2010 IEEE International Conference on Granular Computing. Interval Type-2 Fuzzy Logic for Control Applications

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1 200 IEEE Iteatioal Coeece o Gaula Computig Iteval Type-2 Fuzzy Logic o Cotol pplicatios Osca Castillo Divisio o Gaduate Studies ad Reseach Tiuaa Istitute o Techology Tiuaa, Mexico ocastillo@tectiuaa.mx bstact Type-2 uzzy sets ae used o modelig ucetaity ad impecisio i a bette way. These type-2 uzzy sets wee oigially peseted by Zadeh i 975 ad ae essetially uzzy uzzy sets whee the uzzy degee o membeship is a type- uzzy set. The ew cocepts wee itoduced by Medel ad Liag allowig the chaacteizatio o a type-2 uzzy set with a supeio membeship uctio ad a ieio membeship uctio; these two uctios ca be epeseted each oe by a type- uzzy set membeship uctio. The iteval betwee these two uctios epesets the ootpit o ucetaity (FOU, which is used to chaacteize a type-2 uzzy set. Type-2 uzzy logic ca be used with success i cotol applicatios. Keywods-type-2 uzzy logic, itelliget cotol I. INTRODUCTION O the past decade, uzzy systems have displaced covetioal techology i dieet scietiic ad system egieeig applicatios, especially i patte ecogitio ad cotol systems. The same uzzy techology, i appoximatio easoig om, is esugig also i the iomatio techology, whee it is ow givig suppot to decisio maig ad expet systems with poweul easoig capacity ad a limited quatity o ules. The uzzy sets wee peseted by L.. Zadeh i 965 [,2] to pocess / maipulate data ad iomatio aected by upobabilistic ucetaity / impecisio. These wee desiged to mathematically epeset the vagueess ad ucetaity o liguistic poblems; theeby obtaiig omal tools to wo with itisic impecisio i dieet type o poblems; it is cosideed a geealizatio o the classic set theoy. Itelliget Systems based o uzzy logic ae udametal tools o oliea complex system modelig. The uzzy sets ad uzzy logic ae the base o uzzy systems, whee thei obective has bee to model how the bai maipulates iexact iomatio. Type-2 uzzy sets ae used o modelig ucetaity ad impecisio i a bette way. These type-2 uzzy sets wee oigially peseted by Zadeh i 975 ad ae essetially uzzy uzzy sets whee the uzzy degee o membeship is a type- uzzy set [4,6]. The ew cocepts wee itoduced by Medel ad Liag [8,0] allowig the chaacteizatio o a type-2 uzzy set with a supeio membeship uctio ad a ieio membeship uctio; these two uctios ca be epeseted each oe by a type- uzzy set membeship uctio. The iteval betwee these two uctios epesets the ootpit o ucetaity (FOU, which is used to chaacteize a type-2 uzzy set. The ucetaity is the impeectio o owledge about the atual pocess o atual state. The statistical ucetaity is the adomess o eo that comes om dieet souces as we use it i a statistical methodology. Thee ae dieet souces o ucetaity i the evaluatio ad calculus pocess. The ive types o ucetaity that emege om the impecise owledge atual state ae: Measuemet ucetaity. It is the eo o obseved quatities. Pocess ucetaity. It is the dyamic adomess. Model ucetaity. It is the wog speciicatio o the model stuctue. Estimate ucetaity. It is the oe that ca appea om ay o the pevious ucetaities o a combiatio o them, ad it is called iexactess ad impecisio. Implemetatio ucetaity. It is the cosequece o the vaiability that esults om sotig politics, i.e. icapacity to each the exact stategic obective. II. INTERVL TYPE-2 FUZZY SET THEORY type-2 uzzy set [6,7] expesses the o-detemiistic tuth degee with impecisio ad ucetaity o a elemet that belogs to a set. type-2 uzzy set deoted by, is chaacteized by a type-2 membeship uctio μ ( x, u, whee x X, u J u x [0,] ad 0 μ ( xu, i equatio (. ( x, μ ( x x X deied { } u {( xu,, μ ( xu, x X, u J x [0,] } example o a type-2 membeship uctio costucted i the IT2FLS toolbox was composed by a Pi pimay ad a Gbell secoday type- membeship uctios, these ae depicted i Fig.. ( /0 $ IEEE DOI 0.09/GC

