An Approach to Inverse Fuzzy Arithmetic

Transcription

1 An Appoach to Invese Fuzzy Athmetc Mchael Hanss Insttute A of Mechancs, Unvesty of Stuttgat Stuttgat, Gemany mhanss@mechaun-stuttgatde Abstact A novel appoach of nvese fuzzy athmetc s ntoduced to successfully dentfy the uncetan paametes of lnea and nonlnea models on the bass of uncetan values fo the output vaables of the model The pesented method s based on the tansfomaton method, whch has been poposed as a poweful tool fo the smulaton and analyss of systems wth uncetan model paametes A geneal scheme fo the pactcal mplementaton of the nvese fuzzy-athmetcal appoach s gven, and the effectveness of the method s shown fo two examples, whch consst of a lnea model and a nonlnea model, espectvely 1 Intoducton A vey pactcal appoach fo the smulaton of systems wth uncetan paametes s the numecal epesentaton of the model paametes as fuzzy numbes [6], and then, the use of fuzzy athmetc based on the tansfomaton method [1, 2] to pefom the evaluaton of the model equatons By usng ths method fo the smulaton of fuzzy-paametezed models, the complete nfomaton about the uncetantes n the model can be ncluded, and one can demonstate how these uncetantes ae popagated though the calculaton pocedue Futhemoe, usng the ntnsc analyss pocedue of the tansfomaton method, the degees of nfluence of each fuzzy paamete can be detemned, e one can quantfy the popoton to whch the uncetanty of each model paamete contbutes to the oveall uncetanty of the model output Bascally, the uncetantes of the model paametes can be classfed and assgned to dffeent goups accodng to the type of the ogn In geneal, uncetantes n model paametes may ase due to mpecseness o a lack of nfomaton, due to vaablty, o n consequence of smplfcaton f complex eal-wold systems ae expessed by smplfed models Impecseness of nfomaton, fo example, can be caused by vagueness n vebal chaactezaton of models, such as by the expesson nealy clamped, whch descbes an uncetan bounday condton of a mechancal system A lack of nfomaton often occus f a system s nfluenced by nose sgnals, and vaablty s pesent fo system popetes that may scatte, such as mateal o geomety paametes In these cases, the fuzzyvalued model paametes can usually be defned wthout majo poblems, eg by takng nto account the toleances of the manufactung pocess n case of vayng mateal popetes In the case of smplfcaton, howeve, the uncetantes n the model paametes often eflect effects o dynamcs that have been neglected dung the modelng pocedue o as a consequence of model educton The membeshp functons of those paametes can nomally not be defned n a dect way, they need to be dentfed, nstead, based on the fuzzy-valued output sgnals of the model wth the membeshp functons deved fom expemental data [4, 5] The soluton to ths poblem poves to be non-tval and eques the applcaton of nvese fuzzy athmetc In ths pape, a novel appoach wll be pesented to pactcally pefom nvese fuzzy athmetc fo both lnea and nonlnea poblems Fo the sake of cleaness and completeness, the geneal stuctue of fuzzy-paametezed models as well as the tansfomaton method n ts educed fom [1] wll be ecalled n Secton 2 In Secton 3, the fuzzy athmetcal appoach wll be ntoduced, and a geneal scheme fo ts mplementaton wll be pesented Fnally, the effectveness of the method wll be demonstated by examples fo a lnea and a nonlnea model n Secton 4 2 Smulaton and analyss of fuzzypaametezed models 21 Stuctue of fuzzy-paametezed models In geneal, a fuzzy-paametezed model conssts of thee key components: 1 A set of n ndependent fuzzy-valued model paametes p wth the membeshp functons µ p (x, = 1, 2,, n

