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- 1. F UZZY L OGIC AND F UZZY L OGIC S UN T RACKING C ONTROL RYAN JOHNSON DECEMBER 17, 2002 CALVIN COLLEGE ENGR315AA BSTRACT : Fuzzy logic is a rule-based decision or cold, A-or-not-A. To challenge this type of thinking, consider a half eaten apple. Is it half thereprocess that seeks to solve problems where the system or half gone? Is the glass half full or half empty? Isis difficult to model and where ambiguity or the car going fast or slow? Each of these questionsvagueness is abundant between two extremes. Fuzzy present some shades of gray in this world welogic allows the system to be defined by logic typically describe in black and white.equations rather than complex differential equationsand comes from a thinking that identifies and takes Change is inevitable. There is a danger inadvantage of the grayness between the two extremes. putting definite labels on things. Doing so meansFuzzy logic systems are composed of fuzzy subsets that as changes take place these labels pass fromand fuzzy rules. The fuzzy subsets represent different accurate to inaccurate. Rene Descartes thoughtsubsets of the input and output variables. The fuzzy about change as he pondered a piece of beeswax as itrules relate the input variables to the output variables melted in front of his fireplace. At what point didvia the subsets. Given a set of fuzzy rules, the system the beeswax change from a piece of wax into acan compensate quickly and efficiently. Though the puddle of wax? At some point it had to be both aWestern world did not initially accept fuzzy logic and small puddle and a small piece of wax at the samefuzzy ideas, today fuzzy logic is applied in many time [13]. There is some period between which it issystems. a solid piece and a pure puddle. In this research paper, a solar power sun Grayness is fuzziness. Einstein wonderedtracking system is implemented using fuzzy logic. about the grayness. “So far as the laws ofThe steps of how to create a fuzzy system are mathematics refer to reality, they are not certain.described as well as the description of how the fuzzy And so far as they are certain, they do not refer tosystem works. reality,” he said [13]. Actually, math and science do not fit the world they describe. Math and scienceKeywords: membership function, grayness, fuzzy are neat and organized. They describe the world assubsets, fuzzification, fuzzy rules, defuzzification, neat and organized without any grayness. Math andFuzzy Approximation Theorem (FAT), fuzzy science try to fit every process in the world tonumbers, and fuzzy systems equations and equations are neat and organized. Imagine a world without grayness. It is impossible. The world we live in is very messy and includes I. I NTRODUCTION much grayness. With math and science, we have How do we define the world we live in observed certain tendencies and relationships thattoday? How do we see things around us? Most of us have remained true for a period of time and definedare taught from a very young age to look at the world them as mathematical logic and scientific laws. Thein terms of black and white, A-or-not-A, Boolean 1’s truth of this logic and these laws is only a matter ofand 0’s. Much of science, math, logic, and even degree and could change at any moment [13]. Theyculture assume a world of 1’s and 0’s, true or false, hot could pass from accurate to inaccurate at any time.
- 2. The sun could burn up and never rise again. The This representation seems to most accuratelymoon could stop rotating around the earth. These describe the world that we live in. However, thisneat and organized laws and rules will experience idea challenges Aristotle and his philosophy whichchange. There is an element of grayness present even most of the world has embraced for so long. Thisin math and science. type of thinking is against present scientific thought but is key to fuzzy logic. To further explain the difference between ablack and white scientific or mathematical model Grayness is a key idea of fuzzy logic. Fuzzycompared to a messy real world model, consider when logic is the name given to the analysis that seeks toa person turns from a teen to an adult [13]. Figure 1.1 define the areas of grayness that are so characteristicshows a graph representing an A-or-not-A approach. of the world we live in. Fuzzy logic is an alternativeIt shows that a person is either an adult or non-adult. to the A-or-not-A, Boolean 1 and 0 logic definitionsAristotle’s philosophy was based on A-or-not-A. He built into society. It seeks to handle the concepts offormulated the Law of the Excluded Middle, which partial truth by creating values representing what issays that everything falls into either one group or the between total truth and total falsity. Fuzzy logic canother; it can’t be in both [8]. be used in almost any application and focuses on approximate reasoning while classical logic puts such a large emphasis exact reasoning. II. H ISTORY Fuzzy logic began in 1965 