estimator over a finite horizon, subject to constraints on the sensor’s battery energy governed by an energy harvesting process. We establish structural results on the optimal scheduling which show that, for a given battery energy level and a given harvested energy, the optimal policy is a threshold policy on the error covariance. Similarly, for a given error covariance and a given harvested energy, the optimal policy is a threshold policy on the current battery level. An extension to the problem of transmission scheduling and control with an energy harvesting sensor is also considered.

In this paper, we investigate a class of distortion outage minimization problems for a wireless sensor network in the presence of an eavesdropper. The observation signals transmitted from the sensors to the fusion center (FC) are overheard by the eavesdropper. Both the FC and the eavesdropper reconstruct minimum mean squared error estimates of the physical quantity observed. We address the problem of transmit power allocation to minimize the distortion outage at the FC, subject to an expected total transmit power constraint across the sensor(s) and a secrecy outage constraint at the eavesdropper. Applying a rigorous probabilistic power allocation technique, we derive power policies for the full channel state information (CSI) case. Suboptimal power control policies are studied for the partial CSI case in order to reduce the high computational cost associated with large numbers of sensors or receive antennas. Numerical results show that significantly improved performance can be achieved by adding multiple receive antennas at the FC. In the case of multiple transmit antennas, the distortion outage at the FC can be dramatically reduced and in some cases completely eliminated, by transmitting the observations on the null space of the eavesdropper's channel or deploying an artificial noise technique, in the full CSI and partial CSI cases, respectively.

This paper studies a joint transmission scheduling and controller design problem, which minimizes a linear combination of the control cost and expected energy usage of the sensor. Assuming that the sensor transmission decisions are event-based and determined using the random estimation error covariance information available to the controller, we show a separation in the design of the transmission scheduler and controller. The optimal controller is given as the solution to an LQG-type problem, while the optimal transmission policy is a threshold policy on the estimation error covariance at the controller.

Remote state estimation problems in the presence of an eavesdropper have recently been studied. In this setup, a sensor transmits over a random packet dropping link to a remote estimator, which at the same time can be randomly overheard by an eavesdropper. For i.i.d. packet dropping links to the remote estimator and to the eavesdropper, it has been shown that with unstable systems one can keep the expected estimation error covariance bounded, while the expected eavesdropper error covariance becomes unbounded in the infinite horizon. In this paper we show that the same behaviour can be achieved when transmission of local state estimates occur over a Markovian packet dropping link, and eavesdropping occurs according to another Markovian packet dropping link.

This paper considers a remote state estimation problem with multiple sensors observing a dynamical process, where sensors transmit local state estimates over an independent and identically distributed (i.i.d.) packet dropping channel to a remote estimator. At every discrete time instant, the remote estimator decides whether each sensor should transmit or not, with each sensor transmission incurring a fixed energy cost. The channel is shared such that collisions will occur if more than one sensor transmits at a time. Performance is quantified via an optimization problem that minimizes a convex combination of the expected estimation error covariance at the remote estimator and expected energy usage across the sensors. For transmission schedules dependent only on the estimation error covariance at the remote estimator, this work establishes structural results on the optimal scheduling which show that: 1) for unstable systems, if the error covariance is large then a sensor will always be scheduled to transmit and 2) there is a threshold-type behavior in switching from one sensor transmitting to another. Specializing to the single sensor case, these structural results demonstrate that a threshold policy (i.e., transmit if the error covariance exceeds a certain threshold and don't transmit otherwise) is optimal. We also consider the situation where sensors transmit measurements instead of state estimates, and establish structural results including the optimality of threshold policies for the single sensor, scalar case. These results provide a theoretical justification for the use of such threshold policies in variance based event triggered estimation. Numerical studies confirm the qualitative behavior predicted by our structural results.

In this paper, we investigate the performance of distributed estimation schemes in a wireless sensor network in the presence of an eavesdropper. The sensors transmit observations to the fusion center (FC), which at the same time are overheard by the eavesdropper. Both the FC and the eavesdropper reconstruct a minimum mean-squared error estimate of the physical quantity observed. We address the problem of transmit power allocation for system performance optimization subject to a total average power constraint on the sensor(s), and a security/secrecy constraint on the eavesdropper. We mainly focus on two scenarios: 1) a single sensor with multiple transmit antennas and 2) multiple sensors with each sensor having a single transmit antenna. For each scenario, given perfect channel state information (CSI) of the FC and full or partial CSI of the eavesdropper, we derive the transmission policies for short-term and long-term cases. For the long-term power allocation case, when the sensor is equipped with multiple antennas, we can achieve zero information leakage in the full CSI case, and dramatically enhance the system performance by deploying the artificial noise technique for the partial CSI case. Asymptotic expressions are derived for the long-term distortion at the FC as the number of sensors or the number of antennas becomes large. In addition, we also consider multiple-sensor multiple-antenna scenario, and simulations show that given the same total number of transmitting antennas the multiple-antenna sensor network is superior to the performance of the multiple-sensor single-antenna network.

