Lecture 3 Fuzzy Systems and their Properties Hamidreza Rashidy Kanan Assistant Professor, Ph.D. Electrical Engineering Department, Bu-Ali Sina University h.rashidykanan@basu.ac.ir; kanan_hr@yahoo.com
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3 Fuzzy Systems with Fuzzifier and Defuzzifier Fuzzy Rule Base x in U Fuzzifier Fuzzy Sets in U U U U R 1 n n Fuzzy Inference Engine Fuzzy Sets in V Defuzzifier V R y in V
4 Multi-Input, Single-Output System A multi-output system can always be decomposed into a collection of single-output systems. x1 x2 x3 4-input 1-output fuzzy system 4-input 1-output fuzzy system y1 y2 x4 4-input 1-output fuzzy system y3
5 Structure of Fuzzy Rule Base A fuzzy rule base consists of a set of fuzzy IF-THEN rules. A fuzzy rule base is the heart of the fuzzy system in the sense that all other components are used to implement these rules in a reasonable and efficient manner.
6 Structure of Fuzzy Rule Base We call the rules in the form of (1) canonical fuzzy IF- THEN rules.
7 Lemma 1
8 Properties of Set of Rules Because the fuzzy rule base consists of a set of rules, the relationship among these rules and the rulesas a whole impose interesting questions: Do the rules covers all the possible situations that the fuzzy system may face? Are there any conflicts among these rules?
9 Properties of Set of Rules Definition 1 A set of fuzzy IF-THEN rules is complete if for any x U, there exists at least one rule in the fuzzy rule base, say rule Ru(l) (in the form of (1)), such that: ( ) 0 for all i =1,2,,n. l x A i i
10 Properties of Set of Rules Intuitively, the completeness of a set of rules means that at any point in the input space, there is at least one rule that t fire ; thatt is, the membership value of the IF part of the rule at this point is non- zero.
11 Example
12 Properties of Set of Rules If any rule uein the group goupis missing, then we can find point x* U, at which the IF part proposition of all the remaining rules have zero membership value. For example, if the second rule is missing, i then what htis this x*? The number of rules in a complete fuzzy rule base increases exponentially with the dimension of the input space U.
13 Definitions Definition 2 A set of fuzzy IF-THEN rules is consistent if there are no rules with the same IF parts but different THEN parts. Definition 3 A set of fuzzy IF-THEN rules is continuous if there do not exist such neighboring rules whose THEN part fuzzy sets have empty intersection.
14 Fuzzy Inference Engine In a fuzzy Inference, fuzzy logic principles are used to combine the fuzzy IF-THEN rules in the fuzzy rule base into a mapping from a fuzzy set A A in U to a fuzzy set B B in V. We learned that a fuzzy IF-THEN rule is interpreted as a fuzzy relation lti in the input-output t t product space U V, and we proposed a number of implications that specify the fuzzy relation.
15 Fuzzy Inference Engine If the fuzzy rule baseconsistsofonlyasingle rule, then the generalized modus ponens (GMP) specifies the mapping from fuzzy set A A in U to a fuzzy set B B in V. How to infer with a set of rules? Answer: there are two ways Composition basedinference. Individual-rule based inference.
16 Composition Based Inference In composition based inference, all rules in the fuzzy rule base are combined into a single fuzzy relation UV, which is then viewed as a single fuzzy IF-THEN rule. How to perform this combination?
17 A Set of Rules Means: View the rules as independent conditional statements. union View the rules as strongly coupled conditional statements such that the conditions of all the rules must be satisfied in order for the whole set of rules to have an impact. intersection ti (e.g. Gödel implication)
18 Two Combinations
19 Two Combinations
20 Output of Fuzzy Inference Engine
21 Computational Procedure
22 Details of Inference Engines From previous discussion, we know that there are a variety of choices in the fuzzy inference engine: Composition based inference or individual-rule based inference, and within the composition based inference, Mamdani inference or Gödel inference. Dienes-Rescher implication, Lukasiewicz implication, Zadeh implication, Gödel implication, or Mamdani implications. Different operations for the t-norm and s-norms.
23 How to Select From These Alternatives? Intuitive Appeal The choice should make senses from an intuitive point of view. Computational Efficiency The choice should result in a formula relating B with A, which is simple to compute. Special Properties Some choice hi may result in anif inference engine that has special il properties.
24 Two Commonly Inference Engine
25 Other Inference Engines
26 Other Inference Engines
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28 Fuzzy Systems with Fuzzifier and Defuzzifier Fuzzy Rule Base x in U Fuzzifier Fuzzy Sets in U U U U R 1 n n Fuzzy Inference Engine Fuzzy Sets in V Defuzzifier V R y in V
29 Fuzzifiers In most applications, the input and output of the fuzzy system are real-valued numbers, we must construct interfaces between the fuzzy inference engine and the environment. The fuzzifier is defined as a mapping from a real-valued * n point x U R toafuzzy set A' in U.
30 Criteria to Design Fuzzifier The fuzzifier should consider the fact that the input is at the crisp point x*, that is, the fuzzy set A' should have large membership value at x*. If the input to the fuzzy system is corrupted by noise, then it is desireable that the fuzzifier should help to suppress thenoise. The fuzzifier should help to simplify the computations involved in the fuzzy inference engine.
