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FUZZY LOGIC
WHAT IS FUZZY LOGIC?
In classical set theory members of a given set either have full membership in that set or no membership at all, i.e. the degree of membership is restricted to either TRUE or FALSE or 1 and 0. Linguistic variables (central to fuzzy logic manipulations) hold values that are uniformly distributed between 0 and 1, and are dependent on the linguistic term.
Applications may be computed in either the fuzzy linguistic domain or the conventional crisp domain. Linguistic variables are relevant in applications involving human interface as tasks commonly done by humans cannot easily be described in crisp terms. Fuzzy logic derives its importance from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature-they do not have any definite answers and are dependent on many varied factors.
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In such cases linguistic variables provide a convenient tool to describe a problem.
Fuzzy logic invented by Dr. Lotfi Zadeh in the 160s is used to help computers deal with the approximate. The linguistic variables used conveys relative Information about the object under observation and can convey a surprising amount of information. For this reason, it is essential in the design of expert systems, which apply real-world rules to real-world situations.
Fuzzy Logic is a very powerful concept which, when applied to control systems allows designers to greatly reduce the number of rules necessary to deploy a desired control strategy. It also allows rules to be crafted in a very linguistic syntax and reduces instabilities and hysteresis.
Fuzzy theory and fuzzy systems are used in applications like spellcheckers and palmtop handwriting interpreters-especially in Japan, complicated Kanji strokes can be detected as theyre written using fuzzy methods. It is used in controllers for products such as washing machines, video cameras, and ocean drilling systems; and in pattern recognition and predictive software for financial modeling, medical imaging, and manufacturing.
The first major commercial application was in a cement kiln control, an operation which requires that an operator, monitor four internal states of the kiln, control four sets of operations, and dynamically manage 40 or 50 rules of thumb about their interrelationships, all with the goal of controlling a highly complex set of chemical interactions.
FUZZY SET THEORY and TRUTH VALUES
In Boolean logic, conditions are evaluated to be either TRUE or FALSE. There are no intermediate values. Every element either belongs to the set or it does not belong to it.
This concept is sufficient for many applications, but we can easily find situations where it lacks in flexibility. In fuzzy logic the possible values range from 0.0 to 1.0 (inclusive), not just 0 and 1.
Fuzzy greatly enhances the capability of classical set theory by allowing the degree
of membership or truth value to range over the interval of 0 to 1. Sets in fuzzy logic systems typically describe ranges of operations and are named using linguistic adjectives such as slow, medium or fast. The degree of membership describes how slow or how fast a particular value is.
Let us consider an example where we define "youthness". In this case the set S (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, where each person is assigned a degree of membership in the fuzzy subset YOUNG which answers the question to what degree is person x young? The membership function is based on the persons age.
Eg young(x) = 1, if age(x) = 0.,
=(0-age(x))/10, if 0 age(x) = 0,
=0, if age(x) 0 }
A graph of this looks like
According to the graph if Rohans age is 10 then his degree of youth is 1.00 and if Krishnas age is 8 his degree of youth is 0. However if Edward and Colin are aged 1 and 6 their degree of youth will be 0.0 and 0.40 respectively.
A membership function can be defined as a function that maps each point of fuzzy set A to the real interval [0.0, 1.0] such that as m(A(x)) approaches the grade of membership for x in A increases.
LOGICAL OPERATIONS ON FUZZY SETS
In boolean logic there are the union (or), intersection (and) and not operators. These operators exist in fuzzy logic too, but are defined differently
A AND B = MIN(m(A(x)), m(B(x))) (minimum operator for intersection of two fuzzy sets)
A OR B = MAX(m(A(x)), m(B(x))) (maximum operator for union of two fuzzy sets)
A (NOT A) = 1 - mA(x)
These operations are illustrated with the help of graphs
Let A be a fuzzy interval between 5 and 8 and B be a fuzzy number about 4
The following figure shows the fuzzy set between 5 and 8 AND about 4 (blue line).
The Fuzzy set between 5 and 8 OR about 4 is given below ( blue line).
