Multi-step state-based opacity for unambiguous weighted machines

Opacity is a central concept in the issue of privacy security and has been studied extensively in fields such as finite automata, probabilistic automata, and stochastic automata. Here, we investigate the problem of validating multi-step opaque properties through unambiguous weighted machines from the perspective of cyber-physical systems. First, the notion of multi-step state-based opacity for unambiguous weighted machines is presented and defined. It includes two variants of delays with a finite K and infinite steps. Subsequently, the weighted state estimate with K (infinite)-step delay is established by abstracting the possible state set that the system could have through these weighted observations. Meanwhile, to keep the observable weighted sequence consistent between the bidirectional observers, the unobserved weights of the reverse weighted machine are assumed to be reserved. Subsequently, the existence conditions are developed, and the corresponding algorithms, termed the weighted bidirectional observer, are generalized to verify these properties. Finally, several numerical examples are illustrated to demonstrate the effectiveness of the proposed method. Taken together, the current approach will be conducive to a deep understanding of the security and privacy of cyber-physical systems.