31 Three Fuzzifiers
32 Three Fuzzifiers
33 Remarks About the Three Fuzzifiers The singleton fuzzifier greatly simplifies the computation involved in the fuzzy inference engine for any type of membership functions the fuzzy IF-THEN rules may take. The Gaussian or triangular fuzzifiers also simplify the computation in the fuzzy inference engine, if the membership functions in the fuzzy IF-THEN rules are Gaussian or triangular, respectively.
34 Defuzzifiers The defuzzifier is defined as a mapping from fuzzy set B' in V Rto crisp point y* V. Conceptually, the task of the defuzzifier is to specify a point in V that best represents the fuzzy set B'. This is similar il to the meanvalue of a random variable. ibl However, since the B' is constructed in some special ways, we have a number of choices in determining this representing gpoint.
35 Criteria to Design Defuzzifier Plausibility: The point y* should represent B' from an ititi intuitive point of view; for example, it may lie approximately in the middle of the support of B' or has a high degree of membership in B'. Computational simplicity: This criterion is particularly important for fuzzy control because fuzzy controllers operate in real-time. Continuity: A small change in B' should not result in a large change in y*.
36 Three Defuzzifiers Center of Gravity Defuzzier Center Average Defuzzifier Maximum Defuzzifier
37 Center of Gravity Defuzzifier
38 Pros. and Cons. for Center of Gravity Defuzzifier Advantage: intuitive plausibility. Disadvantage: computationally intensive.
39 Center Average Defuzzifier
40 Graphical Representation of Center Average Defuzzifier
41 Maximum Defuzzifier
42 Three Different Maximum Defuzzifier
43 Graphical Representation of Maximum Defuzzifier
44 Comparison Between Three Defuzzifier Center of Gravity Center Average Maximum Plausibility Yes Yes Yes Computational Simplicity it No Yes Yes Continuity Yes Yes No
45 Example Step 1: Fuzzification
46 Example Step 2: Create Rule Base Rule 1: If x is low AND y is low Then z is high Rule 2: If x is low AND y is high Then z is Low Rule3:IfxishighANDyislowThenzisLow Rule 4: If x is high AND y is high Then z is high
47 Example Step 3: Inference Rule 1: low(x)=0.68, low(y)=0.39 => high(z)=min(0.68,0.39)=0.39h( 0 0 Rule 2: low(x)=0.68, high(y)=0.61 => low(z)=min(0.68,0.61)=0.61 Rule 3: high(x)=0.32, low(y)=0.39 => low(z)=min(0 MIN(0.32,0.39) 0 39)=0.32 Rule 4: high(x)=0 0.32, high(y)=0 0.61 => high(z)=min(0.32,0.61)=0.32
48 Example Step 4:Composition
49 Example Step 5: Defuzzification (Center of Gravity)
50 Example Assume that we need to evaluate student applicants based on their GPA and GRE scores. For simplicity, let us have three categories for each score High (H), Medium (M), Low (L) Let us assume that the decision should be: Excellent (E), Very Good (VG), Good (G), Fair (F), Poor (P) An expert will associate the decisions to the GPA and GRE scores.
51 Fuzzification of GRE GRE 1 Low Medium High 800 1200 1800 GRE GRE { L, M, H }
52 Fuzzification of GPA GPA 1 Low Medium High 2.22 3 3.8 GPA GPA { L, M, H }
53 Fuzzified Decision 1 P F G VG E 60 70 80 90 100 D Decision Index { P, F, G, VG, E }
54 Create Rule Base GRE H M L GPA H E VG F M G G P L F P P
55 Calculate Assume a student with GRE=900 and GPA=3.6 A decision i on the classification i of the applicant is needed. Excellent Very good Etc.
56 GRE=900 GRE 1 Low Medium High 0.8 0.2 GRE 800 1200 1800 GRE 900 { 0.8, 0.2, L M H 0}
57 GPA=3.6 GPA 0.6 0.4 1 Low Medium High 2.22 3.0 3.8 3.6 GPA GPA { 0, 0.6, L M H 0.4}
58 Activated Rules GRE H M L GPA H E VG F M G G P L F P P GPA { 0, 0.6, L M H 0.4} GRE { 0.8, 0.2, L M H 0}
59 Inference GRE GRE 0 0.2 0.8 H M L GPA 0.4 0 0.2 0.4 0.6 0 0.2 0.6 GPA H E VG F M G G P D 0 0 0 0 L F P P D { P, F, G, VG, E} { 0.6, 0.4, 0.2, 0.2, P F G VG E 0}
60 Scaled Fuzzified Decision 1 0.6 0.4 0.2 P F G VG 60 70 80 90 100 Decision Index D { 0.6, 0.4, 0.2, 0.2, P F G VG E 0}
61 Defuzzification (Max Method) Fuzzy set with the largest membership value is selected. Fuzzy decision: D D {,,,, } { P F G VG E { 0.6, 0.4, 0.2, 0.2, P F Final Decision (FD) = Poor Student If two decisions have same membership max, use the average of the two. G VG E 0}
62 Defuzzification (Max Method) 1 0.6 0.4 0.2 P F G VG 60 70 80 90 100 Decision Index
63 Defuzzification (Centroid Method) FD D D E. E VG. D VG E VG 0.0 100 0.2 90 0.2 80 0.4 70 0.6 60 FD 70 0.2 0.2 0.4 0.6 Final Decision (FD) = Fair student
64 Defuzzification (Centroid Method) 1 0.6 P 0.4 F G 0.2 VG 60 70 80 90 100 Decision Index
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66 The Formulas of Some Classes of Fuzzy Systems There are a variety of choices in the fuzzy inference engine, fuzzifier, and defuzzifier modules. dl Fuzzifier Fuzzy Inference Engine Defuzzifier Specifically, we proposed: Five fuzzy inference engines (product, minimum, Lukasiewicz, Zadeh, and Dienes-Rescher), Three fuzzifiers (singleton, Gaussian and triangular) Three types of defuzzifiers (center-of-gravity, center average, g, and maximum).