2 I is a type-2 uzzy Sigleto, the membeship uctio is deied by equatio (6. / si x x μ ( x /0 si x x (6 Figue. FOU o Type-2 Membeship Fuctios. I is cotiuous it is deoted i equatio (2. x ( u/ u/ x (2 x X u u Jx [0,] whee deotes the uio o x ad u. I is discete the it is deoted by equatio (3. N M i μ ( x/ x x ( u / / i i ui xi (3 x X whee deotes the uio o x ad u. I x (u, u [J x u, J x u ] [0,], the type-2 membeship uctio ( x, u μ is expessed by oe type- ieio membeship uctio, J u x μ (x ad oe type- supeio, J u x μ (x (Fig. 2, the it is called a iteval type-2 uzzy set [8] deoted by equatios (4 ad (5. ( xu,, x X, (4 u [ μ ( x, μ ( x] [0,] o / u/ x x X u u u [ Jx, Jx ] [0,] (5 / u/ x x Xu [ μ ( x, μ ( x] [0,] Figue 2. FOU o Gbell Pimay Iteval Type-2 Membeship Fuctios. We ca apply some opeatos to the uzzy sets, o we ca mae some opeatios betwee them [4,0,]. Whe we apply a opeato to oe uzzy set we obtai aothe uzzy set; by the same mae whe we combie a opeatio with two o moe sets we obtai aothe uzzy set. I we have two type-2 uzzy subsets idetiied by the lettes ad B, associated to a liguistic vaiable, we ca deie thee basic opeatios: complemet, uio ad itesectio. The huma owledge is expessed i uzzy ule tems with the ext sytaxes: IF <uzzy popositio> THEN <uzzy popositio> The uzzy popositios ae divided i two types, the ist oe is amed atomic: x is, whee x is a liguistic vaiable ad is a liguistic value; the secod oe is called compouded: x is ND y is B OR z is NOT C, this is a compouded atomic uzzy popositio with the ND, OR ad NOT coectos, epesetig uzzy itesectio, uio ad complemet espectively. The compouded uzzy popositios ae uzzy elatioships. The membeship uctio o the ule IF-THEN is a uzzy elatio detemied by a uzzy implicatio opeato. The uzzy ules combie oe o moe uzzy sets o ety, called atecedet, ad ae associated with oe output uzzy set, called cosequets. The Fuzzy Sets o the atecedet ae associated by uzzy opeatos ND, OR, NOT ad liguistic modiies. The uzzy ules pemit expessig the available owledge about the elatioship betwee atecedet ad cosequets. To expess this owledge 80

3 completely we omally have seveal ules, gouped to om what it is ow a ule base, that is, a set o ules that expess the ow elatioships betwee atecedet ad cosequets. The uzzy ules ae basically IF <tecedet> THEN <Cosequet> ad expesses a uzzy elatioship o popositio. I uzzy logic the easoig is impecise, it is appoximated, that meas that we ca ie om oe ule a coclusio eve i the atecedet does t comply completely. We ca cout o two basic ieece methods betwee ules ad ieece laws, Geealized Modus Poes (GMP [5,6,8,3] ad Geealized Modus Tolles (GMT, that epeset the extesios o geealizatios o classic easoig. The GMP ieece method is ow as diect easoig ad is summaized as: Rule IF x is THEN y is B Fact x is Coclusio y es B Whee,, B ad B ae uzzy sets o ay id. This elatioship is expessed as B o( B. Fig. 3 shows a example o Iteval Type-2 diect easoig with Iteval Type-2 Fuzzy Iputs. Ieece Fuzzy System is a ule base system that uses uzzy logic, istead o Boolea logic utilized i data aalysis [4,0,20]. Its basic stuctue icludes ou compoets (Fig. 4: Fuzziie. Taslates iputs (eal values to uzzy values. Ieece System. pplies a uzzy easoig mechaism to obtai a uzzy output. Type Deuzziie/Reduce. The deuzziie taduces oe output to pecise values; the type educe tasoms a Type-2 Set ito a Type- Fuzzy Set. Kowledge Base. Cotais a set o uzzy ules, ad a membeship uctios set ow as the database. Figue 3. Iteval Type-2 Fuzzy Reasoig. The decisio pocess is a tas that idetiies paametes by the ieece system usig the ules o the ule base data. These uzzy ules deie the coectio