1. Practical examples of Linguistic variables

2. Linguistic variables in Expert Systems
Expert Systems are based on hundreds of “If the Else rules that combine inputs , logic and outputs
Similar to binary logic and state machines.
We can create similar Fuzzy Expert Systems
Looking at the production rules of either a expert system or a fuzzy expert system one can not see any differences
except that the fuzzy system is employing linguistic descriptors rather than absolute numerical values.
However, both parts of fuzzy rules have associated `levels of belief' something lacking in traditional production rules.

3. What is good about fuzzy expert systems
Observe that in traditional production rules even when more than one rule applies only one executes.
With fuzzy rules all applicable rules contribute in calculating the resulting output.
All in all, fuzzy expert systems require fewer production rules since fuzzy rules embody more information.

4. Linguistic variables in Fuzzy Expert Systems
A major reason behind using fuzzy logic is the use of linguistic expressions.
A linguistic variable consists of:
the name of the variable (u),
the term set of the variable (T(u)),
its universe of discourse (U) in which the fuzzy sets are defined,
a syntactic rule for generating the names of values of u, and
a semantic rule for associating with each value its meaning.

5. For example: if u is temperature,
then its term set T(temperature) could be:
T(temperature)={cold, cool, warm, hot} over a universe of discourse U=[0,300].

6. Definitions of Fuzzy Sets in Fuzzy Expert Systems
We can have either continuous or discrete definition of a fuzzy set

7. Example of a Linguistic Variable
…. Terms, Degree of Membership, Membership Function, Base Variable…..

8. Fuzzy Logic Principles and Learning

9. Fuzzy Logic Principles
Fuzzy control produces actions using a set of fuzzy rules based on fuzzy logic
Fuzzy controller involves:
fuzzifying: mapping sensor readings into a set of fuzzy inputs
fuzzy rule base: a set of IF-THEN rules
fuzzy inference: maps fuzzy sets onto other fuzzy sets using membership fncts.
defuzzifying: mapping a set of fuzzy outputs onto a set of crisp output commands

10. Fuzzy Control

11. Fuzzy Control
Fuzzy logic allows for specifying behaviors as fuzzy rules
Such behaviors can be smoothly blended together (e.g., Flakey robot)
Fuzzy rules can be learned
Any learning method can be used

12. Industrial Application of Fuzzy Logic Control

13. History, State of the Art, and Future Development of fuzzy logic systems

14. Uncertainty

15. Types of Uncertainty and the Modeling of Uncertainty
Stochastic Uncertainty:
The Probability of Hitting the Target is 0.8
Three Examples of Lexical Uncertainty:

16. Methods of inference under uncertainty
inference under uncertainty is very important to consider when using expert systems since sometimes data is uncertain (i.e., ambiguous, incomplete, noisy etc.).
A number of theories have been devised to deal with uncertainty.
These include :
classical probability,
Bayesian probability,
Shannon theory,
Dempster-Shafer theory,
other.
A popular method of dealing with uncertainty uses certainty factors

17. Methods of inference under uncertainty
The certainty factor indicates the net belief in the conclusion and premises of a rule based on some evidence.
Certainty factors are hand-crafted by asking potential users questions such as `How much do you believe that opening valve x will start a flooding' and `How much do you disbelieve that opening valve x will start a flooding'.
The degree of certainty is the difference between the two responses.

18. Fuzzy logic representation of Uncertainty using Production Rules

19. Production Rules
Assuming that the knowledge-base module contains knowledge represented in the format of production rules the following sections introduce the following:
the concept of a production rule
the concept of linguistic variables
the fuzzy inference concept
the concept of fuzzification and how to accomplish the crisp to fuzzy transformation
the concept of defuzzification and how to accomplish the fuzzy to crisp transformation

20. Knowledge presentation using production rules
From a philosophical point the concept of knowledge is highly ambiguous and debatable
Knowledge-base builders treat knowledge from a narrower point of view: utilitarian, rules, facts, etc.
This way the knowledge is easier to model and understand.
It remains diverse including:
rules,
facts,
truths,
reasons,
defaults and
heuristics.
The knowledge engineer needs some technique for capturing what is known about the application.

21. Various types of Knowledge representation using production rules
The production rules technique should provide expressive adequacy and notational efficacy.
Knowledge representation is very much under constant research.
Several schemes have been suggested in the literature, namely:
semantic nets,
frames and
logic.
Production rules have also been suggested and are the most popular way of representing knowledge.