2 2 The model tself, whch can be ntepeted as PSfag a set of eplacements N µ p (x geneally nonlnea functons f, = 1, 2,, N, that 1 pefom some opeatons on the fuzzy nput vaables p, = 1, 2,, n p 3 A set of N fuzzy-valued output paametes q wth the membeshp functons µ q (z, = 1, 2,, N, that ae obtaned as the esult of the functons f µ j+1 µ j µ Thus, a fuzzy-paametezed model can n geneal be expessed by a system of equatons of the fom q 1 = f 1 ( p 1, p 2,, p n = (1 q N = f N ( p 1, p 2,, p n As a pe-condton fo the applcaton of nvese fuzzy athmetc, the nvetblty of the system, e ts unque soluton fo the uncetan model paametes p, = 1, 2,, n, has to be guaanteed Fo ths eason, only those models shall be consdeed n ths pape whee the output vaables q 1, q 2,, q N ae stctly monotonc wth espect to each of the model paametes p 1, p 2,, p n Ths allows the uncetan model to be smulated and analyzed by smply applyng the tansfomaton method n ts educed fom, omttng, addtonally, the ecusve elements n the fnal etansfomaton step [1, 2] 22 Smulaton usng the educed tansfomaton method (I Decomposton of the nput fuzzy paametes Fo the smulaton of a fuzzy-paametezed model usng the educed tansfomaton method, each nput fuzzy paamete p s fst beng decomposed nto α-cuts, leadng to a set P = of (m + 1 ntevals X = [ a { X (0, X (1,, X (m } (2, b ], a b, (3 = 1, 2,, n, j = 0, 1,, m Fo the pupose of decomposton nto α-cuts, the µ-axs s subdvded nto m segments, equally spaced by µ = 1/m (Fgue 1, and the (m + 1 levels of membeshp µ j ae then gven by µ j = j m, j = 0, 1,, m (4 (II Tansfomaton of the nput ntevals The ntevals X, = 1, 2,, n, of each level of membeshp µ j, e the µ j -cuts, j = 0, 1,, m, ae now beng 0 a x b Fgue 1 Implementaton of the th uncetan paamete as a fuzzy numbe p decomposed nto ntevals (α-cuts tansfomed nto aays X wth X of the fom 2 1 pas = ( {}}{ ( α, β (, α, β (,, α, β (5 α = ( a,, a }{{} 2 n elements, β Obvously, only the bounday values a ntevals X x = ( b,, b }{{} 2 n elements (6 and b of the, = 1, 2,, n, j = 0, 1,, m, ae consdeed n ths tansfomaton scheme, whch s, howeve, suffcent fo the smulaton of poblems whch ae monotonc wth espect to all the paametes p, as consdeed n ths pape (III Evaluaton of the model Assumng that the fuzzy-paametezed model s gven by the system of equatons n Eq (1, ts estmaton s then caed out by evaluatng the equatons sepaately at each of the columns of the aays, usng the conventonal athmetc fo csp numbes Thus, f the outputs q, = 1, 2,, N, of the system can be expessed n ts decomposed and tansfomed fom by the aays Ẑ, = 1, 2,, N, j = 0, 1,, m, the kth element k ẑ s then gven by whee k ˆx ( kẑ = f k ˆx 1, k ˆx 2,, k ˆx n denotes the kth element of the aay of the aay Ẑ, (7 X (IV Retansfomaton of the output aay The decomposed fom of the fuzzy-valued model output q, expessed by the set Q = { Z (0, Z (1 },, Z (m (8

