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## Fuzzy Systems

This chapter introduces the basic concepts on fuzzy systems that will be used in the thesis, together with information on their use in control and a description of the main representative models. Sections 2.1, 2.2 and 2.3 present the basic concepts of fuzzy sets, fuzzy variables and fuzzy inference respectively. Section 2.4 introduces the general model of a fuzzy system, with details on the two main representative instances: the Mamdani and the Takagi Sugeno models. The different types of partitions of the input space are discussed in section 2.5. Sections 2.6 and 2.7 present representative implementations of fuzzy systems and recurrent fuzzy systems applied to control.

**Fuzzy Sets**

Fuzzy sets were introduced by Zadeh [156, 157, 158] as a generalization of the concept of a regular set. In a regular set, or crisp set, the membership function assigns the value 1 to the elements that belong to the set and 0 to the elements that do not belong to it. In fuzzy sets, elements can have different degrees of membership to the set.

Formally, a fuzzy set A is characterized by a membership function µA that assigns a degree (or grade) of membership to all the elements in the universe of discourse U [101]:

µA(x) : U → [0, 1] (2.1)

The membership value is a real number in the range [0, 1], where 0 denotes definite no membership, 1 denotes definite membership, and intermediate values denote partial mem-bership to the set. In this way, the transition from non membership to membership in a fuzzy set is gradual and not abrupt. Figure 2.1 shows an example of a fuzzy set.

Let us now briefly recall some useful definitions. Let A be a fuzzy set defined in U [21]:

**Definition 1** The support of a fuzzy set is defined as the set of elements that have a non-zero membership value (see figure 2.2): support (A) = {x ∈ U, µA(x) > 0} (2.2)

**Definition 2** A fuzzy set is normal if there is at least a point in its domain with membership one:

(A is normal) ⇔ (∃x ∈ U, µA(x) = 1) (2.3)

**Definition 3** The elements of the universe with a membership value of 0.5 are called crossover points (see figure 2.2): (x ∈ U is a crossover point for A) ⇔ (µA(x) = 0.5) (2.4)

**Definition 4** A fuzzy singleton is a normal fuzzy set with a single point support (see figure 2.2):

(A is a singleton) ⇔ (support(A) = {x}, µA(x) = 1) (2.5)

The operations on fuzzy sets are defined in terms of their membership functions. Let A and B be two fuzzy sets in U , the operations of union, intersection and complement in their most usual form are defined as follows [95](see figure 2.3):

**Definition 5** The union of A and B is defined as: C = A ∪ B µC (x) = max {µA(x), µB (x)} (2.6)

**Definition 6** The intersection of A and B is defined as: C = A ∩ B µC (x) = min {µA(x), µB (x)} (2.7)

### Linguistic variables

A linguistic label is a concept associated to a fuzzy set, defined in terms of its membership function. A linguistic variable is a system variable with its values defined with linguistic labels (or linguistic terms) [158]. As an example, figure 2.4 shows the definition of a linguistic variable with three linguistic terms.

It is desirable, and in most cases implicitly assumed that the linguistic terms associated to a linguistic variable cover the complete input space [95]. Given a linguistic variable V defined in U with terms A1, . . . , An:

**Definition 8** The finite set of fuzzy subsets {A1, . . . , An} of U is a finite fuzzy partition of U

if [120]:

1. i=1 µAi (x) = 1 ∀x ∈ U (2.9)

2. every Ai is normal

Note that the definition of the fuzzy variable angle defined in figure 2.4 defines a finite fuzzy partition of the input space, since the summation of the membership to all different terms for a single point in the domain is always 1, and all fuzzy sets are normal. The concept of finite fuzzy partition is important, since it guarantees that every value in the universe U belongs in the same degree to one or more fuzzy sets when compared with other values, or in other words, the semantic level of representation by linguistic terms is the same for every element in U .

Definition 9 The partition part (V ) defined by V is ǫ-complete if every element in the universe has a membership not smaller than ǫ: (part (V ) is ǫ-complete) ⇔ (∀x ∈ U ∃Ai µAi (x) ≥ ǫ) (2.10)

The partition shown in figure 2.4 is 0.5-complete since no value in the input domain can get a membership smaller than the crossover points.

**Definition 10** The partition part (V ) defined by V is consistent if membership 1 to one fuzzy set implies membership 0 to all other fuzzy sets:

(part (V ) is consistent) ⇔ ((∀x ∈ U µAi (x) = 1) =⇒ (µAj (x) = 0, ∀j = i)) (2.11)

The partition shown in figure 2.4 is consistent since for all input values that have membership 1 to a fuzzy set, the membership is 0 for all other fuzzy sets.