We study transmission power budget minimization of battery-powered nodes in a remote state estimation problem over multi-hop wireless networks. Communication links between nodes are subject to fading, thereby generating random dropouts. Relay nodes help to transmit measurements from distributed sensors to an estimator node. Hopping through each relay node introduces a unit delay. Motivated by the need for estimators with low computational and implementation cost, we propose a jump estimator whose modes depend on a Markovian parameter that describes measurement transmission outcomes over a finite interval. It is well known that transmission power helps to increase the reliability of measurement transmissions, at the expense of reducing the life-time of the nodes’ battery. Motivated by this, we derive a tractable iterative procedure, based on semi-definite programming, to design a finite set of filter gains, and associated power control laws to minimize the energy budget while guaranteeing an estimation performance level. This procedure allows us to tradeoff the complexity of the filter implementation with performance and energy use.

This paper considers a remote state estimation problem with multiple sensors observing a dynamical process, where sensors transmit local state estimates over an independent and identically distributed (i.i.d.) packet dropping channel to a remote estimator. At every discrete time instant, the remote estimator decides whether each sensor should transmit or not, with each sensor transmission incurring a fixed energy cost. The channel is shared such that collisions will occur if more than one sensor transmits at a time. Performance is quantified via an optimization problem that minimizes a convex combination of the expected estimation error covariance at the remote estimator and expected energy usage across the sensors. For transmission schedules dependent only on the estimation error covariance at the remote estimator, this work establishes structural results on the optimal scheduling which show that: 1) for unstable systems, if the error covariance is large then a sensor will always be scheduled to transmit and 2) there is a threshold-type behavior in switching from one sensor transmitting to another. Specializing to the single sensor case, these structural results demonstrate that a threshold policy (i.e., transmit if the error covariance exceeds a certain threshold and don't transmit otherwise) is optimal. We also consider the situation where sensors transmit measurements instead of state estimates, and establish structural results including the optimality of threshold policies for the single sensor, scalar case. These results provide a theoretical justification for the use of such threshold policies in variance based event triggered estimation. Numerical studies confirm the qualitative behavior predicted by our structural results.

In this paper, we investigate network topology reconfiguration in wireless sensor networks for remote state estimation, where sensor observations are transmitted, possibly via intermediate sensors, to a central gateway/estimator. The time-varying wireless network environment is modelled by the notion of a network state process, which is a randomly time-varying semi-Markov chain and determines the packet reception probabilities of links at different times. For each network state, different network configurations can be used, which govern the network topology and routing of packets. The problem addressed is to determine the optimal network configuration to use in each network state, in order to minimize an expected error covariance measure. Computation of the expected error covariance cost function has a complexity of O(2(M Delta max)), where M is the number of sensors and Delta max is the maximum time between transitions of the semi-Markov chain. A sub-optimal method which minimizes the upper bound of the expected error covariance, that can be computed with a reduced complexity of O(2(M)), is proposed, which in many cases gives identical results to the optimal method. Conditions for estimator stability under both the optimal and suboptimal reconfiguration methods are derived using stochastic Lyapunov functions. Numerical results and comparisons with other low complexity approaches demonstrate the performance benefits of our approach., QC 20170118

This paper considers a remote state estimation problem where a sensor observes a dynamical process, and transmits local state estimates over an independent and identically distributed (i.i.d.) packet dropping channel to a remote estimator. The sensor is equipped with energy harvesting capabilities. At every discrete time instant, provided there is enough battery energy, the sensor decides whether it should transmit or not, in order to minimize the expected estimation error covariance at the remote estimator. For transmission schedules dependent only on the estimation error covariance at the remote estimator, the energy available at the sensor, and the harvested energy, we establish structural results on the optimal scheduling which show that for a given battery energy level and a given harvested energy, the optimal policy is a threshold policy on the error covariance, i.e. transmit if and only if the error covariance exceeds a certain threshold. Similarly, for a given error covariance and a given harvested energy, the optimal policy is a threshold policy on the battery level. Numerical studies confirm the qualitative behaviour predicted by our structural results.

In this paper, we investigate the distortion outage minimization problem for a wireless sensor network (WSN) in the presence of an eavesdropper. The observation signals transmitted from the sensors to the fusion center (FC) are overheard by the eavesdropper. Both the FC and the eavesdropper reconstruct minimum mean squared error (MMSE) estimates of the physical quantity observed. We address the problem of transmit power allocation to minimize the distortion outage at the FC, subject to a long-term transmit power constraint among the sensors and a secrecy outage constraint at the eavesdropper. Applying a rigorous probabilistic power allocation technique we derive power policies for the full channel state information (CSI) case. Additional suboptimal power control policies are studied for the partial CSI case in order to reduce the high computational cost as the number of sensors or receive antennas grows. Numerical results show better performance can be achieved by adding multiple receive antennas at the FC.

This note studies the use of relays to improve the performance of Kalman filtering over packet dropping links. Packet reception probabilities are governed by time-varying fading channel gains, and the sensor and relay transmit powers. We consider situations with multiple sensors and relays, where each relay can either forward one of the sensors' measurements to the gateway/fusion center, or perform a simple linear network coding operation on some of the sensor measurements. Using an expected error covariance performance measure, we consider optimal and suboptimal methods for finding the best relay configuration, and power control problems for optimizing the Kalman filter performance. Our methods show that significant performance gains can be obtained through the use of relays, network coding and power control, with at least 30-40% less power consumption for a given expected error covariance specification.