Supervisory Control of Fuzzy Discrete Event Systems: A Formal Approach

Abstract

Fuzzy discrete event systems (DESs) were proposed recently by Lin and Ying [19], which may better cope with the real-world problems with fuzziness, impreciseness, and subjectivity such as those in biomedicine. As a continuation of [19], in this paper we further develop fuzzy DESs by dealing with supervisory control of fuzzy DESs. More specifically, (i) we reformulate the parallel composition of crisp DESs, and then define the parallel composition of fuzzy DESs that is equivalent to that in [19]; max-product and max-min automata for modeling fuzzy DESs are considered; (ii) we deal with a number of fundamental problems regarding supervisory control of fuzzy DESs, particularly demonstrate controllability theorem and nonblocking controllability theorem of fuzzy DESs, and thus present the conditions for the existence of supervisors in fuzzy DESs; (iii) we analyze the complexity for presenting a uniform criterion to test the fuzzy controllability condition of fuzzy DESs modeled by max-product automata; in particular, we present in detail a general computing method for checking whether or not the fuzzy controllability condition holds, if max-min automata are used to model fuzzy DESs, and by means of this method we can search for all possible fuzzy states reachable from initial fuzzy state in max-min automata; also, we introduce the fuzzy n-controllability condition for some practical problems; (iv) a number of examples serving to illustrate the applications of the derived results and methods are described; some basic properties related to supervisory control of fuzzy DESs are investigated. To conclude, some related issues are raised for further consideration.

Lyapunov Stability of Fuzzy Discrete Event Systems

“…The fuzzy languages generated by G is denoted by L G or L for simplicity [28], which is a function from Σ * to [0, 1]. Let s ∈ Σ * .…”

“…From [18,19,28], we know that each fuzzy event is associated with a degree of controllability, so, the uncontrollable set Σ uc and controllable set Σ c are two fuzzy subsets of Σ, and satisfy: for any σ ∈ Σ,…”

“…These issues have been partially investigated in [2,3,28]. Qiu [28] established the supervisory control theory of FDESs, and found a method of checking the existence of supervisors for FDESs; and independently, Cao and Ying [2,3] significantly developed FDESs. As a continuation, this paper is to deal with the failure diagnosis for FDESs.…”

Nondeterministic fuzzy automata

“…In the literature up to now, various variants of fuzzy automata have been proposed in different modeling situations (see, for example, [2,6,7,8,9,10,16,17,22,23,26,27,28,33,38]) and the notions of fuzzy automata and fuzzy languages have proved useful in many areas [1,4,11,12,13,20,21,30,31,32,34,35,37,39]. In terms of fuzzy transition functions, fuzzy automata may be broadly classified into three types: The first type [1,3,6,7,9,11,12,13,16,21,22,23,27,28,32,33,35,37,39] uses fuzzy transition functions like δ : Q × Σ −→ F (Q), where Q represents the state set, Σ is the input alphabet, and F (Q) is the set...…”

“…In terms of fuzzy transition functions, fuzzy automata may be broadly classified into three types: The first type [1,3,6,7,9,11,12,13,16,21,22,23,27,28,32,33,35,37,39] uses fuzzy transition functions like δ : Q × Σ −→ F (Q), where Q represents the state set, Σ is the input alphabet, and F (Q) is the set of all fuzzy subsets of Q. Note that such a fuzzy transition function can be equivalently converted into δ : Q × Σ × Q −→ [0, 1] and can also be represented by fuzzy states and fuzzy transition matrices.…”