A Shape-preserving Affine Takagi-Sugeno Model Based on a Piecewise Constant Nonuniform Fuzzification Transform

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1 A Shape-peseving Affine Takagi-Sugeno Model Based on a Piecewise Constant Nonunifom Fuzzification Tansfom Felipe Fenández, Julio Gutiéez, Juan Calos Cespo and Gacián Tiviño Dep. Tecnología Fotónica, Facultad de Infomática Univesidad Politécnica de Madid, 866 Madid SPAIN Abstact: Fist-ode Takagi-Sugeno systems ae descibed by means of a set of affine functions defined on fuzzy egions usually specified by the coesponding tapezoidal tenso poduct. The standad computational model pesents some shape-failues: 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 altenative efficient model based on an initial tansfomation of the fist-ode tapezoidal TS model into a zeo-ode tiangula TS model followed by a nonunifom fist-ode even B-spline filte applied to the coesponding tiangula antecedent patition. The obtained output is a smooth piecewise multiquadatic C function. Keywods: Affine Takagi-Sugeno model, quadatic ϕ W -splines, nonunifom fuzzification tansfom, shape-peseving appoximation Intoduction Piecewise affine functions ae today a common technique 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 fuzzy contolles. ATS models exploits the chaacteistics of fuzzy patitions in elation to 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 canonical ATS ules of a standad MISO system ae on the fom: R : If x is A, and... and x n is A,n then z=a x+b whee x = (x,..., x n ) T is the multivaiate input vaiable (A,..., A n ) ae tapezoidal membeship functions that belong to the coesponding univaiate Ruspini fuzzy patition [][7] and z=a x+b is the coesponding multivaiate affine output function. These ules can be witten in a multivaiate-closed fom as R : If x is A then z=a x+b whee A = A (x) = A (x )... A n (x n ) is the coesponding multivaiate tapezoidal tenso poduct that belongs to the coesponding multivaiate fuzzy patition. This multivaiate patition of unity satisfies the following constaints [][7]: A (x) and Σ A (x) = The global appoximato obtained by the infeence mechanism gives a blending pocedue of the coesponding local models. Using the standad poduct-sum method, the output function can be witten in matix fom as z m = = A ( x) ( a x + whee A (x) = A (x )... A n (x n ) is the efeed multivaiate tapezoidal tenso poduct of ule. Theefoe, in the coe of multivaiate tenso-poduct tapezoidal membeship function whee A (x)=, the global output function z is equal to he coesponding local affine function z = a x+b. We call these multivaiate egions: multivaiate intepolation intevals of an ATS model. The coesponding univaiate components ae called univaiate intepolation intevals {IP i } (Figue.a). 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 b )

