Local and global identification and interpretation of parameters in Takagi–Sugeno fuzzy models J. Abonyi R. Babuˇ ska Delft University of Technology, Department of Information Technology and Systems Control Engineering Laboratory, P.O.Box 5031 2600 GA Delft, The Netherlands Abstract— This paper addresses the interpretation of parameters in Takagi-Sugeno (TS) fuzzy models. The analysis is presented for the dy- namic gain and steady-state representation, but it holds for parameters related to the dynamics as well. The TS model interpolates between lo- cal 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 consid- ered 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 represen- tation 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). I. I NTRODUCTION Fuzzy identification is an effective tool for the approxi- mation of uncertain nonlinear dynamic systems on the ba- sis of measured input-output data. The Takagi-Sugeno (TS) fuzzy model is often used to represent nonlinear dynamic systems, by interpolating between local linear, time-invariant (LTI) ARX models. One of the great challenges in the identification of nonlinear dynamic systems is the selection of a suitable model structure. This modeling step can be based on the use of a priori knowl- edge about the dynamic behaviour of the process. When no a priori knowledge is available, information about the static characteristic of the system can be extremely useful in order to check the qualitative agreement between the steady-state behaviour of the process and its model [2], [9]. 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. As the Takagi-Sugeno fuzzy model interpolates between local linear models, the steady-state dynamic gain of the fuzzy model can be locally interpreted as the gain of the interpolated linear model. This locally interpreted gain is not identical to the dynamic gain of the fuzzy model. Because of some undesir- able properties of standard TS inference method, it can happen that the globally interpreted gain of the fuzzy model becomes negative even though all the gains of all the local models are positive [4]. For the identification of the consequent LTI models of the fuzzy model, a global or a local approach can be followed. With the global approach the parameters of the rule conse- quents are estimated within one identification problem. There- fore, the generated local models are not necessary the local linearizations of the nonlinear system. The local identification method forces the local linear models to fit the system sepa- rately and locally, resulting in rule consequents that are local linearizations of the nonlinear system [3]. The aim of this paper is to analyze the difference between the locally and globally interpreted steady-state gain with re- gard to the applied global or local identification method. In order to keep the analysis simple and transparent, a fuzzy model of a Hammerstein system is studied. These systems consist of a series connection of a static nonlinearity and a lin- ear dynamics [7], [6]. If the gain of the dynamic part is one, the static nonlinearity represents the steady-state behaviour of the process. Therefore, the easiest way to examine how fuzzy models are able to represent the steady-state behaviour of non- linear processes is to analyze a fuzzy model of a Hammerstein system. The paper is organized as follows. In Section 2, the ap- plied TS fuzzy model of dynamic systems is introduced. Sec- tion 3 addresses the steady-state behaviour of this model. In Section 4, the fuzzy modeling of Hammerstein systems is de- tailed. Section 5 studies the effect of the applied identifica- tion method through an example about modeling a Hammer- stein system with polynomial nonlinearity. In this section two methods will be presented in order to reduce the difference be- tween locally an globally interpreted gains. Conclusions are given in Section 6. II. THE TAKAGI-SUGENO FUZZY MODEL OF A DYNAMIC SYSTEM This paper presents a study of steady-state behaviour of nonlinear dynamic system represented by a fuzzy model as a Nonlinear AutoRegressive with eXogenous input (NARX) model. This model establishes a nonlinear relation between the past inputs and outputs and the predicted output: (1) Here, and are the maximum lags considered for the output and input terms, respectively, is the discrete dead time, and represents the mapping of the fuzzy model.