Predicate-transforms And The Degree Of Control-invariance In Fuzzy Discrete Event Systems

This paper gives the definition of fuzzy predicates in fuzzy discrete event systems. The weakest precondition and the strongest postcondition in discrete event systems are generalized to fuzzy discrete event systems.It proves in classical discrete event systems, for a given predicate, the result of converting it into the weakest precondition then converting it into the strongest postconditions and the result of converting it into the strongest postconditions then converting it into the weakest precondition are equal. they are both the original given predicate. And the above theorem is generalized to fuzzy discrete event systems. It proves that the degree of any fuzzy state belonging to a given predicate, is equal to the degree of this fuzzy state belonging to the predicate which converting the given predicate to the weakest precondition then converting into the strongest postcondition, and it is not less than degree of this fuzzy state belonging to the predicate that converting the given predicate to the strongest postcondition then converting into the weakest precondition.The degree ofu?~-control-inveriance and the degree ofcontrol-invariance are defined in fuzzy discrete event systems. Two theorems are proved. First, for any given predicate, there must be a state feedback controller such that its degree ofu?~-control-inveriance and its degree of control-invariance are equal. Second, for any given predicate,there must be a classical controller such that its degree of control-invariance under the classical controller are maximum.Based on this, the degree of control-invariance under the fuzzy controller is defined in fuzzy discrete event systems. It proves that for any given predicate, there must be a fuzzy controller such that its degree of control-invariance under the classical controller is maximum.