Timed initial-state detectability of discrete-event systems by algebraic method

In this paper, we address the problem of initial-state detectability (I-detectability) for timed discrete-event systems modeled by time-interval automata (TIAs). An I-observer, defined over a timed event set, is developed to check both strong and weak I-detectability. Additionally, an I-detector structure is designed as an alternative method for verifying strong I-detectability, which is more efficient than the I-observer in certain cases. In addition, we are the first to formally define the concepts of strong and weak timed initial-state detectability ($T$-I-detectability) within the framework of timed discrete-event systems. Specifically, I-detectability necessitates that the initial state of a system can be detected after a finite number of observations. From another perspective, $T$-I-detectability entails that the initial state can be ascertained after a delay of $T$ time units. Under the assumption that every cycle in the TIA has strictly positive weight, we establish that a TIA is $T$-I-detectable if and only if it satisfies the condition of being I-detectable. Finally, we introduce an algebraic method to compute the upper bound of time that needs to elapse before the initial state can be determined in an I-detectable TIA.