Fuzzy Methods. Additions to Chapter 5: Fuzzy Arithmetic. Michael Hanss.

Transcription

1 Fuzzy Methods Additions to Chater 5: Fuzzy Arithmetic Michael Hanss Part I: A short review of the Institute of Engineering Comutational Mechanics University of Stuttgart Germany Examle : q = f( ) = 2 2 Examle 2: = tfn(.5,.5,.5).5 for = = [, direct evaluation: f ( ) = [, [, 2 = [-9, 6 OVERESTIMATION (275%) factorized form: f ( ) = (2 ) [, (2 [, ) = [-, 6 OVERESTIMATION (25%) roer result: f ( [, ) = [-, Stard fuzzy arithmetic x q = h( + 2, 2 ) = = tfn(2,, ) 2 = tfn(4.5,.5,.5) 2 x 4 5 x for = = [, 2 = [4, 5 direct evaluation: h (, ) = [, + [4, 5 [, [5, 8 [, 5 = = [, 8 OVERESTIMATION (7%) symbolic simlification: h (, 2 2 ) = + [4, 5 [, = + [, 5 = [, 6 NO OVERESTIMATION 7 roer result: h ( [,, [4, 5) = [, 6 Stard fuzzy arithmetic carries out every arithmetical oeration between fuzzy numbers as an oeration between comletely indeendent oers. Examle : q = f( ) = 2 2 strictly deendent only one variable comletely indeendent Stard fuzzy arithmetic Effect of overestimation strictly deendent oers Examle 2: two indeendent variables 2 q = h(, 2 ) = + 2 symbolic simlification 2 2 q = h(, 2 ) = + = + deendent oers comletely indeendent oers In reality, the oers are not comletely indeendent in most cases!!! stard fuzzy arithmetic stard fuzzy arithmetic stard fuzzy arithmetic OVERESTIMATION OVERESTIMATION NO OVERESTIMATION

2 Stard fuzzy arithmetic Effect of overestimation (deendency roblem, conservatism) Stard fuzzy arithmetic carries out every arithmetical oeration between fuzzy numbers as an oeration between comletely indeendent oers. Solution of the roblem: symbolic rerocessing (not racticable for real-word alications!) new imlementation of fuzzy arithmetic In reality, the oers are not comletely indeendent in most cases!!! The does not exhibit any overestimation! The can be alied for both the simulation analysis of systems uncertain arameters. References: M. Hanss, The for the Simulation Analysis of Systems Parameters. Fuzzy Sets Systems, (): , 22. Uncertain M. Hanss, The Extended for the Simulation Analysis of fuzzy-arameterized models. International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, (6): 7-727, 2. M. Hanss A. Klimke, On the Reliability of the Influence Measure in the of Fuzzy Arithmetic. Fuzzy Sets Systems, 4(): 7-9, 24. U. Gauger, S. Turrin, M. Hanss L. Gaul, A New Uncertainty Analysis for the. Fuzzy Sets Systems, 59(): 27-29, 28. M. Hanss, Alied Fuzzy Arithmetic An Introduction Engineering. Sringer, Berlin, 25. Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, General rocedure: ) Decomosition of the inut fuzzy arameters (x i ) m = j+ j = a i Subdivision of -axis: i intervals (α-cuts) = m = decomosition number i = [ a i, b i m j j = j =,,, m m b i i decomosed into a set of m+ intervals (α-cuts) P i = {[a i, [a i () (),, [a i (m-) (m-), [a i } () (m-) P i General rocedure: ) Decomosition of the inut fuzzy arameters P i = { (), (m-) } 2) Alication of the (x i ) m = j+ j = a i i b i intervals (α-cuts) i = [ a i, b i 2 n elements ^ i = (α i, β i, α i, β i,, α i, β i ) 2 i- airs α i = ( a i,, a i ) β i = ( b i, ) 2 n-i elements 2 n-i elements 2

