fuzzy logic

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FUZZY LOGIC History & applications Fuzzy logic was first proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of "linguistic variables", which in this article equates to a variable defined as a fuzzy set. Other research followed, with the first industrial application, a cement kiln built in Denmark, coming on line in 1975. Fuzzy systems were largely ignored in the U.S. because they were associated with artificial intelligence , a field that periodically oversells itself, especially in the mid-1980s, resulting in a lack of credibility within the commercial domain. The Japanese did not have this prejudice. Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji Miyamoto of Hitachi , who in 1985 provided simulations that demonstrated the superiority of fuzzy control systems for the Sendai railway. Their ideas were adopted, and fuzzy systems were used to control accelerating, braking, and stopping when the line opened in 1987. Another event in 1987 helped promote interest in fuzzy systems. During an international meeting of fuzzy researchers in Tokyo that year, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "inverted pendulum " experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forth. Observers were impressed with this demonstration, as well as later experiments by Yamakawa in which he mounted a wine glass containing water or even a live mouse to the top of the pendulum. The system maintained stability in both cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field.

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FUZZY LOGIC

History & applications

Fuzzy logic was first proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of "linguistic variables", which in this article equates to a variable defined as a fuzzy set. Other research followed, with the first industrial application, a cement kiln built in Denmark, coming on line in 1975.

Fuzzy systems were largely ignored in the U.S. because they were associated with artificial intelligence, a field that periodically oversells itself, especially in the mid-1980s, resulting in a lack of credibility within the commercial domain.

The Japanese did not have this prejudice. Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji Miyamoto of Hitachi, who in 1985 provided simulations that demonstrated the superiority of fuzzy control systems for the Sendai railway. Their ideas were adopted, and fuzzy systems were used to control accelerating, braking, and stopping when the line opened in 1987.

Another event in 1987 helped promote interest in fuzzy systems. During an international meeting of fuzzy researchers in Tokyo that year, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "inverted pendulum" experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forth.

Observers were impressed with this demonstration, as well as later experiments by Yamakawa in which he mounted a wine glass containing water or even a live mouse to the top of the pendulum. The system maintained stability in both cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field.

Following such demonstrations, Japanese engineers developed a wide range of fuzzy systems for both industrial and consumer applications. In 1988 Japan established the Laboratory for International Fuzzy Engineering (LIFE), a cooperative arrangement between 48 companies to pursue fuzzy research.

Japanese consumer goods often incorporate fuzzy systems. Matsushita vacuum cleaners use microcontrollers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. Hitachi washing machines use fuzzy controllers to load-weight, fabric-mix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergent.

As a more specific example, Canon developed an autofocusing camera that uses a charge-coupled device (CCD) to measure the clarity of the image in six regions of its

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field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent overshoot.

The camera's fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires 1.1 kilobytes of memory.

As another example of a practical system, an industrial air conditioner designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an inverter, a compressor valve, and a fan motor. Compared to the previous design, the fuzzy controller heats and cools five times faster, reduces power consumption by 24%, increases temperature stability by a factor of two, and uses fewer sensors.

The enthusiasm of the Japanese for fuzzy logic is reflected in the wide range of other applications they have investigated or implemented: character and handwriting recognition; optical fuzzy systems; robots, including one for making Japanese flower arrangements; voice-controlled robot helicopters, this being no mean feat, as hovering is a "balancing act" rather similar to the inverted pendulum problem; control of flow of powders in film manufacture; elevator systems; and so on.

Work on fuzzy systems is also proceeding in the US and Europe, though not with the same enthusiasm shown in Japan. The US Environmental Protection Agency has investigated fuzzy control for energy-efficient motors, and NASA has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption. Firms such as Boeing, General Motors, Allen-Bradley, Chrysler, Eaton, and Whirlpool have worked on fuzzy logic for use in low-power refrigerators, improved automotive transmissions, and energy-efficient electric motors.

In 1995 Maytag introduced an "intelligent" dishwasher based on a fuzzy controller and a "one-stop sensing module" that combines a thermistor, for temperature measurement; a conductivity sensor, to measure detergent level from the ions present in the wash; a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash; and a magnetostrictive sensor to read spin rate. The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water. It even adjusts for dried-on foods by tracking the last time the door was opened, and estimates the number of dishes by the number of times the door was opened.

Research and development is also continuing on fuzzy applications in software, as opposed to firmware, design, including fuzzy expert systems and integration of fuzzy logic with neural-network and so-called adaptive "genetic" software systems, with the ultimate goal of building "self-learning" fuzzy control systems.

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INTRODUCTION

This is the first in a series of six articles intended to share information and experience in the realm of fuzzy logic (FL) and its application. This article will introduce FL. Through the course of this article series, a simple implementation will be explained in detail.

What Is Fuzzy Logic ? 

Fuzzy logic is a powerful problem-solving methodology with a myriad of applications in embedded control and information processing. Fuzzy provides a remarkably simple way to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making with its ability to work from approximate data and find precise solutions.

Unlike classical logic which requires a deep understanding of a system, exact equations, and precise numeric values, Fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction originating from our knowledge and experience. Fuzzy Logic allows expressing this knowledge with subjective concepts such as very hot, bright red, and a long time which are mapped into exact numeric ranges.

Fuzzy Logic has been gaining increasing acceptance during the past few years. There are over two thousand commercially available products using Fuzzy Logic, ranging from washing machines to high speed trains. Nearly every application can potentially realize some of the benefits of Fuzzy Logic, such as performance, simplicity, lower cost, and productivity.

