4. Lecture Fuzzy Systems

Transcription

1 Soft Control (AT 3, RMA) 4. Lecture Fuzzy Systems Fuzzy Inference

2 4. Outline of the Lecture. Introduction to 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. Fuzzy-quantities 2. Fuzzy-Inference 4. Connective Systems: Neural networks Application: Identification and neural control 5. Genetic algorithms: Stochastic optimization application: Optimization 6. Summary & Literature 84

3 Contents of the 4th Lecture. Relations. Logical Close 2. Fuzzy-logic Close 2. Fuzzy-Linguistics. Linguistic variables and Terms 2. Linguistic rules (fuzzy implication) 3. Fuzzy-Inference. Premise evaluation 2. Activation 3. Accumulation 4. Summary 85

4 Logical Close (relations) Example: Relation between color and ripeness of a tomato: Quantity of colors: X = (green, yellow, red) vectors: green = ( ); yellow = ( ); red = ( ) Amount of maturity Grade: Y = (immature, half ripe, ripe) vectors: immature = ( ); half ripe = ( ); ripe = ( ) Color ripeness ratio: R given by relations table or matrix R: x \ y Green Yellow Red Immature Half Mature Mature Interpretation of Relationsmatrix: IF a tomato is green, then it is immature (grün R = immature) IF a tomato is yellow, then it is half ripe (yellow R = half ripe) IF a Tomato is Red, then it is ripe R (Red R = ripe) Relations are suitable for modeling of IF THEN rules normal Matrix multiplication 86

5 Close Fuzzy logic (fuzzy relations) Example: Relation between color and ripeness of a tomato: Color ripeness Relation: This is the fuzzy relationship as given by R2 relations table or matrix R2: x \ y Green Yellow Red Immature,3 Half mature,5,6 Mature,3 R2,3,5,6,3 Interpretation of Relation matrix: IF a Tomato is Green, THEN it is probably immature, but not exceeding half mature Green R2 = (,5 ) IF a Tomato is Yellow, THEN it is likely half mature, but may also be mature or immature Yellow R2 = (,3,3) IF a Tomato is RED, THEN it is probably ripe, but at least half mature Red R2 = (,6 ) 87

6 Close fuzzy logic with fuzzy input values For example you can tell that an enhancement to fuzzy input values is possible Assumption: A choice between green and yellow can not be taken : Color ranges from green to yellow: x = (.5.5 ) (,5,5 ) R2 = (,5,5,3) Fuzzy matrix multiplication e.g.: Product = MIN, Summe = MAX The tomato is very likely half mature or immature, but it could also be mature Fuzzy implication But first, some definitions 88

7 Linguistic variables and Terms Linguistics Variables: Size, whose values are linguistic Terms (VDI / VDE 355) Objective: Problem, to transfer the verbal description, into algorithmic computation method. Conventional (sharp, exact) variable X Presentable form of X = numerical value * Unit Example: Profit = 25 ; Temperature = 2.73 C; distance m = The quantity of numerical values is generally not finally Linguistics Variable X Presentable form of X = linguistic Term Beispiele: Profit = small; Temperature = medium; Distance = short The amount of linguistic Term is final (even with unlimited basic quantity) Each linguistic term could set out a fuzzy quantity Linguistic Term: Of course language has the properties to characterize size (such as "high", "warm") (VDI / VDE 355) 89

8 Example: Linguistic variable temperature μ Very low low medium high Very high 5 T/ C Linguistic Variable: Temperature Linguistic Terms: Very low, low, medium, high, very high 9

9 Definition of Linguistic variables In general, the fuzzy quantity at the bottom of the definition frame is accepted as a trapezium In the intermediate area, Delta shaped fuzzy quantities are often used The number of linguistic terms depends on the application case, typical values range from 3 to 7 the less terms, the easier is the definition and the subsequent establishment of rules the more values, the more difficult is the determination, one must have more knowledge about the system available (high granularity of knowledge) In general, the fuzzy quantities overlaps so that it comes a sharp signal value can simultaneously belong to several quantities Reasonable way is that for each value there should be a exact definition of the degree of affiliation with at least one fuzzy amount greater than It is often also required that the sum of all membership levels for a sharp value is always 9

