Detectability measure on state estimation of discrete event systems

In this paper, we investigate how to quantitatively evaluate the performance of state estimation in a discrete event system. We adopt stochastic automaton to quantitatively describe a discrete event system. With the stochastic automaton, we can calculate the probability of occurrence of any event sequence. We say that an event sequence (with sufficient length) is detectable if we can determine the current state and subsequent states of the system after the occurrence of the sequence. We then define the sum of probabilities of all detectable event sequences as a quantitative state estimation indicator. This indicator is the limit when the length of sequence goes to infinite. In order to calculate the limit, we augment the discrete event system into a larger automaton which includes the information of the discrete event system and all its state estimates. The augmented automaton is also a stochastic automaton and can be converted into a Markov Chain by removing all the event labels. The calculation of the limit is then translated into the calculation of the sum of probabilities of some states in the Markov Chain, which can be done efficiently.