Impovement of Fist-ode Takagi-Sugeno Models Using Local Unifom B-splines Felipe Fenández, Julio Gutiéez, Gacián Tiviño and Juan Calos Cespo Dep. Tecnología Fotónica, Facultad de Infomática Univesidad Politécnica de Madid, 866 Madid, Spain felipe.fenandez @es.bosch.com, jg@dtf.fi.upm.es, gtivino@dtf.fi.upm.es, cespozj@dtf.fi.upm.es Abstact Fist-ode Takagi-Sugeno models ae mainly based on the intepolation between seveal local affine functions usually defined on tapezoidal fuzzy patitions. The standad computational model pesents some shapefailues: the appoximation does not peseve the positivity, monotony o convexity of the data that belong to the coesponding antecedent tem coes. Moeove the standad output does not have a continuous deivative. This pape pesents an impoved model that pimaily tansfoms the oiginal fist-ode tapezoidal TS system into an equivalent zeoode tiangula TS one. Futhemoe, fo each univaiate tansition egion: two equidistant tiangula labels ae added, one at each end of the coesponding inteval, to captue the infomation of deivatives of the affine functions. Finally in each tansition egion, a local even box filte is applied to the coesponding fou tiangula labels in ode to obtain a local unifom quadatic B-spline patition. This tansfom peseves the oiginal affine functions in the educed coes of the oiginal fuzzy patition and convets the intemediate C piecewise linea-multilinea output function into a C piecewise lineamultiquadatic one. Keywods: Takagi-Sugeno model, unifom quadatic B-splines, shape-peseving splines Intoduction Models usually have a limited ange of validity and to emphasize this aspect, they ae called local models on an opeating egime, as opposed to global models that ae valid in the full ange of opeation. Howeve, by means of a piecewise appoach, the full ange of opeation is coveed by a numbe of possibly ovelapping egimes. Within each opeating egime the system is modelled by a local model, and they can be blended in the tansition egions using suitable weighting functions, taken into account that he opeating egimes ae not had boundaies egions. This means that thee will be a gadual tansition between local models when the system is in intemediate egimes. Piecewise affine functions ae poweful tool to descibe nonlinea systems. Affine Takagi-Sugeno (ATS) models extend the capability of piecewise based modelling to fuzzy systems and give a simple fomal language to chaacteize and analyze complex systems. ATS models can exploit fuzzy patitions and piecewise affine systems by combining simple local models, each valid within a cetain opeating egime. Each ule antecedent defines a fuzzy local egion and the associated ule consequent descibes the coesponding affine function that specifies the coesponding local model. The est of the pape is oganized as follows: Section eviews ATS model, Section analyses the shape failue of standad ATS model, Section 4 pesents the main chaacteistics of the fuzzy patition tansfomations intoduced, Section 5 and 6 espectively depict the linea-multilinea and lineamultiquadatic ATS models, and Section 7 descibes the elevant popeties of the local linea-quadatic B- spline patitions obtained. Some pactical examples ae shown in sections 8 and 9, and finally Section concludes the pape. Review of ATS model The consideed ATS ules of a standad MISO system ae on the conjunctive canonical fom: R : If x is A, and... and x n is A,n then z = p T x+q whee x = (x,..., x n ) T is the multivaiate input vaiable, (A,..., A n ) ae tapezoidal membeship This eseach has been suppoted by CICYT TIC 6-9 HiSCoP
functions that belong to the coesponding univaiate patition of unity o Ruspini fuzzy patition [][4][8] and z= p T x+q is the coesponding multivaiate affine output function. Each univaate tapezoidal patition is defined by a knot sequence (t, t, t,, t m ). A simple univaate tapezoidal patition with m= 5 it is shown in Figue.a. Multivaiate ATS ules can also be witten in a multivaiate-closed fom as R : If x is A then z= p T x+q whee A = A (x) = A (x )... A n (x n ) is the coesponding multivaiate tenso poduct. The coesponding multivaiate patition of unity satisfies the following constaints: A (x) and Σ A (x) = The global appoximato obtained by the infeence mechanism gives a blending pocedue of the coesponding local affine models. Using the standad poduct-sum method, the output function can be witten in