10 Linguistics Rules IF THEN rule with assumption(condition IF part ) and Conclusion (conclusion THEN-part ), at least the assumption must be linguistic (VDI/VDE 355) Generally speaks at the closing of An Implication (IF-THEN-Rule) a given fact (current value of the assumption) a final (resulting value of the Conclusion) Example Implication: IF the tomato is red then it is ripe Fact: given is that the tomato is red Conclusion: Tomato is ripe 92

11 Fuzzy-Implication General Description of the implication: IF the statement A, then is the statement B Or A B Required: Truth of the conclusion should not be greater than that of the assumption Membership function normally R : A B Discrete case: μ R (x, y) = μ x y (x, y) = μ (x) μ 2 (y) = μ T (x) μ 2 (y) (x, y) G G 2 This is a fuzzy matrices product (e.g. MIN MAX) Continuous case μ R (x, y) = μ x y (x, y) = μ (x) μ 2 (y) (x, y) G G 2 μ R (x, y) = μ x y (x, y) = min(μ (x), μ 2 (y)) (x, y) G G 2 93

12 Linguistic rule (formal) Generally speaks at the closing of An Implication (IF-THEN-Rule) a given fact (current value of the assumption) a final (resulting value of the Conclusion) Formal (discrete) Implication: μ R (x, y) = μ x y (x, y) = μ x (x) μ y (y) = μ xt (x) μ y (y) Fact: μ x (x) End: μ y (y) = μ x (x) μ R (x, y) = Fuzzy-Inferenzbild Formal (Continuous Bsp.: MIN-MAX) Implication: μ R (x, y) = min(μ x (x), μ y (y)) Fact: μ x (x) End: μ y (y) = max(min(μ x (x), μ R (x, y))) Maximum over all x 94

13 Linguistic rule: For example heating water () For example, heating water according to the rule R R: IF temperature T = low THEN W = high μ T μ W niedrig hoch,5,5 T/ C W/% 95

14 Linguistic rule: For example heating water (2) μ T μ W low hoch,5,5 T/ C W/% Discretization of the basic quantities and the fuzzy-terms: G = {, 2, 3, 4, 5} G 2 = {6, 7, 8, 9, } μ T (T) = (,5,5 ) μ W (W) = (,5,5 ) Relationsmatrix: μ R (T, W) = μ TT (T) μ W (W) = min(μ T (T), μ W (W)) T \ W ,5,5,5 3,5,5 4,5,5,5 5 96

15 Linguistic rule: For example heating water (3) μ T μ W Incorrect Low High,5,5 Discretization of facts of G = {, 2, 3, 4, 5} μ T (T) = (,5 ) Calculation of the results : μ W (W) = μ T (T) μ R (T,W) = max(min(μ T (T), μ R (T,W))) = (,5,5,5 ) T \ W ,5,5,5 Maximum of all T 8,5,5 T/ C 9,5,5,5 6 8 W/% 97

16 Linguistic rule: For example heating water (4) μ T μ W incorrect niedrig high H =,5 3 5 T/ C Graphical Interpretation: The result of inference in a rule is the "truncated" fuzzy quantity of conclusion, the amount by which the degree of compliance of premise is given. (NOT α-cut),5,3 6 8 W/% Let H is the degree of compliance with the premise, then μ W (W) = H μ W (W) = min(h, μ W (W)) Can we also constructed from the result? 98