3 Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, General rocedure: ) Decomosition of the inut fuzzy arameters P i = { (), (m-) } 2) Alication of the (x i ) m = j+ j = general form a i i b i ^ i = ( (γ,i, γ 2,i,, γ (m+ j),i ),, (γ,i, γ 2,i,, γ (m+ j),i ) ) γ l,i = ( c l,i,, c l,i ) (m + j) n i elements a i c l,i = ( c (j+) l,i + c (j+) 2 l,i b ) i (m + j) i (m + j) -tules for l = for l = 2,,, m j for l = m j + (m + j) n elements j =,,, m j =,,, m 2 j =,,, m General rocedure: ) Decomosition of the inut fuzzy arameters P i = { (), (m-) } 2) Alication of the ^ i = ( x^ i, 2 x^ i,, r x^ i ) r = 2 n r = (m + j) n general form ) Evaluation of the model Z ^ = ( z ^, 2 z ^,, k z ^,, r z ^ ) k ^z = F ( k x^, k x^ 2,, k x^ n ) where F is a known mathematical function or system or F 2 Black box n q Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, Simulation of system uncertain arameters System n uncertain arameters i, i =,2,, n outut q = F (, 2,, General rocedure: ) Decomosition of the inut fuzzy arameters P i = { (), (m-) } 2) Alication of the ^ i = ( x^ i, 2 x^ i,, r x^ i ) r = 2 n r = (m + j) n general form ) Evaluation of the model Z ^ = ( z ^, 2 z ^,, k z ^,, r z ^ ) k z ^ = F ( k x^, k x^ 2,, k x^ n ) 4) Retransformation of the outut array Q = {Z, Z (),, Z (m-), Z } Z i = [ f, g where f = min (f (j+), k z ^ ) g = max (g (j+), k z ^ ) for j =,,, m f = min ( k z ^ ) = max ( k z ^ ) = g General rocedure: ) Decomosition of the inut fuzzy arameters P i = { (), (m-) } 2) Alication of the ^ i = ( x^ i, 2 x^ i,, r x^ i ) r = 2 n r = (m + j) n general form ) Evaluation of the model Z ^ = ( z ^, 2 z ^,, k z ^,, r z ^ ) k ^z = F ( k x^, k x^ 2,, k x^ n ) 4) Retransformation of the outut array Q = {Z, Z (),, Z (m-), Z } 5) Recomosition of the outut intervals Z i = [ f, g

4 Geometric interretation of the transformation scheme Geometric interretation of the transformation scheme x x general form x x 2 x2 exact solution for monotonic behavior recommended for non-monotonic behavior Analysis of system uncertain arameters The result q only shows the overall effect of all uncertain arameters together. Can one determine the ercentages to which the n uncertain arameters contribute to the overall uncertainty of the result? Yes! The information in the array Z ^ (values arrangement of the elements) is used before its retransformation into the interval Z Absolute measure of influence: η i Exresses the absolute effect of the uncertainty of the i-th arameter i on the uncertainty of the outut q of the roblem at the membershi level j. Relative measure of influence: ρ i Exresses the relative effect of the uncertainty of the i-th arameter i on the uncertainty of the outut q (non-dimensional form of the measure of influence). Absolute measure of influence: η i η i = 2 n (b i a i ) s = k + ( l ) 2 n i + general form Analysis of system uncertain arameters 2 n i 2 i ( s 2 z s z ) k = l = s 2 = k + ( 2l ) 2 n i (m j + ) n i (m j + ) i η i = (m j + ) n (b i a i ) k = ( s 2 z s z ) l = s = k + ( l ) ( m j + ) n i + s 2 = k + [ ( m j + ) l ( m j + ) n i Relative measure of influence: ρ i ρ i = n ρ i = i = n m j η i ( a i + b i ) j = m j η q ( a q + b q ) q = j = s z = s-th element of the array Z 4

5 Overview of in Various Fields of the Engineering Sciences Mechanical Engineering / Automotive Engineering Part II: of fuzzy arithmetical uncertainty management (comrehensive modeling) in various fields of the engineering sciences Vibrations of an engine hood Vibrations of a circuit board Simulation 2 - uncertain suort arameters: exonent of sring constants Circuit board suorted on two sides Eigenfrequencies Eigenfrequencies: relative measures of influence Simulation - uncertain material arameters: E, ρ, t Eigenfrequencies Frequency resonse Frequency resonse Frequency resonse: relative measures of influence 5

6 Engine hood (Mercedes SLK) Simulation - modal analysis Eigenfrequencies Fuzzy inut arameters:. Thickness of uer sheet metal 2. Thickness of lower sheet metal. Metal mass density Analysis: absolute relative measures of influence To view Bottom view Second modal shae Overview of in Various Fields of the Engineering Sciences Mechanical Engineering / Automotive Engineering Sixth modal shae Crash analysis of structures CARS Main chassis beam Helmet BIKES 6

7 Crash of a helmet against a rigid wall (roject in collaboration Università degli Studi di Firenze ) Fuzzy inut arameters:. Thickness of the comonent 4, T4 2. Thickness of the comonent 9, T9. Thickness of the comonent 2, T2 4. Thickness of the comonent 22, T22 5. Strain-rate factor for the comonent 4, M4 Main chassis beam of a vehicle (Mercedes SLK) crashing against a rigid wall 5 arameters: T4, T9, T2, T22, M4 7

8 Overview of in Various Fields of the Engineering Sciences Control Engineering Robust control (against uncertainties) 8

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