Fuzzy Logic has been found to be very suitable for embedded control applications. Several manufacturers in the automotive industry are using fuzzy technology to improve quality and reduce development time. In aerospace, fuzzy enables very complex real time problems to be tackled using a simple approach. In consumer electronics, fuzzy improves time to market and helps reduce costs. In manufacturing, fuzzy is proven to be invaluable in increasing equipment efficiency and diagnosing malfunctions.

Fuzzy logic is a superset of conventional (Boolean) logic that has beenextended to handle the concept of partial truth -- truth values between"completely true" and "completely false". It was introduced by Dr. LotfiZadeh of UC/Berkeley in the 1960's as a means to model the uncertaintyof natural language. (Note: Lotfi, not Lofti, is the correct spellingof his name.)

Zadeh says that rather than regarding fuzzy theory as a single theory, weshould regard the process of ``fuzzification'' as a methodology togeneralize ANY specific theory from a crisp (discrete) to a continuous

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(fuzzy) form (see "extension principle" in [2]). Thus recently researchershave also introduced "fuzzy calculus", "fuzzy differential equations",and so on .

What is a fuzzy expert system?

A fuzzy expert system is an expert system that uses a collection offuzzy membership functions and rules, instead of Boolean logic, toreason about data. The rules in a fuzzy expert system are usually of aform similar to the following:

if x is low and y is high then z = medium

where x and y are input variables (names for know data values), z is anoutput variable (a name for a data value to be computed), low is amembership function (fuzzy subset) defined on x, high is a membershipfunction defined on y, and medium is a membership function defined on z.The antecedent (the rule's premise) describes to what degree the ruleapplies, while the conclusion (the rule's consequent) assigns amembership function to each of one or more output variables. Most toolsfor working with fuzzy expert systems allow more than one conclusion perrule. The set of rules in a fuzzy expert system is known as the rulebaseor knowledge base.

The general inference process proceeds in three (or four) steps.

1. Under FUZZIFICATION, the membership functions defined on the input variables are applied to their actual values, to determine the degree of truth for each rule premise.

2. Under INFERENCE, the truth value for the premise of each rule is computed, and applied to the conclusion part of each rule. This results in one fuzzy subset to be assigned to each output variable for each rule. Usually only MIN or PRODUCT are used as inference rules. In MIN inferencing, the output membership function is clipped off at a height corresponding to the rule premise's computed degree of truth (fuzzy logic AND). In PRODUCT inferencing, the output membership function is scaled by the rule premise's computed degree of truth.

3. Under COMPOSITION, all of the fuzzy subsets assigned to each output variable are combined together to form a single fuzzy subset for each output variable. Again, usually MAX or SUM are used. In MAX composition, the combined output fuzzy subset is constructed by taking

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the pointwise maximum over all of the fuzzy subsets assigned tovariable by the inference rule (fuzzy logic OR). In SUM composition, the combined output fuzzy subset is constructed by taking the pointwise sum over all of the fuzzy subsets assigned to the output variable by the inference rule.

4. Finally is the (optional) DEFUZZIFICATION, which is used when it is useful to convert the fuzzy output set to a crisp number. There are more defuzzification methods than you can shake a stick at (at least 30). Two of the more common techniques are the CENTROID and MAXIMUM methods. In the CENTROID method, the crisp value of the output variable is computed by finding the variable value of the center of gravity of the membership function for the fuzzy value. In the MAXIMUM method, one of the variable values at which the fuzzy subset has its maximum truth value is chosen as the crisp value for the output variable.

Extended Example:

Assume that the variables x, y, and z all take on values in the interval[0,10], and that the following membership functions and rules are defined:

low(t) = 1 - ( t / 10 ) high(t) = t / 10

rule 1: if x is low and y is low then z is high rule 2: if x is low and y is high then z is low rule 3: if x is high and y is low then z is low rule 4: if x is high and y is high then z is high

Notice that instead of assigning a single value to the output variable z, eachrule assigns an entire fuzzy subset (low or high).

Notes:

1. In this example, low(t)+high(t)=1.0 for all t. This is not required, but it is fairly common.

2. The value of t at which low(t) is maximum is the same as the value of t at which high(t) is minimum, and vice-versa. This is also not required, but fairly common.

3. The same membership functions are used for all variables. This isn't required, and is also *not* common.

In the fuzzification subprocess, the membership functions defined on the

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input variables are applied to their actual values, to determine thedegree of truth for each rule premise. The degree of truth for a rule'spremise is sometimes referred to as its ALPHA. If a rule's premise has anonzero degree of truth (if the rule applies at all...) then the rule issaid to FIRE. For example,

x y low(x) high(x) low(y) high(y) alpha1 alpha2 alpha3 alpha4------------------------------------------------------------------------------0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.00.0 3.2 1.0 0.0 0.68 0.32 0.68 0.32 0.0 0.00.0 6.1 1.0 0.0 0.39 0.61 0.39 0.61 0.0 0.00.0 10.0 1.0 0.0 0.0 1.0 0.0 1.0 0.0 0.03.2 0.0 0.68 0.32 1.0 0.0 0.68 0.0 0.32 0.06.1 0.0 0.39 0.61 1.0 0.0 0.39 0.0 0.61 0.010.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 1.0 0.03.2 3.1 0.68 0.32 0.69 0.31 0.68 0.31 0.32 0.313.2 3.3 0.68 0.32 0.67 0.33 0.67 0.33 0.32 0.3210.0 10.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 1.0

In the inference subprocess, the truth value for the premise of each rule iscomputed, and applied to the conclusion part of each rule. This results inone fuzzy subset to be assigned to each output variable for each rule.

MIN and PRODUCT are two INFERENCE METHODS or INFERENCE RULES. In MINinferencing, the output membership function is clipped off at a heightcorresponding to the rule premise's computed degree of truth. Thiscorresponds to the traditional interpretation of the fuzzy logic ANDoperation. In PRODUCT inferencing, the output membership function isscaled by the rule premise's computed degree of truth.