22. Knowledge presentation using production rules
Production rules are small chunks of knowledge expressed in the form of if..then statements.
The left hand side (IF) represents the antecedent or conditional part.
The right hand side (THEN) represents the conclusion or action part.
A number of rules collectively define a modularized know-how system.
The principal use of production rules is in the encoding of empirical associations between incoming patterns of data and actions that the system should perform as a consequence.
The production rules are either expressed by an expert of the field, or derived using induction.

23. Fuzzy logic Control

24. Fuzzy Logic Control
Fuzzy controller design consist of turning intuitions, and any other information about how to control a system, into set of rules.
These rules can then be applied to the system.
If the rules adequately control the system, the design work is done.
If the rules are inadequate, the way they fail provides information to change the rules.
Analyze experimentally, change

25. Example: Control a Plant
A valve in an internal combustion engine that regulates the amount of vaporized fuel entering the cylindres

26. FLAKEY ROBOT = Fuzzy Logic for Autonomous Vehicle Motion Planning
EXAMPLE
FLAKEY ROBOT = Fuzzy Logic for Autonomous Vehicle Motion Planning

27. Using Fuzzy Logic for Autonomous Vehicle Motion Planning
Findings of Stanford Research Institute (SRI)
Based on the performance of the robot “Flakey” circa 1993
Discussion of autonomous navigation and path planning in an uncertain environment
Paper: “Using Fuzzy Logic for Autonomous Vehicle Motion Planning”

28. Difficulties of a classical robot control problem
Autonomous operation of a mobile robot in a real-world unstructured environment poses a series of problems:
knowledge about the environment is usually:
incomplete
uncertain, and
approximate
Perceptually acquired information is not reliable:
noise introduces uncertainty and imprecision
limited range and visibility introduces incompleteness
errors in interpretation

29. More Difficulties with classical robot Problem
Real world environments have complex and largely unpredictable dynamics
objects can move
the environment may be modified
features may change
– Vehicle action execution is not reliable:
the results produced by sending a given command to an effector can only be approximately estimated
action execution may fail entirely

30. Robot Architecture using Fuzzy Controller
Flakey
Map of the rooms
LPS
Key is “Local
Perceptual
Space”:
LPS is a data structure
providing
geometric
picture around
vehicle
Local Perceptual Space (LPS)
Goals are represented in the LPS by means of control structures
Camera,etc

31. The Fuzzy Controller
Flakey
Physical motion of the robot is based on a complex fuzzy controller
The controller provides a layer of robust high-level motor skills.
Basic building block of controller is a “behavior”:
A behavior is defined as implementing an atomic motor skill aimed at achieving or maintaining a give goal situation
– e.g. follow a wall.

32. Implementing Behaviors
Flakey
desirability function
Each behavioral skill is represented by means of a “desirability function” that expresses preferences over possible actions with reference to the goal:
e.g. a behavior aimed at following a given wall prefers actions that keep the agent parallel to the wall at a safe distance

33. Behavior through Fuzzy Rules
Flakey
Each behavior was implemented by a set of fuzzy rules of the form
IF A THEN C
A is composed of fuzzy predicates and connectives, and
C is a fuzzy set of control vectors
An example of a “keep off” behavior rule is:
IF obstacle-close-in-front AND NOT obstacle-close-on-left THEN turn-sharp-left

34. Fast Reactive Behaviors
Flakey
Purely reactive behaviors, intended to provide quick simple reactions to potential dangers typically use sensor data
This sensor data has undergone little or no interpretation.
Since quick response is necessary to avoid disaster, little processing can be done.
Rules can be implemented in an FPGA

35. Control Structures
Flakey
Purposeful behavior like attempting to reach a certain location must take explicit goals into consideration.
Goals are represented in the LPS by means of control structures.
Control structure is a triple
S = (A,B,C)
A is a virtual object (artifact) in the LPS
B is a behavior that specifies the way to react to the presence of this object, and
C is a fuzzy predicate expressing the context where the control structure is relevant

36. Control Structure Example
S = (A,B,C)
A is a virtual object (artifact) in the LPS
B is a behavior that specifies the way to react to the presence of this object, and
C is a fuzzy predicate expressing the context where the control structure is relevant
Flakey
An example control structure is
S1=(CP1, go-to-CP, near(CP1)
CP1 is a control-point (marker for a location), together with a heading and a velocity
go-to-CP reacts to the presence of S1 in the LPS by generating the commands to reach the location, heading and velocity specified by CP1.
go-to-CP includes rules like:
IF facing(CP1) AND too-slow-for(CP1) THEN accelerate-smooth-positive