17 Drawer: calculating the conclusion Calculation of results: μ W (W) = μ T (T) μ R (T,W) = max( min(μ T (T), μ R (T,W))) über T μ W (W) = μ T (T) (μ TT (T) μ W (W)) = max( min(μ T (T), min(μ T (T), μ W (W)) ) ) über T μ W (W) = μ T (T) μ TT (T) μ W (W) = max ( min(μ T (T), μ T (T), μ W (W) ) ) über T μ W (W) = (μ T (T) μ TT (T)) μ W (W) = min( max( min(μ T (T), μ T (T))), μ W (W) ) über T μ W (W) = H μ W (W) = min(h, μ W (W)) H = μ T (T) μ TT (T) = max( min(μ T (T), μ T (T))) über T 99

18 Linguistic rule: For example heating water (6) μ T μ W low high H =,5,5,3 T/ C Limit for the reception of exact value: for example T = 2 C W/% Interesting: Various facts at the implication that lead to the same conclusion can be found It is only the degree of compliance with the premise that differs

19 Intermediate-state and the Way Forward So far achieved: Figure, verbal statements and fuzzy logic Possibility of processing easier IF THEN rules Inputs and outputs are fuzzy variables Problems: The simultaneous processing of several rules for the treatment of complex issues Some rules must also use the compound statements (IF A AND B AND C THEN D) The size of input and output of technical systems (Fuzzy Control) are exact (no linguistic) Way Forward: Extension of rules-processing Fuzzy-Inference Definition of Systems, the exact size and supply determination Fuzzy-System

20 Inference Analysis of the rule base, allowing input of fuzzifizierten magnitudes, and producing the output as a fuzzy quantity. The steps involved are the inference, the premise evaluation, the activation and the accumulation (VDI / VDE 355) engl.: inference Inference Premise evaluation (Aggregation) Activation (Composition) Accumulation 2

21 Premise evaluation Determining the degree of linguistic membership of a premise rule, by relating the membership of all levels of linguistic premises using fuzzy operators (VDI / VDE 355) engl.: aggregation Synonym: Aggregation Premise evaluation (Aggregation) Activation (Composition) Accumulation 3

22 Linguistic premise and partial premise Linguistic premise: condition (IF part) a of linguistic rule, can results from the combination of several linguistic partial premises together (VDI/VDE 355) engl.: premise, linguistic condition Synonym: complex linguistic statement Example: Temperature is warm and pressure is high Linguistic part premise: Partial statement in a premise is a linguistic rule, in which only a linguistic variable and a linguistic term is present (VDI / VDE 355) engl.: linguistic subcondition Synonym: linguistic Elementary declaration Example.: Temperatur is warm 4

23 Several premises in a rule IF A AND B THEN C Let HA be the degree of compliance of Part A premise and HB be the degree of compliance of Part B premise then they can have a connection through fuzzy AND operator to the degree of compliance premise Example: MIN operator; the minimum levels of compliance provides several premises in the event of the degree of compliance rule IF A OR B C THEN Let HA be the degree of compliance of Part A premise and HB be the degree of compliance of Part B premise then they can have a connection through fuzzy OR operator to the degree of compliance premise Example: MAX operator; The maximum levels of compliance provides several premises in the event of the degree of compliance rule Simplification: rules whose premises are associated with OR will be split up and can be used in several rules 5

24 Example of premise evaluation 6

25 Activation Determining the identity of a degree of linguistic rule concluded from the degree of belonging and any weighting factor of premise (VDI / VDE 355) engl.: activation, composition Synonym: Composition Common features: minimum, product Premise Evaluation (Aggregation) Activation (Composition) Accumulation 7

26 Example of activation μ T μ W incorrect low high,5,5 H =,3,3 3 5 T/ C Let H be of the degree of compliance with the premise, then with MIN μ W (W) = H μ W (W) = min(h, μ W (W)) 6 8 W/% Alternative: use of the product in the activation μ W (W) = H μ W (W) = H μ W (W) μ T μ W incorrect low high,5,5 H =,3,3 T/ C W/% 8

27 Accumulation Summary of degree of belonging of the conclusions of all linguistic rules to the output of fuzzy quantity (VDI / VDE 355) engl.: accumulation The accumulation, the conclusions of the individual rules (fuzzy quantities) combined (association, OR) using one of the OR-defined functions; usual: Max Algebraic Sum Sum (if after the conversion to a sharp value is, it is intolerable that the resulting membership function may accept a higher values) Premise Evaluation (Aggregation) Activation (Composition) Accumulation 9

