1. Misconceptions about Fuzzy concepts

2. Fuzziness is not Vague we shall have a look at some propositions.
Dimitris is six feet tall
The first proposition (traditional) has a crisp truth value of either TRUE or FALSE.
He is tall
The second proposition is vague.
It does not provide sufficient information for us to make a decision, either fuzzy or crisp.
We do not know the value of the pronoun.
Is it Dimitris, John or someone else?

3. Fuzziness is not Vague Andrei is tall
This proposition is a fuzzy proposition.
It is true to some degree depending in the context, i.e., the universe of discourse.
It might be SomeWhat True if we are referring to basketball players or it might be Very True if we are referring to horse-jockeys.

4. Fuzziness is not Multi-valued logic
The limitations of two-valued logic were recognised very early.
A number of different logic theories based on multiple values of truth have been formulated through the years.
For example, in three-valued logic three truth values have been employed.
These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and 0.5 respectively.
In 1921 the first N-valued logic was introduced.
The set of truth values Tn were assumed to be evenly divided over the closed interval [0,1].
Fuzzy logic may be considered as an extension of multi-valued logic but they are somewhat different.
Multi-valued logic is still based on exact reasoning whereas fuzzy logic is approximate reasoning.

5. Fuzziness is not Probability!!!

6. Fuzziness is not Probability
Let X be the set of all liquids (i.e., the universe of discourse) .
Let L be a subset of X which includes all suitable for drinking liquids. A B
Bottle A label is marked as membership of L is 0.9.
The label of bottle B is marked as probability of L is 0.9.
Which one would you drink?

7. Fuzziness is not Probability
This is better explained using an example.
Let X be the set of all liquids (i.e., the universe of discourse) .
Let L be a subset of X which includes all suitable for drinking liquids.
Suppose now that you find two bottles, A and B.
The labels do not provide any clues about the contents.
Bottle A label is marked as membership of L is 0.9.
The label of bottle B is marked as probability of L is 0.9.
Given that you have to drink from the one you choose, the problem is of how to interpret the labels.

8. Fuzziness is not Probability
Well, membership of 0.9 means that the contents of A are fairly similar to perfectly potable liquids.
If, for example, a perfectly liquid is pure water then bottle A might contain, say, tonic water.
Probability of 0.9 means something completely different.
You have a 90% chance that the contents are potable and 10% chance that the contents will be unsavoury, some kind of acid maybe.
Hence, with bottle A you might drink something that is not pure but with bottle B you might drink something deadly. So choose bottle A.

9. Fuzziness is not Probability
Opening both bottles you observe beer (bottle A) and hydrochloric acid (bottle B).
The outcome of this observation is that the membership stays the same whereas the probability drops to zero.
All in all:
probability measures the likelihood that a future event will occur,
fuzzy logic measures the ambiguity of events that have already occurred.
In fact, fuzzy sets and probability exist as parts of a greater Generalized Information Theory.
This theory also includes:
Dempster-Shafer evidence theory,
possibility theory,
and so on.

10. Applications of Fuzzy concepts

11. Fuzzy inference for practical control by Mamdani
The most commonly used fuzzy inference technique is the so-called Mamdani method.
In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination.
He applied a set of fuzzy rules supplied by experienced human operators.

12. Mamdani versus Sugeno Models
Most of our examples were for Mamdani Model.
Another famous model comes from Sugeno.
We will discuss and compare both models.Mo

13. Sugeno fuzzy inference
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Sugeno fuzzy inference
n Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient.
n Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent. A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else.

14. IF x is A AND y is B THEN z is f (x, y)
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Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule consequent. Instead of a fuzzy set, he used a mathematical function of the input variable. The format of the Sugeno-style fuzzy rule is
IF x is A AND y is B THEN z is f (x, y)
where x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; and f (x, y) is a mathematical function.

15. IF x is A AND y is B THEN z is k
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The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form:
IF x is A AND y is B THEN z is k
where k is a constant.
In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented by singleton spikes.