This figure gives an example for a negation. The blue line is the NEGATION of the fuzzy set A.
THE FUZZY PROCESS
.In real-world applications we typically deal with crisp numbers. Fuzzy systems provide a layer of abstraction similar to the spoken language when implementing a control strategy. However, gauges, meters and control devices don't understand such concepts as cold and hot or slow and fast, so there is a mapping process that needs to take place to convert crisp values into fuzzy values and vice versa. We call these mapping operations fuzzification and de-fuzzification.
Fuzzification It is the process of mapping input variables to their respective fuzzy set. Suppose we have a range of temperatures over which we assign three fuzzy sets cold, warm, and hot.
Here from 0 to about 0 degrees the temperature is cold with a truth value of 1. As we move into the transition zone,Temperature is still cold but with a decreasing truth value. At the same time, Temperature is also warm with a truth value that continues to increase until about 7.5 degrees. Between 7.5 and 55 degrees we transition again into decreasing values of warm and increasing values of hot. Finally, beyond 55 degrees Temperature is entirely hot (truth value is 1).
De-Fuzzification Once a fuzzy result is produced, it typically needs to be broken down into a crisp value to be useful.This process is called "de-fuzzification".
Suppose we are given a fuzzy result that is slow with a truth value of 0.5 and the following fuzzy function definition for slow
To resolve the given information into a crisp value we take the following steps -Define a region bounded by the fuzzy function with y-values less than the truth value. -Calculate the centroid (center of area) for this region.-The x-coordinate of the centroid is the crisp value from this fuzzy set.
In this example the crisp output is 1444 RPM.
EXPERT SYSTEMS AND ARTIFICIAL INTELLIGENCE
Expert Systems are computer programs that are derived from a branch of computer science research called Artificial Intelligence (AI). AIs scientific goal is to understand intelligence by building computer programs that exhibit intelligent behavior. It is concerned with the concepts and methods of symbolic inference, or reasoning, by a computer, and how the knowledge used to make those inferences will be represented inside the machine.
The term intelligence covers many cognitive skills, including the ability to solve problems, learn, and understand language. But most progress to date in AI has been made in the area of problem solving -- concepts and methods for building programs that reason about problems rather than calculate a solution.
AI programs that achieve expert-level competence in solving problems in task areas by bringing to bear a body of knowledge about specific tasks are called knowledge-based or expert systems. The term expert systems is reserved for programs whose knowledge base contains the knowledge used by human experts, in contrast to knowledge gathered from textbooks. The area of human intellectual endeavor to be captured in an expert system is called the task domain. Task refers to some goal-oriented, problem-solving activity. Domain refers to the area within which the task is being performed. Typical tasks are diagnosis, planning, scheduling, configuration and design.
Building an expert system is known as knowledge engineering.The knowledge engineer should ensure that (1) The computer has all the required knowledge to solve the problem. () One or more forms to represent required knowledge as symbol patterns in memory of the computer(knowledge representation). () The computer should be able to use the knowledge efficiently by selecting from a handful of reasoning methods.
The Building Blocks of Expert Systems
Every expert system consists of two principal parts knowledge base and the reasoning, or inference engine.
The knowledge base of expert systems contains both factual and heuristic knowledge.
1.Factual knowledge is that knowledge of the task domain that is widely shared, typically found in textbooks or journals.
.Heuristic knowledge is the less rigorous, more experiential, more judgmental knowledge of performance. In contrast to factual knowledge, heuristic knowledge is rarely discussed, and is largely individualistic. It is the knowledge of good practice, good judgment, and plausible reasoning in the field. It is the knowledge that underlies the art of good guessing".
Inference engineThe problem-solving model, or paradigm, organizes and controls the steps taken to solve the problem. One common but powerful paradigm involves chaining of IF-THEN rules to form a line of reasoning. The problem-solving methods are built into program modules called inference engines or inference procedures that manipulate and use knowledge in the knowledge base to form a line of reasoning.
Though an expert system consists primarily of a knowledge base and an inference engine, a couple of other features are worth mentioning reasoning with uncertainty, and explanation of the line of reasoning.
