2. Fuzzy Systems Simulation Session 5
Course : T0423-Current Popular IT III
Year : 2013
Fuzzy Systems Simulation Session 5

3. Learning Outcome
After taking this course, students should be expected to explain and discuss the Fuzzy Systems Simulation
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4. Lecture Outline Presentation and discussion about :
What is Fuzzy Rule-based Systems
What is FAM
Describing Example 6.1
Describing Example 8.2
Describing Example 8.4
Presenter : Alexander Auditya, Andre Lauzardi, Diah Rostanti and Harris Kristanto
(discussion 30 minutes )
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5. Introduction
The real world is complex; complexity in the world generally arises from uncertainty in the form of ambiguity.
How can humans reason about real systems, when the complete description of a real system often requires more detailed data than a human could ever hope to recognize simultaneously and assimilate with understanding?
The answer is that humans have the capacity to reason approximately, a capability that computers currently do not have.
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6. Fuzzy rule-based systems
Fuzzy rules are linguistic IF-THEN- constructions that have the general form "IF A THEN B" where A and B are (collections of) propositions containing linguistic variables. A is called the premise and B is the consequence of the rule.
In effect, the use of linguistic variables and fuzzy IF-THEN- rules exploits the tolerance for imprecision and uncertainty.
In this respect, fuzzy logic mimics the crucial ability of the human mind to summarize data and focus on decision-relevant information.
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7. Introduction (cont.)
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8. Example
Example 8.1
For the nonlinear function y = x2, we can formulate a matrix relation to model the mapping imposed by the function. Discretize the independent variable x (the input variable) on the domain x = −2, −1, 0, 1, 2. We find that the mapping provides for the dependent variable y (the output variable) to take on the values y = 0, 1, 4. This mapping can be represented by a matrix relation, R, or
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9. Relational Equations (cont.)
Fuzzy
Relational Equations (cont.)
Suppose our knowledge concerning a certain nonlinear process is not algorithmic, like the algorithm y = x2 in Example 8.1, but rather is in some other more complex form.
This more complex form could be data observations of measured inputs and measured outputs.
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10. Canonical
rule-based form
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11. Example
Example 8.2 page 256
For the nonlinear function y = 10 sin x1, we will develop a fuzzy rule-based system using four simple fuzzy rules to approximate the output y. The universe of discourse for the input variable x1 will be the interval [−180◦, 180◦], and the universe of discourse for the output variable y is the interval [−10, 10].
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12. Example (cont.)
First, we will partition the input space x1 into five simple partitions on the interval [−180◦, 180◦]
Second, we develop four simple rules, listed in Table 8.3, that we think emulate the dynamics of the system (in this case the system is the nonlinear equation y = 10 sin x1
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13. Example (cont.)
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14. Example (cont.)
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15. Automated Methods for Fuzzy Systems
Most of the algorithms used in the book incorporate Gaussian membership functions for the inputs μ(x),
where xi is the i th input variable, ci is the i th center of the membership function (i.e., where the membership function achieves a maximum value), and σi is a constant related to the spread of the i th membership function.
Typical Gaussian
Membership Function
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16. Regression Vector
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17. Rules
A typical example of a rule for a multiple-input, single-output system is as follows:
These rules are developed by the algorithms to predict and/or govern an output for the system with given inputs. Most importantly, the algorithms incorporate the use of fuzzy values rather than fuzzy linguistic terms in these rules (hence the membership functions).
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18. Rules (cont.)
As mentioned, most of our examples use Gaussian membership functions for the premise and delta functions for the output, resulting in the following equation to predict the output given an input data-tuple xj :
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19. Rules (cont.)
where R is the number of rules in the rule-base and n is the number of inputs per data-tuple. For instance, the system of Figure 7.1 has two inputs (x1 and x2); thus, n = 2 and if there were two rules in the rule-base, R = 2.
The parameter R is not known a priori for some methods, but is determined by the algorithms. The symbol θ is a vector that includes the membership function parameters for the rule-base, ci , σi, and bi .
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20. Rules (cont.)
The data-tuples we shall use for our examples are the same as those used in Passino and Yurkovich [1998].
Table contain these data, which are presumably a representative portion of a larger nonlinear data set, Z = {([x1, x2], y)}.
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21. Recursive Least Squares Algorithm
The RLS algorithm makes updating much easier.
It operates without using all the training data and most importantly without having to compute the inverse of T each time the is updated. RLS calculates (k) at each time step k from the past estimate (k − 1) and the latest data pair that is received, xk, yk. The following
example demonstrates the training of a fuzzy model using the RLS algorithm given data set Z.
For the RLS algorithm, we use a covariance matrix to determine , which is calculated using the regression vector and a previous covariant
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22. Recursive Least Squares Algorithm (cont.)
Next we set our initial conditions for , at time step 0 (k = 0)
If these values are not readily available, another set of values may be used but more cycles may be needed to arrive at good values. As mentioned previously, this example only demonstrates the training of the fuzzy model using one cycle.
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23. Recursive Least Squares Algorithm (cont.)
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24. Recursive Least Squares Algorithm (cont.)
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25. Recursive Least Squares Algorithm (cont.)
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26. Recursive Least Squares Algorithm (cont.)
We have now calculated the vector parameters based on the three inputs needed to model the system. For this example, performing another cycle with the training data set changes very little; this is left as an exercise at the end of the chapter. Now that has been determined it is used in conjunction with ξ in Equation (7.7) to calculate the resulting output values for the training data-tuples:
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27. TM 1 (Group) Submit on session 6
Please explain about Mamdani and Tsukamoto Methods in Fuzzy Logic.
Exercise 8.11 page 275
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28. References Fuzzy Logic with Engineering Applications, Chapter 7 and 8
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