with a paper called “Fuzzy Sets” by a man named Lotfi Zadeh. Zadeh is an Iranian immigrant and professor from UC Berkeley’s electrical engineering and computer science department. The first historical connection to fuzzy logic Figure 1.1: Scientific Representation can be seen in the thinking of Buddha, the founder of Buddhism around 500 B.C. He believed that theFigure 1.2 shows the same graph with the shade of world was filled with contradictions and everythinggray principle, the A-and-not-A principle. It does not contained some of its opposite. Contrary tofollow Aristotle’s law of bivalence. Chances are Buddha’s thinking, the Greek philosopher Aristotlesomeone will have some adult characteristics and created binary logic through the Law of thesome non-adult characteristics. To some degree they Excluded Middle. Much of the Western worldare an adult and to some degree they are not an adult. accepted his philosophy and it became the base of scientific thought. Still today, if something is proven to be logically true, it is considered scientifically correct [7]. Prior to Zadeh, a man named Max Black published a paper in 1937 called “Vagueness: An exercise in Logical Analysis” [13]. The idea that Black missed was the correlation between vagueness and functioning systems. Zadeh, on the other hand, saw this connection and began to develop his “fuzzy” ideas and fuzzy sets. Because fuzzy thinking challenges Aristotelian thinking and therefore scientific logical Figure 2: Grayness Representation thinking, Zadeh’s ideas experienced much opposition from the Western world. There were three main criticisms. The first was that people 2
- 3. wanted to see fuzzy logic applied. This didn’t happen degree each of these devices is a vehicle. Somefor sometime since new ideas take time to apply. The represent a vehicle more than others but all fall insecond criticism came from probability schools. the grayness between a vehicle and non-vehicle.Fuzzy logic uses numbers between 0 and 1 to describe The point is that the word vehicle stands for a fuzzyfuzzy degrees. Probabilists felt that they did the same set and things belong to this set to some degree.thing [13]. The third criticism was the largest. In The actual fuzzy emblem is the yin-yangorder for fuzzy logic to work, people had to agree that symbol [13]. A thing is most fuzzy when it isA-and-not-A was correct. This threatened modern equally a thing and a non-thing. If it is more a thingscience and math ideas. As a result, the Western than a non-thing, it is less fuzzy. If it is more a non-world rejected fuzzy logic for a period of time. thing than a thing, it is less fuzzy. The yin-yang The Eastern world, however, embraced fuzzy symbol, shown in Figure 2.1 is equally black andthinking. By 1980, Japan had over 100 successful white. It is in its most fuzzy state.fuzzy logic devices [13]. According to Zadeh, in 1994,the United States was only ranked third in fuzzyapplication behind Japan and Germany [2]. Stilltoday, the United States is some years behind in fuzzylogic development and implementation. Zadeh recalls that he chose the word “fuzzy”because he “felt it most accurately described what wasgoing on in the theory” [2]. Other words that hethought about using to describe the theory but didn’taccurately describe it included soft, unsharp, blurred,or elastic. He chose the term “fuzzy” because “it tiesto common sense” [13]. Figure 2.1: Yin-yang symbol III. F UZZY L OGIC To further see how fuzzy sets contain There are many benefits to using fuzzy logic. smaller sets and so forth, consider an off-roadFuzzy logic is conceptually easy to understand and has vehicle. An off road vehicle is a smaller set ofa natural approach [8]. Fuzzy logic is flexible and can vehicles. Every off-road vehicle is a vehicle, but notbe easily added to and adjusted. It is very tolerant of every vehicle is an off-road vehicle. The question isimprecise data and can model complex nonlinear raised: when is a vehicle an off-road vehicle? Oncefunctions with little complexity. It can also be mixed again it is a matter of degree. An off-road truck withwith conventional control techniques. There are raised suspension stands for an even smaller set ofthree major components of a fuzzy system: fuzzy sets, vehicles, a subset of off-road vehicles. These fuzzyfuzzy rules, and fuzzy numbers. sets combined with fuzzy rules build a fuzzy system. Fuzzy sets can be created out of anything. Fuzzy logic and fuzzy thinking occur in sets.Consider an example of a vehicle. We all speak The second component of a fuzzy system isvehicle the same, but we think of vehicles on a the fuzzy rules. Fuzzy rules are based on humandifferent, personal level. It is a noun. It describes knowledge. Consider how a human reasons withsomething. There is a group of devices that we call this simple example: should you bring an umbrellavehicles. These devices might include a semi-truck, a with you to work? First, you have the knowledge ofplane, a bus, a car, a bike, a scooter, or a skateboard. the forecast: about a 70% chance of rain. Second,What I consider a vehicle to be could be something you have the knowledge of the function of anvery different from what someone else considers a umbrella: to keep you dry when it is raining. Fromvehicle to be. Which is really a vehicle and which is this knowledge, you create rules that guide younot? Some seem closer to our idea of a vehicle than through your decision. If it rains, you will get wet.others. Aristotle would say that there is only a If you get wet, you will be uncomfortable at work.vehicle and a non-vehicle. Fuzzy logic says that to a If you use an umbrella, you will stay dry. Therefore, 3