Knowledge is almost always incomplete and uncertain. To deal with uncertain knowledge, a rule may have associated with it a confidence factor or a weight. The set of methods for using uncertain knowledge in combination with uncertain data in the reasoning process is called reasoning with uncertainty. An important method for easoning with uncertainty is called fuzzy logic, and the systems that use them are known as fuzzy systems. At this stage we would like to present to you the relation between control systems and fuzzy logic.
FUZZY EXPERT SYSTEMS
A fuzzy expert system is an expert system that uses a collection of fuzzy membership
functions and rules, instead of Boolean logic, to reason about data.
The rules in a fuzzy expert system are usually of a form 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 an output variable (a name for a data value to be computed), low is a membership function (fuzzy subset) defined on x, high is a membership function defined on y, and medium is a membership function defined on z.
The antecedent (the rules premise) describes to what degree the rule applies, while the conclusion (the rules consequent) assigns a membership function to each of one or more output variables. Most tools for working with fuzzy expert systems allow more than one conclusion per rule. The set of rules in a fuzzy expert system is known as the rulebase or knowledge base.
The general inference process proceeds as follows
1.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.
.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.
.COMPOSITION, all of the fuzzy subsets assigned to each output variable are combined together to form a single fuzzy subset for each output variable. Usually MAX or SUM are used. In MAX composition, the combined output fuzzy subset is constructed by taking the point wise maximum over all of the fuzzy subsets assigned to variable by the inference rule (fuzzy logic OR). In SUM composition, the combined output fuzzy subset is constructed by taking the point wise 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. 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.
APPLICATION OF FUZZY LOGIC IN EXPERT CONTROL SYSTEMS
FUZZY LOGIC CONTROL OF INSPIRED OXYGEN CONCENTRATION IN VENTILATED NEWBORN INFANTS
The control of oxygen delivery to mechanically ventilated newborn infants is a time intensive process that must balance adequate tissue oxygenation against possible toxic effects of oxygen exposure. Oxygen toxicity plays a role in the development of chronic lung disease in newborn infants requiring mechanical ventilation. In premature infants, varying levels of oxygen exposure are implicated in the development of retinopathy of prematurity. Because of these effects, control of oxygen delivery to ventilated newborns has become a priority in neonatal intensive care. In this case study we will discuss the implementation of a fuzzy controller for the adjustment of inspired oxygen concentration (FIO) in ventilated newborns.
Drawbacks of the conventional controller Manual control of the FIO, however, may lag the clinical condition of the patient. That is, a patient may have an increased oxygen requirement as demonstrated by a lower oxygen saturation, but the manual increase of FIO may be delayed by human response times. Conversely, a patient may have a decreased oxygen requirement as clinical conditions improve, yet the amount of oxygen delivered may not be immediately decreased. The latter scenario may be more common because of the perception that a patient with high oxygen saturation is doing well and does not require immediate intervention.
These drawbacks led to the implementation of microcomputer based system to automatically and continuously control the FIO delivered to mechanically ventilated newborn infants. This system utilizes a fuzzy logic controller based on rules generated by neonatologists who routinely provide care for ventilated infants. In the context of FIO control in the newborn infant, a fuzzy logic approach can simplify the many complex factors and interactions that determine patient oxygenation. For example, a ventilated infant may exhibit decreased oxygen levels in the blood (as measured by SaO) for any of the following reasons failure to make respiratory effort, a plug in the endotracheal tube, or an increase in pulmonary shunting. Each cause may require differing changes in FIO to maintain target SaO levels, and many other factors may influence oxygenation. At different times, the same magnitude of change in FIO may result in completely different oxygenation states, even within the same patient.
SYSTEM DESCRIPTION
FIOcontroller
In this expert controller SaO is the . measurement parameter and FIO is the control parameter for the operational model of maintaining patient oxygenation. The error between the patients SaO and the target SaO ([[Delta]]SaO), and the slope of SaO (SaO-slope) are the specific inputs to the fuzzy controller. The design of the fuzzy controller then follows standard methods, with fuzzification of the input parameters, construction of fuzzy inference rules, and defuzzification or calculation of a crisp output value that represents the controllers action.