3 of (m + 1 ntevals Z = [ c, d ], c d, j = 0, 1,, m, can be obtaned by etansfomng the aays Ẑ accodng to c = mn k kẑ, d = max k = 1, 2,, N, j = 0, 1,, m (9 kẑ, (10 (V Recomposton of the output ntevals By ecomposng the ntevals Z, j = 0, 1,, m, of the sets Q, = 1, 2,, N, accodng to the levels of membeshp µ j, an appoxmaton of the fuzzy-valued model output q can be acheved 23 Analyss usng the educed tansfomaton method To detemne the popotons to whch the n uncetan paametes p of the system sepaately contbute to the oveall uncetanty of the system outputs q, the nfomaton gven by the values and the aangement of the elements n Ẑ can be used [1, 3] In case of the educed tansfomaton method, the coeffcents η, = 1, 2,, n, j = 0, 1,, (m 1, = 1, 2,, N, can be detemned accodng to η = wth 1 2 n 1 (b a 2 n k=1 2 1 l=1 ( s2ẑ s 1 (k, l = k + (l 1 2 n +1 s 2 (k, l = k + (2l 1 2 n s1 ẑ (11 (12 Agan, the values a and b denote the lowe and uppe bounds of the nteval X, and k ẑ s the kth element of the aay Ẑ The coeffcents η can be ntepeted as gan factos that expess the effect of the uncetanty of the th paamete p on the uncetanty of the th output vaable q of the poblem at the membeshp level µ j Moe explctly, wthn the ange of uncetanty coveed at the membeshp level µ j, devatons z fom the cental value z of the output fuzzy numbe q can be consdeed as beng elated to the coespondng devatons x fom the cental values x of the nput fuzzy paametes p by the appoxmaton [3] n z η x, (13 =1 j = 0, 1,, (m 1, = 1, 2,, N In case of possbly nonlnea dependences of the outputs q on the paametes p, the method of system analyss can be extended by the ntoducton of sngle-sded gan factos and η whch expess the effect of the uncetanty of the th paamete p on the uncetanty of the th output vaable q when only postve devatons fom the peak value x, and negatve devatons, espectvely, ae consdeed The sngle-sded gan factos can be detemned on the bass of Eqs (11 and (12 by usng, n tun, only the ght banch, and the left banch, espectvely, of the membeshp functon of the th fuzzy paamete p whle the othe fuzzy paametes eman unmodfed To obtan a non-dmensonal fom of the nfluence measues wth espect to the usually dffeent physcal dmensons of p, some standadzed mean gan factos κ can be detemned, and futhemoe, as a elatve measue of nfluence, some nomalzed values ρ, = 1, 2,, n [2] Fo the poblems consdeed n ths pape, howeve, only η + the defnton of the gan factos η ± 3 Invese fuzzy athmetc s mpotant Wth egad to the stuctue of fuzzy-paametezed models as defned n Eq (1, the man poblem of nvese fuzzy athmetc conssts n the dentfcaton of the uncetan model paametes p 1, p 2,, p n on the bass of gven values fo the output vaables q 1, q 2,, q N Wheeas n the case N < n the dentfcaton poblem s unde-detemned, ts soluton eques the applcaton of an optmzaton pocedue fo N > n In the pesent pape, howeve, only the case N = n shall be consdeed, whee the numbe of avalable output vaables s dentcal to the numbe of uncetan model paametes At fst vew, the soluton to the nvese fuzzy athmetcal poblem wth N = n appeas to be athe staghtfowad and easy to acheve at least fo lnea systems by evaluatng the nveted model equatons, solved fo the paametes p 1, p 2,, p n, usng the tansfomaton method wth q 1, q 2,, q n as nput vaables Ths way of poceedng, howeve, clealy fals and leads to an enomous oveestmaton of the uncetanty of the model paametes p 1, p 2,, p n (see example n Secton 41 The eason fo ths falue must be seen