Verification of Approximate Prognosability via Barrier Certificates

In this paper, we investigate the verification of approximate prognosability for discrete-time control systems with continuous state set in the context of fault prognosis. Existing works on this topic rely on constructing finite abstractions, which lead to significant computation burden. To address this challenge, we propose an abstraction-free method via barrier certificates. Specifically, we consider a notion of socalled approximate $(M, \delta)$-prognosability requiring that every fault, characterized by entering a fault region, can be predicted before its occurrence under observation precision $\delta$ and once an alarm is issued, fault will occur for sure within M time instants. Then, we propose a verification scheme based on an M-deterministic finite automaton over an augmented system of the original system. Then, we reduce the verification of (the lack of) approximate ($M, \delta$)-prognosability to a safety verification problem, which can be checked effectively by barrier certificates. Furthermore, a counter-example guided inductive synthesis framework is proposed to compute these barrier certificates.