- 4. you decide to carry an umbrella with you. The rules There are several ways to associate a fuzzythat guided you to your decision relate one thing or number to a description in words. The associationevent or process to another thing or event in the form takes place in the form of a certain shape. Thisof if-then statements [13]. The knowledge of the shape is called a membership function. There arechance of rain led to rules that made you decide the four shapes that are mainly used. These include away you did. This is how fuzzy rules are created, triangle, a trapezoid, a Gaussian shape, and athrough human knowledge. Singleton. Figure 2.3 shows the possible shapes to use for subset definition. Fuzzy rules define fuzzy patches. Fuzzypatches, along with grayness, are key ideas in fuzzylogic. “These patches tie common sense to simplegeometry and help get the knowledge out of ourheads and onto paper and into computers,” says BartKosko, a world-renowned proponent and populizer offuzzy logic [13]. The patches are defined by how thefuzzy system is built and cover an output line definedby the system. Figure 2.2 shows fuzzy patches thatcover an output line. A concept designed by Koskocalled Fuzzy Approximation Theorem (FAT) statesthat a finite number of patches can cover a curve [13].If the patches are large, the rules are large and sloppy.If the patches are small, the rules are precise. Trying Figure 2.3: Membership Function Shapesto make rules that are too precise builds muchcomplexity in to a fuzzy system. Each of these membership functions are convex in shape meaning as the domain increases, that the shapes rising edge starts at zero, rises to a maximum value, and the decreases to zero again. IV. B UILDING A F UZZY S YSTEM Figure 2.2: Fuzzy Patches Covering a Line To apply the above ideas, consider a two- axis sun tracking system for a stand-a-lone photovoltaic system. The system details are as Fuzzy numbers are fuzzy sets on real follows:numbers [9]. More simply, they are ordinary • The sun tracker is a pole mount system.numbers whose precise value is not known. Anyfuzzy number is a function whose domain is a • The panel will rotate counter-clockwise orspecified set. Fuzzy numbers allow approximate clockwise depending on the sun position withcomparisons [3]. This approximation allows the the pole as a pivot point. In general, as the sunrepresentation of numbers in form of “about n” or travels from the east to the west on a given day,“roughly n” and is useful when data is imprecise or the panels will follow it from the east to the westwhen it is important not to reject a value because it is by a counter-clockwise rotation.very close but not right on “n”. Consider an object • The second axis of the panel will havemoving at a speed that is approximately equal to 50 predetermined settings that require manualmph. It is going “about 50 mph.” Fuzzy numbers are adjustment depending on the season of the year.useful in that they allow us to ignore the rigidity of This means that only the east-to-west rotation isaccepting that the speed is actually 50.1 mph or even actuator controlled.51 mph. From this an approximate comparison can bemade to another object going “about 50 mph.” 4
- 5. • At night, the tracker will rotate the panels to the or clockwise. This value will be supplied to the morning position and rest there for the duration actuators that will turn the panels. A counter- of the night. clockwise rotation will result in the panels following the sun from east to west as a day progresses. A• Attached to each side of the panel is a light clockwise rotation will be compensation for any intensity sensor. The right sensor (from the sun’s overshoot upon panel adjustment. The actuators perspective) will tell if there is more light need to be able to make fine adjustments as well as intensity to the right and the left sensor (from the rough adjustments as the day continues. sun’s perspective) will tell if there is more light intensity to the left. The second step in building a fuzzy system is to define the fuzzy subsets. Subsets are created for• Both light intensity sensor signals feed into a each variable. Often they are named by common comparator where the signals are compared to see sense names. The number and size of the subsets to which side is getting more sun. This information