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Fault Diagnosis of Manufacturing Systems Using Finite State Automata
Published in Javier Campos, Carla Seatzu, Xiaolan Xie, in Manufacturing, 2018
Stéphane Lafortune, Richard Hill, Andrea Paoli
Automaton: A deterministic automaton, denoted by G, is a five-tuple G = (X, E, f, x0, Xm) where X is the set of states; E is the finite set of events associated with the system modelled by G; f : X × E → X is the transition function: f(x, e) = y means that there is a transition labelled by event e from state x to state y, also denoted by the transition triple (x, e, y) (in general, f is a partial function on its domain); x0 is the initial state; and Xm ⊆ X is the set of marked states.
Standard Mathematical Models
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
William S. Levine, James T. Gillis, Graham C. Goodwin, Juan C. Agüero, Juan I. Yuz, Harry L. Trentelman, Richard. Hill
The current state of a deterministic automaton model can be completely determined by knowledge of the events that have occurred in the model’s past. Specifically, the automaton begins in its initial state, denoted graphically here by a short arrow (see Figure 5.16). When an event occurs, the transition structure of the model indicates the next state of the model that is entered. With a deterministic model, the occurrence of an event completely determines the successive state. In other words, an event may not lead to multiple states from a given originating state. In the graphical model, states with double circles are marked to indicate successful termination of a process. Mathematically, a deterministic automaton can be denoted by the five-tuple G = (Q, Σ, δ, q0, Qm), where Q is the set of states, Σ is the set of events, δ : Q × Σ → Q is the state transition function, q0 ∈ Q is the initial state, and Qm ⊆ Q is the set of marked states. The notation Σ* will represent the set of all finite strings of elements of Σ, including the empty string ε, and is called the Kleene-closure of the set Σ. The empty string ε is a null event for which the system does not change state. In this presentation the function δ is also extended to δ : Q × Σ* → Q. The notation δ(q,s)! for any q ∈ Q and any s ∈ Σ* denotes that δ(q, s) is defined. An event σ is said to be feasible at a state q if δ(q, σ)!. It is sometimes appropriate to employ a nondeterministic automaton model where there may be multiple initial states and an event σ may transition the model to any one of several different states from the same originating state q. All automata employed here may be assumed to be deterministic unless otherwise noted.
Relative predictability of failure event occurrences and its opacity-based test algorithm
Published in International Journal of Control, 2019
Rui Zhao, Fuchun Liu, Jianxin Tan
A DES is modelled as a deterministic automaton G = (X, Σ, δ, x0), where X is the finite state space, Σ is the set of events, δ : X × Σ → X is the transition function and x0 ∈ X is the initial state of the system. The behaviour of the system is described by a prefix-closed language. The model G accounts for the normal and failed behaviour of the system. Given the event set Σ, Σ* denotes the Kleene closure of Σ. A member of Σ* is called a trace and any subset of Σ* is called a language. The language generated by G is defined as i.e. it includes all traces executed from the initial state x0 of G. Given a trace s originating from x0, denote L/s as the post language of L after s, i.e. All events of G are partitioned as , where Σo denotes the set of observable events and Σuo denotes the set of unobservable events. Given an event σ ∈ Σ and L, ψ(σ, L) is the set of strings in L that end with σ. Formally,
Diagnosability of composite automata based on semi-tensor product
Published in Systems Science & Control Engineering, 2021
Zengqiang Chen, Yingrui Zhou, Zhipeng Zhang, Yongyi Yan
G is a deterministic automaton, denoted by a six-tuple , where X is the set of states, is the set of input events with observable events and unobservable events , Y is the set of outputs and is the initial state. is the partial state transition function, where denotes that state can be reached from x by event σ. For any , , , where is the set of input strings upon Σ, including ϵ. Note that there exists a projection defined as There are two kinds of automata according to output function: Moore automaton and Mealy automaton. Moore automata are automata with (state) outputs, where , denoted by Figure 1(a) and Mealy automata are input/output automata, where , denoted by Figure 1(b). There are many physical systems modelled as Moore automata, where the output depends only on a current state (Cassandras & Lafortune, 2009). Notice that if any state from X can be an initial state and it has nothing to do with outputs, G can be abbreviated as . All in all, the number of variants for a deterministic automaton can vary with needed.