n the fundamental pe-condton of the tansfomaton method whch eques ts fuzzy-valued nput paametes to be stctly ndependent, e to ndependently ntate the oveall uncetanty n the system by uncetan paametes of dffeent ogn Ths condton, of couse, can neve be fulflled fo the use of nveted model equatons snce the nput paametes q, = 1, 2,, n, of ths pocedue do all featue a functonal dependency on the model paametes p, = 1, 2,, n (see Eq (1 To stll successfully solve the nvese fuzzy athmetcal poblem, the followng scheme can be appled, consstng of

4 an appopate combnaton of the smulaton and the analyss pat of the tansfomaton method: 1 Detemnaton of the peak values ˇ x 1, ˇ x 2,, ˇ x n : Owng to Eq (1, the peak values x of the eal model paametes p, = 1, 2,, n and the peak values z of the output vaables q, = 1, 2,, n, ae elated by the system of equatons z 1 = f 1 ( x 1, x 2,, x n = (14 z n = f n ( x 1, x 2,, x n Statng fom the n gven values z n the nvese poblem, the n peak values ˇ x of the unknown fuzzy-valued model paametes ˇ p, = 1, 2,, n can be detemned ethe by analytcally solvng the equatons (14 fo x, = 1, 2,, n, as one can easly do fo lnea systems, o by numecally solvng the system of equatons usng a cetan teaton pocedue 2 Computaton of the gan factos: Fo the detemnaton of the sngle-sded gan factos η + and η, the model has to be smulated fo some assumptve uncetan paametes p, = 1, 2,, n, usng the tansfomaton method as defned above The peak values of p have to be set equal to the just computed values ˇ x, = 1, 2,, n, and the assumed uncetanty should be set to a lage enough value, so that the expected eal ange of uncetanty n ˇ p s pefeably coveed 3 Assembly of the uncetan paametes ˇ p 1, ˇ p 2,, ˇ p n : Recallng the epesentaton of a fuzzy numbe n ts decomposed fom (Secton 22, the lowe and uppe bounds of the ntevals of the fuzzy paametes ˇ p at the (m + 1 levels of membeshp µ j shall be defned as ǎ and ˇb, and the bounds of the gven output values q as c and d The nteval bounds ǎ and ˇb, whch fnally povde the membeshp functons of the unknown model paametes ˇ p, = 1, 2,, n, can then be detemned on the bass of Eq (13 though ǎ 1 ˇ x 1 ˇb 1 ˇ x 1 ǎ 2 ˇ x 2 ˇb 2 ˇ x 2 ǎ n ˇb n ˇ x n ˇ x n = H 1 c d c 2 z 2 d 2 z 2 c n d n z n z n (15 wth H = and H H 11 H 12 H 1n H 21 H 22 H 2n H n1 H n2 H nn (16 = (17 1 η (1 + sgn(η η + (1 sgn(η + 2 η (1 sgn(η η + (1 + sgn(η +, = 1, 2,, n, j = 0, 1,, m 1 The values ǎ (m (m = ˇb, = 0, 1,, n, ae aleady detemned by the peak values ˇ x To vefy the dentfed model paametes ˇ p 1, ˇ p 2,, ˇ p n, the model equatons (1 can be e-smulated by means of the tansfomaton method, usng ˇ p 1, ˇ p 2,, ˇ p n as the fuzzy nput paametes The degee of confomty of the soobtaned output fuzzy numbes ˇ q 1, ˇ q 2,, ˇ q n wth the ognal output values q 1, q 2,, q n can seve as a measue of the qualty of the dentfcaton Fnally, to clafy the fom of Eqs (15 to (17, the specal case n = 1 shall be consdeed n the ensung Fom Eqs (15 and (16 then follows [ c d ] = H 11 [ ǎ 1 ˇ x 1 ˇb 1 ˇ x 1 whch, afte the ncluson of Eq (17 leads to ( c = η ǎ 1 ˇ x 1 and The case whee η + d = η + f c = η (ˇb + d = η f ], (18 (19 (ˇb 1 ˇ x 1, (20 η +, η > 0, 1 ˇ x 1 (21 ( ǎ 1 ˇ x 1, (22 η +, η < 0 and η have dffeent algebac sgns cannot occu snce monotoncty of the outputs q wth espect to the model paametes p has ntally been postulated fo the poblem

