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## Über dieses Buch

During the past few years two principally different approaches to the design of fuzzy controllers have emerged: heuristics-based design and model-based design. The main motivation for the heuristics-based design is given by the fact that many industrial processes are still controlled in one of the following two ways: - The process is controlled manually by an experienced operator. - The process is controlled by an automatic control system which needs manual, on-line 'trimming' of its parameters by an experienced operator. In both cases it is enough to translate in terms of a set of fuzzy if-then rules the operator's manual control algorithm or manual on-line 'trimming' strategy in order to obtain an equally good, or even better, wholly automatic fuzzy control system. This implies that the design of a fuzzy controller can only be done after a manual control algorithm or trimming strategy exists. It is admitted in the literature on fuzzy control that the heuristics-based approach to the design of fuzzy controllers is very difficult to apply to multiple-inputjmultiple-output control problems which represent the largest part of challenging industrial process control applications. Furthermore, the heuristics-based design lacks systematic and formally verifiable tuning tech­ niques. Also, studies of the stability, performance, and robustness of a closed loop system incorporating a heuristics-based fuzzy controller can only be done via extensive simulations.

## Inhaltsverzeichnis

### 1. Fuzzy Identification from a Grey Box Modeling Point of View

Abstract
The design of mathematical models of complex real-world (and typically nonlinear) systems is essential in many fields of science and engineering. The developed models can be used, e.g., to explain the behavior of the underlying system as well as for prediction and control purposes
P. Lindskog

### 2. Constructing Fuzzy Models by Product Space Clustering

Abstract
There are several different approaches to modeling of complex nonlinear systems. The main distinction can be made between global and local methods. Global methods describe the system under study using nonlinear functional relationships between the system’s variables. Examples are nonlinear state space models or input-output black-box models such as the popular NARX (Nonlinear AutoRegressive with eXogenous input) structure used often in connection with neural or wavelet networks. Local approaches, on the other hand, attempt to cope with complexity and nonlinearity of systems by decomposing the modeling problem into a number of simpler, in most cases, linear sub-problems (Johansen and Foss, 1993; Banerjee et al., 1995). These methods are conceptually simple and intuitively appealing, as they are close to the way human solve problems. Local models are usually more easily interpretable than complicated global models.
R. Babuška, H. B. Verbruggen

### 3. Identification of Takagi-Sugeno Fuzzy Models via Clustering and Hough Transform

Abstract
In this chapter we consider the identification of a Takagi-Sugeno fuzzy model (TS fuzzy model) [2]. This type of fuzzy model is especially useful in the area of fuzzy model-based control [10]. The TS fuzzy model is a nonlinear system model represented by fuzzy rules of the type
$${R^i}:If{x_1}isA_1^iand...and{x_m}isA_m^ithen{y^i} = a_0^i + a_1^i{x_1} + ... + a_m^i{x_m}$$
(1.1)
where R i ( i = 1, 2, …,n denotes that the i-th fuzzy rule, x j (j = 1,2,…, m) are input variables and y i is an output. Furthermore, a j i are the parameters contained in the consequent (then-part) of the i-th rule, and the A 1 i ,A 2 i ,…,A m i are the linguistic values taken by the input variables in the antecedent (if-part) of the i-th rule. The meaning of these linguistic values is defined by corresponding membership functions. As shown in (1.1) and (1.2), this fuzzy model describes a nonlinear input-output relation.
Min-Kee Park, Seung-Hwan Ji, Eun-Tai Kim, Mignon Park

### 4. Rapid Prototyping of Fuzzy Models Based on Hierarchical Clustering

Abstract
Throughout this chapter we consider the identification of a MISO (multi-input/single-output) system. The vector of system inputs x has p components, i.e., x = (x 1,…,x p ) ∈ X = (X 1 × X 2 × … × X p ) ⊆ ℝ P and for the system output y we have that yY ⊆ ℝ. The input-output data space is the product space Z = (X × Y). Furthermore, a set of sample I/O pairs is denoted as Ω = {(x t1,x t2,…x tp ), y t ), t = 1,…,N} = {(x t ,y t ) ∈ Z, t = 1,…,N}.
M. Delgado, M. A. Vila, A. F. Gomez-Skarmeta

### 5. Fuzzy Identification Using Methods of Intelligent Data Analysis

Abstract
For complex nonlinear systems like industrial processes, the creation of mathematical models for system identification is a difficult and tedious task. Moreover, many model parameters have to be measured in a painstaking and expensive effort to account for fabrication variations and specific environmental conditions. While complete analytical knowledge is rare in complex technical environments, process measurements provide a powerful source of information about their dynamic behavior.
J. Hollatz

### 6. Identification of Singleton Fuzzy Models via Fuzzy Hyperrectangular Composite NN

Abstract
In this chapter we present a method for the identification of fuzzy singleton models based on a class of Fuzzy HyperRectangular Composite Neural Networks (FHRCNNs). The prior knowledge required, besides the available input-output data, is knowledge about the input and output variables and the number of partitions dividing the ranges of the output variables.
Mu-Chun Su

### 7. Identification of Linguistic Fuzzy Models by Means of Genetic Algorithms*

Abstract
In this chapter, we deal with the identification of linguistic fuzzy models (or Mamdani fuzzy models) for multiple-input/single-output (MISO) systems. We consider a variant of the classical linguistic fuzzy model in which there does not exist a pre-determined relationship between the linguistic values of the input and output variables and the membership functions used to define the meaning (semantics) of these linguistic values. We call this type of linguistic fuzzy model an approximative linguistic fuzzy model [4, 10].
Oscar Cordón, Francisco Herrera

### 8. Optimization of Fuzzy Models by Global Numeric Optimization

Abstract
In this chapter we deal with the identification of an optimal linguistic fuzzy model via the use of an evolutionary search technique., e.g., genetic algorithms. First some theoretical aspects of linguistic fuzzy models will be presented. Second, the identification problem in the context of fuzzy models will be discussed, and the structure identification and parameter estimation methods that form the subject of this chapter will be presented. Third, we illustrate the use of the identification method proposed by an example and discuss the results obtained. Finally, practical aspects concerning the implementation of the proposed method are examined.
V. Vergara, C. Moraga

### 9. Identification of Linguistic Fuzzy Models Based on Learning

Abstract
The learning method presented in this chapter is aimed at the identification of linguistic fuzzy models, also called Mamdani-type fuzzy models ([1, 19, 26, 30, 34]). The learning of this type of fuzzy models is, in general, done by dividing the learning process into two subtasks which are solved independently from each other: the construction of the reference fuzzy sets (fuzzy partition of the input and output spaces) on the one hand, and on the other hand, the generation of the rule base. The partition of the input and output spaces is normally obtained from human experts [27], by a clustering technique [21, 35], or simply by using a classical fuzzy partition such as triangular membership functions [4]. The rule base is then determined, either by performing a matching procedure on the set of training points (training input-output data) [13, 26, 30], by solving a fuzzy relational equation [22, 24], or by training a fuzzy neural network that represents the underlying logic of the input-output relationships in terms of fuzzy if-then rules [12, 25]. However, because of the close interdependence between the fuzzy partion and the rule base, the division of the global learning strategy into two independent subtasks may not be adequate.
Y. Nakoula, S. Galichet, L. Foulloy
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