An effective method developed by Dr. This method is an important component of the toolbox. Fuzzy logic is all about the relative importance of precision: How important is it to be exactly right when a rough answer will do?

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Fuzzy logic is a fascinating area of research because it does a good job of trading off between significance and precision — something that humans have been managing for a very long time. In this sense, fuzzy logic is both old and new because, although the modern and methodical science of fuzzy logic is still young, the concepts of fuzzy logic relies on age-old skills of human reasoning. Fuzzy logic is a convenient way to map an input space to an output space. Mapping input to output is the starting point for everything.

Consider the following examples:. With information about how good your service was at a restaurant, a fuzzy logic system can tell you what the tip should be. With your specification of how hot you want the water, a fuzzy logic system can adjust the faucet valve to the right setting. With information about how far away the subject of your photograph is, a fuzzy logic system can focus the lens for you. With information about how fast the car is going and how hard the motor is working, a fuzzy logic system can shift gears for you.

Determining the appropriate amount of tip requires mapping inputs to the appropriate outputs. Between the input and the output, the preceding figure shows a black box that can contain any number of things: fuzzy systems, linear systems, expert systems, neural networks, differential equations, interpolated multidimensional lookup tables, or even a spiritual advisor, just to name a few of the possible options.

Clearly the list could go on and on. Of the dozens of ways to make the black box work, it turns out that fuzzy is often the very best way.

Why should that be? As Lotfi Zadeh, who is considered to be the father of fuzzy logic, once remarked: "In almost every case you can build the same product without fuzzy logic, but fuzzy is faster and cheaper. The mathematical concepts behind fuzzy reasoning are very simple. Fuzzy logic is a more intuitive approach without the far-reaching complexity.

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## International Journal of Fuzzy Systems

With any given system, it is easy to layer on more functionality without starting again from scratch. Everything is imprecise if you look closely enough, but more than that, most things are imprecise even on careful inspection. Fuzzy reasoning builds this understanding into the process rather than tacking it onto the end. You can create a fuzzy system to match any set of input-output data. In direct contrast to neural networks, which take training data and generate opaque, impenetrable models, fuzzy logic lets you rely on the experience of people who already understand your system.

Fuzzy systems don't necessarily replace conventional control methods.

In many cases fuzzy systems augment them and simplify their implementation. The basis for fuzzy logic is the basis for human communication. This observation underpins many of the other statements about fuzzy logic. Because fuzzy logic is built on the structures of qualitative description used in everyday language, fuzzy logic is easy to use. The last statement is perhaps the most important one and deserves more discussion. Natural language, which is used by ordinary people on a daily basis, has been shaped by thousands of years of human history to be convenient and efficient.

Sentences written in ordinary language represent a triumph of efficient communication. Fuzzy logic is not a cure-all. When should you not use fuzzy logic? The safest statement is the first one made in this introduction: fuzzy logic is a convenient way to map an input space to an output space. If you find it's not convenient, try something else. If a simpler solution already exists, use it. Fuzzy logic is the codification of common sense — use common sense when you implement it and you will probably make the right decision. Many controllers, for example, do a fine job without using fuzzy logic.

However, if you take the time to become familiar with fuzzy logic, you'll see it can be a very powerful tool for dealing quickly and efficiently with imprecision and nonlinearity. You can create and edit fuzzy inference systems with Fuzzy Logic Toolbox software. You can create these systems using graphical tools or command-line functions, or you can generate them automatically using either clustering or adaptive neuro-fuzzy techniques.

The toolbox also lets you run your own stand-alone C programs directly. You can customize the stand-alone engine to build fuzzy inference into your own code.

## An introduction to fuzzy systems.

This theory proposed making the membership function or the values False and True operate over the range of real numbers [0. New operations for the calculus of logic were proposed, and showed to be in principle at least a generalization of classic logic.

It is this theory which we will now discuss. The notion central to fuzzy systems is that truth values in fuzzy logic or membership values in fuzzy sets are indicated by a value on the range [0. For example, let us take the statement:.

If Jane's age was 75, we might assign the statement the truth value of 0. The statement could be translated into set terminology as follows:. This statement would be rendered symbolically with fuzzy sets as:. At this juncture it is important to point out the distinction between fuzzy systems and probability.

Both operate over the same numeric range, and at first glance both have similar values: 0. By contrast, fuzzy terminology supposes that Jane is "more or less" old, or some other term corresponding to the value of 0. Further distinctions arising out of the operations will be noted below. Before we can do this rigorously, we must state some formal definitions:. Definition 1 : Let X be some set of objects, with elements noted as x.

Definition 2 : A fuzzy set A in X is characterized by a membership function. Operationally, the differences are as follows:.

### Volume 26, Number 2, April 2018

For independent events, the probabilistic operation for AND is multiplication, which it can be argued is counterintuitive for fuzzy systems. The probabilistic calculation yields a result that is lower than either of the two initial values, which when viewed as "the chance of knowing" makes good sense. However, in fuzzy terms the two membership functions would read something like "Bob is very smart" and "Bob is very tall.

The probabilistic calculation would yield the statement.

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If Bob is very smart, and Bob is very tall, then Bob is a quite tall, smart person. The fuzzy calculation, however, would yield.

## Introduction to Fuzzy Systems

Another problem arises as we incorporate more factors into our equations such as the fuzzy set of heavy people, etc. We find that the ultimate result of a series of AND's approaches 0. Fuzzy theorists argue that this is wrong: that five factors of the value 0. Fuzzy theorists argue that a sting of low membership grades should not produce a high membership grade instead, the limit of the resulting membership grade should be the strongest membership value in the collection. Other values have been established by other authors, as have other operations.