matix fom as Analysis of shape-failues of ATS models The output function of a standad ATS model does not have continuous deivatives and is not shape peseving [9] since it exhibits wild wiggles that ae not inheent in the coe data [][]. In this section we biefly analysed these undesiable popeties of standad ATS systems. Fo this pupose, the following basic SISO ATS is consideed (see a paticula example in Figue ): R : If x is A then z = ; R : If x is A then z = a (x-) + b whee the tapezoidal fuzzy patition {A, A } is defined by the knot sequence (t,,, t ). The standad output function z of this system in the tansition inteval TI =(, ) is given by z (x) = (-x) + (x ) (a (x-)+ b ) = a x +(b -a )x Its deivative is given by R z = = A ( x) ( p T x + q ) z = (x-.4) whee A (x) = A (x )... A n (x n ) is the efeed multivaiate tenso poduct of ule, and R = (m +) (m +), (m n +)/ n is the numbe ules. Theefoe, in the coes of multivaiate tenso-poduct tapezoidal membeship function whee A (x)=, the global output function z is equal to he coesponding local affine function z= p T x+q. We call these multivaiate egions: multivaiate affine intevals of an ATS model. The coesponding univaiate components ae called univaiate affine intevals AI i =[t i, t i+ ]. They ae shown as white intevals in Figue.b. In the multivaiate egions whee the tenso-poduct <A (x)<, the global output function z is given by a local convex combination of the coesponding local affine functions z= Σ A (x) (a x+b.) We call such egions multivaiate tansition intevals of an ATS model. The coesponding univaiate components ae called univaiate tansition intevals intevals TI i = (t i+, t i+ ). They ae shown as gey intevals in Figue.b. z=5 (x-) + z= z= x (5 (x-) +) -x x A A -,5,5,5 Figue : Example of output function and its deivative of a standad ATS system. z (x) = a x +b -a =a (x /(-b / a )) which is not equal in the cones of the tansition inteval (z ()= (b -a ); z ()= (b +a ) ) to the coesponding deivatives of the specified affine functions (z =, z =a ). This deivative becomes negative in the inteval [,] when x</(-b /a )). If (-<(b /a )<-) then the output deivative is negative in the inteval [, /(-b /a ))], even though the slopes of the local affine models consideed ae nonnegative. Theefoe, the global output function of an ATS system does not have continuous deivative and it is
not shape peseving [9] since it does not conseve the positivity, monotony and convexity of the defined data by the coesponding multidimensional antecedent tem coes. A geneal eason of this objectionable behaviou is that the same membeship functions ae used to blend the constant pat b i and linea pat a i x of the affine function a i x+b i. Notice that classical Hemite spline intepolation [5] uses completely diffeent types of blending functions fo these two components. The appoach pesented in this pape is in a sense analogous to Bezie splines appoximatos [5], i.e. we tansfom the infomation of the slopes (deivatives) of the affine output functions into new additional contol points. Next section pesents an equivalent linea-multilinea ATS model, which has a shape-peseving behaviou. 4 Fuzzy patition tansfomations potfolio In ode to avoid the shape failues of standad ATS schema, this pape pesents a tansfomational pocess that pimaily convet the oiginal fist-ode tapezoidal TS system into an equivalent zeo-ode tiangula TS one. To accomplish this pocess, each oiginal univaiate tapezoidal antecedent fuzzy patitions hee consideed ae fist tansfomed into equivalent tiangula one (Figue.c) in ode to obtain a shape-peseving C piecewise lineamultilinea appoximato. In this pape it is taken into account the usual pactical hypothesis that the lengths of affine intevals ae longe than the lengths addition of the adjacent tansition intevals: AI i TI i- + TI i. This consideation allows a subsequent local independent filteing that geatly simplifies the coesponding analysis and involved computations. In ode to obtain an output function with C continuity, we need to captue the infomation of deivatives of the affine functions in the cones of each multivaiate tansition inteval. To accomplish this taget, in this pape, two equidistant tiangula labels ae added, one at each end of each univaiate tansition inteval (Figue.d). The coesponding expanded tansition egion is called in this pape, enlaged tansition inteval. Finally, to impove the