16. Sugeno-style rule evaluation

17. Sugeno-style aggregation of the rule outputs
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Sugeno-style aggregation of the rule outputs

18. Weighted average (WA):
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Weighted average (WA):
Sugeno-style defuzzification

19. How to make a decision on which method to apply – Mamdani or Sugeno?
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How to make a decision on which method to apply – Mamdani or Sugeno?
n Mamdani method is widely accepted for capturing expert knowledge. It allows us to describe the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference entails a substantial computational burden.
n On the other hand, Sugeno method is computationally effective and works well with optimisation and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems.

20. Example 4 (Mamdani Fuzzy Model)
Single input single output Mamdani fuzzy model with 3 rules:
If X is small then Y is small  R1
If X is medium then Y is medium  R2
Is X is large then Y is large  R3
X = input [-10, 10] Y = output [0,10]

21. Single input single output antecedent & consequent MFs
Using centroid defuzzification, we obtain the following overall input-output curve
Overall input-output curve
Single input single output antecedent & consequent MFs

22. Example 5 (Mamdani Fuzzy model)
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Two input single-output Mamdani fuzzy model with 4 rules: If X is small & Y is small then Z is negative large If X is small & Y is large then Z is negative small If X is large & Y is small then Z is positive small If X is large & Y is large then Z is positive large

23. Two-input single output antecedent & consequent MFs
X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min
composition & centroid defuzzification, we can
determine the overall input output surface
Two-input single output antecedent & consequent MFs

24. Overall input-output surface
Intelligent Systems and Soft Computing
X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min
composition & centroid defuzzification, we can
determine the overall input output surface
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Overall input-output surface

25. Overall input-output surface
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Overall input-output surface
X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min composition & centroid defuzzification,
we can determine the overall input output surface

26. Example of Mamdani: Cement Kiln Example

30. Examples for Sugeno Fuzzy Models
Single output-input Sugeno fuzzy model with three rules
If X is small then Y = 0.1X + 6.4
If X is medium then Y = -0.5X + 4
If X is large then Y = X – 2
If “small”, “medium” & “large” are nonfuzzy sets then the overall input-output curve is a piece wise linear

31. If X is small then Y = 0.1X + 6.4
If X is medium then Y = -0.5X + 4
If X is large then Y = X – 2

32. However, if we have smooth membership functions (fuzzy rules) the overall input-output curve becomes a smoother one

33. Examples for Sugeno Fuzzy Models
Two-input single output fuzzy model with 4 rules
R1: if X is small & Y is small then z = -x +y +1
R2: if X is small & Y is large then z = -y +3
R3: if X is large & Y is small then z = -x +3
R4: if X is large & Y is large then z = x + y + 2
Overall input-output surface

34. Building a fuzzy expert system: case study
EXAMPLE 8
Building a fuzzy expert system: case study

35. Building a fuzzy expert system: case study
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Building a fuzzy expert system: case study
n A service centre keeps spare parts and repairs failed ones.
n A customer brings a failed item and receives a spare of the same type.
n Failed parts are repaired, placed on the shelf, and thus become spares.
n The objective here is to advise a manager of the service centre on certain decision policies to keep the customers satisfied.

36. Process of developing a fuzzy expert system
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Process of developing a fuzzy expert system
1. Specify the problem and define linguistic variables.
2. Determine fuzzy sets.
3. Elicit and construct fuzzy rules.
4. Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system.
5. Evaluate and tune the system.

37. Step 1: Specify the problem and define linguistic variables
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Step 1: Specify the problem and define linguistic variables
There are four main linguistic variables:
average waiting time (mean delay) m,
repair utilization factor of the service centre r (is the ratio of the customer arrival day to the customer departure rate) ,
number of servers s,
initial number of spare parts n.

38. Linguistic variables and their ranges

39. Step 2: Determine fuzzy sets
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Step 2: Determine fuzzy sets
Fuzzy sets can have a variety of shapes.
However,
a triangle or a trapezoid can often provide an adequate representation of the expert knowledge,
and at the same time, significantly simplifies the process of computation.