37. Blending of Behaviors Many behaviors can be simultaneously active
Flakey
Many behaviors can be simultaneously active
Fuzzy controller selects the controls that best satisfy the active behaviors
Satisfaction is weighted by each behavior’s relevance to the current situation.
e.g. can’t follow a wall if there isn’t one
Context dependent blending of behaviors is implemented by combining the output of all the behaviors using context rules

38. Generating a plan: simple goal-regressing planner used:
S = (A,B,C)
A is a virtual object (artifact) in the LPS
B is a behavior that specifies the way to react to the presence of this object, and
C is a fuzzy predicate expressing the context where the control structure is relevant
Flakey
simple goal-regressing planner used:
based on a topological map annotated with approximate measurements (no obstacles) working backwards from goal.
An example plan might be:
S1 = (Obstacle, keep-off, near(Obstacle))
S2 = (Corr1, follow,~near(obstacle) AND at(Corr2) AND
~near(Corr2))
S3 = (Corr2, follow,~near(obstacle) AND at(Corr2) AND
~near(Door5))
S4 = (Door5, cross,~near(Obstacle) AND near(Door5))
Control structure

39. Executing the Plan Flakey S1 = (Obstacle, keep-off, near(Obstacle))
S2 = (Corr1, follow,~near(obstacle) AND at(Corr2) AND
~near(Corr2))
S3 = (Corr2, follow,~near(obstacle) AND at(Corr2) AND
~near(Door5))
S4 = (Door5, cross,~near(Obstacle) AND near(Door5))

40. More FUZZY LOGIC SYSTEMS APPLICATIONS

41. More FUZZY LOGIC SYSTEMS APPLICATIONS
1. Discuss a Fuzzy Logic Control System
2. Steps in Designing a Fuzzy Logic Control System
3. Design of a Fuzzy Logic Control System:
Input membership function,
Fuzzy logic rules table,
Output membership function.

42. Components of Fuzzy system:
The components of a conventional expert system and a fuzzy system are the same.
Fuzzy systems though contain `fuzzifiers’.
Fuzzifiers convert crisp numbers into fuzzy numbers,
Fuzzy systems contain `defuzzifiers',
Defuzzifiers convert fuzzy numbers into crisp numbers.

43. Fuzzification
The function of the fuzzification component is to convert crisp numbers to equivalent fuzzy sets.
Please notice that the inputs might require some pre-processing in order to fit the range of the fuzzy system.

44. Defuzzification
The output of the combined operation is defuzzified before being broadcast to the external world.
This implies the conversion of a fuzzy set to a crisp number.
There are several techniques of defuzzification.

45. Conventional vs Fuzzy system

46. Sections of a Fuzzy Logic Control System
Fuzzifier section
System inputs with range of values
Mapped to membership function can be non-linear in correspondence
Fuzzy control section
Input fuzzy values
Through fuzzy rules
Produce the fuzzy output values
De-fuzzifier section
Fuzzy output values are combined together in values needed for output

47. Cruise Control Example of Fuzzy Logic Controller

48. Example design of a Fuzzy Logic Control System – Cruise Control
The block diagram of the intelligent cruise control system.

49. Input membership functions for Cruise Control
Car’s Speed Relative to Cruise Setting Membership Function
Car’s Speed Membership Functions
Distance from Car Ahead
Difference in Car’s Speed Relative to Cruise Setting
Car’s Speed in Miles per Hour
Space Distance in Feet
Shows fuzzification of three inputs
Membership values in Y axis
The three input membership functions

50. Fuzzy Logic Rules Table for Cruise Control
speed
slow / distance from car ahead
fast / distance from car ahead 15 30 45 60 75
+10
0.15 +5
0.35
0.50 -5
0.65
0.85
-10
All combinations in this particular case are specified using a 3-Dimensional Table.
For more dimensions you can use Kmap or Marquand Chart.