For example, let's look at rule 1 for x = 0.0 and y = 3.2. As shown in thetable above, the premise degree of truth works out to 0.68. For this rule, MIN inferencing will assign z the fuzzy subset defined by the membershipfunction:

rule1(z) = { z / 10, if z <= 6.8 0.68, if z >= 6.8 }

For the same conditions, PRODUCT inferencing will assign z the fuzzy subsetdefined by the membership function:

rule1(z) = 0.68 * high(z) = 0.068 * z

Note: The terminology used here is slightly nonstandard. In most texts,

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the term "inference method" is used to mean the combination of the thingsreferred to separately here as "inference" and "composition." Thusyou'll see such terms as "MAX-MIN inference" and "SUM-PRODUCT inference"in the literature. They are the combination of MAX composition and MINinference, or SUM composition and PRODUCT inference, respectively.You'll also see the reverse terms "MIN-MAX" and "PRODUCT-SUM" -- thesemean the same things as the reverse order. It seems clearer to describethe two processes separately.

In the composition subprocess, all of the fuzzy subsets assigned to eachoutput variable are combined together to form a single fuzzy subset for eachoutput variable.

MAX composition and SUM composition are two COMPOSITION RULES. In MAXcomposition, the combined output fuzzy subset is constructed by takingthe pointwise maximum over all of the fuzzy subsets assigned to theoutput variable by the inference rule. In SUM composition, the combinedoutput fuzzy subset is constructed by taking the pointwise sum over allof the fuzzy subsets assigned to the output variable by the inferencerule. Note that this can result in truth values greater than one! Forthis reason, SUM composition is only used when it will be followed by adefuzzification method, such as the CENTROID method, that doesn't have aproblem with this odd case. Otherwise SUM composition can be combinedwith normalization and is therefore a general purpose method again.

For example, assume x = 0.0 and y = 3.2. MIN inferencing would assign thefollowing four fuzzy subsets to z:

rule1(z) = { z / 10, if z <= 6.8 0.68, if z >= 6.8 }

rule2(z) = { 0.32, if z <= 6.8 1 - z / 10, if z >= 6.8 }

rule3(z) = 0.0

rule4(z) = 0.0

MAX composition would result in the fuzzy subset:

fuzzy(z) = { 0.32, if z <= 3.2 z / 10, if 3.2 <= z <= 6.8 0.68, if z >= 6.8 }

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PRODUCT inferencing would assign the following four fuzzy subsets to z:

rule1(z) = 0.068 * z rule2(z) = 0.32 - 0.032 * z rule3(z) = 0.0 rule4(z) = 0.0

SUM composition would result in the fuzzy subset:

fuzzy(z) = 0.32 + 0.036 * zSometimes it is useful to just examine the fuzzy subsets that are theresult of the composition process, but more often, this FUZZY VALUE needsto be converted to a single number -- a CRISP VALUE. This is what thedefuzzification subprocess does.

There are more defuzzification methods than you can shake a stick at. Acouple of years ago, Mizumoto did a short paper that compared about tendefuzzification methods. Two of the more common techniques are theCENTROID and MAXIMUM methods. In the CENTROID method, the crisp value ofthe output variable is computed by finding the variable value of thecenter of gravity of the membership function for the fuzzy value. In theMAXIMUM method, one of the variable values at which the fuzzy subset hasits maximum truth value is chosen as the crisp value for the outputvariable. There are several variations of the MAXIMUM method that differonly in what they do when there is more than one variable value at whichthis maximum truth value occurs. One of these, the AVERAGE-OF-MAXIMAmethod, returns the average of the variable values at which the maximumtruth value occurs.

For example, go back to our previous examples. Using MAX-MIN inferencingand AVERAGE-OF-MAXIMA defuzzification results in a crisp value of 8.4 forz. Using PRODUCT-SUM inferencing and CENTROID defuzzification results ina crisp value of 5.6 for z, as follows.

Earlier on in the FAQ, we state that all variables (including z) take onvalues in the range [0, 10]. To compute the centroid of the function f(x),you divide the moment of the function by the area of the function. To compute the moment of f(x), you compute the integral of x*f(x) dx, and to compute thearea of f(x), you compute the integral of f(x) dx. In this case, we wouldcompute the area as integral from 0 to 10 of (0.32+0.036*z) dz, which is

(0.32 * 10 + 0.018*100) = (3.2 + 1.8) = 5.0

and the moment as the integral from 0 to 10 of (0.32*z+0.036*z*z) dz, which is

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(0.16 * 10 * 10 + 0.012 * 10 * 10 * 10) = (16 + 12) = 28

Finally, the centroid is 28/5 or 5.6.

Where are fuzzy expert systems used?fuzzy expert systems are the most common use of fuzzy logic. Theyare used in several wide-ranging fields, including: o Linear and Nonlinear Control o Pattern Recognition o Financial Systems o Operation Research

Where is fuzzy logic used?

Fuzzy logic is used directly in very few applications. The Sony PalmTopapparently uses a fuzzy logic decision tree algorithm to performhandwritten (well, computer lightpen) Kanji character recognition.Most applications of fuzzy logic use it as the underlying logic systemfor fuzzy expert systems.

What is fuzzy control?

The purpose of control is to influence the behavior of a system bychanging an input or inputs to that system according to a rule orset of rules that model how the system operates. The system beingcontrolled may be mechanical, electrical, chemical or any combinationof these.

