Local and global identification and interpretation of parameters in Takagi–Sugeno fuzzy models

This paper addresses the interpretation of parameters in Takagi-Sugeno (TS) fuzzy models. The analysis is presented for the dynamic gain and steady-state representation, but it holds for parameters related to the dynamics as well. The TS model interpolates between local linear models. The overall gain obtained by interpolating the gains of the local models can be interpreted as the local dynamic gain of the entire fuzzy model. This locally interpreted gain is not identical to the dynamic gain obtained by linearization of the fuzzy model at the considered equilibrium. We analyze the origin of this difference with regard to the applied identification method. In order to keep the analysis simple and transparent, a fuzzy model of a Hammerstein system is studied. The results show that fuzzy models obtained by local identification (weighted least squares for each rule) typically yields a poor steady-state representation and the model can only be locally interpreted. On the contrary, a fuzzy model obtained by global identification (one least-square solution for the entire model) can result in a qualitatively bad local interpretation of the gain even though approximates the real process well. Therefore, this model can only be used for prediction or local linearization through Taylor expansion. It is shown that the difference between the globally and locally interpreted gain can be reduced by using a priori knowledge in global identification. The steady-state representation of fuzzy models obtained by local identification can be improved by using inference based on the smoothed maximum operator (instead of the weighted mean).