Introduction to Fuzzy Systems
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Abstract
The following chapter describes the basic concepts of fuzzy systems and approximate reasoning. The study focuses mainly on fuzzy models based on Zadeh’s compositional rule of inference. The presentation begins with an introduction of fundamental ideas of fuzzy conditional (ifthen) rules. A collection of fuzzy ifthen rules formulates the socalled knowledge base, which formally represents the knowledge to be processed during approximate reasoning. The subsequent sections present formal definitions related to the compositional rule of inference and approximate reasoning using a knowledge base. Theoretical considerations are supplemented with practical examples of fuzzy systems as the foundation of many modern structures. The description includes fuzzy systems proposed by Mamdani and Assilan, Takagi, Sugeno and Kang, and Tsukamoto.
2.1 Introduction
The main inspiration behind the introduction of fuzzy sets theory was the necessity for modeling realworld phenomena, which are inherently vague and ambiguous. Human knowledge about complex problems can be successfully represented using the imprecise terms of natural language. The theories of fuzzy sets and fuzzy logic provide formal tools for mathematical representation and efficient processing of such information.
The typical structure of a fuzzy system (Fig. 2.1) consists of four functional blocks: the fuzzifier, the fuzzy inference engine, the knowledge base, and the defuzzifier. Both linguistic values (defined by fuzzy sets) and crisp (numerical) data can be used as inputs for a fuzzy system. If crisp data are applied, then the inference process is preceded by fuzzification, which assigns the appropriate fuzzy set to the nonfuzzy input. The values of input variables are mapped into linguistic values of the output variable by means of the appropriate method of approximate reasoning (inference engine) using expert knowledge, which is represented as a collection of fuzzy conditional rules (knowledge base). In addition to the linguistic values, the numerical data may be required as the fuzzy system output. In such cases defuzzification methods are used, which assign the representative crisp data to the resultant output fuzzy set.
Practical applications of fuzzy systems include problems for which the complete mathematical description is unavailable, or where the usage of the precise (nonfuzzy) model is uneconomical or highly inconvenient. The ability to process inaccurate information makes a fuzzy system an excellent tool, for example, for control processes [12, 19], system identification [11, 20], decision support [24, 33], and signal and image processing [4, 23].
In the following sections only static fuzzy systems (i.e., systems where the outputs are determined only on the basis of the current input values) are considered. Included are concepts of knowledge representation in the form of fuzzy conditional rules, the idea of approximate reasoning, and the description of basic structures of fuzzy systems.
2.2 Fuzzy Conditional Rules
 P1:
\(\quad \text {if } a\le c, \text {then } \varPsi \left( a,b\right) \ge \varPsi \left( c,b\right) \),
 P2:
\(\quad \text {if } b\le c, \text {then } \varPsi \left( a,b\right) \le \varPsi \left( a,c\right) \),
 P3:
\(\quad \varPsi \left( 0,b\right) = 1\),
 P4:
\(\quad \varPsi \left( a,1\right) = 1\),
 P5:
\(\quad \varPsi \left( 1,0\right) = 0\),

by using knowledge of a human expert or based on the physical laws describing the phenomenon (white box modeling),

by automatically extracting the rules based on numerical data representing the relationship between inputs and outputs of the phenomenon (black box modeling),

mixed, where part of the knowledge is derived from a human expert and part from automated extraction (grey box modeling).
The possible applications of a fuzzy system depend, however, not only on the properly defined knowledge base, but also on the appropriate design of an inference engine.
2.3 Approximate Reasoning
\(\text {Premise I (fact):}\)  p 
\(\text {Premise II (rule):}\)  \(p \Longrightarrow q\) 
\(\text {Conclusion:}\)  q 
\(\text {Premise I (fact):}\)  \(p^{\prime }\) 
\(\text {Premise II (rule):}\)  \(p \Longrightarrow q\) 
\(\text {Conclusion:}\)  \(q^{\prime }\) 
\(\text {Premise I (fact):}\)  X \(\mathbf {is}\) \(L_{A^{\prime }}\) 
\(\text {Premise II (rule):}\)  \(\mathbf {if}\) X \(\mathbf {is}\) \(L_A\), \(\mathbf {then}\) Y \(\mathbf {is}\) \(L_B\) 
\(\text {Conclusion:}\)  Y \(\mathbf {is}\) \(L_{B^{\prime }}\) 
2.3.1 Compositional Rule of Inference
Equations (2.30) and (2.31) define the membership function of a fuzzy set representing the resulting conclusion of an inference using only one fuzzy ifthen rule. For a knowledge base consisting of many fuzzy conditional statements it is necessary to combine conclusions from all individual rules.
2.3.2 Approximate Reasoning with Knowledge Base