Classic control theory uses a mathematical model to define a relationshipthat transforms the desired state (requested) and observed state (measured)of the system into an input or inputs that will alter the future state ofthat system.

reference----->0------->( SYSTEM ) -------+----------> output

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^ | | | +--------( MODEL )<--------+feedback

The most common example of a control model is the PID (proportional-integral-derivative) controller. This takes the output of the system and comparesit with the desired state of the system. It adjusts the input value basedon the difference between the two values according to the followingequation. output = A.e + B.INT(e)dt + C.de/dt

Where, A, B and C are constants, e is the error term, INT(e)dt is theintegral of the error over time and de/dt is the change in the error term.

The major drawback of this system is that it usually assumes that the systembeing modelled in linear or at least behaves in some fashion that is amonotonic function. As the complexity of the system increases it becomesmore difficult to formulate that mathematical model.

Fuzzy control replaces, in the picture above, the role of the mathematicalmodel and replaces it with another that is build from a number of smallerrules that in general only describe a small section of the whole system. Theprocess of inference binding them together to produce the desired outputs.

That is, a fuzzy model has replaced the mathematical one. The inputs andoutputs of the system have remained unchanged.

The Sendai subway is the prototypical example application of fuzzy control.

Fuzzy sets

The input variables in a fuzzy control system are in general mapped into by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification".

A control system may also have various types of switch, or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another.

Given "mappings" of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take based on a set of "rules", each of the form:

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IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased.

In this example, the two input variables are "brake temperature" and "speed" that have values defined as fuzzy sets. The output variable, "brake pressure", is also defined by a fuzzy set that can have values like "static", "slightly increased", "slightly decreased", and so on. This rule by itself is very puzzling since it looks like it could be used without bothering with fuzzy logic, but remember the decision is based on a set of rules:

All the rules that apply are invoked, using the membership functions and truth values obtained from the inputs, to determine the result of the rule.

This result in turn will be mapped into a membership function and truth value controlling the output variable.

These results are combined to give a specific ("crisp") answer, the actual brake pressure, a procedure known as "defuzzification".

This combination of fuzzy operations and rule-based "inference" describes a "fuzzy expert system".

Traditional control systems are based on mathematical models in which the control system is described using one or more differential equations that define the system response to its inputs. Such systems are often implemented as "PID controllers" (proportional-integral-derivative controllers). They are the products of decades of development and theoretical analysis, and are highly effective.

If PID and other traditional control systems are so well-developed, why bother with fuzzy control? It has some advantages. In many cases, the mathematical model of the control process may not exist, or may be too "expensive" in terms of computer processing power and memory, and a system based on empirical rules may be more effective.

Furthermore, fuzzy logic is well suited to low-cost implementations based on cheap sensors, low-resolution analog-to-digital converters, and 4-bit or 8-bit one-chip microcontroller chips. Such systems can be easily upgraded by adding new rules to improve performance or add new features. In many cases, fuzzy control can be used to improve existing traditional controller systems by adding an extra layer of intelligence to the current control method.

Logical interpretation of fuzzy control

In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. Obviously, if we consider systems of rules in which the class antecedent define a partition such a paradoxical

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phenomenon does not arise. In any case there is sometime of unsatisfactory in considering two variables x and y in a rule without some kind of functional dependence. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic. Also in Gerla 2005 a logical approach to fuzzy control is proposed based on the following idea. Denote by f the fuzzy function associated with the fuzzy control system, i.e., given the input r, s(y) = f(r,y) is the fuzzy set of possible outputs. Then given a possible output 't', we interpret f(r,t) as the truth degree of the claim "t is a good answer given r". More formally, any system of IF-THEN rules can be translate into a fuzzy program in such a way that the fuzzy function f is the interpretation of a vague predicate Good(x,y) in the associated least fuzzy Herbrand model. In such a way fuzzy control becomes a chapter of fuzzy logic programming. The learning process becomes a question belonging to inductive logic theory.

FuzzyCLIPS

FuzzyCLIPS is a fuzzy logic extension of the CLIPS (C Language Integrated Production System) expert system shell from NASA. It was developed by the Integrated Reasoning Group of the Institute for Information Technology of the National Research Council of Canada and has been widely distributed for a number of years. It enhances CLIPS by providing a fuzzy reasoning capability that is fully integrated with CLIPS facts and inference engine allowing one to represent and manipulate fuzzy facts and rules. FuzzyCLIPS can deal with exact, fuzzy (or inexact), and combined reasoning, allowing fuzzy and normal terms to be freely mixed in the rules and facts of an expert system. The system uses two basic inexact concepts, fuzziness and uncertainty. It has provided a useful environment for developing fuzzy applications but it does require significant effort to update and maintain as new versions of CLIPS are released.

Neuro-fuzzy

In the field of artificial intelligence, neuro-fuzzy refers to combinations of artificial neural networks and fuzzy logic. Neuro-fuzzy was proposed by J. S. R. Jang. Neuro-fuzzy hybridization results in a hybrid intelligent system that synergizes these two techniques by combining the human-like reasoning style of fuzzy systems with the learning and connectionist structure of neural networks. Neuro-fuzzy hybridization is widely termed as Fuzzy Neural Network (FNN) or Neuro-Fuzzy System (NFS) in the literature. Neuro-fuzzy system (the more popular term is used henceforth) incorporates the human-like reasoning style of fuzzy systems through the use of fuzzy sets and a linguistic model consisting of a set of IF-THEN fuzzy rules. The main strength of neuro-fuzzy systems is that they are universal approximators with the ability to solicit interpretable IF-THEN rules.

The strength of neuro-fuzzy systems involves two contradictory requirements in fuzzy modeling: interpretability versus accuracy. In practice, one of the two properties

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prevails. The neuro-fuzzy in fuzzy modeling research field is divided into two areas: linguistic fuzzy modeling that is focused on interpretability, mainly the Mamdani model; and precise fuzzy modeling that is focused on accuracy, mainly the Takagi-Sugeno-Kang (TSK) model.

