Interpolation in structured fuzzy rule bases

Fuzzy rule based systems are important in many control engineering applications. In real problems, often the number of input variables is very high. Time complexity of the fuzzy control algorithms based on either the CRI or Mamdani's technique is too high for real time control. In the real situations, often a few or maximally a few dozen variables dominate the system in a certain area of the state space. This is why an experienced operator is able to control rather well, although overview of all the variables is virtually impossible. The subset of variables changes when the working point changes. Hierarchically structured rule bases are introduced where 'sub-rule bases' contain only subsets of the input variables. Fuzzy partition of the state space is applied for the structuring of the total input universe of discourse. While application of the corresponding sub-rule base in the 'middle' of a fuzzy subset of X is unambiguous, the 'twilight zones', i.e. areas belonging to different subsets to a greater than 0 degree form a serious theoretical problem. The idea of interpolation based on cylindric extensions of the subspaces is introduced and the necessary formulas are presented. Some examples are shown.