Multilayer Optimization Approach for Fuzzy Systems

The design of fuzzy inference systems comes along with several decisions taken by the designers since is necessary to determine, in a coherent way, the number of membership functions for the inputs and outputs, and also the specification of the fuzzy rules set of the system, besides defining the strategies of rules aggregation and defuzzification of output sets. The need to develop systematic procedures to assist the designers has been wide because the trial and error technique is the unique often available (Figueiredo & Gomide, 1997). In general terms, for applications involving system identification and fuzzy modeling, it is convenient to use energy functions that express the error between the desired results and those provided by the fuzzy system. An example is the use of the mean squared error or normalized mean squared error as energy functions. In the context of systems identification, besides the mean squared error, data regularization indicators can be added to the energy function in order to improve the system response in presence of noises (from training data) (Guillaume, 2001). In the absence of a tuning set, such as happens in parameters adjustment of a process controller, the energy function can be defined by functions that consider the desired requirements of a particular design (Wan, Hirasawa, Hu & Murata, 2001), i.e., maximum overshoot signal, setting time, rise time, undamped natural frequency, etc. From this point of view, this article presents a new methodology based on error backpropagation for the adjustment of fuzzy inference systems, which can be then designed as a three layers model. Each one of these layers represents the tasks performed by the fuzzy inference system such as fuzzification, fuzzy rules inference and defuzzification. The adjustment procedure proposed in this article is performed through the adaptation of its free parameters, from each one of these layers, in order to minimize the energy function previously specified. In principle, the adjustment can be made layer by layer separately. The operational differences associated with each layer, where the parameters adjustment of a layer does not influence the performance of other, allow single adjustment of each layer. Thus, the routine of fuzzy inference system tuning acquires a larger flexibility when compared to the training process used in artificial neural networks. This methodology is interesting, not only for the results presented and obtained through computer simulations, but also for its generality concerning to the kind of fuzzy inference system used. Therefore, such methodology is expandable either to the Mandani architecture or also to that suggested by Takagi-Sugeno.