**Fuzzy inference**

Relations between fuzzy variables are represented by fuzzy rules of the form:

if antecedents then consequents (2.12)

where generally the antecedents consist of propositions of the form x is A, where x is a fuzzy variable and A one of its terms, and the consequents consist of proposition of the form y is B, where y is a variable and B one of its terms. It is possible to draw conclusions from a fuzzy rule by applying a rule of inference, a concept that is defined below.

Let A1, . . . , An be fuzzy sets defined respectively in U1, . . . , Un:

**Definition 11** The Cartesian product [95] of A1, . . . , An is a fuzzy set defined in U1 × . . . × Un with membership function: µA1 ×…×An (x) = min {µA1 (x), . . . , µAn (x)} (2.13)

A fuzzy relation denotes a degree of presence or absence of a combination of elements belonging to different sets. Formally:

**Definition 12** A n-ary fuzzy relation R between A1, . . . , An is a fuzzy set defined in U1 ×. . . ×Un with membership function µR(x1, . . . , xn) [95].

Definition 13 Given the fuzzy relations R and S defined respectively in U × V and V × W , the fuzzy max-min composition R ◦ V is the fuzzy relation in U × W defined with the following membership function [95]: µR◦V (u, w) = supv∈V (min (µR(u, v), µS (v, w))) (2.14)

where sup selects the maximum value (supremum) of a set of elements and min computes the minimum value of the two arguments. The composition is formed by pairs of elements of U × W with their membership computed as the maximum of the minimum values of the membership of the individual elements when combined with all the elements from V .

The compositional rule of inference [95, 158] is defined as follows:

**Definition 14** If R is a fuzzy relation in U × V and A is a fuzzy set in U , then the fuzzy set B in V induced by R is defined as: B=A◦R

**An example of fuzzy inference**

Since the concepts of fuzzy inference and composition of fuzzy relations are central to the operation of fuzzy systems, this section presents an example intended to clarify them. In this example, two fuzzy relations defined in the context of a postal packages transportation com-pany will be introduced, relating the size of the packages with their weight and transportation cost.

The size of a postal package will be described with the linguistic variable size defined with the three linguistic labels small, medium and large, referring each one of them to the concept of a package being of small size, medium size and large size respectively. In the same way, the fuzzy variable weight will be described with the three linguistic labels light, medium and heavy to represent three different grades of weight. Finally, a third fuzzy variable labeled cost defined with the linguistic labels cheap, medium and expensive, will be used to represent three different categories of prices.

#### Fuzzy Systems

A fuzzy system [159] is a computing framework based on the concepts of the theory of fuzzy sets, fuzzy rules and fuzzy inference. Figure 2.5 illustrates its four main components: a knowledge base, a fuzzification interface, an inference engine (or decision making unit) and a defuzzification interface [8].

The knowledge base consists of a rule base defined in terms of fuzzy rules, and a data base that contains the definitions of the linguistic terms for each input and output linguistic variable. The fuzzification interface transforms the (crisp) input values into fuzzy values, by computing their membership to all linguistic terms defined in the corresponding input do-main. The inference engine performs the fuzzy inference process, by computing the activation degree and the output of each rule. The defuzzification interface computes the (crisp) output values by combining the output of the rules and performing a specific transformation (more details are presented in following sections).

Fuzzy systems are expected to be able to produce an output for every input. This property is called completeness [95] and is defined as follows:

**Definition 15** A fuzzy system is complete, or it has the completeness property, if its rule base covers all possible combination of input values.

Depending on the fuzzy sets used in the definition of the rules in the rule base, the fuzzy systems can be classified as descriptive or approximative fuzzy systems [63]:

**Definition 16** A fuzzy system is a descriptive fuzzy system if its rule base refers to fuzzy sets externally defined in the database and shared by all the rules.

**Definition 17** A fuzzy system is an approximative fuzzy system if each rule defines its own fuzzy sets.

**Fuzzy Systems**

Approximative fuzzy systems can potentially reach more accuracy in the approximation when different fuzzy sets are used for each rule since they provide more degrees of freedom. How-ever, descriptive fuzzy systems have better linguistic interpretation providing higher knowl-edge comprehensiveness.

Fuzzy systems can be classified in different categories, depending on the shape of the rules and the type of operators used for implementing the modules. The most widely used are the Mamdani and the Takagi-Sugeno models, being also relevant the SAM from Kosko [36, 89, 112] and the Tsukamoto model [8]. All of them can be implemented as approximative or descriptive fuzzy systems. Next sections introduce the characteristics of the main models in terms of a MISO (multiple input single output) structure. There is no loss of generality since a MIMO (multiple input multiple output) fuzzy system can always be decomposed in a group of MISO fuzzy systems [159, 95], as will be shown in section 2.4.3.

