Precise Fuzzy Inference Systems

Precise Fuzzy Inference Systems

I. INTRODUCTION
Fuzzy inference is the process of mapping an input to an output using fuzzy logic. It consists of membership functions which are responsible for the fuzzification of inputs, fuzzy operators and IF-THEN rules. Fuzzy inference systems have been successfully applied in many different fields, such as automatic control, data classification, decision analysis, expert systems, and computer vision. And because of this multidisciplinary characteristic, fuzzy inference systems can also be mentioned as fuzzy rule based systems, fuzzy expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic controllers, and fuzzy systems.
Fuzzy inference systems are usually divided into two areas: linguistic fuzzy model which focus on interpretability, mainly Mamdani models, and precise fuzzy models which focus on accuracy, mainly Takagi-Sugeno-Kang (TSK) model. Compared with Mamdani model, the first two step of fuzzy inference process of TSK method, fuzzifying inputs and applying the fuzzy operator, are exactly the same. The most fundamental difference between these two models are the way the crisp output is generated from the fuzzy inputs [1]. TSK network uses weighted average to compute the crisp output instead of the defuzzification step in Mamdani model which is time consuming. Then together with the fact that in most cases TSK model has less fuzzy rules than Mamdani method make TSK model a more computational efficient fuzzy inference system. On the other hand, known as an universal approximator [2], TSK can do smooth piece-wise linear approximation for a nonlinear function which makes it a precise fuzzy inference system.
There are two types of learning algorithms of TSK fuzzy inference system: offline and online. Using offline algorithm, the structure of system (like the number of rules) is prefixed before the learning process. The inputs are in a batch mode and can be repeatedly accessed for training parameters. After

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