40. Fuzzy sets of Mean Delay m
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Fuzzy sets of Mean Delay m

41. Fuzzy sets of Number of Servers s

42. Fuzzy sets of Repair Utilisation Factor r

43. Fuzzy sets of Number of Spares n

44. Step 3: Elicit and construct fuzzy rules
To accomplish this task, we might ask the expert to describe how the problem can be solved using the fuzzy linguistic variables defined previously.
Required knowledge also can be collected from other sources such as books, computer databases, flow diagrams and observed human behavior.
The matrix form of representing fuzzy rules is called fuzzy associative memory (FAM).

45. The square FAM representation

46. The rule table

47. Rule Base 1

48. Cube FAM of Rule Base 2

49. Step 4: Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system
To accomplish this task, we may choose one of two options: to build our system using a programming language such as C/C++ or Pascal, or to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox, Fuzzy Clips, or Fuzzy Knowledge Builder.

50. Step 5: Evaluate and tune the system
The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our fuzzy system meets the requirements specified at the beginning.
Several test situations depend on the mean delay, number of servers and repair utilization factor.
The Fuzzy Logic Toolbox can generate surface to help us analyze the system’s performance.

51. Three-dimensional plots for Rule Base 1

52. Three-dimensional plots for Rule Base 1

53. Three-dimensional plots for Rule Base 2

54. Three-dimensional plots for Rule Base 2

55. However, even now, the expert might not be satisfied with the system performance.
To improve the system performance, we may use additional sets - Rather Small and Rather Large – on the universe of discourse Number of Servers, and then extend the rule base.

56. Modified fuzzy sets of Number of Servers s

57. Cube FAM of Rule Base 3

58. Three-dimensional plots for Rule Base 3

59. Three-dimensional plots for Rule Base 3

60. Tuning fuzzy systems
1. Review model input and output variables, and if required redefine their ranges.
2. Review the fuzzy sets, and if required define additional sets on the universe of discourse.
The use of wide fuzzy sets may cause the fuzzy system to perform roughly.
3. Provide sufficient overlap between neighboring sets. It is suggested that triangle-to-triangle and trapezoid-to-triangle fuzzy sets should overlap between 25% to 50% of their bases.

61. 4. Review the existing rules, and if required add new rules to the rule base.
5. Examine the rule base for opportunities to write hedge rules to capture the pathological behaviour of the system.
6. Adjust the rule execution weights. Most fuzzy logic tools allow control of the importance of rules by changing a weight multiplier.
7. Revise shapes of the fuzzy sets. In most cases, fuzzy systems are highly tolerant of a shape approximation.

62. Example: Simulation of accident

67. Fuzzy Logic is NOT fuzzy thinking

69. Where is fuzzy logic used?
Fuzzy logic is a powerful problem-solving methodology.
It is used directly and indirectly in a number of applications.
Fuzzy logic is now being applied all over Japan, Europe and more recently in the United States of America.
It is true though that all we ever hear about is Japanese fuzzy logic.
Products such as:
the Panasonic rice cooker,
Hitachi's vacuum cleaner,
Minolta's cameras,
Sony's PalmTop computer,
and so on.
This is not unexpected since Japan adopted the technology first.

70. Where is fuzzy logic used?
whereas in the West companies might keep their fuzzy development secret because of the implication of the word `fuzzy',
or because companies want to preserve competitive advantage,
or because fuzzy logic is embedded in products without advertisement.
Most applications of fuzzy logic use it as the concealed logic system for expert systems.

71. Where is fuzzy logic used?
The areas of potential fuzzy implementation are numerous and not just for control:
Speech recognition,
fault analysis,
decision making,
image analysis,
scheduling
and many more are areas where fuzzy thinking can help.
Hence, fuzzy logic is not just control but can be utilized for other problems.

72. Example: Fraudulent Behavior
One business problem, namely that of fraud detection, was recently addressed using fuzzy logic.
The system detects probable fraudulent behavior:
by evaluating all the characteristics of a provider's claim data in parallel,
against the normal behavior of a small ( in demographic terms ) community.
An all-American success story is the use of fuzziness on keeping a commercial refrigerator thermally controlled (0.1 C).
The excitement comes due to the fact that this refrigerator has flown on several space shuttle flights.

