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 s
ensors. 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 filter 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. The resilience of the secured filter with detection is verified by an explicit relationship between the upper bound of the estimation error and the number of detected attacked sensors. Moreover, for the noise-free case, we prove that the state estimate of each sensor asymptotically converges to the system state under certain conditions. Numerical simulations are provided to illustrate the developed results.
We consider the design of a fair sensor schedule for a number of sensors monitoring different linear time-invariant processes. The largest average remote estimation error among all processes is to be minimized. We first consider a general setup for t
he max-min fair allocation problem. By reformulating the problem as its equivalent form, we transform the fair resource allocation problem into a zero-sum game between a judge and a resource allocator. We propose an equilibrium seeking procedure and show that there exists a unique Nash equilibrium in pure strategy for this game. We then apply the result to the sensor scheduling problem and show that the max-min fair sensor scheduling policy can be achieved.
This paper considers the problem of sensory data scheduling of multiple processes. There are $n$ independent linear time-invariant processes and a remote estimator monitoring all the processes. Each process is measured by a sensor, which sends its lo
cal state estimate to the remote estimator. The sizes of the packets are different due to different dimensions of each process, and thus it may take different lengths of time steps for the sensors to send their data. Because of bandwidth limitation, only a portion of all the sensors are allowed to transmit. Our goal is to minimize the average of estimation error covariance of the whole system at the remote estimator. The problem is formulated as a Markov decision process (MDP) with average cost over an infinite time horizon. We prove the existence of a deterministic and stationary policy for the problem. We also find that the optimal policy has a consistent behavior and threshold type structure. A numerical example is provided to illustrate our main results.
This paper considers optimal attack attention allocation on remote state estimation in multi-systems. Suppose there are $mathtt{M}$ independent systems, each of which has a remote sensor monitoring the system and sending its local estimates to a fusi
on center over a packet-dropping channel. An attacker may generate noises to exacerbate the communication channels between sensors and the fusion center. Due to capacity limitation, at each time the attacker can exacerbate at most $mathtt{N}$ of the $mathtt{M}$ channels. The goal of the attacker side is to seek an optimal policy maximizing the estimation error at the fusion center. The problem is formulated as a Markov decision process (MDP) problem, and the existence of an optimal deterministic and stationary policy is proved. We further show that the optimal policy has a threshold structure, by which the computational complexity is reduced significantly. Based on the threshold structure, a myopic policy is proposed for homogeneous models and its optimality is established. To overcome the curse of dimensionality of MDP algorithms for general heterogeneous models, we further provide an asymptotically (as $mathtt{M}$ and $mathtt{N}$ go to infinity) optimal solution, which is easy to compute and implement. Numerical examples are given to illustrate the main results.
Jointly optimal transmission power control and remote estimation over an infinite horizon is studied. A sensor observes a dynamic process and sends its observations to a remote estimator over a wireless fading channel characterized by a time-homogene
ous Markov chain. The successful transmission probability depends on both the channel gains and the transmission power used by the sensor. The transmission power control rule and the remote estimator should be jointly designed, aiming to minimize an infinite-horizon cost consisting of the power usage and the remote estimation error. A first question one may ask is: Does this joint optimization problem have a solution? We formulate the joint optimization problem as an average cost belief-state Markov decision process and answer the question by proving that there exists an optimal deterministic and stationary policy. We then show that when the monitored dynamic process is scalar, the optimal remote estimates depend only on the most recently received sensor observation, and the optimal transmission power is symmetric and monotonically increasing with respect to the innovation error.
The problem of quickest change detection with communication rate constraints is studied. A network of wireless sensors with limited computation capability monitors the environment and sends observations to a fusion center via wireless channels. At an
unknown time instant, the distributions of observations at all the sensor nodes change simultaneously. Due to limited energy, the sensors cannot transmit at all the time instants. The objective is to detect the change at the fusion center as quickly as possible, subject to constraints on false detection and average communication rate between the sensors and the fusion center. A minimax formulation is proposed. The cumulative sum (CuSum) algorithm is used at the fusion center and censoring strategies are used at the sensor nodes. The censoring strategies, which are adaptive to the CuSum statistic, are fed back by the fusion center. The sensors only send observations that fall into prescribed sets to the fusion center. This CuSum adaptive censoring (CuSum-AC) algorithm is proved to be an equalizer rule and to be globally asymptotically optimal for any positive communication rate constraint, as the average run length to false alarm goes to infinity. It is also shown, by numerical examples, that the CuSum-AC algorithm provides a suitable trade-off between the detection performance and the communication rate.