28 Example of Accumulation Two Rules: IF T = low IF T = mittel Fact: T = 45 C THEN W= high THEN W = mittel

29 Rule base The completeness of the linguistic rules, describes the existing knowledge to achieve certain objectives (VDI / VDE 355) Synonym: Regulations (VDI / VDE 355) engl.: rule base General form: R: IF x = A......AND xn = An THEN y = B Rj: IF x = Aj......AND xn = Ajn THEN y = Bj Rm: IF x = Am AND xn = Amn THEN y = Bm Input sizes: x,..., xn Output Size: y Terms linguistic input size xi: Ai, A2i,..., Ami Terms linguistic the original size y: B, B2,..., Bm

30 Analysis of the rule base General Form: R: IF x = A......AND xn = An THEN y = B Rj: IF x = Aj......AND xn = Ajn THEN y = Bj Rm: IF x = Am......AND xn = Amn THEN y = Bm Input sizes: x,..., xn Output Size: y Terms linguistic input size xi: Ai, A2i,..., Ami Terms linguistic the original size y: B, B2,..., Bm Let Hi is the degree of compliance of Rule Ri, then (MAX-MIN): yi = min(hi, Bi) degree of membership of Conclusion Ri y = max(yi) degree of membership of output size i =...m y = max(min(hi, Bi)) i =...m 2

31 Characterization of inference methods To describe an inference method in three steps premise evaluation, aggregation and accumulation; operators to be used must be determined Premise evaluation: Operators for AND and OR (T-standard and the S-standard) Activation: operator for the conclusion of premise to conclusion (T-standard) Accumulation: operator for the summary of the individual outputs (OR, the standard) Simplification In general, it is assumed that the premises are only linked by AND Establishment of the OR operator for the premise evaluation is not applicable t- standard is typically used in the evaluation for the AND operator, also used in activation Conclusion The determination of the operators for activation and accumulation is sufficient in most cases Common methods are MAX MIN inference, MAX-Prod-inference and Sum-Prodinference 3

32 MAX-MIN-Inference Inference that the minimum operator is used in the activation and the maximum operator is used in the accumulation (VDI / VDE 355) Activation on MIN Accumulation on MAX y = max(min(hi, Bi)) i =...m Usually the minimum operator is used for the premise evaluation Maximum and minimum operator belonging together form a pair of t- standard and the standard 4

33 MAX-Prod-Inference Inference that the product operator is used in the activation and the maximum operator is used in the accumulation used (VDI / VDE 355) Activation on Product Accumulation on MAX y = max(hi Bi)) i =...m Usually the product operator is used for the premise evaluation This is a combination of t-standard and the s-standard BUT maximum operator and product operator belonging together do not form a pair of t-standard and the standard 5

34 Sum-Prod-Inference Inference that the product operator is used in the activation and the sum operator is used in the accumulation (VDI / VDE 355) Activation on Product Accumulation on Sum y = Hi Bi i =...m Usually the product operator is used for the premise evaluation the sum operator is not the s-standard However (anticipation of the application): A use of the Sum-Prodinference arises when appropriate choice of membership functions and defuzzification method a piecewise linear characteristic, this can be a benefit of fuzzy controller 6

35 Outlook The pros and cons of each inference methods are visible in the application to concrete problems. Especially in fuzzy controllers, these can be illustrated by determining a corresponding characteristic field Introduction of fuzzy controllers in the next lecture Comparison of different methods with a concrete example in the next lecture (exercise) 7

36 Summary and learning of the 4th Lecture Familiar concepts of fuzzy linguistics Can set up a rule base Able to explain the procedure for the inference Premise evaluation (aggregation) Activation (composition) Accumulation Inference can apply different methods MAX-MIN MAX-PROD SUM-PROD 8