compositionbased inference (first aggregate then infer: FATI), where first a combination of all rules from the knowledge base is constructed, and then inference using the supremumstar composition is conducted,

individual rulebased inference (first infer then aggregate: FITA), in which the first step involves inference using the supremumstar composition for each of the rules individually and then, a combination of inference results is performed.
2.3.3 Fuzzification and Defuzzification
2.4 Basic Types of Fuzzy Systems
Due to a wide range of possible applications there are many different types of fuzzy systems that have been proposed in the literature thus far [4, 16, 22, 23, 31]. But new solutions characterized by decreased computation complexity, improved modeling quality, or greater ease of the linguistic interpretation of the inference results are still the topic of research. The model proposed by E.H. Mamdani and S. Assilan [19] is generally regarded as the first fuzzy system presented in the literature. Currently, it can be considered as the foundation of the fuzzy models family based on ifthen rules with fuzzy sets in antecedents as well as consequents.
2.4.1 Mamdani–Assilan Fuzzy Model
Approximate reasoning without the defuzzification necessity was presented in papers by Takagi and Sugeno [27] and Sugeno and Kang [25]. The proposed model, called the Takagi–Sugeno–Kang fuzzy system (TSKFS), is described in the following subsection.
2.4.2 Takagi–Sugeno–Kang Fuzzy System
Equation (2.58) can be interpreted as a mixture of experts, each modeled by a single fuzzy rule. Each rule defines the relationship between outputs and inputs of the system in the relevant input range. The weighted average of statements from all local experts (rules) determines the reasoning result. The weight, represented by the firing strength of the rule, specifies the influence level of a single expert on the final inference outcome.
An example of TSKFS inference with two inputs and two conditional fuzzy rules is shown in Fig. 2.4. The main advantage of the TSKFS is the low computational effort required to determine the numerical output of the system as the inference process does not involve defuzzification. However, it does not allow for the application of different interpretations of the fuzzy rules and different types of aggregation operators. This is due to the application of singletons in the rules consequents. The artificial neural network based fuzzy inference system (ANNBFIS) [17] is devoid of such disadvantages. The ANNBFIS combines the benefits of the usage of a fuzzy set in the rule consequent (as in the MAFS) together with the dependency of the consequent location on system inputs (as in the TSKFS) [4, 15, 16]. Another extension of the TSKFS is the Tsukamoto fuzzy system (TFS) [28]. The main difference between TSKFS and TFS is the method of determining the singleton location in the consequent of the fuzzy rule. In TFS it is defined using a monotonic function as well as a firing strength of the rule.
2.4.3 Tsukamoto Fuzzy System
The TFS is rarely used due to the difficulty in obtaining the conditional fuzzy rules from a human expert in the form (2.63). For the same reasons the Baldwin fuzzy system (BFS) [1, 2] is difficult to apply in practice. The BFS represents a different approach to fuzzy modeling, which is not based on Zadeh’s compositional rule of inference but on reasoning using fuzzy truth value restrictions. The literature describes many other interesting proposals of fuzzy models, including those based on intervalvalued fuzzy sets and type2 fuzzy sets. A detailed overview can be found, for example, in [13, 18, 29, 30].
2.5 Summary
In this chapter we discussed basic problems related to the idea of fuzzy systems based on the Zadeh compositional rule of inference. The presentation started with explaining the concepts of the linguistic variable and fuzzy conditional statement. Next, different types of the fuzzy ifthen rules and various methods of their mathematical representation were presented. Also, an overview of the compositional rule of inference proposed by Zadeh was introduced. General theoretical considerations on approximate reasoning were supplemented with examples of elementary fuzzy models. We described the basic solutions being the foundation of many modern constructions including fuzzy systems of Mamdani–Assilan, Takagi–Sugeno–Kang, and Tsukamoto.
Notes
Acknowledgements
This work was supported by the Ministry of Science and Higher Education funding for statutory activities (BK220/RAu3/2016).
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