3. Lecture Fuzzy Systems

Transcription

1 Soft Control (AT 3, RMA) 3. Lecture Fuzzy Systems Fuzzy Knowledge

2 3. Outline of the Lecture 1. Introduction of Soft Control: definition and limitations, basics of "smart" systems 2. Knowledge representation and knowledge processing (Symbolic AI) Application: expert systems 3. Fuzzy systems: Dealing with Fuzzy knowledge Application: Fuzzy Control 1. Fuzzy-quantities 4. Connective Systems: Neural Networks Applications: Identification and neural control 5. Genetic algorithms: Stochastic optimization Application: Optimization 6. Summary & Literature 50

3 Fuzzy Systems Core Idea (Natural Model) Dealing with fuzzy (non-crisp) knowledge History In the mid-1960s Zadeh fuzzy logic In the mid-1970s Mandani Fuzzy Control Application in Automation Engineering First industrial applications in the early 1980s Fuzzy controller Examples Drying processes Gas heater Fuzzy control of an inverted pendulum Washing machine (AEG) Fuzzy control of a hammer drill 51

4 Contents of the 3rd Lecture 1. Classical quantities 1. Definition and essential terms 2. Problems 2. Fuzzy-Quantities Definition and terms Operations on quantities and classical connection with the logic Expansion of operations on fuzzy quantities 3. Summary 52

5 The Classical Concept of Quantity A Quantity M is a Summary of wohlbestimmten and wohlunterschiedenen Objects unserer Anschauung oder unseres Denkens zu einem Ganzen. These objects are elements of so-called M. If an object belongs to M, The we write x M, if not, then x M Similar Quantities: M1 M2 (x M1 x M2) Dissimilar Quantities: M1 M2 M1 is a Sub-set of quantity M2: M1 M2 (x M1 x M2) M1 is a genuine Sub-set of quantity M2: M1 M2, if M1 M2 und M1 M2 Blank Quantity: 53

6 Description of classical quantities 54

7 Problems in dealing with classical quantities Main problem is the binary decision on the affiliation of a quantity (elements are not always well-differentiated) Especially critical for continuous measurement (usually given in the Automatic Control) Example: for the interval of temperature from 0 C to 100 C following applies : "temperature is high" 1 μ μ T=hoch T/ C for T = 60,00 C "the temperature is high" valid for T = 59,99 C "the temperature is high" not valid For use with control based systems, we have to give steps (jumps) e.g.: R1: If temp. is high, then Heating-systems turns off R2: if temp. is NOT high, then Heating system turns on Solution: Fuzzy Quantity 55

8 Fuzzy Quantities 56

9 Affiliation Function (ZGF) Function, every element X from a general basic numerical area, has a G degree of belonging to a fuzzy-quantity, is assigned as μ(x) (VDI/VDE 3550) The affiliation level is 0 or 1 μ(x) = 1 means, that x completely belongs to Fuzzy-quantity μ(x) = 0 means, that x does not belong to Fuzzy-quantity Values from 0 to 1 mean that x partly belongs to the fuzzy quantity Finally, If G have many Elements discreet representation of ZGF Indication of the value pairs {x, μ(x)} If there are many elements in G or G is a continuum, for example cont. Measurement parametric representation of ZGF Functions determined by a few parameters Advantage: low memory consumption, fine resolution Disadvantage may be complicated calculation 57

10 Parametric Representation (1): step linear Indication of the interpolation function Spezialfall: trapezoide Funktionen 58

11 Parametric Representation (2): trapezoid or triangular form For Special case b=c we obtain, triangular form ZGF 59

12 Parametric Representation (3): Normalized Gaussian function 60

13 Parametric Representation (4): Sigmoid difference functions 61

14 Parametric Representation (5): generalized bell function 62

15 Parametric Representation (6): LR-Fuzzy-quantity Given the parametric presentation of their flanks (separately for right and left flank) Between the flanks (m1 <x <m2), μ (x) = 1 63

16 Parametric Representation (7): Singleton (Also discreet) 64

17 Terms for the description of fuzzy quantities General adaptation of term Quantity (for two quantities A and B over a basic quantity G) Equality of Fuzzy quantities: A = B μ A (x) = μ B (x) x G Blank quantity : μ (x) = 0 x G Universal quantity: μ U (x) = 1 x G Further terminologies High Normality Support Core -cut Fuzzy-subset Fuzzy-similarity 65