Although generally assumed to be the realization of a fuzzy system through connectionist networks, this term is also used to describe some other configurations including:

Deriving fuzzy rules from trained RBF networks. Fuzzy logic based tuning of neural network training parameters.

Fuzzy logic criteria for increasing a network size.

Realising fuzzy membership function through clustering algorithms in unsupervised learning in SOMs and neural networks.

Representing fuzzification, fuzzy inference and defuzzification through multi-layers feed-forward connectionist networks.

It must be pointed out that interpretability of the Mamdani-type neuro-fuzzy systems can be lost. To improve the interpretability of neuro-fuzzy systems, certain measures must be taken, wherein important aspects of interpretability of neuro-fuzzy systems are also discussed.[

Pseudo outer-product-based fuzzy neural networks

Pseudo outer-product-based fuzzy neural networks ("POPFNN") are a family of neuro-fuzzy systems that are based on the linguistic fuzzy model.[2]

Three members of POPFNN exist in the literature:

POPFNN-AARS(S), which is based on the Approximate Analogical Reasoning Scheme[3]

POPFNN-CRI(S), which is based on commonly accepted fuzzy Compositional Rule of Inference[4]

POPFNN-TVR, which is based on Truth Value Restriction

The "POPFNN" architecture is a five-layer neural network where the layers from 1 to 5 are called: input linguistic layer, condition layer, rule layer, consequent layer, output linguistic layer. The fuzzification of the inputs and the defuzzification of the outputs are respectively performed by the input linguistic and output linguistic layers while the fuzzy inference is collectively performed by the rule, condition and consequence layers.

The learning process of POPFNN consists of three phases:

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1. Fuzzy membership generation2. Fuzzy rule identification

3. Supervised fine-tuning

Various fuzzy membership generation algorithms can be used: Learning Vector Quantization (LVQ), Fuzzy Kohonen Partitioning (FKP) or Discrete Incremental Clustering (DIC). Generally, the POP algorithm and its variant LazyPOP are used to identify the fuzzy rules.

Type-2 fuzzy sets and systems

Type-2 fuzzy sets and systems generalize (type-1) fuzzy sets and systems so that more uncertainty can be handled. From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Prof. Lotfi A. Zadeh [27], when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a type-2 fuzzy set. A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes.

In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. In the following discussions it is names may be used, they are the same fuzzy set). When the latter is done, the resulting type-2 fuzzy set is called a general type-2 fuzzy set (to distinguish it from the special interval type-2 fuzzy set).

Prof. Zadeh didn’t stop with type-2 fuzzy sets, because in that 1976 paper [27] he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, … . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy.

The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its footprint of uncertainty (FOU).

For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of

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an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model.

The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions , a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets.

Work on type-2 fuzzy sets languished during the 1980’s and early-to-mid 1990’s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990’s as a result of Prof. Jerry Mendel and his student’s works on type-2 fuzzy sets and systems . Since then, more and more researchers around the world are writing articles about type-2 fuzzy sets and systems.

Interval Type-2 Fuzzy Sets

Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily interval arithmetic—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. So, the literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world.

Formulas for the following have already been worked out for interval type-2 fuzzy sets:

Fuzzy set operations : union, intersection and complement ([6], [12]) Centroid (a very widely used operation by practitioners of such sets, and also an

important uncertainty measure for them) ([7], [12])

Other uncertainty measures [fuzziness, cardinality, variance and skewness [22] and uncertainty bounds [26]

Similarity ([1], [24], [25])

Subsethood [21]

Fuzzy set ranking [25]

Fuzzy rule ranking and selection [31]

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Type-reduction methods ([7], [12])

Firing intervals for an interval type-2 fuzzy logic system ([3], [8], [12])

Fuzzy weighted average [9]

Linguistic weighted average [23]

Synthesizing an FOU from data that are collected from a group of subject [10]

Fuzzy Subsets:

Just as there is a strong relationship between Boolean logic and theconcept of a subset, there is a similar strong relationship between fuzzylogic and fuzzy subset theory.

In classical set theory, a subset U of a set S can be defined as amapping from the elements of S to the elements of the set {0, 1},

U: S --> {0, 1}

This mapping may be represented as a set of ordered pairs, with exactlyone ordered pair present for each element of S. The first element of theordered pair is an element of the set S, and the second element is anelement of the set {0, 1}. The value zero is used to representnon-membership, and the value one is used to represent membership. Thetruth or falsity of the statement

x is in U

is determined by finding the ordered pair whose first element is x. Thestatement is true if the second element of the ordered pair is 1, and thestatement is false if it is 0.

Similarly, a fuzzy subset F of a set S can be defined as a set of orderedpairs, each with the first element from S, and the second element fromthe interval [0,1], with exactly one ordered pair present for eachelement of S. This defines a mapping between elements of the set S andvalues in the interval [0,1]. The value zero is used to representcomplete non-membership, the value one is used to represent completemembership, and values in between are used to represent intermediateDEGREES OF MEMBERSHIP. The set S is referred to as the UNIVERSE OFDISCOURSE for the fuzzy subset F. Frequently, the mapping is describedas a function, the MEMBERSHIP FUNCTION of F. The degree to which thestatement

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x is in F

is true is determined by finding the ordered pair whose first element isx. The DEGREE OF TRUTH of the statement is the second element of theordered pair.

In practice, the terms "membership function" and fuzzy subset get usedinterchangeably.