5 As one can see fom Eqs (19 to (22, the pesented fomulaton of nvese fuzzy athmetc guaantees that a postve vaaton fom the peak value z 1 s nduced by a postve vaaton fom ˇ x 1 f the gan factos ae postve, and by a negatve vaaton fom ˇ x 1 f the gan factos ae negatve Vce vesa, ths apples fo a negatve vaaton fom z 1 smlaly Futhemoe, the mpotance of the sngle-sded gan factos s ponted out by the equatons: the ght-hand gan factos ae assgned to postve vaatons fom the peak values ˇ x 1, and the left-hand gan factos to negatve ones 4 Examples 41 Lnea model As a fst example, a statc model of ode n = 2 s consdeed whch s lnea wth espect to ts uncetan model paametes The system of equatons s gven by q 1 = f 1 ( p 1, p 2 = 4 p 1 + p 2 (23 q 2 = f 2 ( p 1, p 2 = 3 p 1 2 p 2, (24 whch can be ewtten n the matx fom [ ] [ ] [ ] q1 4 1 p1 = (25 q p 2 }{{} A To povde model outputs q 1 and q 2, whch wll seve as nput values fo the subsequent nvese poblem, the model shall be evaluated fo some gven model paametes p 1 and p 2 defned by the membeshp functons pesented n Fgue 2 The model paamete p 1 s gven by a symmetc fuzzy numbe of quas-gaussan shape [1] wth the peak value x 1 = 10 and the standad devaton σ 1 = 5% x 1 = 005, the model paamete p 2 by a tangula fuzzy numbe wth the peak value x 2 = 20 and the wost-case devatons α 2L = 15% x 2 = 03 to the left-hand sde and α 2R = 10% x 2 = 02 to the ght-hand sde, espectvely As a esult, fuzzy-valued model outputs q 1 and q 2 wth the peak values z 1 = 20 and z 2 = 10 can be obtaned Followng the steps of the scheme of nvese fuzzy athmetc n Secton 3, one obtans: PSfag eplacements As a esult of the analyss of the system, usng the educed tansfomaton method, the followng gan factos can be acheved: η 11+ = η 11 = 40, η 12+ = η 12 = 10, η 21+ = η 21 = 30 η 12+ = η 12 = 20, j = 0, 1,, m 1 (27 Due to the smplcty of the consdeed model and the pesence of an analytcal fom, the coectness of ths numecally obtaned gan factos can easly be vefed Futhemoe, as a chaactestc popety of lnea systems, the dentty of the left-hand and the ght-hand gan factos as well as the ndependence of the membeshp level µ j can be obseved 3 Assembly of ˇ p 1 and ˇ p 2 : The unknown model paametes ˇ p 1 and ˇ p 2 can fnally be assembled on the bass of Eqs (15 to (17 and wth the esults of Eq (27 whch yeld H = , ( j = 0, 1,, m 1 Snce the ognal model paametes p 1 and p 2 ae avalable fo ths test example, the estmated model paametes ˇ p 1 and ˇ p 2 can dectly be compaed to the ognal ones, and t shows that the membeshp functons ae dentcal (Fgue 2 Thus, an exta esmulaton of the system fo the pupose of compang the ognal and the e-smulated output vaables s not equed µ p (x p 1 p 2 1 Peak values ˇ x 1 and ˇ x 2 : Based on the peak values of q 1 and q 2, z 1 = 20 and z 2 = 10, the peak values ˇ x 1 and ˇ x 2 can be calculated by means of Eq (25 though [ ] [ ] [ ] ˇ x 1 = A ˇ x 1 z1 1 = (26 2 z Gan factos η + and η,, = 1, 2: p 1 p x Fgue 2 Ognal model paametes p 1 and p 2 [ ], estmated paametes ˇ p 1 and ˇ p 2 (dentcal to p 1 and p 2, and oveestmated paametes p 1 and p 2 [ ] of the lnea model