18 High Normality The amount of a fuzzy quantity is the maximum value of their affiliation to function H(M) = max{μ M (x) x G} A fuzzy-stock M is normal,ifh(m) = 1 gilt, Otherwise subnormal Here and normally in practice, only normal fuzzy quantities are considered 66

19 Support The support of a fuzzy set is the part of the definition frame in which the affiliation values greater than 0 are accepted (VDI/VDE 3550) Synonym: Medium (VDI / VDE 3550), influence width English: support Calculation: Let G is the basic quantity and M belongs to G, the support of M defined as a fuzzy quantity by supp(m) = {x G μ M (x) > 0} given 1 μ μ M supp(m) = {x G a < x < d} 0 a b c d supp(m) x 67

20 Core The core of a fuzzy set is the part of the definition frame in which the affiliation function accepts the value 1 (VDI/VDE 3550) Synonyms: Tolerance (VDI/VDE 3550) English: core, tolerance Calculation: Let G is the basic quantity and M belongs to G, then core of M is the is defined as fuzzy quantity core(m) = {x G μ M (x) = 1} given μ 1 μ M core(m) = {x G b < x < c} 0 a b c d core(m) x 68

21 -cut Der - cut a fuzzy quantity is the part of the definition frame in which the affiliation function values greater then 1 are accepted (VDI/VDE 3550) Synonyms: -Cut (VDI/VDE 3550), -Level Englisch: cut Calculation: Let G is the basic quantity and M belongs to G, then the -cut of M is defined as fuzzy a quantity -Schnitt(M) = {x G μ M (x) > } given μ 1 ½ μ M ½-Schnitt(M) = {x G e < x < f} = {x G (a+b)/2 < x < (d+c)/2 } 0 a b c d e ½-Schnitt(M) f 69 x

22 Context: Support, -cut, Core, Basic quantity NOTE: basic quantity, support, core and -cut a lot of fuzzy quantities are classical quantities Venn-Diagram Core -Cut Support Basic Quantity 70

23 Fuzzy subset A fuzzy quantity μ 1 is called Fuzzy-Subset of a Fuzzy quantity μ 2 on the Basic quantity G (Notation: μ 1 μ 2 ), is valid if: μ 1 (x) μ 2 (x) x G 1 μ μ 1 μ 2 μ 1 μ 2 0 x 71

24 Fuzzy Similarity Two fuzzy quantities A and B are fuzzy-similar if core (A) = core (B) and supp (A) = supp (B) Two Fuzzy quantities are exactly fuzzy-similar if they only differ in their forms of left and right flank Conclusion 1: Major changes in the description of a fuzzy set achieved by amendment of support. Conclusion 2: It is generally sufficient to use trapezoid or triangular membership functions. 1 μ 0 a b c d x 72

25 Operations of classical set theory and relationship to the logic Average of quantities (AND): x is part of the intersection of M1 and M2 x is part of M1 AND x is part of M2 Association of quantities (OR): x is part of the union of M1 and M2 x is part of M1 OR x element of M2 Complement of quantities (NOT): x is the element complementary set of M1 x is NOT the element of M1 73

26 Enhancement on fuzzy quantities by Zadeh 74

27 Average of fuzzy quantities 75

28 Association of fuzzy quantities 76

29 Complement of fuzzy quantities 77

30 Problems with the NOT operator Classical: A AND NOT A = 0 A OR NOT A = 1 78

31 Validity of equivalencies 79

32 T-standard and S-standard T-Standard Generalization of the logical AND links the membership degrees of input sizes from the interval [0, 1] into the original size density of 0 to 1 membership degree, with the figure monotonous, associative and commutative. S-Standard (Synonym: t-conorm) Generalization of the logical OR links the membership degrees of input sizes from the interval [0, 1] into the original size density of 0 to 1 membership degree, with the figure monotonous, associative and commutative. Operator pair If a t-standard,and S-standard are applied together then De-Morgan' laws are met, and they both together provide a Operator pair. 80

33 Other operators VDI / VDE

34 Summary and learning of the 3rd Lecture Know how of elementary notions of classical quantities Why classical knowledge is problematic to describe quantities of continuous partial facts Fuzzy terminologies of quantities and possibilities to display them Calculation of characteristic values of fuzzy quantities (support, core, height, cut) Know how of relationship between quantity and logic Know how of elementary operators of fuzzy quantities and fuzzy logic and how they can be applied 82