5.12 Fuzzy ControL SySteM 145

Figure 5.14 shows fuzzy rules and their relationship in the portfolio

management for a ﬁnancial application (Sajja, 2006).

5.12 FUZZY CONTROL SYSTEM

A fuzzy control system is based on fuzzy logic. Given membership func-

tions that map crisp values to fuzzy values and vice versa, the machine can

interact with the human being in friendlier way. Fuzzy logic systems address

the imprecision of the input and output variables by deﬁning fuzzy numbers

and fuzzy sets, which can be expressed in linguistic variables (e.g., high, low,

and average). Fuzzy controllers are the most important applications of fuzzy

theory. They work rather differently from conventional controllers because

the experts’ knowledge is used for more effective and human-like control

instead of a pure, crisp, mathematical approach (like differential equations)

to describe a system.

This knowledge can be expressed in a natural way using linguistic variables,

which are described by fuzzy sets. Fuzzy logic-based control systems are also

useful for some of the control applications where a systematic mathematical

model does not exist. Instead of describing control strategy as mathematical

equations, control is expressed as a set of linguistic rules. A control system can

be abstracted as a box with inputs ﬂowing into it and outputs emerging from

Figure 5.14

Fuzzy rules and

relationships

510152025303540

20

40

60

80

100

120

1

0.5

0

0 0.5 1

Less Moderate More

Low

Average

High

Investment amount in thousands

If profit is high

then invest more

Profit in percentage

Membership

degree

Membership degree

76473_CH05_Akerkar.indd 145 8/11/09 10:16:58 AM

146 Chapter 5 Fuzzy LogiC

it. The process of designing a fuzzy control system can be described using fol-

lowing steps:

Step 1: Identify the principal input, output, and process tasks.

Step 2: Identify linguistic variables used, and deﬁne fuzzy sets and mem-

bership accordingly.

Step 3: Use these fuzzy sets and linguistic variables to form procedural

rules.

Step 4: Determine the defuzziﬁcation method.

Step 5: Test the system and modify if necessary.

The components of a typical control system are shown in Figure 5.15.

Mamdani and Assilian (1975) designed a fuzzy control system for a

steam engine. The purpose was to maintain a constant speed by control-

ling the pressure on pistons by adjusting the heat supplied to a boiler. After

that, fuzzy controllers were developed for air conditioners, video cameras,

washing machines, and so on. The fuzzy control system can be considered a

nonlinear static function that maps controller inputs on controller outputs.

Fuzzy controllers are used for systems where a preferred response must be

maintained based on whatever inputs are received. Naturally, inputs to the

system can alter the state of the system, which causes a change in response.

Thus, the duty of the controller is to take appropriate action by providing a

set of inputs to ensure the preferred response.

A fuzzy controller consists of four main components:

Fuzzy knowledge base: The knowledge base consists of fuzzy rules for 1.

the system that represent the knowledge and experience of a human

expert. For example, “If the temperature is fairly high and the pressure is

very low, then the output is medium.”

Figure 5.15

A fuzzy controller

Output

Input

Process

Fuzzy

rules

Fuzzy sets and

definitions of

membership

functions

User

interface

Machine

interface

Fuzzy values for

human interface

Crisp values for

machine interface

Interpretation

User

Instrument

76473_CH05_Akerkar.indd 146 8/11/09 10:16:58 AM

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