That's a lot of mathematical baggage, so here's an example. Let'stalk about people and "tallness". In this case the set S (theuniverse of discourse) is the set of people. Let's define a fuzzysubset TALL, which will answer the question "to what degree is personx tall?" Zadeh describes TALL as a LINGUISTIC VARIABLE, whichrepresents our cognitive category of "tallness". To each person in theuniverse of discourse, we have to assign a degree of membership in thefuzzy subset TALL. The easiest way to do this is with a membershipfunction based on the person's height.

tall(x) = { 0, if height(x) < 5 ft., (height(x)-5ft.)/2ft., if 5 ft. <= height (x) <= 7 ft., 1, if height(x) > 7 ft. }

A graph of this looks like:

1.0 + +------------------- | / | /0.5 + / | / | /0.0 +-------------+-----+------------------- | | 5.0 7.0

height, ft. ->

Given this definition, here are some example values:

Person Height degree of tallness--------------------------------------Billy 3' 2" 0.00 [I think]Yoke 5' 5" 0.21Drew 5' 9" 0.38Erik 5' 10" 0.42Mark 6' 1" 0.54

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Kareem 7' 2" 1.00 [depends on who you ask]

Expressions like "A is X" can be interpreted as degrees of truth,e.g., "Drew is TALL" = 0.38.

Note: Membership functions used in most applications almost never have assimple a shape as tall(x). At minimum, they tend to be triangles pointingup, and they can be much more complex than that. Also, the discussioncharacterizes membership functions as if they always are based on asingle criterion, but this isn't always the case, although it is quitecommon. One could, for example, want to have the membership function forTALL depend on both a person's height and their age (he's tall for hisage). This is perfectly legitimate, and occasionally used in practice.It's referred to as a two-dimensional membership function, or a "fuzzyrelation". It's also possible to have even more criteria, or to have themembership function depend on elements from two completely differentuniverses of discourse.

Logic Operations:

Now that we know what a statement like "X is LOW" means in fuzzy logic,how do we interpret a statement like

X is LOW and Y is HIGH or (not Z is MEDIUM)

The standard definitions in fuzzy logic are:

truth (not x) = 1.0 - truth (x) truth (x and y) = minimum (truth(x), truth(y)) truth (x or y) = maximum (truth(x), truth(y))

Some researchers in fuzzy logic have explored the use of otherinterpretations of the AND and OR operations, but the definition for theNOT operation seems to be safe.

Note that if you plug just the values zero and one into thesedefinitions, you get the same truth tables as you would expect fromconventional Boolean logic. This is known as the EXTENSION PRINCIPLE,which states that the classical results of Boolean logic are recoveredfrom fuzzy logic operations when all fuzzy membership grades arerestricted to the traditional set {0, 1}. This effectively establishesfuzzy subsets and logic as a true generalization of classical set theoryand logic. In fact, by this reasoning all crisp (traditional) subsets AREfuzzy subsets of this very special type; and there is no conflict betweenfuzzy and crisp methods.

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Some examples -- assume the same definition of TALL as above, and in addition,assume that we have a fuzzy subset OLD defined by the membership function:

old (x) = { 0, if age(x) < 18 yr. (age(x)-18 yr.)/42 yr., if 18 yr. <= age(x) <= 60 yr. 1, if age(x) > 60 yr. }

And for compactness, let

a = X is TALL and X is OLD b = X is TALL or X is OLD c = not (X is TALL)

Then we can compute the following values.

height age X is TALL X is OLD a b c------------------------------------------------------------------------3' 2" 65 0.00 1.00 0.00 1.00 1.005' 5" 30 0.21 0.29 0.21 0.29 0.795' 9" 27 0.38 0.21 0.21 0.38 0.625' 10" 32 0.42 0.33 0.33 0.42 0.586' 1" 31 0.54 0.31 0.31 0.54 0.467' 2" 45 1.00 0.64 0.64 1.00 0.003' 4" 4 0.00 0.00 0.00 0.00 1.00

For those of you who only grok the metric system, here's a dandylittle conversion table:

Feet+Inches = Meters -------------------- 3' 2" 0.9652 3' 4" 1.0160 5' 5" 1.6510 5' 9" 1.7526 5' 10" 1.7780 6' 1" 1.8542 7' 2" 2.1844

Fuzzy logic has rapidly become one of the most successful of today's technologies for developing sophisticated control systems. The reason for which is very simple. Fuzzy logic addresses such applications perfectly as it resembles human decision making with an ability to generate precise solutions from certain or approximate information. It fills an important gap in engineering design methods left vacant by purely mathematical

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approaches (e.g. linear control design), and purely logic-based approaches (e.g. expert systems) in system design.

While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models.

What does it offer?

The first applications of fuzzy theory were primaly industrial, such as process control for cement kilns. However, as the technology was further embraced, fuzzy logic was used in more useful applications. In 1987, the first fuzzy logic-controlled subway was opened in Sendai in northern Japan. Here, fuzzy-logic controllers make subway journeys more comfortable with smooth braking and acceleration. Best of all, all the driver has to do is push the start button! Fuzzy logic was also put to work in elevators to reduce waiting time. Since then, the applications of Fuzzy Logic technology have virtually exploded, affecting things we use everyday.Take for example, the fuzzy washing machine . A load of clothes in it and press start, and the machine begins to churn, automatically choosing the best cycle. The fuzzy microwave, Place chili, potatoes, or etc in a fuzzy microwave and push single button, and it cooks for the right time at the proper temperature. The fuzzy car, manuvers itself by following simple verbal instructions from its driver. It can even stop itself when there is an obstacle immedeately ahead using sensors. But, practically the most exciting thing about it, is the simplicity involved in operating it.

Degrees of truth

Fuzzy logic and probabilistic logic are mathematically similar – both have truth values ranging between 0 and 1 – but conceptually distinct, due to different interpretations -- see interpretations of probability theory. Fuzzy logic corresponds to "degrees of truth", while probabilistic logic corresponds to "probability, likelihood"; as these differ, fuzzy logic and probabilistic logic yield different models of the same real-world situations.

Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 0.7 empty and 0.3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might equally well design a set membership function where the glass would be considered full for all values down to 50 ml. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of randomness. A probabilistic setting would first define a scalar variable for the fullness of the glass, and second, conditional distributions describing the probability that someone

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would call the glass full given a specific fullness level. This model, however, has no sense without accepting occurrence of some event, e.g. that after a few minutes, the glass will be half empty. Note that the conditioning can be achieved by having a specific observer that randomly selects the level for the glass, a distribution over deterministic observers, or both. Consequently, probability has nothing in common with fuzziness, these are simply different concepts which superficially seem similar because of using the same unit interval of real numbers [0,1]. Still, since theorems such as De Morgan's have dual applicability and properties of random variables are analogous to properties of binary logic states, one can see where the confusion might arise.

Applying truth values

A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled.

Linguistic variables

While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts.[4]

A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. For example, L. A. Zadeh proposed to take the square of the membership function. This model, however, does not work properly.

Example

Fuzzy set theory defines fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.

Rules are usually expressed in the form:IF variable IS property THEN action

For example, a simple temperature regulator that uses a fan might look like this:

IF temperature IS very cold THEN stop fanIF temperature IS cold THEN turn down fan

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IF temperature IS normal THEN maintain levelIF temperature IS hot THEN speed up fan

There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees.

The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y:

NOT x = (1 - truth(x))x AND y = minimum(truth(x), truth(y))x OR y = maximum(truth(x), truth(y))

There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat", which modify the meaning of a set using a mathematical formula.

Propositional fuzzy logics

The most important propositional fuzzy logics are:

Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are prelinear commutative bounded integral residuated lattices.

Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras.

Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras.

Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras.

Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras.

Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation.

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Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ.

Predicate fuzzy logics

These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal (resp. existential) quantifier in t-norm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula.

Decidability issues for fuzzy logic

The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla showed that such a definition is not adequate and therefore proposed the following one. Ü denotes the set of rational numbers in [0,1]. A fuzzy subset s : S [0,1] of a set S is recursively enumerable if a recursive map h : S×N Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is proposed in Gerla 2006. The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property).

Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to give supports for a Church thesis for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper). Another open question is to start from this notion to find an extension of Gödel’s theorems to fuzzy logic.

Fuzzy databases

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Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankova's dissertation. Later, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.M. Medina, M.A. Vila et al. In the context of fuzzy databases, some fuzzy querying languages have been defined, highlighting the SQLf by P. Bosc et al. and the FSQL by J. Galindo et al. These languages define some structures in order to include fuzzy aspects in the SQL statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels and so on.

Application areas

Fuzzy logic is used in the operation or programming of:

Air conditioners Automobile and such vehicle subsystems as automatic transmissions, ABS and

cruise control

Tokyo monorail

Cameras

Digital image processing , such as edge detection

Dishwashers

Elevators

Some microcontrollers and microprocessors (e.g. Freescale 68HC12)

Hydrometeor classification algorithms for polarimetric weather radar

Language filters on message boards and chat rooms for filtering out offensive text

The Massive engine used in the Lord of the Rings films, which allowed large-scale armies to enact random yet orderly movements

Mineral Deposit estimation

Pattern recognition in Remote Sensing

Rice cookers

Video game artificial intelligence

Home appliances (e.g. washing machine)

Comparison to probability

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Fuzzy logic and probability are different ways of expressing uncertainty. While both fuzzy logic and probability theory can be used to represent subjective belief, fuzzy set theory uses the concept of fuzzy set membership (i.e., how much a variable is in a set), probability theory uses the concept of subjective probability (i.e., how probable do I think that a variable is in a set). While this distinction is mostly philosophical, the fuzzy-logic-derived possibility measure is inherently different from the probability measure, hence they are not directly equivalent. However, many statisticians are persuaded by the work of Bruno de Finetti that only one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. On the other hand, Bart Kosko argues that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty. He also claims to have proven a derivation of Bayes' theorem from the concept of fuzzy subsethood. Lotfi Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to what is called possibility theory.

WHERE DID FUZZY LOGIC COME FROM?

The concept of Fuzzy Logic (FL) was conceived by Lotfi Zadeh, a professor at the University of California at Berkley, and presented not as a control methodology, but as a way of processing data by allowing partial set membership rather than crisp set membership or non-membership. This approach to set theory was not applied to control systems until the 70's due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that people do not require precise, numerical information input, and yet they are capable of highly adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement. Unfortunately, U.S. manufacturers have not been so quick to embrace this technology while the Europeans and Japanese have been aggressively building real products around it.

HOW IS FL DIFFERENT FROM CONVENTIONAL CONTROL METHODS?

FL incorporates a simple, rule-based IF X AND Y THEN Z approach to a solving control problem rather than attempting to model a system mathematically. The FL model is empirically-based, relying on an operator's experience rather than their technical understanding of the system. For example, rather than dealing with temperature control in terms such as "SP =500F", "T <1000F", or "210C <TEMP <220C", terms like "IF (process is too cool) AND (process is getting colder) THEN (add heat to the process)" or "IF (process is too hot) AND (process is heating rapidly) THEN (cool the process quickly)" are used. These terms are imprecise and yet very descriptive of what must actually happen. Consider what you do in the shower if the temperature is too cold: you will make the water comfortable very quickly with little trouble. FL is capable of mimicking this type of behavior but at very high rate.

HOW DOES FL WORK?