smoothness and continuity ode of the coesponding output function, an even box filte is locally applied to the fou tiangula labels of each univaiate tansition egion in ode to obtain a local unifom quadatic B-spline patition []. This filteing tansfom (moe deeply descibed in sections 6 and 7) peseves the oiginal affine functions in the educed coes of the oiginal multivaiate fuzzy patition and convets the intemediate C piecewise linea-multilinea function into a C piecewise lineamultiquadatic one. The computational complexity of the filteed ATS systems in the tansition egions is augmented, since the incease in one unit of the antecedent-ovelapping facto. Within each local univaiate filte inteval it changes fom two to thee. This is the involved penalty paid fo augmenting the continuity ode and smoothness of the coesponding output function. A A A t t t t t 4 t 5 (a) AI TI AI TI AI t t t t t 4 t 5 (b) B B B B B 4 B 5 t t t t t 4 t 5 (c) B B B B B 4 B 5 B 6 B 7 B 8 B 9 t t t t t 4 t 5 t 6 t 7 t 8 t 9 (d) Figue : Fuzzy patition tansfomation: (a) Oiginal tapezoidal fuzzy patition. (b) Coesponding affine and tansition intevals. (c) Equivalent tiangula patition. (d) Extended tiangula patition
Howeve, the definition of local unifom quadatic B- spline patitions in the tansition egions also makes easie the coesponding computation. The obtained output function is a piecewise lineamultiquadatic appoximato that peseves the affine functions on the educed coes of the initial multivaiate antecedent tems. Standad B-spline patitions have also been used in diffeent fuzzy models [][][6][7] as a high-level hybidisation between fuzzy and spline techniques. In this pape a low-level hybidisation is pesented that deeply combines synegies between cisp and B-spline patitions, in ode to obtain a shape-peseving fuzzy model. Moeove, the obtained quadatic B-splines ae locally unifom within each tansition intevals, which imply less computational load than the efeed appoaches in the coesponding fuzzy algoithms. 5 Linea-multilinea ATS model In the pevious section, in ode to impove the ATS model and to obtain a shape-peseving piecewise output function, each oiginal univaiate tapezoidal antecedent fuzzy patitions {A ji } of a domain U j, defined by the knots sequence T i =(t jo, t j,... t jm ), has been fist tansfomed into an equivalent univaiate tiangula one {B jk } using the same knots sequence. In the SISO example of Figue, the output function is defined by the ule: If x is A then z = 5(x-) + else z = Fo the antecedent knots sequence T=(-,,, ), the coesponding outputs of tiangula antecedent ules ae espectively c = c =, c = c =6. The global output of this system is a piecewise affine function also shown by a dashed line in Figue. In a geneal MIMO case, the deived zeo-ode TS model has a ule fo each cone of the oiginal multidimensional antecedents coe intevals (Figue ). The new equivalent zeo-ode TS system gives the following non-smooth C linea-multilinea output function R z = B ( x) c = whee R =(m +)(m +) (m n +) is the numbe of new deived ules and c ae the constant output functions of the coesponding ules. This output function is a C shape-peseving piecewise lineamultilinea intepolato. Cone function values c k Multilinea egions Linea egions 6 Linea-multiquadatic ATS model Fo each tansition inteval (t j, t j+ ) of each input vaiable, two additional tansfomations ae caied out: ) The efeed enlagement of each tansition inteval using two exta equidistant knots, one at each end of the tansition inteval: (t j, t j+ ) (t j -w, t j, t j+, t j+ +w) whee w= t j, t j+ =(t j+ -t j ) is the width of the oiginal tansition inteval. To simplify the analysis, in each enlaged tansition inteval a local vaiable u is defined: u= (x-t j ) / (t j+ -t j ) = (x-t j )/w In this tansfomed local domain, the coesponding unifom tiangula patition is defined by the knots sequence (-,,, ). ) Each pevious univaiate tiangula fuzzy patition {B j : j=-,,,} (unifom B-spline patition of ode two []) is tansfomed into a linea-quadatic spline fuzzy patition {B j : j=-,,,} (spline patition of ode thee) using the coesponding local filte. Fo each univaiate tiangula tem B j the following box filte is applied B j (u) = B Uj X x Figue : Intemediate zeo-ode ATS model j ( v ) N ( u v) dv whee N (.) is a even box filte o fist-ode B-spline [] defined by N (u)= { u ½ ; } This box filte has a unit width in the local tansfomed domain and a width w in the oiginal domain. x