51. Output membership function for Cruise Control
The output is the accelerator percentage change needed to keep the car a safe distance behind the car ahead and to keep the car at cruising speed.
Membership values in X axis

52. Fuzzy Inference: creating fuzzy output

53. Fuzzy Inference: creating fuzzy output
At the end of the fuzzification step the working memory module contains the values of the fuzzified input.
Each production rule is examined and all the rules that have their premises satisfied `fire'.
Hence, the only rules which do not fire are those that at least one of their premises has a membership degree of zero.
In the case that more than one rule fires, this is common and desirable, the system generates a single fuzzy output.
This is achieved by combining all fuzzy outputs.
The single fuzzy output is then passed to the defuzzification module which generates a crisp value.
Then the system is ready to start the entire process all over again.

54. SIMPLE EXAMPLE: a four-rule system

55. SIMPLE EXAMPLE: a four-rule system
Assumptions
Let X,Y and Z be the linguistic variables.
Let the membership functions be low and high.
Let the membership functions be the same for all linguistic variables.
Define the membership functions as:
low(linguistic) = 1 - t
high(linguistic) = t
where t is a value in the interval [0,1].

56. A Simple Example (Membership Functions)
Memberships functions in both X and Y axes
LOW
HIGH

57. A Simple Example (Rules)
The rule base contains the following four rules:
if X is low and Y is low then Z is high (rule-1)
if X is low and Y is high then Z is low (rule-2)
if X is high and Y is low then Z is low (rule-3)
If X is high and Y is high then Z is high (rule-4)

58. A Simple Example (Calculations)
Next assume that the inputs are 0 and 0.32 for the linguistic variables X and Y respectively.
The problem then is to find if Z is high or low and its crisp value.
The next steps are followed:
Step-1: Find the membership grade for the premises of each rule. We have 8 premises but due to replication only 4 are needed.
Low(X) = 1- t = 1-0 = 1
Low(Y) = 1- t = = 0.68
High(X) = t = 0
High(Y) = t = 0.32
This is our decision

59. A Simple Example (Calculations)
Step-2: Identify the rules that can fire. In our case only the first two rules can fire since the last two rules have a premise with a zero degree of membership.
Step-3: For each rule that can fire find the strength of the firing.
Rule-1 = min(1,0.68) = 0.68
Rule-2 = min(1,0.32) = 0.32
Note that min is used because the premises are connected with a logical AND.

60. A Simple Example
Rule-2 = min(1,0.32) = 0.32

61. A Simple Example (Calculations)
Step-4: All the fuzzy outputs are combined together to form a single fuzzy subset.
One way is to take the pointwise maximum value over all fuzzy outputs.
In our case we have two fuzzy outputs, so for each point we take the maximum value.

62. A Simple Example The rule base contains the following four rules:
if X is low and Y is low then Z is high (rule-1)
if X is low and Y is high then Z is low (rule-2)
if X is high and Y is low then Z is low (rule-3)
If X is high and Y is high then Z is high (rule-4)
In our case we have two fuzzy outputs, so for each point we take the maximum value.

63. A Simple Example (Calculations)
Step-5: The fuzzy combined output needs to be converted to a single crisp value.
Using one method which takes the Average-of-Maxima.
With this method one finds the maximum peak of the fuzzy combined output - in our case this is 0.68.
Then one collects all the t values for which the maximum value occurs - in our case there are 42 cases.
Finally, the crisp value is the average of such variables - in our case 0.84.

64. Development Cycle
Do not discuss in the class – this is home reading

65. Development Cycle

66. Development Cycle
Step 1: The goal is to establish the characteristics of the system, and also to define the specific operating properties of the proposed fuzzy model.
Traditional systems analysis and knowledge engineering techniques can be employed at this stage. The designer must identify the relevant and appropriate inputs to the system; the basic transformations, if any, that are performed on the inputs,; and what output is expected from the system.
The designer must also decide if the fuzzy system is a subsystem of a `global' system and if so to define where the fuzzy subsystem fits into the `global architecture', or if it is the `global' system itself.
The numerical ranges of inputs and outputs must also be specified. This step applies to the development of all systems. Be prepared to throw away original versions at this stage.

67. Development Cycle
Step 2: The goal is to decompose each control variable (i.e., inputs and outputs) into fuzzy sets and to give unique names to them.
It has been reported in the literature that the number of labels associated with a control variable should generally be odd and between five and nine.
Also, in order to obtain a smooth transition from a state to another each label should overlap somewhat with its neighbours. Overlapping of 10 to 50 percent is advised.
Finally, the density of the fuzzy sets should be highest around the optimal control point of the system and should thin out as the distance from that point increases.