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FL requires some numerical parameters in order to operate such as what is considered significant error and significant rate-of-change-of-error, but exact values of these numbers are usually not critical unless very responsive performance is required in which case empirical tuning would determine them. For example, a simple temperature control system could use a single temperature feedback sensor whose data is subtracted from the command signal to compute "error" and then time-differentiated to yield the error slope or rate-of-change-of-error, hereafter called "error-dot". Error might have units of degs F and a small error considered to be 2F while a large error is 5F. The "error-dot" might then have units of degs/min with a small error-dot being 5F/min and a large one being 15F/min. These values don't have to be symmetrical and can be "tweaked" once the system is operating in order to optimize performance. Generally, FL is so forgiving that the system will probably work the first time without any tweaking.

What are fuzzy numbers and fuzzy arithmetic?Fuzzy numbers are fuzzy subsets of the real line. They have a peak orplateau with membership grade 1, over which the members of theuniverse are completely in the set. The membership function isincreasing towards the peak and decreasing away from it.

Fuzzy numbers are used very widely in fuzzy control applications. A typicalcase is the triangular fuzzy number

1.0 + + | / \ | / \0.5 + / \ | / \ | / \0.0 +-------------+-----+-----+-------------- | | | 5.0 7.0 9.0

which is one form of the fuzzy number 7. Slope and trapezoidal functionsare also used, as are exponential curves similar to Gaussian probabilitydensities.

How are membership values determined?Determination methods break down broadly into the following categories:

1. Subjective evaluation and elicitation

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As fuzzy sets are usually intended to model people's cognitive states, they can be determined from either simple or sophisticated elicitation procedures. At they very least, subjects simply draw or otherwise specify different membership curves appropriate to a given problem. These subjects are typcially experts in the problem area. Or they are given a more constrained set of possible curves from which they choose. Under more complex methods, users can be tested using psychological methods.

2. Ad-hoc forms

While there is a vast (hugely infinite) array of possible membership function forms, most actual fuzzy control operations draw from a very small set of different curves, for example simple forms of fuzzy numbers (see [7]). This simplifies the problem, for example to choosing just the central value and the slope on either side.

3. Converted frequencies or probabilities

Sometimes information taken in the form of frequency histograms or other probability curves are used as the basis to construct a membership function. There are a variety of possible conversion methods, each with its own mathematical and methodological strengths and weaknesses. However, it should always be remembered that membership functions are NOT (necessarily) probabilities. See [10] for more information.

4. Physical measurement

Many applications of fuzzy logic use physical measurement, but almost none measure the membership grade directly. Instead, a membership function is provided by another method, and then the individual membership grades of data are calculated from it .

What is the relationship between fuzzy truth values and probabilities?

This question has to be answered in two ways: first, how does fuzzy theory differ from probability theory mathematically, and second, how does it differ in interpretation and application.

At the mathematical level, fuzzy values are commonly misunderstood to be probabilities, or fuzzy logic is interpreted as some new way of handling probabilities. But this is not the case. A minimum requirement of probabilities is ADDITIVITY, that is that they must add together to one, or the integral of their density curves must be one.

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But this does not hold in general with membership grades. And while membership grades can be determined with probability densities in mind (see [11]), there are other methods as well which have nothing to do with frequencies or probabilities.

Because of this, fuzzy researchers have gone to great pains to distancethemselves from probability. But in so doing, many of them have lost trackof another point, which is that the converse DOES hold: all probabilitydistributions are fuzzy sets! As fuzzy sets and logic generalize Booleansets and logic, they also generalize probability.

In fact, from a mathematical perspective, fuzzy sets and probability exist as parts of a greater Generalized Information Theory which includes many formalisms for representing uncertainty (including random sets, Demster-Shafer evidence theory, probability intervals, possibility theory, general fuzzy measures, interval analysis, etc.). Furthermore, one can also talk about random fuzzy events and fuzzy random events. This whole issue is beyond the scope of this FAQ, so please refer to the following articles, or the textbook by Klir and Folger (see [16]).

Semantically, the distinction between fuzzy logic and probability theory has to do with the difference between the notions of probability and a degree of membership. Probability statements are about the likelihoods of outcomes: an event either occurs or does not, and you can bet on it. But with fuzziness, one cannot say unequivocally whether an event occured or not, and instead you are trying to model the EXTENT to which an event occured. This issue is treated well in the swamp water example used by James Bezdek of the University of West Florida (Bezdek, James C, "Fuzzy Models .

Are there fuzzy state machines?

Yes. FSMs are obtained by assigning membership grades as weights to thestates of a machine, weights on transitions between states, and then acomposition rule such as MAX/MIN or PLUS/TIMES (see [4]) to calculate newgrades of future states. Refer to the following article, or to SectionIII of the Dubois and Prade's 1980 textbook (see [16]).

What is possibility theory?

Possibility theory is a new form of information theory which is relatedto but independent of both fuzzy sets and probability theory.Technically, a possibility distribution is a normal fuzzy set (at least

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one membership grade equals 1). For example, all fuzzy numbers arepossibility distributions. However, possibility theory can also bederived without reference to fuzzy sets.

The rules of possibility theory are similar to probability theory, butuse either MAX/MIN or MAX/TIMES calculus, rather than the PLUS/TIMEScalculus of probability theory. Also, possibilistic NONSPECIFICITY isavailable as a measure of information similar to the stochasticENTROPY.

Possibility theory has a methodological advantage over probability theoryas a representation of nondeterminism in systems, because the PLUS/TIMEScalculus does not validly generalize nondeterministic processes, whileMAX/MIN and MAX/TIMES do.

How can I get a copy of the proceedings for ?

This is rough sometimes. The first thing to do, of course, is to contact the organization that ran the conference or workshop you are interested in. If they can't help you, the best idea mentioned so far is to contact the Institute for Scientific Information, Inc. (ISI), and check with their Index to Scientific and Technical Proceedings (ISTP volumes).