Gustafson-kessel (G-K) clustering approach of T-S fuzzy model for nonlinear processes

The dynamics of pH process is highly nonlinear, time-varying with change in gain of several orders. It is very difficult to investigate the dynamic behavior of such systems using conventional modeling techniques. An effective approach is to partition the available data into subsets and approximate each subset by a simple piecewise linear model. Fuzzy clustering can be used as tool to partition the data where transitions between the subsets are gradual. In this paper, Takagi-Sugeno (T-S) model is developed for a nonlinear function and a pH process using fuzzy c-means and Gustafson-Kessel (G-K) clustering techniques. The result shows that G-K algorithm gives satisfactory results compared to c-means algorithm. The performance of the proposed model based on G-K algorithm is also compared with the results obtained by NARX and conventional fuzzy modeling techniques. The comparison shows the superiority of the proposed model.

1 INTRODUCTION

Modeling of the pH process is considered to be a difficult task because one needs to have knowledge about the components and their nature in the process stream in order to model its dynamics using conventional techniques. In the modeling aspect, rigorous models from first principles involving the material balance and equilibrium equations were established in [1] and later extended in [2] through the concept of the reaction invariant, and more complicated situations were considered in [3]. Due to the susceptibility to change in operating point, varying gain and load disturbances, the performance of the practical processes deviates from conventional modeling output [4]. Fuzzy identification is an effective tool for the approximation of uncertain nonlinear systems on the basis of measured data [5].