Download Design of Fuzzy Systems

‫طراحي يك سيستم فازي‬
Designing of a Fuzzy System
Vali Derhami
Yazd University, Computer Department
[email protected]
FIS is Universal Approximator
 Theorem 9.1 (Universal Approximation
Theorem). Suppose that the input universe of
discourse U is a compact set in Rn. Then, for any
given real continuous function g(x) on U and
arbitrary > 0, there exists a COG fuzzy system
with Gaussian membership function such that:
 Also study 14.1 pp.278-287 from Tanaka’s book
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Author: Vali Derhami
Design of Fuzzy Systems
 Structure and Parameters (Ch. 5.1 Babuska’s book):
 structure selection involves:
 Input and output variables.
 Structure of the rules: This choice involves the model type
(linguistic, singleton, relational, Takagi-Sugeno) and the
antecedent form
 Number and type of membership functions for each variable.
 Type of the inference mechanism, connective operators,
defuzzification method.
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Author: Vali Derhami
Design of Fuzzy Systems (Cont.)

After the structure is fixed, the performance of a fuzzy model can be finetuned by adjusting its parameters. Tunable parameters: parameters of
antecedent & consequent membership functions, (determine their shape
and position) and the rules (determine the mapping between the antecedent
and consequent fuzzy regions).
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Author: Vali Derhami
Design of Fuzzy Systems
 Design methods:
 Domain Knowledge ( as a Fuzzy Expert System) (Ch. 5.1 Babuska’s book)
 Numerical Data (Rule Extraction)
 Learning Algorithms
 Hybrid
 Mathematical Modeling
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Author: Vali Derhami
Domain Knowledge
1- Select the input and output variables, the structure of the
rules, and the inference and defuzzification methods.
2. Decide on the number of linguistic terms for each
variable and define the corresponding membership
functions.
3. Formulate the available knowledge in terms of fuzzy ifthen rules.
4. Validate the model (typically using data). If the model
does not meet the expected performance, iterate on the
above design steps.
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Author: Vali Derhami
Design based on numerical data (Look-Up
scheme) (Ch. 12, wang)
 The analytic formula of g(x) is unknown,
 a limited number of input-output pairs
 [xk, yk; zk] , k=1, 2, …, N
 Step1: Define fuzzy sets to cover the input and output
spaces.
 X=[min(xk) max(xk)]; Y=[min(yk) max(yk)]; Z=[min(zk)
max(zk)];
 Define (nx,ny, nz) normal fuzzy sets in each space.
 Tx, Ty, Tz : completeness and consistency
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Author: Vali Derhami
 Step 2: Generate one rule from one input-output pair.
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Author: Vali Derhami

Step 3. Assign a degree to each rule generated in Step 2 to resolve
conflicts
 keep only one rule from a conflicting group that has the
maximum degree.
D( Rulek )  ai ( xk )b j ( yk )cl ( zk )
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Author: Vali Derhami
 Step 4. Create the fuzzy rule base.
 The rules generated in Step 2 that do not conflict with any other
rules.
 The rule from a conflicting group that has the maximum degree,
where a group of conflicting rules consists of rules with the
same IF parts.
 Linguistic rules from human experts (due to conscious
knowledge).
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Author: Vali Derhami
 Step 5. Construct the fuzzy system based on the fuzzy rule base.
 For example, fuzzy systems with product inference engine,
singleton fuzzifier, and center average defuzzifier.
 Study Ch. 3.4 Tanaka’s book
 Study “Truck upper-backer control” in page 157 and
“Time series prediction” in page 162 from Wanag’s
 MATLAB command:
 FIS = GENFIS1(DATA, NUMMFS, INPUTMF, OUTPUTMF)
 The designed FIS can be improved by learning algorithms (See Ch.
5.3 Babuska’s book
Author: Vali Derhami
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Learning Algorithms
‫ الگوريتمهاي آموزشي مي توانندبراي تنظيم سيستم استفاده شوند‬
 Learning Structure
 Learning Parameters:
yˆ  f (u; )
Methods:
Supervised Learning
Unsupervised Learning
Reinforcement Learning
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Author: Vali Derhami
‫مثال‬
 Example
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Mathematical Modeling
 Read 2.2.1 from Tanaka’s book
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Author: Vali Derhami