Takagi-Sugeno Fuzzy Models

Takagi and Sugeno (1985) proposed models where the consequent part of the rule is described by a linear regression model. These models are easier to identify because each rale describes a fuzzy region in which the output depends on the inputs in a linear maimer. An example of such a model is shown in Eqn. (28.3)  

The model as shown in Eqn. (28.4) can represent mnlti-inpnt, mnlti-outpnt static and 

N is the number of rules in the rule base, is the degree of fulfillment (or degree of membership) of the antecedent of rule i. If the antecedent is a multidimensional proposition, such as shown in Eqn. (28.6)  

In Fig. 28.2 an example with three TS rules is shown constituting a local linear description of the functional relationship between 7 and x. 

Rather than assuming three rules, one could also decide to make a more accurate approximation by dividing the x-region into five sections and create a linear model in each of these sections. In addition, one could also assume different membership functions this will be illustrated in section 28.4. 

---

Let us assume that we like to approximate the function z = shown in Fig. 28.3, in the regions = [-2, 2] by a Takagi-Sugeno fuzzy model. 

Dragan K (2002) Design of adaptive takagi-sugeno-kang fuzzy models. Appl Soft Comput 2(2) 89-103... 

The algorithms summarized in the previous two sections can easily be applied to identify the consequence part of a fuzzy Takagi-Sugeno model. 

Tavanai et al (2005) developed a fuzzy system to model the color yield k/s) of polyester (PET) yarns as a function of time, temperature, and dispersed dye concentration. A typical rule for the color yield was if (temperature is low) and (time is low) and (concentration is low), then (color yield is very low). Only a small number of rules were necessary to create a viable model. In a later paper, Nasiri et al (2011) developed this model by further offering a compromise between the accuracy of classic Takagi-Sugeno systems and the interpretability of Mamdani models by applying genetic algorithms to improve the fuzzy rules. 

The output of the model of the Takagi-Sugeno inference is therefore a linear combination of input values with coefficients non-linearly dependent on these values. In a simple manner, this extends the model to nonlinear ARX, where nonlinearity is determined by the membership function of the fuzzy inputs. A similar approach may be used with neural MLP instead of linear ARX local models (Fig. 3.8). 

Takagi, T. and SuGENO, M., 1985, Fuzzy identification of systems and its apvplications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, SMC-15,... 

---