Introduction
Fuzzy
logic is a buzzword that has been heard throughout the science and engineering
industry in recent years. Although it may seem to be a new fad, fuzzy logic had
its beginnings in 1965 with a paper in the journal of Information and Control called “Fuzzy Sets” by Lotfi Zadeh. Zadeh
was then chair of the department of electrical engineering and computer
sciences at the University of California at Berkeley. The paper was the first
of many he would write about fuzzy logic and fuzzy systems.
Fuzzy logic is a fundamentally different science. This paper is meant to be a simple introduction. I will begin with some of the foundations and principles. Then I will discuss some of the math that is used in dealing with fuzzy systems. Finally I will discuss some of the applications and current implementations of fuzzy logic.
Principles
East vs. West: The Foundations of
Science
Fuzzy
Logic is a fundamentally different approach to viewing the world with a
scientific discipline. The foundations of modern science and fuzzy logic both
began thousands of years ago.
Very few
people think about how we think about science. Western science and philosophy,
at the fundamental level, are based on the notion of a “single truth”. This
means that there is a concrete, black and white answer to every question that
can be asked. In the world of computers, this is easy to see because every
piece of information is described in combinations of 1’s and 0’s. This is
called bivalence. Fuzzy logic is mutivalence. The answers to questions in fuzzy
logic lie on a continuum between 1 and 0. This means that answers are not
(always) 100% true or 100% false, but are at the same time partially true and
partially false.
It is
easy to draw parallels between the ideas of bivalence and multivalence and
religions of the eastern and western world. Many battles have been fought
because of the belief in a single truth. These are ideas mainly found in
western religions. Eastern religions, Zen Buddhism for example, accept the idea
of many truths. Zen Buddhism is one of the major religions of Japan. This is
part of the reason why the concepts of fuzzy logic were much more widely
accepted in Japan compared to the U.S. Japan has national organizations that
were set up to research and fund the research of fuzzy logic. The Laboratory
for International Fuzzy Engineering Research and the Fuzzy Logic Systems
Institute are two such organizations.
Western
people are permeated with the ideas of western science at a very early age.
This has stunted the growth of fuzzy logic research is western countries. Japan
has embraced the idea and has hundreds of companies that produce fuzzy products
today. Of the few U.S. patented fuzzy logic devices, Japan holds the patents of
about 90% of them.
Fuzzy
logic has brought to the forefront the idea that all thinking, including
scientific thinking, is based on a fundamental set of values, faith, and
philosophy. Accepting fuzzy logic means accepting a new philosophy.
Fuzzy Logic and Fuzziness
The
idea of “fuzziness” comes from the idea of vagueness. Humans view the world in
a subjective way. The dividing lines that we create to make our decisions are
vague. The answer to the question “what is art?” is a good example. This idea
is different for every person. The dividing line between art and non-art is
therefore a vague one. Other examples are: young and old, hot and cold, slow
and fast, and light and dark. These are all words in our language that have
slightly different meanings to different people.
The
information we interpret every day from the real world is very complex, yet we
are still able to form our own ideas and make decisions about it. Each person
uses their own set of “rules” and human reasoning to make decisions concerning
complex information. Fuzzy Logic is a mathematical discipline that tries to
model this reasoning and therefore make machines reason the way we do.
Fuzzy
Math
I will now introduce a few of the basic concepts of the mathematical discipline of fuzzy logic.
Fuzzy Sets
The
first and simplest concept of fuzzy logic is fuzzy sets. We think in fuzzy sets
every day. It is easier to start with an example. There are people who are tall
and people who are short. In our own minds, we formulate the idea of who is
tall and who is short. The decision of who belongs to the set of short people
and who belongs to the set of tall people is very subjective. I am 6’8” tall.
That is tall by most people’s standards, but someone who is 7’2” might not see
it that way. Every time a person looks at a crowd and decides who is tall, they
are creating what fuzzy logician’s call a fuzzy set of tall people. Membership
to the set also comes in varying degrees. People who are “really” tall and
“fairly” tall both belong to the fuzzy set of tall people, but they do not
belong to the set in the same degree. The line between tall and not tall is
fuzzy.
Many
nouns in our language actually describe a fuzzy set in our own minds. A person
may write the word house and many different such structures could come to mind.
The word chair could be anything from a crate to a lazy-boy. Each word
describes a fuzzy set. The members of the set are different for every person.
Below is a graphical representation of the example of tall and short.
As you can see from the graph, a person of a certain
height gets a certain degree of membership to a set. Some people may belong to
two sets at the same time. Some 6ft tall belongs about 40% to the tall fuzzy
set and about 40% to the short fuzzy set. Members of fuzzy sets belong to their
opposites to some degree. A person who is tall to some degree is also short to
some degree.
This shows that lines between sets are fuzzy. Some fuzzy sets have more concrete dividing lines than others. The degree of fuzziness of the lines between sets is a measure of fuzzy entropy.