The consideed box filte is not applied in the whole inteval (t j -w, t j+ +w) but in a educed one (t j -w/, t j+ +w/) called filte inteval, which is defined by the inteval (-.5,.5) in the tansfomed local domain of vaiable u. Outside this filte inteval the coesponding output affine function is peseved. Figue 4 shows an example of the final lineaquadatic spline patition obtained fom a oiginal aveage value, fo two symmetical points of this window, is equal to the coesponding cental value. Next section analyses the main chaacteistics of the local B-spline patitions obtained. 7 Local Linea-quadatic B-spline patitions, w w,8,6,4, 4 5 6 7 8 9 Figue 4: Linea-quadatic spline fuzzy patition geneated using two local unifom box filtes (box filte widths: w =, w =.6). tapezoidal one, defined by the knot sequence (,,, 7, 7.6, ), whose tansition inteval widths ae w o = and w =.6. The intemediate enlaged tiangula patition is defined by knot sequence (,,,, 4, 6.4 7, 7.6, 8. ). In the filte intevals (.5,.5) and (6.7, 7.9) a local box filte was applied to the coesponding unifom tiangula patition (box filte widths: w o = and w =.6). The local unifom box filte descibed inceases the smoothness and continuity ode of the coesponding geneated output function and tansfoms the intemediate C piecewise multilinea function into a C piecewise multiquadatic one. The following popety justifies the affine-peseving natue of the local even box filte applied. Affine invaiance popety. Affine functions ae fixed points of even B-spline filtes: (a v + b) N ( u v) dv = au + U Poof. It is a consequence of the even and unit-aea popety of B-spline functions N (u) used. In each position of the coesponding filte, the coesponding b In each local tansfomed inteval (-, ), lineaquadatic B-spline patitions {B j j=-,,, } ae deived by means of the efeed even box (fist-ode B-spline) filte N applied to each unifom tiangula fuzzy patition {B j }: N {B j } = {B j } j=-,,, Figue 5 depicts this linea-quadatic patition in the tansfomed domain, which is unifom and piecewise quadatic in the filte inteval (-.5,.5), and symmetical in elation to the value.5. This local B- spline patition is linea outside this filte inteval. Popeties. Local unifom linea-quadatic B-splines {B j } on the tansfomed inteval (-, ) have the following popeties of inteest fo the shapepeseving ATS model consideed:. Suppot width: supp(b - ) = supp(b ) =.5; supp(b ) = supp(b ) =.5;. Coe: coe(b - ) = -; coe(b ) =;. Continuity class: {B j } C 4. Polynomial fom: {B j } ae fomed by piecewise linea and quadatic functions. They ae piecewise quadatic functions within the filte inteval (.5,
,9,8,7,6,5,4,,, B - B B B /8 /4 - -.5.5.5 Figue 5: Piecewise linea-quadatic B-spline patition {B j : j=-,,, } defined ove the enlaged tansition inteval (-, )..5) and ae piecewise linea functions outside this filte inteval. 5. Linea-quadatic B-splines {B j } ae vaiation diminishing, meaning that thee ae no moe local exteme points in the linea combination of splines than thee ae in the data set of the coesponding weighting points. 6. Linea-quadatic B-splines {B j } ae shapepeseving, they etain geometic popeties of the initial data set, such as positivity, monotonicity, convexity, linea and plana egions. 7. Univaiate linea-quadatic B-splines {B j } ae diectly extended to an abitay numbe of dimensions, while peseving the above popeties, by means of the coesponding tenso poduct. Sketch of poof. These popeties ae a diect esult of fist-ode B-spline filte popeties applied to the local unifom tiangula fuzzy patition consideed. To compute the quadatic B-splines {B j : j=,-,,,}, we apply the coesponding box filte: B j (u) = u + / u / B j ( v) dv within the filte inteval [-.5,.5]. Outside this filte inteval, the esultant splines ovelap the oiginal tiangula splines B j. Ove this filte inteval, the concened ovelapping quadatic splines have the following expessions: If (-.5 u<.5) then B - (u)= (u-.5) / B (u) = (u+.5) / B (u)= - B - - B + =/4-u If (-.5 u<.5) then B (u)= (u-.5) / B (u) = (u-.5) / B (u)= - B - B =/4-(u-) These fomulas give the thee quadatic segments of the coesponding unifom quadatic B-splines ove the intevals [-.5,.5] and [.5,.5]. Outside the filte inteval [-.5,.5] the coesponding local unifom tiangula splines ae not filteed, because they belong to the same affine egion, and theefoe an even box filte has no influence on the output function (affine invaiance popety). 8 Example of a SISO ATS model The paticula univaiate ATS system consideed, shown in Figue 6, is specified on a tapezoidal fuzzy patition {A, A, A } defined by the knot sequence: T=(t,t,...,t 5 ) = (,,, 7, 7.6, )