create is based on how robust the system is to be. is supplied to the control system. Creating much overlap between sets creates a more This system was implemented using the robust system. Fuzzy sets can be number based orFuzzy Logic Toolbox found in MATLAB 6.1. This description based. Number based fuzzy sets are setsfuzzy logic tool is quite easy to use and allows many that reference to a number. They ask the questionengineering adjustments to be made to the system. “How much?” Description based fuzzy sets are setsMathematica also has a fuzzy logic tool. This tool, that focus on categories. They ask the questionhowever, is only tested in version 2.2, which was “What is it?” [3]. An example would be a descriptioncurrent around 1997. Mathematica has included the set “color” that might have subsets of red, orange, orfuzzy tool operations in versions since this time; yellow.however they require a fix file that can be First, consider the input variable. To trackdownloaded at Mathematica’s website. Visit the the sun, the system needs to know which side of theMathematica website to see a working example of its panel is receiving more sunlight. The system isfuzzy logic tool. The example shows how a truck can supplied a single input of the difference in lightback itself into a parking spot with the use of fuzzy intensity between the sensors. Subsets shouldlogic [14]. describe in common language which sensor is There are three main steps in creating a fuzzy measuring more light intensity and how much lightsystem: intensity that sensor is reading. If the sensors are1. Choose the input and output variables. measuring equal light, the subset should reflect that.2. Choose the subsets of the variables and create The subset for this situation will be called EQUAL. their membership functions. If it is mostly in the right sensor, the subset will be3. Create the fuzzy rules that will relate the input called MOST RIGHT. This should be done for each variables to the output variables via each subset. input variable. The resultant input subsets are as follows: MOST LEFT, MORE LEFT, LITTLE LEFT, The first step is to choose the variables. EQUAL, LITTLE RIGHT, MORE RIGHT, and MOSTUltimately, these variables become the inputs and RIGHT relating to which sensor is measuring moreoutputs. For the tracking system, the first variable or light intensity.input would be the signal coming out of thecomparator. The comparator will supply the fuzzy Seven subsets were chosen to represent thesystem with a difference in light intensity between input variable. This number of subsets willthe sensors. Though there are two sensors, the only adequately cover each sun-tracking situation for nowthing that needs to be known is the different light and may need to be changed depending on how theintensities between the sensors making this system a system reacts.single input system. Having a single input greatly The same process is required for the outputreduces the complexity of the system. variable. In simple language, the output subsets The second variable or output is the number should describe the number of degrees to turn theof degrees to turn the panels either counter-clockwise system either clockwise or counter-clockwise with 5
- 6. reference to its current location. The output subsets Figure 3.2 shows the output membershipare as follows: MORE COUNTER-CLOCKWISE functions. The triangles were created to be the same(CCW), SOME COUNTER-CLOCKWISE, LITTLE size as the input membership functions. In FigureCOUNTER-CLOCKWISE, RIGHT-ON, LITTLE 3.2, the X-axis units are the degrees to move theCLOCKWISE (CW), SOME CLOCKWISE, and MORE panel. Moving in the counter-clockwise direction isCLOCKWISE. Once again, seven subsets were chosen defined by a negative magnitude of degrees withto represent the output variable. This number of reference to the current panel location. Moving insubsets will adequately cover the rotation of the the clockwise direction is defined by a positivepanels. The same subset principles apply to the output magnitude of degrees with reference to the currentsubsets as the input subsets. panel location. For the fuzzy system, these subsets are drawnto some shape creating membership functions. Theseshapes allow a way to go back and forth between thedescription of the variable in numbers and thedescription of the variable in words. Triangles werechosen for this system. This is an area whereengineering is needed. Any other membership shapecould have been used. Triangular shapes will be usedfor an initial design. A key point is that the shapesmust overlap. The overlapping of the shapes willcreate robustness as mentioned before. This systemhas adequate overlapping and therefore is adequatelyrobust. Figure 3.2: Output Subsets Figure 3.1 shows the input membershipfunctions. Notice that the bases of the triangles are The third step in building a fuzzy system isdifferent widths. The widest sets are least important to define the fuzzy rules. The fuzzy rules associateand give rough