68. Development Cycle
If possible begin with an exhaustive list of production rules
Deal with redundant, impossible and implausible rules and/or conditions later.
When the number of rules increases dramatically, may be several rule-bases might be constructed.
Each rule-base to deal with a particular situation/condition of the system to be modelled.

69. Development Cycle
Step 3: The goal is to obtain the production rules that tie the input values to the output values.
Since each production rule `declares' a small chunk of knowledge, the order in the knowledge base is unimportant.
Nevertheless, in order to maintain the knowledge base one should group the production rules by their premise variable.
How many rules is obviously dependent on the application and is related to the number of control variables.

70. Development Cycle
Step 4: The goal is to decide on the way that is going to be used in order to convert a an output fuzzy set into a crisp solution variable.
There are many ways to perform the conversion but usually process control applications use the centroid technique.
The rest of the steps are similar to any modelling exercise.
For instance, at the end of the fuzzy system construction the process of simulation commences.
The model is compared against known test cases and the results are validated.
When the results are not as desired changes are made until the desired performance is achieved.

71. Popular Fuzzy Inference Methods

72. Popular Fuzzy Inference Methods
Under the fuzzy inference process the grade of each premise is found and applied to the conclusion part of each rule.
MIN or PRODUCT are two popular inference methods.
With MIN inferencing the output of the conclusion part is clipped off at a height equal to the rule's degree of firing.
This was used in the simple example before and the MIN inference for the first rule is re-shown below
(Recall that the rule's degree of firing was 0.68):

73. MIN
0.68

74. PRODUCT
With PRODUCT inferencing the output membership function is scaled by the rule's degree of firing.
Using the first rule of the simple example and the PRODUCT inference we get the following

75. Product

76. Popular methods for combining outputs

77. Popular methods for combining outputs
Under the combining outputs process all outputs are combined together to produce a single fuzzy output.
MAX and SUM are two popular techniques for combining outputs.
With MAX the pointwise maximum for all fuzzy outputs is taken.
For the first rule of the simple example we used the MAX techniques which resulted in the following diagram:

78. Max

79. SUM With SUM the pointwise sum for all fuzzy outputs is taken.
Again for the first rule of the simple example we get:
Observe that we operate in fuzzy space, all inputs and outputs are in fuzzy interval [0, 1]

80. Defuzzification

81. Popular defuzzification methods
Under the defuzzification process a fuzzy output is converted to a crisp number.
Two of the more popular techniques are the MAXIMUM and the CENTROID.
With MAXIMUM one selects the maximum value of the fuzzy output as the crisp value.
There are several variations to the MAXIMUM theme.
One such variation is the AVERAGE-OF-MAXIMA which was used in the simple example and gave us a crisp value of 0.84.
With the CENTROID method, the center of gravity of the fuzzy output gives the crisp value.
Using PRODUCT inferencing and the SUM combination the CENTROID results in a crisp value of 0.56.

82. Example of calculation of Maximum
where n is the total number of fuzzy points (strips).

83. Calculating the output of a rule

84. Ways of Combining Fuzzy Logic Output Values
There are four different
techniques to combine the fuzzy logic rules output
values.
They are:
Maximizer
Average
Centroid
Singleton
Observe that we no longer operate here in fuzzy interval [0, 1]
Temperature not fuzzy value

85. With MAXIMUM one selects the maximum value of the fuzzy output as the crisp value.
With the CENTROID method, the center of gravity of the fuzzy output gives the crisp value.

86. Another approach to Calculating the output of a rule
Rules have promises which are usually combined together by the connective and.
The output is calculated by taking the degree of membership of the lesser of all premises as the value of the combination and truncating the output fuzzy set at that level.

87. Calculating the output of a rule (cont)
Truncated at 0.2
Rules have promises which are usually combined together by the connective and.
The output is calculated by taking the degree of membership of the lesser of all premises as the value of the combination and truncating the output fuzzy set at that level.

88. Combining rule outputs
When all rules have been evaluated a single fuzzy set is calculated by combining all outputs.
The combination involves the connective or.
The single output is calculated by taking the maximum of their respective output fuzzy sets grades at each point along the horizontal axis.

