Parallel and multistage fuzzy inference based on families of α-level sets

In this paper, methods for parallel fuzzy inference and multistage-parallel fuzzy inference are studied on the basis of families of α-level sets. The parallel fuzzy inference is characterized by the unification of inference consequences obtained from a number of conditional propositions. Thus, in this paper, the methods for the unification of inference consequences via α-level sets are presented first. It is found that the unification approximated by using fuzzy convex hull is efficient in the case where the unification is performed by the maximum operation. The methods for defuzzification are also examined via α-level sets for the unified consequences. The computational efficiency is evaluated in order to show the effectiveness of the unification and defuzzification via α-level sets. Moreover, it is studied by computer simulations how the approximation by fuzzy convex hull affects the performance in fuzzy control. The results indicate that this approximation does not degrade the control performance. Next, the multistage-parallel fuzzy inference is considered from the operational point of view via α-level sets. The multistage-parallel fuzzy inference is characterized by passing the unified consequence of parallel fuzzy inference in each stage to the next stage as a fact. Hence, the studies are focused on this consequence passing in this paper. It is clarified that the straightforward way of inference operations via α-level sets is time consuming because of the non-convexity in the unified inference consequence in each stage. In order to solve the problem, the multistage-parallel fuzzy inference is formulated into a form of linguistic-truth-value propagation. As a result, the inference operations in middle stages can be conducted by convex fuzzy sets and then efficient computations for inference is provided. The computational efficiency is also evaluated to show the effectiveness of the formulation. Finally, this paper concludes with some brief discussions.