73. Example: Automatic Focusing system
Another interesting application has been reported by Aptronix Inc.
Fuzzy logic was there used as a means of determining correct focus distance for cameras with automatic focusing system.
Traditionally, such a camera focuses at the middle of the view finder.
This can be inaccurate though when the object of interest is not at the center.
Using fuzzy logic, three distances are measured from the view finder; left, center and right.
For each measurement a plausibility value is calculated and the measurement with the highest plausibility is deemed as the place where the object of interest is located.

74. When to use Fuzzy Logic?
If the system to be modelled is a linear system which can be represented by a mathematical equation or by a series of rules then straightforward techniques should be used.
Alternatively, if the system is complex, fuzzy logic may be the technique to follow.

75. When to use Fuzzy Logic? We define a complex system :
when it is nonlinear, time-variant, ill-defined;
when variables are continuous;
when a mathematical model is either too difficult to encode or does not exist or is too complicated and expensive to be evaluated;
when noisy inputs;
and when an expert is available who can specify the rules underlying the system behavior.
Use fuzzy logic for complex systems

76. What about our homeworks and projects?
We learned about the following models for “robot brains”:
Finite State Machine (deterministic)
Probabilistic State Machine
Pseudo-quantum State Machine
Boolean Function and any other mapping.
EACH OF THESE MODELS CAN BE COMBINED WITH FUZZY LOGIC IN YOUR HOMEWORK AND PROJECT
There are various sensors that you can find in the lab.
Many sensors can be also purchased inexpensively on internet.
Camera and Kinect (Real-Sense, etc) are excellent sensors
EACH OF THESE SENSORS CAN BE COMBINED WITH FUZZY LOGIC IN YOUR PROJECT

77. What to do with a camera? (case 1)
1. Camera looks at you
you
camera
You do gestures (body, facial, mouse-simulating, other interface simulating)
Camera looks at you
Your software uses your motions as controls for “robot brain”
Robot mimicks you, you find in data-base, etc.

78. What to do with a camera? (case 2)
2. Camera looks at all robots , obstacles and humans (you) from the ceiling perspective
camera
Planning of motions of all agents.
Recognition of motions of random agents like humans and animals.
Corrective actions, like in case of robot falling.
Soccer and other robot group games.

79. What to do with a camera? (case 3)
Camera located on a robot, robot looks about himself as a feedback.
camera
robot
Universal sensor of positions
Universal sensor of conflicts

80. What to do with a camera? (case 4)
Many cameras, located on various parts of the robot, look at the robot parts or at the environment.
camera
camera
camera
robot
camera
camera
Universal sensor of positions
Hand-shaking

81. Summary Fuzzy logic captures intuitive, human expressions.
Fuzzy sets, statements, and rules are the basis of control.
The technique is extremely powerful, and is used in systems at a growing rate.
It is combined with other methods and is the base of soft computing
Used much in Intelligent Robotics
Can be added to any project from this class and two next classes.
Plenty of software is available.
Easy to modify

83. Questions and Problems (1)
Complete Example 4. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Complete Example 5. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Complete Example 6. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Complete Example 7. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Complete Example 8. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Complete Example 9. Write code, explain the robot details. Show results. Explain your methodology of working with this example.
Explain and illustrate the concepts of Static, Versus Adaptive, versus Self-Organizing Fuzzy Systems.
Discuss the misconceptions about fuzzy logic: fuzzy logic is fuzzy thinking, fuzzy logic is probabilistic, fuzzy logic means no math, fuzziness is vague.
Explain Mamdani’s Controller for Cement Kiln. How to go from Operator’s Manual to Fuzzy Rules. How to select values of parameters?
Explain the Takagi-Sugeno Fuzzy Control System.
Give and explain five examples of Fuzzy Systems and invent a new fuzzy system by yourself.

84. Sources Paul and Mildred Burkey Saba Rahimi Weilin Pan Xuekun Kou
A. Ferworn
Kevin Morris
Dr Dimitris Tsaptsinos
Kingston University, Mathematics