The set-based estimation has gained a lot of attention due to its ability to guarantee state enclosures for safety-critical systems. However, collecting measurements from distributed sensors often requires out-sourcing the set-based operations to an aggregator node, raising many privacy concerns. To address this problem, we present set-based estimation protocols using partially homomorphic encryption that pre-serve the privacy of the measurements and sets bounding the estimates. We consider a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Sets are represented by zonotopes and constrained zonotopes as they can compactly represent high-dimensional sets and are closed under linear maps and Minkowski addition. By selectively encrypting parameters of the set repre-sentations, we establish the notion of encrypted sets and intersect sets in the encrypted domain, which enables guaranteed state estimation while ensuring privacy. In particular, we show that our protocols achieve computational privacy using the cryptographic notion of computational indistinguishability. We demonstrate the efficiency of our approach by localizing a real mobile quadcopter using ultra-wideband wireless devices.

This paper studies how to estimate the weighted adjacency matrix of a network out of the state sequence of a model with binary-valued states, by using a recursive algorithm. In the considered system, agents display and exchange these binary-valued states generated from intrinsic quantizers. It is shown that stability of the model and identifiability of the system parameters can be guaranteed under continuous random noise. Under standard Gaussian noise, the problem of estimating the real-valued weighted adjacency matrix and the unknown quantization threshold is transformed to an optimization problem via a maximum likelihood approach. It is further verified that the unique solution of the optimization problem is the true parameter vector. A recursive algorithm for the estimation problem is then proposed based on stochastic approximation techniques. Its strong consistency is established and convergence rate analyzed. Numerical simulations are provided to illustrate developed results. 

We study distributed estimation of a high-dimensional static parameter vector through a group of sensors whose communication network is modeled by a fixed directed graph. Different from existing time-triggered communication schemes, an event-triggered asynchronous scheme is investigated in order to reduce communication while preserving estimation convergence. A distributed estimation algorithm with a single step size is first proposed based on an event-triggered communication scheme with a time-dependent decaying threshold. With the event-triggered scheme, each sensor sends its estimate to neighbor sensors only when the difference between the current estimate and the last sent-out estimate is larger than the triggering threshold. Different sensors can have different step sizes and triggering thresholds, enabling the parameter estimation process to be conducted in a fully distributed way. We prove that the proposed algorithm has mean-square and almost-sure convergence, respectively, under an integrated condition of sensor network topology and sensor measurement matrices. The condition is satisfied if the topology is a balanced digraph containing a spanning tree and the system is collectively observable. The collective observability is the possibly mildest condition, since it is a spatially and temporally collective condition of all sensors and allows sensor measurement matrices to be time-varying, stochastic, and nonstationary. Moreover, we provide estimates for the convergence rates, which are related to the step size as well as the triggering threshold. Furthermore, as an essential metric of sensor communication intensity in the event-triggered distributed algorithms, the communication rate is proved to decay to zero with a certain speed almost surely as time goes to infinity. In addition, we show that it is feasible to tune the threshold and the step size such that requirements of algorithm convergence and communication rate decay are satisfied simultaneously. We also show that given the step size, adjusting the decay speed of the triggering threshold can lead to a tradeoff between the convergence rate of the estimation error and the decay speed of the communication rate. Specifically, increasing the decay speed of the threshold would make the communication rate decay faster but reduce the convergence rate of the estimation error. Numerical simulations are provided to illustrate the developed results.

We study how to secure distributed filters for linear time-invariant systems with bounded noise under false-data injection attacks. A malicious attacker is able to arbitrarily manipulate the observations for a time-varying and unknown subset of the sensors. We first propose a recursive distributed filter consisting of two steps at each update. The first step employs a saturation-like scheme, which gives a small gain if the innovation is large corresponding to a potential attack. The second step is a consensus operation of state estimates among neighboring sensors. We prove the estimation error is upper bounded if the parameters satisfy a condition. We further analyze the feasibility of the condition and connect it to sparse observability in the centralized case. When the attacked sensor set is known to be time-invariant, the secured filter is modified by adding an online local attack detector. The detector is able to identify the attacked sensors whose observation innovations are larger than the detection thresholds. Also, with more attacked sensors being detected, the thresholds will adaptively adjust to reduce the space of the stealthy attack signals.

