Approximate current state observability of discrete-time nonlinear systems under cyber-attacks

Widespread use of the Internet and other means of communication has increased the risk of cyber-attacks that can affect the warranted operation of a system. To prevent damages from attacks, an assessment of its security plays an important role in providing fast and reliable solutions. This paper focuses on state estimation of nonlinear systems under attacks. We consider a plant represented by a discrete-time nonlinear system and an attacker modeled as a finite state machine. We propose a novel notion of observability, called approximate current state observability under attacks, which corresponds to the possibility of identifying the current state of the plant after a given transient despite the malicious action of an attacker that can replace the plant output symbols in the communication network infrastructure. To provide conditions for this property to hold, we resort to the use of formal methods and in particular, of symbolic models. Symbolic models provide an abstract description of purely continuous systems, where a symbolic state corresponds to an aggregate of continuous states in the original system. The approach we propose is particularly useful when dealing with cyber-security of continuous processes in that it offers a framework to deal with the heterogeneous models of the plant and the attacker. An academic example showing the applicability of the results presented is included.& COPY; 2023 Elsevier Ltd. All rights reserved.