betwee the iput ad output uzzy vaiables. uzzy ule has the om: IF <tecedet> THEN <Cosequet>, whee atecedet is a compoud uzzy logic expessio o oe o moe simple uzzy expessios coected with uzzy opeatos; ad the cosequet is a expessio that assigs uzzy values to output vaiables. The ieece system evaluates all the ules o the ule base ad combies the weights o the cosequets o all elevat ules i oe uzzy set usig the aggegate opeatio. This opeatio is aalog i uzzy logic to the S-om opeato. Figue 4. Type-2 Ieece Fuzzy System Stuctue. III. INTERVL TYPE-2 FUZZY SYSTEM DESIGN The Mamdai ad Taagi-Sugeo-Kag (TSK Iteval Type-2 Fuzzy Ieece Models [0] ad the desig o Iteval Type-2 membeship uctios ad opeatos ae implemeted i the IT2FLS Toolbox (Iteval Type-2 Fuzzy Logic Systems eused om the Matlab commecial Fuzzy Logic Toolbox. The Iteval Type-2 Fuzzy Ieece Systems (IT2FIS stuctue is the MTLB obect that cotais all the iteval type-2 uzzy ieece system iomatio. This stuctue is stoed iside each GUI tool. ccess uctios such as getiistype2 ad setiistype2 mae it easy to examie this stuctue. ll the iomatio o a give uzzy ieece system is cotaied i the IT2FIS stuctue, icludig vaiable ames, membeship uctio deiitios, ad so o. The implemetatio o the IT2FLS GUI is aalogous to the GUI used o Type- FLS i the Matlab Fuzzy Logic Toolbox, thus pemittig the expeieced use to adapt easily to the use o IT2FLS GUI. Fig. 5 ad Fig. 6 show the mai scees o the Iteval Type-2 Fuzzy Ieece Systems Stuctue Edito called IT2FIS (Iteval Type-2 Fuzzy Ieece Systems. 8

4 Figue 5. IT2FIS Edito. Figue 6. Iteval Type-2 MF s Edito. The Mamdai IT2FIS, is desiged with iputs, m outputs ad ules. The th ule with iteval type-2 uzzy atecedets, iteval type-2 uzzy cosequet C { μ } i, il, i, ad iteval type-2 uzzy acts { σ }, l, i ae ieed as a diect easoig [0]. The evaluatio o this type o easoig is as ollows: R C ( C ( C, th,,,,,, ule. H, acts. C H R [ ( C ] i i / α Y α [ μ ( y, μ ( y] [0,] C C ( ( * ( * ( * ( μ x C, i i μ x i i C * ( * ( * ( μ x C, i i μ x i i C C C [ [ ( C ]] i i / α Y α [ μ ( y, μ ( y ] [0,] C C ( μ y C ( ( μ y C ( ( C, * μ ( xi* μ ( xi * μ ( y i i C ( C, * μ ( xi* μ ( xi * μ ( y i i C The deuzziicatio o the iteval type-2 uzzy aggegated output set ideuzztype2(, type C C is: whee type is the ame o the deuzziicatio techique. I i ae iteval type-2 uzzy sigletos the: C [ μ ( xi ] μ ( i, C y / α Y α [ μ ( y, μ ( y] [0,] C C [* μ ( x ˆ ] C i i, C [* μ ( x ˆ ] C i i, C [ C C, μ ( ] xi μ i, C [ ( y ]] / α Y α [ μ ( y, μ ( y] [0,] C C 82

5 [ ( ] [* ( ˆ ] ( C C μ xi i, C [ ( ] [* ( ˆ ] ( C C μ xi i, C The IT2FIS de Taagi-Sugeo-Kag system is desiged with iputs, m outputs ad ules. The th ule with { μ } iteval type-2 uzzy atecedets i, il, i, iteval, type- uzzy set ae used o the cosequets sets,, θ 0, + θ i, x i ad eal acts ae ieed as a diect easoig [0]. The evaluatio o this easoig is: α [ α, α] [ μ i, ( xˆ ] ( μ xˆ ( ˆ i μ x i i, i i, * (,* ( whee α [ α, α] is the iig set o the iteval type- uzzy atecedet o the th ule. ˆ, θ0, + θi, xi l [ ] whee,,, is a eal uctio o the iteval cosequets o the th ule. I i, i, i, i, i, i 0,...,, whee c i, is the cete ad s i, deotes the l spead, the,, is expessed as: i θ [ c s, c + s ] l, ci, xi + c0, si, xi s0,, ci, xi + c0, + si, xi + s0, With the Kai ad Medel algoithm [0] the l α ad IV. SIMULTION RESULTS I this sectio we show simulatio esults o iteval type-2 uzzy cotol o two bechma cases. The ist oe is the poblem o showe cotol that has aleady bee solved with type- uzzy cotol by may authos [3], ad ow we show bette esults with type-2 uzzy cotol. The secod poblem is the tuc bace-uppe cotol situatio that also has solutios with type- ad ow we show the esults with type-2 uzzy cotol.. Showe Cotol Simulatio We applied a iteval type-2 uzzy cotol scheme to the showe cotol poblems ad i Fig. 7 ad Fig. 8 the cotol esults. Figue 7. Tempeatue iteval type-2 uzzy cotol. α ae evaluated to obtai the FIS output vaiables, these ae expessed as L l l l l α, α, + α, l L+ L l α α + α L+ R α, α, + α, R+ R α α + α R+ l ˆ + y 2 Figue 8. Iteval type-2 uzzy cotol low. I this expeimet we evaluate the system espose to compae the type- ad type-2 cotols with the ISE (Itegal o Squae Eo, IE (Itegal o bsolute value o the Eo ad ITE (Itegal o the Time multiplied by the 83

