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This paper deals with the state estimation problem in discrete-event systemsmodeled with nondeterministic finite automata, partially observed via a sensormeasuring unit whose measurements (reported observations) may be vitiated by amalicious attacker. The attacks considered in this paper include arbitrarydeletions, insertions, or substitutions of observed symbols by taking intoaccount a bounded number of attacks or, more generally, a total cost constraint(assuming that each deletion, insertion, or substitution bears a positive costto the attacker). An efficient approach is proposed to describe possiblesequences of observations that match the one received by the measuring unit, aswell as their corresponding state estimates and associated total costs. Wedevelop an algorithm to obtain the least-cost matching sequences byreconstructing only a finite number of possible sequences, which wesubsequently use to efficiently perform state estimation. We also develop atechnique for verifying tamper-tolerant diagnosability under attacks thatinvolve a bounded number of deletions, insertions, and substitutions (or, moregenerally, under attacks of bounded total cost) by using a novel structureobtained by attaching attacks and costs to the original plant. The overallconstruction and verification procedure have complexity that is of O(|X|^2C^2),where |X| is the number of states of the given finite automaton and C isthe maximum total cost that is allowed for all the deletions, insertions, andsubstitutions. We determine the minimum value of C such that the attacker cancoordinate its tampering action to keep the observer indefinitely confusedwhile utilizing a finite number of attacks. Several examples are presented todemonstrate the proposed methods.

Error- and Tamper-Tolerant State Estimation for Discrete Event Systems under Cost Constraints