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为了进一步证明分层模糊系统在很好地解决了多变量模糊系统的规则数随输入变量个数指数增长的问题的同时,是否能够解决系统的参数个数随输入变量指数式的增长,推导一类二叉树型分层模糊系统和一类典型多输入单输出(MISO)模糊系统的输入输出表达式,证明了两者的等效性条件。得出结论:此类二叉树型分层模糊系统要达到典型MISO模糊系统同样的逼近精度,其参数个数至少要与典型模糊系统一样多。这一特例对分层模糊系统是否能从根本上解决MISO模糊系统的“维数灾”问题提出了质疑。
In order to further prove that the hierarchical fuzzy system can well solve the problem that the number of rules of multivariable fuzzy system increases exponentially with the number of input variables and whether it can solve the exponential growth of the number of parameters of the system with input variables The input-output expressions of a class of binary tree hierarchical fuzzy system and a class of typical multiple input single output (MISO) fuzzy systems prove the equivalence condition of the two. The conclusion is that this kind of binary tree hierarchical fuzzy system should reach the same approximation precision of the typical MISO fuzzy system, the number of parameters should be at least as many as typical fuzzy system. This particular case poses the question whether hierarchical fuzzy systems can fundamentally solve the problem of “dimensionality” of MISO fuzzy systems.