67 The Formulas of Some Classes of Fuzzy Systems Therefore, we have at least 5 * 3 * 3 = 45 types of fuzzy systems by combining these different types of inference engines, fuzzifiers, and defuzzifiers. The center-of-gravity defuzzifier is computationally expensive and the center average defuzzifier is a good approximation of it. So, we classify the fuzzy systems to be considered into two groups: Fuzzy systems with center average defuzzifier. Fuzzy systems with maximum df defuzzifier.
68 Fuzzy Systems with Center Average Defuzzifier l First system: suppose that the fuzzy set B is normal with l center y. Then the fuzzy systems with product inference engine, eg e,singletonfuzzifier e and center e average defuzzifier e are of the following form:
69 Remarks About the Mentioned Fuzzy Systems The mentioned fuzzy systems are the most commonly used fuzzy systems in the literature. They are computationally simple and intuitively appealing. The output of the fuzzy system is a weighted average of the centers of the fuzzy sets in the THEN parts of the rules, where the weights equal the membership values of the fuzzy sets in the IF parts of the rules at the input point. Consequently, the more the input point agrees with the IF part of a rule, the larger weight ihtthisrule is given; thismakes sense intuitively. iti
70 The Dual Role of Fuzzy Systems On one hand, fuzzy systems are rule-based systems that are constructed t dfrom a collection of linguisticrules. i l On the other hand, fuzzy systemsstems are nonlinear mappings that in many cases can be represented by precise and compact formulas such as the mentioned fuzzy system. An important contribution of fuzzy systems theory is to provide a systematic procedure for transforming a set of linguistic rules into a nonlinear mapping. Because nonlinear mappings are easy to implement, fuzzy systems have found their way into a variety of engineering applications.
71 Different Subclasses of the Mentioned Fuzzy Systems By choosing different forms of membership functions for inputs and output, we obtain different subclasses of the mentioned fuzzy systems. If we choose the following Gaussian membership function for inputs and output: Then the mentioned fuzzy systems become:
Fuzzy Systems with Center Average Defuzzifier 72 l Second system: suppose that the fuzzy set B is normal l with center y. Then the fuzzy systems with minimum inference engine, singleton fuzzifier and center average defuzzifier are of the following form:
73 Fuzzy Systems with Center Average Defuzzifier l Third system: suppose that the fuzzy set B is normal with l center y. Then the fuzzy systems with product if inference engine, Gaussian fuzzifier with *=product, center average l defuzzifier and Gaussian membership functions (with a i 1 ) are of the following form:
74 Fuzzy Systems with Center Average Defuzzifier l Forth system: suppose that the fuzzy set B is normal with l center y. Then the fuzzy systems with Lk Lukasiewicz i or Dienes-Rescher inference engine, singleton or Gaussian or triangular fuzzifie and center average defuzzifier are of the following form: The above fuzzy system gives a constant output no matter what the input is. Therefore, the combinations of fuzzy inference engine, fuzzifier, and defuzzifier in do not result in useful flfuzzy systems.
75 Fuzzy Systems with Maximum Defuzzifier First system: suppose that the fuzzy set l B is normal with l center y. Then the fuzzy systems with product if inference engine, singleton fuzzifier and maximum defuzzifier are of the following form:
76 Remarks About the Fuzzy Systems with Maximum Defuzzifier The mentioned fuzzy systems are simple functions, that is, they are piece-wise constant functions, and these constants are the centers of the membership functions in the THEN parts of the rules. As long as the product of membership values of the IF-part fuzzy sets of the rule is greater than or equal to those of the other rules, the output of the fuzzy system remains unchanged. Therefore, these kinds of fuzzy systems are robust to noise and small disturbances in the input and in the membership functions.
77 Remarks About the Fuzzy Systems with Maximum Defuzzifier However, these fuzzy systems are not continuous, that is, when l* changes from one number to the other, f(x) changes in a discrete fashion. If the fuzzy systems are used in decision i making or other open-loop applications, this kind of abrupt change may be tolerated, but it is usually unacceptable in closed-loop control.
78 An Example of Function Approximation Using Fuzzy System