9 8 7 6 5 4 z=- (x-)+ z=. (x-7)+.8 z=.5 (x-7.6)+ 4 5 6 7 8 9 Figue 6: Example of a SISO Shape-peseving ATS system using local unifom quadatic B-splines. The specified affine fuzzy ules ae: R: IF x is A then z = (x-) + R: IF x is A then z =. (x-7) +.8 R: IF x is A then z =.5 (x-7.6) + Using the augmented knot sequence (,,,, 4, 6.4, 7, 7.6, 8., ), a tiangula fuzzy patition {B, B, B, B, B 4, B 5, B 6, B 7, B 8, B 9 } is deived. The coesponding zeo-ode TS fuzzy ules ae: R: If x is B then z = 9 R5: If x is B5 then z =.68 R: If x is B then z = 6 R6: If x is B6 then z =.8 R: If x is B then z = R7: If x is B7 then z = R: If x is B then z = R8: If x is B8 then z = 4.5 R4: If x is B4 then z =. R9: If x is B9 then z = 9 The two filte intevals ae shown in Figue 6, whee the coesponding box filte widths ae and.6. Figue 6 also depicts the piecewise affine function C obtained using the standad ATS model and the piecewise linea-quadatic output function C computed using the shape-peseving model peviously descibed. The obtained output function povides a suitable shape-peseving smooth appoximation and has a local affine behaviou outside the filte intevals. 9 Example of a MISO ATS model The second system consideed is a bivaiate ATS system specified on a tenso poduct of tapezoidal fuzzy patitions: {A, A } {A, A } that ae specified by the following knots sequences: T = (t, t, t, t ) = (,, 5, 8) T = (t, t, t, t ) = (,, 5, 8) The coesponding bivaiate ATS model is defined by the single ule: R: If x is A and x is A then z = 4+.8x +x /8 else z=; The coesponding enlaged tiangula patitions ae (,,, 5, 7, 8) and (,,, 5, 7, 8). The additional local filte applied is a bivaiate fist-ode even B- spline tenso-poduct N N of width. A zoom of the obtained smooth shape-peseving contol suface is depicted in Figue 7.b, whee it is possible to appeciate that the lineaity of the oiginal suface is also peseved excluding the coesponding bivaiate filte intevals. Figue 7.a shows the coesponding contol suface of the standad tapezoidal ATS model, whee it is possible to appeciate some shape failues. Conclusions In this pape a new shape-peseving affine Takagi- Sugeno system has been intoduced based on the local use of unifom quadatic B-spline patitions in the tansition egions, automatically deived fom the oiginal tapezoidal patition.
z = 4+.8x +x /8 z = 4 8 6 4-4 8 6 4 (a) (b) Figue 7: Bivaiate output function: (a) Standad ATS model (b) Shape-peseving ATS model The stability of the coesponding ATS contolle can be impoved taking into account the shape-peseving natue of the descibed model. The deduced piecewise linea-quadatic fuzzy model povides a suitable appoximation method that can be applied to system identification, signal pocessing and contol design poblems Refeences [] R. Babuska, C. Fantuzzi, U. Kaymak, and H. B. Vebunggen, Impoved infeence fo Takagi- Sugeno models, In Poc. Fifth IEEE Intenational Confeence on Fuzzy Systems, New Oleans, USA, pp. 7-76, 996. [] R. Babuska, Fuzzy Modeling fo Contol, Kluwe Academic Publishe, 998. [] C. K. Chui, Wavelets: A mathematical Tool fo Signal Analysis, SIAM, 997. [4] Diankov, H. Hellendoon, M. Reinfank, An Intoduction to Fuzzy Contol, Spinge-Velag, 99. [5] G. Fain, Cuves and Sufaces fo Compute- Aided Geometic design, Fouth edition, Academic Pess, 998. [6] F. Fenández and J. Gutiéez, Stuctued Design of an Extended TS contolle Using Global Fuzzy Paametes and Fuzzification Tansfom, Poc. of 8th Intenational Confeence IPMU,Vol.II, pp. 74-7. [7] F. Fenández and J. Gutiéez, A Takagi-Sugeno model with fuzzy inputs viewed fom multidimensional inteval analysis, Fuzzy Sets and Systems, vol. 5.,, pp. 9-6. [8] G. Kli, B. Yuan, Fuzzy Sets and Fuzzy Logic, Pentice Hall, 995. [9] B. I. Kvasov, Methods of Shape-Peseving spline appoximation, Wod Scientific,. [] J. Zhang and A. Knoll, Constucting Fuzzy Contolles with B-spline Models, IEEE Inten. Confeence on Fuzzy Systems, 996. [] J. Zhang and A. Knoll, Unsupevised Leaning of Contol Sufaces Based on B-spline Models, IEEE Intenational Confeence on Fuzzy Systems, 997, pp. 75-7.