6 bsolute value o the Eo uctioality citeia. The best esults wee i type-2 cotols, as show i Table I. TBLE I. COMPRISON OF FUZZY CONTROLLERS TYPE FLS ISE IE ITE VRIBLE Type Tempeatue Type Tempeatue Type Flow Type Flow B. Tuc bace-uppe cotol simulatio We apply the iteval type-2 uzzy cotol scheme to the bace-uppe cotol poblem ad i Fig. 9 ad Fig. 0 the cotol esults o the ca taectoies ae show. I this case study we use a SNR28 db sigal-to-oise-atio to geeate ucetaity i the plat output vaiables. We compae the type- ad type-2 cotols usig the mea uctioality citeia o each taectoy, obtaiig the ollowig esults: ISE2.2053, IE y ITE6.209 o type- ad ISE2.0386, IE2.830 y ITE o type-2. The type-2 uzzy cotolle was bette. V. CONCLUSION We have peseted i this pape the basic cocepts o iteval type-2 uzzy logic. lso, the use o a developed toolbox o iteval type-2 uzzy logic that ca be used to apply the theoy i solvig eal-wold poblems is illustated. Simulatio esults i cotol applicatios show the easibility o the iteval type-2 uzzy logic appoach i achievig itelliget cotol. The desig ad implemetatio o the iteval type-2 uzzy logic toolbox is potetially impotat o eseach i the iteval type-2 uzzy logic aea, thus solvig complex poblems o the dieet applied aeas. Ou utue wo icludes impovig the type-2 uzzy logic Toolbox with a bette gaphics use iteace (GUI, itegatig a leaig techique Toolbox to optimize the owledge base paametes o the iteval type-2 uzzy ieece system, ad the desig o iteval type-2 uzzy eual etwo hybid models. CKNOWLEDGMENT The authos would lie to tha DGEST ad CONCYT o the iacial suppot povided to this eseach wo. REFERENCES Figue 9. Taectoy Iteval type-2 uzzy cotol. Figue 0. Taectoy 2 iteval type-2 uzzy cotol. [] L.. Zadeh, Fuzzy sets, Iomatio ad Cotol, vol. 8, pp , 965. [2] L.. Zadeh, Outlie o a ew appoach to the aalysis o complex systems ad decisio pocesses, IEEE Tas. o Systems, Ma, ad Cybeetics, Vol. 3, No., pp , Ja [3] L.. Zadeh, The cocept o a liguistic vaiable ad its applicatio to appoximate easoig, Iomatio Scieces, vol. 8 pp , 975. [4] L.. Zadeh, Fuzzy logic, Compute, vol., pp , 988. [5] L.. Zadeh, Kowledge epesetatio i uzzy logic, IEEE Tas. o Kowledge ad Data Egieeig, vol., pp , 989. [6] N.N. Kai ad J.M. Medel. Itoductio to Type-2 Fuzzy Logic Systems, Uiv. o Southe Cali., Los geles, C, Jue 998. [7] L.. Zadeh, Fuzzy logic computig with wods, IEEE Tas. o Fuzzy Systems, vol. 2, pp. 03, 996. [8] Q. Liag ad J. Medel, Iteval type-2 uzzy logic systems: Theoy ad desig, IEEE Tas. o Fuzzy Systems, vol. 8, pp , [9] D. Dubois, ad H. Pade, Fuzzy sets ad systems: theoy ad applicatios, cademic Pess, New Yo 980. [0] J. M. Medel, Ucetai ule-based uzzy logic systems: itoductio ad ew diectios, Petice-Hall, Uppe Saddle Rive, New Jesey, 200. [] M. Mizumoto, ad K. Taaa, Some popeties o uzzy sets o type- 2, Iomatio ad Cotol, vol. 3, pp , 976. [2] L.-X. Wag, daptive uzzy systems ad cotol: desig ad stability aalysis, Petice Hall, Uppe Saddle Rive, New Jesey, 994. [3] O. Castillo, ad P. Meli, Type-2 uzzy logic: theoy ad applicatios, Spige-Velag, Heidelbeg, Geay, [4] J.R. Casto, O. Castillo, P. Meli, ad. Rodiguez-Diaz, Buildig uzzy ieece systems with a ew iteval type-2 uzzy logic toolbox, Tas. o Computatioal Sciece, vol., pp. 04-4,