Detectability of Finite-State Automata
A finite-state automaton (FSA) (see Sect. 2.3) is obtained from a standard nondeterministic finite automaton (NFA, see Sect. 1.3) by removing all accepting states and also replacing the unique initial state by a set of initial states. Such a modification can be used to better describe discrete-event systems (DESs) (Wonham and Cai 2019; Cassandras and Lafortune 2010). DESs usually refer to as event-driven processes that cannot be described by differential equations, where the latter usually represent time-driven processes. The FSA-based DESs are usually simple and many properties of FSAs are decidable, so FSAs are rather welcome in engineering-oriented studies. However, despite being simple, FSAs can also perform many good properties that are of both theoretical and practical importance. During the past three decades, plenty of properties and their verification or synthesis techniques on DESs in the framework of FSAs have been proposed and developed, e.g., controllability and observability (The notion of observability in the supervisory control framework is totally different from that studied in this book (see Chaps. 4, 5, 7).) (Ramadge and Wonham 1987; Ramadge 1986; Lin and Wonham 1988), diagnosability (Lin 1994; Sampath 1995), detectability (Shu et al. 2007; Shu and Lin 2011; Zhang 2017), opacity (Saboori and Hadjicostis 2014, 2012, 2011, 2013; Lin 2011), etc., where controllability and observability are defined on formal languages, diagnosability is defined on events, but the others are defined on states (except for that the result of Lin (2011) is defined on formal languages).
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