In this chapter, we study how to design secure centralized and distributed filters for linear time-invariant systems with bounded noise under false data injection attacks in sensor networks. An adversary is able to compromise a subset of sensors and manipulate the measurements arbitrarily. We provide two motivating examples for this problem setup from smart buildings and autonomous vehicles. Then we design a centralized filter based on a saturation method, which gives a small gain if the innovation is large enough, indicating the high likelihood of compromised measurements. The estimation error of the secure centralized filter is proved to be asymptotically upper-bounded. Moreover, a secure two-time-scale distributed filter is obtained by modifying the centralized filter and employing an estimate consensus approach. Boundedness of the estimation error of the distributed filter is proved. Numerical simulations are provided in the end to show the usefulness of the two filters.

We study how to design a secure observer-based distributed controller such that a group of vehicles can achieve accurate state estimates and formation control even if the measurements of a subset of vehicle sensors are compromised by a malicious attacker. We propose an architecture consisting of a resilient observer, an attack detector, and an observer-based distributed controller. The distributed detector is able to update three sets of vehicle sensors: the ones surely under attack, surely attack free, and suspected to be under attack. The adaptive observer saturates the measurement innovation through a preset static or time-varying threshold, such that the potentially compromised measurements have limited influence on the estimation. Essential properties of the proposed architecture include: (1) The detector is fault-free, and the attacked and attack-free vehicle sensors can be identified in finite time; (2) The observer guarantees both real-time error bounds and asymptotic error bounds, with tighter bounds when more attacked or attack-free vehicle sensors are identified by the detector; (3) The distributed controller ensures closed-loop stability. The effectiveness of the proposed methods is evaluated through simulations by an application to vehicle platooning.

We study distributed filtering for a class of uncertain systems over corrupted communication channels. We propose a distributed robust Kalman filter with stochastic gains, through which upper bounds of the conditional mean square estimation errors are calculated online. We present a robust collective observability condition, under which the mean square error of the distributed filter is proved to be uniformly upper bounded if the network is strongly connected. For better performance, we modify the filer by introducing a switching fusion scheme based on a sliding window. It provides a smaller upper bound of the conditional mean square error. Numerical simulations are provided to validate the theoretical results and show that the filter scales to large networks.

In order to protect smart distribution grids from intrusions, it is important to understand possible risks and impacts of attacks. We study the worst-case attack strategy of a power injection attack against the physical layer of a smart distribution grid with a high penetration of photovoltaic resources. We derive both the worst attack signal and worst attack location: The worst attack signal is a step function which switches its sign at the final stage, and the worst attack location is the node with the largest impedance to the grid substation. Numerical examples on a European benchmark model verify the developed results. Finally, both theoretical and numerical results are used to discuss feasible defense strategies against power injection attacks.

A new game theory based framework is proposed for path tracking and stabilization of autonomous vehicles. In the developed framework, vehicle body and corner traction control strategies are formulated in terms of players in a differential game. An integrated stability and path tracking control based on a non-cooperative differential game is developed. It includes bidirectional slip effect and wheel dynamics, which reflect more accurate longitudinal and lateral dynamics in harsh maneuvers and scenarios with sudden changes in the path planners trajectories. The open-loop and closed-loop Nash equilibrium control strategies are obtained by solving a two-player linear-quadratic differential game for the dynamical system of the overall tracking error. The performance of the proposed control strategy is validated with software simulations in various driving conditions.

Social sampling is a novel randomized message passing protocol inspired by social communication for opinion formation in social networks. In a typical social sampling algorithm, each agent holds a sample from the empirical distribution of social opinions at initial time, and it collaborates with other agents in a distributed manner to estimate the initial empirical distribution by randomly sampling a message from current distribution estimate. In this paper, we focus on analyzing the theoretical properties of the distributed social sampling algorithm over random networks. First, we provide a framework based on stochastic approximation to study the asymptotic properties of the algorithm. Then, under mild conditions, we prove that the estimates of all agents converge to a common random distribution, which is composed of the initial empirical distribution and the accumulation of quantized error. Besides, by tuning algorithm parameters, we prove the strong consistency, namely, the distribution estimates of agents almost surely converge to the initial empirical distribution. Furthermore, the asymptotic normality of estimation error generated by distributed social sample algorithm is addressed. Finally, we provide a numerical simulation to validate the theoretical results of this paper.