Connectionist Fuzzy Relational Systems

There have been an enormous number of fuzzy systems developed so far. For excellent surveys the reader is referred to [1][3][5][7][9][10][18][19]. The most commonly used systems are linguistic models and functional models introduced by Takagi and Sugeno. The linguistic systems store an input-output mapping in the form of fuzzy IF-THEN rules with linguistic terms both in antecedents and consequents. The functional fuzzy systems use linguistic values in the condition part of rules, but the input-output mapping is expressed by functions of inputs in a rule consequent part. The above models are used in all fields of machine learning and computational intelligence. They all have advantages and drawbacks. The linguistic systems use intelligible and easy to express IF-THEN rules with fuzzy linguistic values. Functional systems allow modeling of input-output mapping but they suffer from lack of interpretability. Another approach, rarely studied in the literature, is based on fuzzy relational systems (see e.g. Pedrycz [9]). This relates input fuzzy linguistic values to output fuzzy linguistic values thanks to fuzzy relations. It allows the setting fuzzy linguistic values in advance and fine-tuning model mapping by changing relation elements. They were used in some areas, e.g. to classification [16] and control [2]. In this paper we propose a new neuro-fuzzy structure of the relational system (Section 3 and 5), allowing relation elements to be fine-tuned by the backpropagation algorithm. It will be also shown that relational fuzzy systems are under specific assumptions equivalent to linguistic systems with rule weights (Section 4). Moreover, another new class of neuro-fuzzy systems, based on a relational approach with a fuzzy certainty degree, will be suggested in Section 5. Finally, the systems are tested on problems of truck backer-upper nonlinear control and nonlinear function approximation (Section 6).