This paper provides a novel and better understanding of the performance potential of a nonsingleton (NS) fuzzy system over a singleton (S) fuzzy system. It is done by extending sculpting the state space works from S to NS fuzzification and demonstrating uncertainties about measurements, modeled by NS fuzzification: first, fire more rules more often, manifested by a reduction (increase) in the sizes of first-order rule partitions for those partitions associated with the firing of a smaller (larger) number of rules—the coarse sculpting of the state space; second, this may lead to an increase or decrease in the number of type-1 (T1) and interval type-2 (IT2) first-order rule partitions, which now contain rule pairs that can never occur for S fuzzification—a new rule crossover phenomenon—discovered using partition theory; and third, it may lead to a decrease, the same number, or an increase in the number of second-order rule partitions, all of which are system dependent—the fine sculpting of the state space. The authors' conjecture is that it is the additional control of the coarse sculpting of the state space, accomplished by prefiltering and the max_min (or max-product) composition, which provides an NS T1 or IT2 fuzzy system with the potential to outperform an S T1 or IT2 system when measurements are uncertain.

中文翻译：

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从塑造状态空间的角度比较单例和非单例 1 类和区间 2 类模糊系统的性能潜力


本文提供了对非单例 (NS) 模糊系统相对于单例 (S) 模糊系统的性能潜力的新颖且更好的理解。这是通过将状态空间的雕刻工作从 S 模糊化扩展到 NS 模糊化，并展示由 NS 模糊化建模的测量的不确定性来完成的：首先，更频繁地触发更多规则，表现为一阶规则分区大小的减少（增加）对于那些与触发较小（较大）数量的规则相关的分区——状态空间的粗略雕刻；其次，这可能会导致类型 1 (T1) 和区间类型 2 (IT2) 一阶规则分区的数量增加或减少，这些分区现在包含 S 模糊化永远不会出现的规则对——一条新规则交叉现象——利用分配理论发现；第三，它可能导致二阶规则分区数量的减少、相同或增加，所有这些都依赖于系统——状态空间的精细雕刻。作者的猜想是，它是通过预过滤和 max_min（或最大乘积）组合实现的对状态空间粗略雕刻的额外控制，为 NS T1 或 IT2 模糊系统提供了超越 S 的潜力当测量结果不确定时，采用 T1 或 IT2 系统。