Something
that people wouldn’t think of as fuzzy is a number. Numbers have a fuzzy
representation as well. Zero, for example, belongs 100% to the set of zero.
What about numbers close to zero?
What numbers are almost zero? In the
fuzzy representation of zero, numbers close to zero belong to the zero fuzzy
set to a certain degree. Again, words such as close to or almost are described
using fuzzy sets.
Fuzzy Systems and Rules
Fuzzy
sets can be grouped together and rules
can be applied to them. Rules are a series of if-then statements that specify
actions to take based on fuzzy input data. Again, this concept is best
explained with an example.
Say we
are trying to control an air conditioner. This is one of the most frequently
used examples of fuzzy control of a system. There are two “variables” in the
system. One variable is the input to the system and is simply the temperature
of the room. The other is the output fan speed, which must be faster to cool
off the room, and slower of the air is cool enough. Here is where we use
language and fuzzy sets to describe the system. Temperature ranges have certain
words that describe them. We might say that the room is “warm” or “really warm”
or “kind of warm”. All these terms can be described in the fuzzy set of warm
temperatures. The temperature range we call comfortable can be about 62 -
68 degrees. The motor speeds can be “fast” or “slow” or “really slow”,
etc. Below is the graphical representation of the fuzzy sets just described.
Notice
the middle area, where the temperature is comfortable and the motor speed is
correct, the drawn triangles are narrower. The reason is that we want finer
control near in the correct temperature range. The areas out of the correct
range need only rougher control. Now that we have sets to work with, we apply
fuzzy rules to correlate the two groups of sets. Here is a set of common sense
rules for the system:
1) If cold then stop.
2) If cool then slow.
3) If comfortable then medium.
4) If warm then fast.
5) If hot then full.
These are fuzzy rules because they correlate fuzzy
ideas like “cool” and “slow”. To get from the fuzzy input to the output, you associate
the input with the output as shown below.
Notice the squares that cover a linear path. The squares
represent the rules. They are called patches.
They cover a linear path because this is a linear system. When a specific input
comes in, it triggers the all the rules to a certain degree. In this system,
some rules will always be triggered at 0%, but this is not always the case.
Each input value does not map to a specific output value, but maps to a fuzzy
set or group of sets. Inputs map to a group of sets when the sets overlap. The
above picture shows that the input of 65 degrees belongs 100% to the medium
fuzzy set of the motor. The representation of the output set is the medium
triangle. Finally to get a specific output to feed to a device like a motor, an
average value is taken from the curve that traces the triangle. In this case,
the motor speed would be 50. This step is called defuzzification. The process
to get an output value if the input happens to belong, in some degree, to two
or more sets is as follows: Get the percentage of membership to each fuzzy set.
Graphically reduce the heights of the triangles to their respective percentage
of membership. Then, add the two sets together and take and average value of
the curve that traces the two summed triangles. This is a weighted average and
is the basis for most fuzzy systems.
So the
output of a fuzzy system is a weighted average. Below is a more generic model
of a fuzzy system.
In simplest terms, a fuzzy system takes an input,
fuzzifies it, adds the terms, defuzzifies, and outputs a value. To some, in may
look like a filter of some sort. This basic fuzzy system leads to more complex
systems where “Fuzzy Experts” make decisions and are able to modify the rules
according to data. These systems are called adaptive fuzzy systems.
Applications
Fuzzy logic perhaps has its widest use in control systems. Using fuzzy
logic, especially in the control of mechanical systems, often results in
smoother control than a binary system can offer. Another obvious use for fuzzy
logic is language recognition and other such interactions between humans and
computers. As we saw before, fuzzy sets are based on language. Artificial
intelligence and neural networks research is leaning toward fuzzy logic
concepts because that is the way our brains behave. Other applications include:
imaging, signal processing, and even data encryption.
Finally,
here is a list of everyday products to give you an idea of the range of devices
that are using fuzzy logic today:
Air
conditioners Palm PC’s
(language recognition)
Anti-lock brakes Yamaichi Stock Index (interesting)
Washer-Dryers Camcorders
Elevators Still cameras
Refrigerators Trains
Industrial
Controls Copiers
Microwaves Televisions
Conclusion
Fuzzy
logic is everywhere today. It is becoming more widely accepted, especially as
companies start to make money from fuzzy products. It will have a far-reaching
impact on artificial intelligence, machine IQ, and way we interact with
computers. Fuzzy logic is a mathematical discipline that tries to view the
world the way we do. I have only scratched the surface on this topic. Fuzzy
logicians are finding new applications for fuzzy logic in math, science, and
technology every day.
References
1) Kosko, B. Fuzzy
Thinking: The New Science Of Fuzzy Logic. New York: Hyperion Books, 1993.
2) http://www-isis.ecs.soton.ac.uk/research/nfinfo/fztut.shtml Fuzzy Logic
Tutorials.
3) http://www.usc.edu/dept/ee/ USC EE Department.