adjustment. The thin sets give fine the sun intensity measurements with the panelcontrol. This is another area for engineering. position. The rules will form the patches that willChanging the size of the triangles requires system cover the output line. Common sense was used totweaking and testing. In Figure 3.1, the X-axis define the rules. If the sun is more intense to therepresents the intensity difference between the right of the panels, then the panel should movesensors and the Y-axis represents the fuzzy degree that clockwise toward the sun. Therefore, if the sensorthat subset is true. difference is MORE RIGHT, then the panel movement is MOST CW. Figure 3.3 shows the remaining rules. The rules are all weighted the same in this example. Figure 3.3: Fuzzy Rules Defined Figure 3.1: Input Subsets 6
- 7. V. S YSTEM F UNCTIONALITY & THE is found to be the degree to move output value. This is called an additive fuzzy system because the F UZZY P ROCESS triangles are added to get the output set. The fuzzy system is now complete. Mostfuzzy systems are controlled by fuzzy chips. Thesechips walk through the fuzzy process millions oftimes per second in fuzzy logical inferences persecond or FLIPS [13]. Fuzzy chips aremicroprocessors that are designed to store and processfuzzy rules [11]. The first digital fuzzy chip wascreated in 1985 and processed 16 rules in 12.5microseconds, a rate of 0.08 million fuzzy logicalinferences per second. Today there are fuzzy chipsthat process up to two mil1ion rules per second [11]. The fuzzy process has three main stages:1. Fuzzification2. Rule check and degree of truth determination3. DefuzzificationConsider Figure 4.1. Figure 4.1 shows an overview of Figure 4.2: Panel Centered on Sun Outputthe fuzzy process. First, there is an input X that isfuzzified into A. A is considered with each fuzzy ruleto see which rules are true and to what degree. B Next, consider the case where the panel hasprime represents the degree that each rule is true. All overshot the sun position by a few degrees. Thethe B primes are added and then sent through the right sensor now sees more light than the left sensor.defuzzier, which in the case of this example finds the Now, there are two triangles affected by the input,average or center of mass of the summed B primes as EQUAL and LITTLE RIGHT. This can be seen inthe value to be outputted, the value Y. Figure 4.2. There are two rules that are each true to some degree. This gives two output triangles that are each true to some degree. To find the distance to move the panels, the triangles are added together and the average or center of mass of the figure is found. Figure 4.1: Fuzzy process Now, consider the sun tracking system. To seehow this system finds an output value, considerFigure 4.2. Figure 4.2 shows how the input subsetsand output subsets are related. The input in this caseshows equal light intensities in each sensor. TheEQUAL triangle is the only subset that is affected andis 100% true. This means that the rule if EQUAL,then RIGHT ON is 100% true and the RIGHT ONtriangle in the output is 100% true also. The outputtriangles are added and the average or center of mass 7
- 8. FINE_COUNTER-CLOCKWISE and FINE_CLOCKWISE. Figure 4.3: Panel Offset With the addition of the membership functions, new rules must be created. Figure 4.5 Figure 4.3 shows that will little sun position shows the rules with the new rules added.shift (a difference in magnitude of only .235), thedegrees to move the panel is already 27.1 degrees.This is a bit much seeing as the sun tracker will onlymove a total of about 270 degrees on the longest dayof the year. This may mean that the system designthus far does not have fine enough adjustment. Sincethe sun in continuously moving, there will be verylittle change in sun position each time it is checked.Therefore, it is desirable to have it move only a fewdegrees for little differences in sun intensity. Figure 4.5: New Membership Rules Added To try to fix this, consider changing the inputmembership functions to a Gaussian shape. Thewidths will remain the same for the time being. With the new rules and new membershipFigure 4.4 shows that this helped a little. Now, when functions, the system now has good fine adjustment.there is a sun intensity difference of .213, the panel Figure 4.6 shows that with a sun intensity differenceshould move about 22 degrees. Unfortunately, the of .213, the panel should move about 6.4 degrees.fine tune adjustment needs to be even better yet. This is sufficient for typical sun movement throughout the day. Once again, Figure 4.6 shows how there were about three rules and in this case three output membership functions that were true to some degree when the sun intensity difference was inputted into the system. The average of the addition of the degrees of truth of each output membership function was found to be the degrees to move the panel to line up with the sun. Figure 4.4: Gaussian Input Membership Function Shapes Next, consider adding two more inputmember functions and two more output memberfunctions. These will be added to surround the inputmember function EQUAL for fine adjustment and tosurround the output member function RIGHT_ONfor fine adjustment. The input membership functions Figure 4.6: Fine Adjustmentwill be called FINE_RIGHT and FINE_LEFT. Theoutput membership functions will be called 8
