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

Abstract

This paper deals with the state estimation problem in discrete-event systems modeled with nondeterministic finite automata, partially observed via a sensor measuring unit whose measurements (reported observed symbols) may be vitiated by a malicious attacker. The attacks considered in this paper include arbitrary deletions, insertions, or substitutions of observed symbols by taking into account a bounded number of attacks or, more generally, a total cost constraint (assuming that each deletion, insertion, or substitution bears a positive cost to the attacker). We provide a characterization of the sequences of symbols that match the received sequence of possibly corrupted symbols, and subsequently use them to recursively perform tamper-tolerant state estimation subject to cost constraints. Each step of the recursive state estimation procedure has complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(|X||\Sigma |C)$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$|X|$</tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$|\Sigma |$</tex-math></inline-formula> ) is the number of states (events) of the given finite automaton and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C$</tex-math></inline-formula> is the maximum total cost that is allowed for all deletions, insertions, and substitutions.