6 Fnally, to llustate the above-mentoned dsablty of dectly evaluatng the nveted model equatons (26 wth q 1 µ p and q 2 as the fuzzy-valued nput paametes of the tansfomaton method, the so-obtaned oveestmated esults p 1 (x and p 2 fo the model paametes p 1 and p 2 ae plotted PSfag n eplacements Fgue p 1 p 2 ˇ p 1 ˇ p 2 42 Nonlnea model As a second example, a statc model of ode n = 2 s consdeed whch s nonlnea wth espect to ts uncetan model paametes The system of equatons s gven by q 1 = g 1 ( p 1, p 2 = 4 p p 2 (29 q 2 = g 2 ( p 1, p 2 = 3 p 1 2 p 2 (30 Fo the povson of output values q 1 and q 2, the model shall be evaluated fo the same paametes p 1 and p 2 as defned n Secton 41 fo the lnea model Followng agan the steps of the scheme of nvese fuzzy athmetc n Secton 3, one obtans: 1 Peak values ˇ x 1 and ˇ x 2 : Based on the peak values of q 1 and q 2, z 1 = 20 and z , the peak values ˇ x 1 and ˇ x 2 can be calculated by means of Eq (29 and (30 ethe though teatve soluton o dectly though ˇ x 1 = 1 ( 2 4 z z 1 3 z 2 (31 ˇ x 2 = 1 4 ( 9ˇ x 1 2 6ˇ x 1 z 2 + z 2 2 (32 2 Gan factos η + and η,, = 1, 2: As a esult of the analyss of the system, usng the educed tansfomaton method, the gan factos can be acheved Due to the nonlneaty of the model, the left-hand and the ght-hand gan factos ae n geneal not dentcal and not ndependent of the level of membeshp µ j 3 Assembly of ˇ p 1 and ˇ p 2 : The unknown model paametes ˇ p 1 and ˇ p 2 can fnally be assembled on the bass of Eqs (15 to (17 and the computed gan factos Fo easons of compason, the membeshp functons of the estmated model paametes ˇ p 1 and ˇ p 2 as well as those of the ognal model paametes p 1 and p 2 ae plotted n Fgue 3 Wheeas the membeshp functons ae nealy dentcal fo the model paamete p 1, those of p 2 show a slght dffeence fo lowe levels of membeshp x Fgue 3 Ognal model paametes p 1 and p 2 [ ] and estmated paametes ˇ p 1 and ˇ p 2 [ ] of the nonlnea model 5 Conclusons The pesented appoach of nvese fuzzy athmetc has demonstated ts ablty to pactcally dentfy the uncetan paametes of both lnea and nonlnea models on the bass of gven uncetan values fo the output vaables of the model The effectveness of the method has exemplaly been shown fo two statc models, whee the esults could be ated as extemely good, despte of the nonlnea chaacte of one model and the athe lage uncetanty of ts model paametes wth espect to eal-wold applcatons Exceedng the athe academc examples n ths pape, the method has aleady been appled vey successfully to dynamc poblems of mechancal engneeng and geomechancs [4, 5] Refeences [1] M Hanss The tansfomaton method fo the smulaton and analyss of systems wth uncetan paametes Fuzzy Sets and Systems, 130(3: , 2002 [2] M Hanss Smulaton and analyss of fuzzy-paametezed models wth the extended tansfomaton method In Poc of the 22nd NAFIPS Int Conf, Chcago, IL, USA, 2003 [3] M Hanss and A Klmke On the elablty of the nfluence measue n the tansfomaton method of fuzzy athmetc Fuzzy Sets and Systems, 2003 (n pess [4] M Hanss, S Oexl, and L Gaul Identfcaton of a boltedjont model wth uncetan paametes loaded nomal to the contact nteface Mech Res Comm, 29(2-3: , 2002 [5] M Hanss and A P S Selvadua Influence of fuzzy vaablty on the estmaton of hydaulc conductvty of tansvesely sotopc geomateals In Poc of the NUMOG VIII Intenatonal Symposum on Numecal Models n Geomechancs, pages , Rome, Italy, 2002 [6] A Kaufmann and M M Gupta Intoducton to Fuzzy Athmetc Van Nostand Renhold, New Yok, 1991