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This paper addresses quantized output feedback stabilization under Denial-of-Service (DoS) attacks. First, assuming that the duration and frequency of DoS attacks are averagely bounded and that an initial bound of the plant state is known, we propose an output encoding scheme that achieves exponential convergence with finite data rates. Next we show that a suitable state transformation allows us to remove the assumption on the DoS frequency. Finally, we discuss the derivation of state bounds under DoS attacks and obtain sufficient conditions on the bounds of DoS duration and frequency for achieving Lyapunov stability of the closed-loop system.

We consider a multi-adversary version of the supervisory control problem for discrete-event systems, in which an adversary corrupts the observations available to the supervisor. The supervisor's goal is to enforce a specific language in spite of the opponent's actions and without knowing which adversary it is playing against. This problem is motivated by applications to computer security in which a cyber defense system must make decisions based on reports from sensors that may have been tampered with by an attacker. We start by showing that the problem has a solution if and only if the desired language is controllable (in the Discrete event system classical sense) and observable in a (novel) sense that takes the adversaries into account. For the particular case of attacks that insert symbols into or remove symbols from the sequence of sensor outputs, we show that testing the existence of a supervisor and building the supervisor can be done using tools developed for the classical DES supervisory control problem, by considering a family of automata with modified output maps, but without expanding the size of the state space and without incurring on exponential complexity on the number of attacks considered.Discrete event systems (DESs) are non-deterministic transition systems defined over a typically finite statespace. The DESs supervisory control problem refers to the design of a feedback controller -called a supervisor -that restricts the set of possible sequences of transitions (typically represented by strings over an alphabet of transitions) to a desired set K. The supervisor's task is complicated by the fact that (i) only a subset of transitions can be inhibited (the so called "controllable" transitions) and (ii) the supervisor only has partial information about the state of the system, which it gathers by observing a string of "output symbols." This basic problem is motivated by a wide range of applications that include manufacturing systems, chemical batch plants, power grids, transportation systems, database management, communication protocols, logistics, and computer security. The latter is the key motivating application for the work reported here.

Abstract-We introduce a scalable observer architecture to estimate the states of a discrete-time linear-time-invariant (LTI) system whose sensors can be manipulated by an attacker. Given the maximum number of attacked sensors, we build on previous results on necessary and sufficient conditions for state estimation, and propose a novel multi-modal Luenberger (MML) observer based on efficient Satisfiability Modulo Theory (SMT) solving. We present two techniques to reduce the complexity of the estimation problem. As a first strategy, instead of a bank of distinct observers, we use a family of filters sharing a single dynamical equation for the states, but different output equations, to generate estimates corresponding to different subsets of sensors. Such an architecture can reduce the memory usage of the observer from an exponential to a linear function of the number of sensors. We then develop an efficient SMT-based decision procedure that is able to reason about the estimates of the MML observer to detect at runtime which sets of sensors are attack-free, and use them to obtain a correct state estimate. We provide proofs of convergence for our algorithm and report simulation results to compare its runtime performance with alternative techniques. Our algorithm scales well for large systems (including up to 5000 sensors) for which many previously proposed algorithms are not implementable due to excessive memory and time requirements. Finally, we illustrate the effectiveness of our algorithm on the design of resilient power distribution systems.

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