89. Combining rule outputs
Think about the set of fuzzy rules as a network with fuzzy literals and operators MIN, MAX, etc. In primary inputs you have fuzzification, in primary outputs you have defuzzification. The network analogy is very useful in your thinking about fuzzy logic, mv logic, or any other kind of data structure to store knowledge.
Shifted horizontally
Do in class a complete example of designing a fuzzy logic network for a simple robot control with about 6 rules. Discuss various operators and fuzzifiers, defuzzifiers.

90. STATIC, ADAPTIVE, SELF-ORGANISING SYSTEMS

91. STATIC, ADAPTIVE, SELF-ORGANISING SYSTEMS
Up to now we have seen that fuzzy logic supports the task of designing a control system.
What we examined in previous sections is known as the `static' fuzzy system.
In such a system, conventional techniques are employed in order to elicit or induce production rules.
Such a system, receives inputs which are then normalised and converted to fuzzy representations, the knowledge base is executed, an output fuzzy set is generated which is then converted to a crisp value.

92. “static” fuzzy system.

93. ADAPTIVE, SELF-ORGANISING SYSTEMS
The rules do not change, except if modified by the hand of the designer, for all the knowledge base lifetime.
Static systems are fine for applications in which the environment is known and predictable.
But because they can not adapt to gradual changes in their environments they can lead to disaster when the assumptions upon which they are built are violated.
An adaptive system adjusts to time or process conditions.
This means that such a system modifies:
- the charactersitics of the rules,
the topology of the fuzzy sets, and
the method of defuzzification
among others.

94. An adaptive system

95. ADAPTIVE, SELF-ORGANISING SYSTEMS
A performance metric, often another expert system or an algorithm, compares the current and the stored array of past solutions and the comparison result is passed to an adaptation machine..
The adaptation machine, often another expert system, decides what changes to make in the underlying fuzzy model.
For instance:
the contribution weights connected with each rule can be modified or
the membership functions re-drawn.

96. Misconceptions about Fuzzy concepts

97. 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?

98. 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.

99. 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.

100. Fuzziness is not Probability!!!

101. 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.

102. 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.

103. 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.

104. Applications of Fuzzy concepts

105. Example: Cement Kiln Example

109. Example: Simulation of accident

114. Fuzzy Logic is NOT fuzzy thinking

116. 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.

117. 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.

118. 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.

119. 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.

120. 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.

121. 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.

122. 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

123. Summary Fuzzy logic captures intuitive, human expressions.
Fuzzy sets, statements, and rules are the basis of control.
The technique is extremely powerful, and appears in mills at a growing rate.
It is combined with other methods and is the base of soft computing
Used much in Intelligent Robotics

125. Questions and Problems (1)
Explain Linguistic Variables in Expert Systems.
Explain and illustrate these concepts: fuzzifying, fuzzy rule base, fuzzy inference, defuzzifying
Give various methods to explain the concept of uncertainty in robotics.
Uncertainty versus fuzziness.
Explain Fuzzy Control in robot Flakey.
What is LPS?
Blending of behaviors and generating a plan in Flakey. What is the advantage of fuzzy logic?
Compare, subsystem after subsystem, a fuzzy controller with a classical controller.
Cruise Control Example of Fuzzy Logic Controller, how can you improve it for more sensors, more motors and more advanced rules?
Give an example how to create a fuzzy rule table.
Read the Four-Rules example. Explain how to use graphics to calculate values of operators. Use this example to explain how a more complicated Braitenberg Vehicle would work. Assume that it has to execute both Aggressive and Shy behaviors and we use another sensor or combination of sensors to change from one to another behavior.

126. Questions and Problems (2)
Explain stages of the development cycle using fuzzy logic for a simple Braitenberg- Vehicle like robot with more sensors than 5.
Explain graphical methods for MIN, MAX, SUM, PRODUCT and other operators.
Explain on example how these methods to calculate outputs work: Centroid, Maximum, Average-of-Maxima.
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.

127. HOMEWORK 3 (not this year)
This homework should be returned by December 1.
Homework 3 should involve Fuzzy Logic control for a robot. (besides this constrain you have a complete freedom to make software of your interest, you can use also ANN, kinematics, inverse kinematics, etc).
This can be a simulated robot or a robot from your project.
Robot should have some kind of sensors and some kind of effectors.
The robot should respond with its motions to the changes of the environment (mobile robot avoids obstacles, follow a line, follow a corridor, etc, robot arm puts blocks on top of one another, etc).
The control of the robot should use fuzzy logic.
Example of good robot arm is here

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