- 9. VI. C ONCLUSION huge, involved equations. Sometimes it is just common sense and a little fuzzy thinking. Fuzzy logic seeks to define the areas ofgrayness that are so characteristic of the world we livein. Fuzzy logic is an alternative to the A-or-not-A, R EFERENCESusing the idea that A-and-not-A is okay. It seeks tohandle the concepts of partial truth by creating fuzzy [1] Aziz, Shahariz Abdul. “You Fuzzyin’ With Me?”numbers representing what is between total truth and 1996. Online posting. 13 Dec. 2002.total falsity. It allows control with little math. <http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vSimple human knowledge and thinking can create a ol1/sbaa/article1.html >reliable and quickly adjusting control system. It isimportant to understand the thinking behind fuzzy [2] Blair, Betty. “Interview with Lotfi Zadeh.”logic and to see that the world is not just black and Azerbaijan International. 2.4 (1994). 4 Dec. 2002.white. It is important to see the grayness. <http://www.azer.com/aiweb/categories/magazine/2 4_folder/24_articles/24_fuzzylogic.html> Fuzzy systems are created with three mainsteps. The first is to define the input and output [3] “Chapter 1: Fuzzy Mathematics: Fuzzy Logic,variables. The second is to define the fuzzy subsets of Fuzzy Sets, Fuzzy Numbers.” 14 Dec. 2002.each input and output variable and create <http://members.aol.com/wsiler/chap01.htm>membership functions. The third is to define fuzzyrules that relate each input membership function toeach output membership function. Upon the [4] Conti, S., G. Tina, C. Ragusa. “Optimal Sizingcompletion of a fuzzy system, the fuzzy process will Procedure for Stand-Alone Photovoltaic Systems byfuzzify an input, check each rule to find a degree of Fuzzy Logic.” Journal of Solar Energy Engineering.truth, and then defuzzify the result into an output Feb. 2002, vol. 124. 77-82.value. Fuzzy logic can be applied to more thingsthan just control systems. For example, it can be used [5] Cruz, Adriano. “Extension Principle.” 2002.for optimization. Using fuzzy logic for stand-alone UFRJ. 12 Dec. 2002.photovoltaic system size determination is a relatively <http://equipe.nce.ufrj.br/adriano/fuzzy/transparencinew application. Fuzzy Logic “has proven to be an as/extension.pdf>efficient tool for defining decision making schemes inmulti-objective optimization problems: the designer [6] “Fuzzy Arithmetic.” 4 Oct. 2000. Online posting.can specify the rules underlying the system behavior Everything 2. 12 Dec, 2002.and the fuzzy sets that represent the characteristics of <http://www.everything2.com/index.pl?node=fuzzyeach variable” [4]. It has also been used in some %20arithmetic>earthquake prediction processes. [7] “Fuzzy Logic.” Online posting. 12 Dec. 2002. Further research on fuzzy logic could be <http://www.ch172.thinkquest.hostcenter.ch/fuzzy-done on fuzzy arithmetic. Fuzzy logic does not logic7.html>always prove to be completely accurate because notall mathematical functions will work with fuzzy [8] “Fuzzy Logic Toolbox.” The MathWorks. Onlinenumbers. There is much research being done in this posting. 2002. 6 Dec. 2002.area and there are many proposed solutions. <http://www.mathworks.com/access/helpdesk/help/t All in all, fuzzy logic is another way to look oolbox/fuzzy/fuzzy.shtml >at the world. It is another way of thinking andchallenges our current scientific thought. It presents [9] Giachetti, Ronald E., Robert E. Young. “Aan easy and practical way to solve many problems. Parametric Representation of Fuzzy Numbers andSometimes it is important to step back and consider a their Arithmetic Operators.” Online posting. 14 Dec.problem from a different angle. Not all solutions are 2002. 9
- 10. <http://citeseer.nj.nec.com/cache/papers/cs/9060/http:zSzzSzwww.nist.govzSzmsidstaffzSzgiachettizSzpfn-fss.pdf/giachetti96parametric.pdf>[10] Hanns, Michael. “A Nearly Strict FuzzyArithmetic for Solving Problems with Uncertainties.”Online posting. 14 Dec. 2002.<http://www.mecha.uni-stuttgart.de/Mitarbeiter/Hanss/papers/nafips00a.pdf >[11] Isaka, Satoru, Bart Kosko. “Fuzzy Logic.” -Scientific American. Online posting. July 1993. 14Dec. 2002.<http://www.fortunecity.com/emachines/e11/86/fuzzylog.html>[12] Jacob, Christian. Chapter 4: Fuzzy Systems.Online posting. 11 Dec. 2002.<http://pages.cpsc.ucalgary.ca/~jacob/Courses/Winter2001/CPSC533/Slides/04-Fuzzy-6up.pdf>[13] Kosko, Bart. “Fuzzy Thinking: The New Scienceof Fuzzy Logic.” Hyperion. New York. 1993.[14] “Tour Of Fuzzy Logic Functions.” WolframReasearch, Inc. 28 Nov. 2002.<http://library.wolfram.com/examples/FuzzyLogic/> 10

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