Fuzzy models can be divided into three classes: Linguistic Models (Mamdani Models), Fuzzy Relational Models, and Takagi-Sugeno (TS) Models. Both linguistic and fuzzy relational models are linguistically interpretable and can incorporate prior qualitative knowledge provided by experts (Zadeh, 2008). TS models are able to accurately represent a wide class of nonlinear systems using a relatively small number of parameters. TS models perform an interpolation of local models, usually linear, by means of a fuzzy inference mechanism. Their functional rule base structure is well-known to be intrinsically favorable for control applications.

This chapter deals with Takagi-Sugeno (T-S) fuzzy models because of their capability to approximate a large class of static and dynamic nonlinear systems. In T-S modeling methodology, a nonlinear system is divided into a number of linear or nearly linear subsystems. A quasi-linear empirical model is developed by means of fuzzy logic for each subsystem. The whole process behavior is characterized by a weighted sum of the outputs from all quasi-linear fuzzy implication. The methodology facilitates the development of a nonlinear model that is essentially a collection of a number of quasi-linear models regulated by fuzzy logic. It also provides an opportunity to simplify the design of model predictive control. In such a model, the cause-effect relationship between control u and output y at the sampling time n is established in a discrete time representation. Each fuzzy implication is generated based on a system step response (Andone&Hossu, 2004), (Hossu et al., 2010) , (Huang et al. 2000).

IF y(n) is A0, y(n -1) is A1,….,y(n — m +1) is A’m_a,

and u(n) is B0, u(n -1) is B,…, u(n -1 +1) is Bj-1 (1)

T

THEN yl (n +1) = y(n) + ^ hjAu(n +1 — j)

j=1

where:

Aij fuzzy set corresponding to output y(n-j) in the ith fuzzy implication

Bj fuzzy set corresponding to input u(n-j) in the ith fuzzy implication

hij impulse response coefficient in the ith fuzzy implication

T model horizon

Au(n) difference between u(n) and u(n-1)

A complete fuzzy model for the system consists of p fuzzy implications. The system output y(n +1) is inferred as a weighted average value of the outputs estimated by all fuzzy implications.

£аУ (n+1)

y(n+1) = ——- (2)

TMj

j=1

where

Mj =*Aj kB’ (3)

i k

considering

®j =~^~ (4)

ZMj

j=1

p

y{n +1) = ^a’y’ (n +1)

j=i