**The Mamdani Fuzzy System**

A standard MISO Mamdani fuzzy system [95, 8, 53] has n input variables x1, x2, . . . , xn and one output variable v. Each input variable xi is fuzzified by pi fuzzy sets Aij (pi ≥ 1, 1 ≤ i ≤ n, 1 ≤ j ≤ pi) whose membership functions µiki (ki = 1, 2, . . . , pi) are arbitrarily defined. The output variable is defuzzified by q fuzzy sets Bi (q ≥ 1). A complete Mamdani fuzzy system has one rule for each combination of input fuzzy sets. As an example, the definition of the k-th (1 ≤ k ≤ ω) rule Rk follows:

if x1 is Ak1 and . . . and xn is Akn then v is Bk where the Aki are the antecedent fuzzy sets (or antecedent linguistic terms) associated to each input variable xi, with Aki ∈ {Ai1, . . . , Aipi } and Bk ∈ {B1, . . . , Bq }.

The linguistic terms for the input variables define regions on the input space where the propositions defined by the consequents apply. Usually the output is computed by the max-min inference, which establishes that the activation degree of each rule is computed by ap-plying the min operator to the antecedents, and the operator max to aggregate the outputs to get the final output value. Figure 2.6 shows an example of the application of the max-min inference. Formally, the degree of activation wk of the rule Rk is computed as the minimum membership value of all antecedent linguistic terms:

wk = min µAk (x) (2.18)

**Table of contents :**

**Introduction **

**1 Evolutionary Computation **

1.1 Overview

1.2 History

1.2.1 Genetic algorithms

1.2.2 Evolution strategies

1.2.3 Evolutionary programming

1.2.4 Genetic programming

1.3 Representations

1.3.1 Bit strings

1.3.2 Vectors of real numbers

1.3.3 Permutations

1.3.4 Parse trees

1.3.5 Production rules

1.3.6 Specific representations

1.4 Selection

1.4.1 Deterministic selection

1.4.2 Proportional selection

1.4.3 Rank selection

1.4.4 Tournament selection

1.5 Variation operators

1.5.1 Recombination

1.5.2 Mutation

1.5.3 Parameters setting

1.6 Conclusions

**2 Fuzzy Systems **

2.1 Fuzzy Sets

2.2 Linguistic variables

2.3 Fuzzy inference

2.3.1 An example of fuzzy inference

2.4 Fuzzy Systems

2.4.1 The Mamdani Fuzzy System

2.4.2 The Takagi-Sugeno Fuzzy System

2.4.3 MIMO vs. MISO fuzzy systems

2.5 Partition of the Input Space

2.6 Fuzzy Controllers

2.6.1 The ANFIS model

2.6.2 The GARIC model

2.6.3 The genetic learning model of Hoffmann

2.6.4 The SEFC model

2.7 Recurrent fuzzy controllers

2.7.1 Fuzzy temporal rules

2.7.2 The RFNN model

2.7.3 The RSONFIN model

2.7.4 The DFNN model

2.7.5 The TRFN model

2.8 Conclusions

**3 The Voronoi Approximation **

3.1 Basic Computational Geometry Concepts

3.1.1 Voronoi Diagrams

3.1.2 Delaunay Triangulations

3.1.3 Barycentric Coordinates

3.2 The Voronoi approximation

3.2.1 Examples

3.3 Evolution of Voronoi approximators

3.3.1 Representation

3.3.2 Recombination

3.3.3 Mutation

3.3.4 Experimental Study of the Variation Operators

3.4 Function approximation

3.4.1 First experiment

3.4.2 Second experiment

3.5 Conclusions

**4 Voronoi-based Fuzzy Systems **

4.1 The Voronoi-based Fuzzy System

4.2 Properties

4.2.1 Fuzzy finite partition

4.2.2 ǫ-completeness property

4.3 Evolution of Voronoi-based fuzzy systems

4.3.1 Representation

4.3.2 Properties

4.4 Control experiments

4.4.1 Cart-pole system

4.4.2 Evolutionary robotics

4.5 Conclusions

**5 Recurrent Voronoi-based Fuzzy Systems **

5.1 Recurrent Voronoi-based fuzzy systems

5.2 Properties

5.2.1 Recurrent rules representation

5.2.2 Recurrent units with semantic interpretation

5.3 Evolution of recurrent Voronoi-based fuzzy systems

5.4 Experiments

5.4.1 System identification

5.4.2 Evolutionary robotics

5.5 Conclusions

**Conclusions **

**Bibliography **

Index