To fuzzify the input parameters, the values of [[Delta]]SaO and SaO-slope were divided into fuzzy regions, with 7 regions chosen for [[Delta]]SaO and 5 regions chosen for the SaO-slope.
Using the fuzzy input parameters, the inference rules that form the body of the controller were constructed in the standard declarative form IF situation THEN action. The combination of 7 [[Delta]]SaO fuzzy regions and 5 SaO-slope fuzzy regions yields 5 rules. The logic of these inference rules are based on the expert knowledge of the neonatologists.
Some rules governing the expert controller
Rule IF the [[Delta]]SaO is small-negative AND the SaO-slope is medium-negative (situation)
THEN increase the FIO by a medium-positive amount. (action)
Rule IF the [[Delta]]SaO is large-negative AND the SaO-slope is large-negative THEN increase the FIO by an extremely-large-positive amount.
Rule IF the [[Delta]]SaO is small-negative AND the SaO-slope is small-positive THEN do nothing.
For any pair of [[Delta]]SaO and SaO-slope inputs, we apply each of the inference rules in turn. Each rule will yield an action value. The defuzzification step then involves choosing a method to combine all the action values into a final value (a crisp value) that represents the controller output.
The actual operation of the FIO controller is as follows
1) SaO values are obtained for the patient every 1- seconds.
) Every 10 seconds, the [[Delta]]SaO and the SaO-slope are calculated. [[Delta]]SaO = (ave. SaO values over last 10seconds) - (target SaO)SaO-slope = least squares regression of SaO values over last 10 seconds.) The calculated [[Delta]]SaO and SaO-slope are used as indices for the compiled fuzzy controller look-up table. A suggested FIO change is returned as the controller output.
CONCLUSION
The use of expert control systems using fuzzy logic is now becoming very popular. Expert control systems can take over tedious and time consuming jobs otherwise done by humans and are almost always more reliable than their human counterparts.
70% of Indians earn their livelihood from agriculture. Expert control systems have been tried and tested successfully in many parts of the world and in different applications. We propose the use of expert control systems in horticulture in India to take a big load off our farmers shoulders.
The amount of water and manure to be given to the plants depends on the soil and air humidity and also the time of day. The amount of water is also dependent on the type of crop that is being grown and on the kind of soil.A system that administers this will prove very beneficial to farmers especially those with large plantations and different varieties of crops.In our design we propose to have two fuzzy input variablesthe amount of humidity in the air and the time of day. We propose to use the drip irrigation system where the output fuzzy variable is the amount of water to be released.
HUMIDITYTIME OF DAYAMOUNT OF WATER
Very lowVLMorningMZeroZ
LowLAfternoonAVery little VL
NormalNEarly eveningEEModerateM
HighHNightNExtraE
The algorithm for such a system is derived from the block given below which can be interpreted as
If the humidity is Very low AND (it is morning or early evening)
THEN, The amount of water will be Extra.
VLLNH
MEMMVL
AVLZZZ
EEELNVL
NVLVLZZ
This kind of system will require sensors to test the humidity of the atmosphere and it will also require a sensor to determine the time of day. The width and centre of the membership functions can be easily changed and configured according to the type of crop growing in that region and the soil content. For example if we are implementing this system in the Thar desert region or on the Fertile plains of the Ganges the value of the humidity function will differ.i.e in the Thar desert a humidity of 0% may be considered as normal(N)while on the Ganges plain the same amount is considered very low(VL).
It is also observed that in fuzzy logic control the transition from one fuzzy subset to another provides a smooth transition from one control action to another, thus the need to overlap these fuzzy subsets. However too much overlap would blur the distinction in the control action. A heuristic approach is to overlap the fuzzy subsets by about 5%.
Books
A First Course in Fuzzy Logic
By Hung T. Nguyen, Elbert A. Walker
Websites
www.cms.dmu.ac.uk
www.abo.fi
http//citeseer.nj.nec.com
www.generation5.org
http//coel.ecgf.uakron.edu
www.emsl.pnl.gov.080
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