Discussion on Accuracy of Approximation with Smooth Fuzzy Models

The structure of fuzzy model impacts how well it approximates the nonlinear function, and how many rules are required to gain the desired accuracy. The most of the earlier works rely on diminishing the higher derivation of the fuzzy model in front of the higher derivatives of the real system. However, the smooth compositions are m-time differentiable and will not diminish. This has motivated to derive the relation of required fuzzy rules with the arbitrary accuracy for function approximation through the smooth fuzzy model. The originality of the work is that the approximation error and the number of required fuzzy rules in this paper, rely on the structure of the fuzzy model and the involved s-t compositions, beside the nonlinear properties of the real plant, through a reliable mathematical formulation. Hence, we have presented a prediction-correction algorithm to include all the main factors. It is proved that number of the required rules are lower than those of the earlier works to gain the same level of model accuracy.

I. Introduction

Fuzzy models have been used for function approximation during the years and we are witnessing variety applications of fuzzy modeling based techniques in engineering for identification and control of nonlinear and complex systems [1], [2]. Although the primary motivation of interest in the fuzzy models has been their capacity to employ the knowledge and experience of operator for improvement of the system's operation [3], however, when the number of systems states, systems' dimension and number of fuzzy rules increases, the linguistic interpretation of fuzzy systems fades out and they mainly become interesting for their approximation properties [2], [4].