2 affine functions z= Σ A (x) (a x+b.) We call such egions multivaiate appoximation intevals of an ATS model. The coesponding univaiate components ae called univaiate appoximation intevals {AP i } (Figue.a) ATS schema pesents some shape failues as is pointed out in [][]. In ode to avoid this poblem, the oiginal univaiate tapezoidal antecedent fuzzy patitions hee consideed ae tansfomed into A A A IP AP IP AP IP (a) B B B B B 4 B 5 (b) Figue : Tansfomation of a univaiate tapezoidal fuzzy patition into an equivalent tiangula one. equivalent univaiate tiangula ones (Figue.b) in ode to obtain a shape-peseving C piecewise multilinea appoximato. An additional nonunifom fist-ode B-spline fuzzification (descibed in Section ) is applied to the peviously obtained multivaiate tenso poduct tiangula fuzzy patition, to impove the smoothness of the coesponding output function o fo taking into account the additional fuzziness of the consideed system. Notice that, the computational complexity of an ATS system is popotional to the maximum multivaiate ovelapping facto ov [5] of the coesponding multivaiate fuzzy patition (ov = ov... ov n ). The ovelapping facto of the intemediate univaiate nomal tiangula fuzzy patitions is equal two. This ovelapping facto is inceased in ode to augment the continuity ode and smoothness of the coesponding output function. In this pape, with the aim of having an efficient computational model, we limit to thee this ovelapping facto of the fuzzified univaiate patitions obtained. The coesponding fuzzified output function is a piecewise multiquadatic appoximato that peseves the affine functions on the coes of the multivaiate fuzzified antecedent tems. Shape failues of Affine TS Models A SISO affine Takagi-Sugeno fuzzy system is specified by a set of ules of the fom: R i : If x is A i then z=a i x+b i whee x is the input vaiable, z is the output vaiable, A i is the coesponding antecedent tem and a i x+b i is the elated affine output function. The univaiate antecedent tems {A i } usually consideed fom a fuzzy patition o patition of unity and have a nomal tapezoidal fom [][5]. To analyse the shape-failues of a SISO affine T-S scheme the following basic nomalised system (Figue ) is consideed, with a slope a : If x is A then z = If x is A then z= a (x-) + The standad output function z of this system in the input inteval [ ] is given by z (x) = x (a(x-)+) = ax (x-(-/a)) The output deivative z (x) = a x - a (-/a) = a (x- (a-)/a) becomes negative fo slopes a>, in the input inteval [, (a-)/a], even though the gain o slopes of the local affine models consideed ae nonnegative. Theefoe, this appoximation is not shape peseving [8] and does not conseve the positivity, z= A z =4 (x-/4) -x x z= (x-) + z=x (x-/) A -,5,5,5 Figue : Output function and its deivative of the nomalized standad ATS system consideed (a=).

3 monotony and convexity of the defined data in the coesponding 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. Note that classical Hemite spline intepolation [4] use completely diffeent blending functions fo these two components. The appoach pesented in this pape is in a sense analogous to Bezie splines appoximatos [4], i.e. to tansfom the infomation of the slopes (deivatives) of the equied output function into new additional contol points. Next section pesents a multiquadatic fuzzified shape-peseving affine Takagi-Sugeno (SPATS) model, which has a shape-peseving behaviou. Multiquadatic SPATS Model The consideed SPATS model is obtained by means of a two-phase tansfomation ) Tansfomation of each univaiate tapezoidal fuzzy patition {A i } into an equivalent tiangula one {B j } using the same knots sequence, and a simultaneous deivation of the coesponding output functions c k fo each cone of the oiginal multidimensional intepolation intevals o multidimensional antecedent tem coe intevals (Figue ). The new obtained zeoode TS system has the following fom m = z B ( x) c = whee m = m n is the numbe of new deived ules. Cone function values c k Multilinea egions Affine egions X x quadatic spline fuzzy patition {B j } (spline of ode thee) called ϕ w -spline patition (moe deeply descibed in Section 4), using the following nonunifom fuzzification tansfom fo each univaiate tiangula tem B j B j, w (u) = B Uj j ( v) /(w) N (( u v) /(w)) dv whee N (.) is an even fist-ode B-spline[] defined by N (u)= { u ½ ; } and w is the ectangula window width applied, whee w=w(u) is a piecewise constant function that peseves the value within each fuzzification inteval Iw j = [x j -w j, x j +w j ] coesponding to each intemediate knot x j. Fo both knots of each univaiate appoximation inteval AP i, the window width value w i is defined as popotional to the width of coesponding appoximation intevals w i = k Wi AP i whee k Wi /. These paametes k wi epesents the coesponding local tension paametes of the coesponding splines. Outside these fuzzification intevals, the window width value is zeo (Figue 4). In this pape, it is accepted that the oiginal univaiate appoximation intevals widths AP i ae less o equal than the widths of the coesponding adjacent univaiate intepolation intevals IP i and IP i+. This hypothesis simplifies the analysis of the coesponding ϕ w -splines and is a natual constaint in many pactical applications. The efeed nonunifom fuzzification tansfom is a genealization of the unifom one [6]. This nonunifom tansfom 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.,5,5 Appoximation inteval Fuzzification inteval w (u) Figue : Intemediate zeo-ode TS model x ) Ulteio tansfomation of each univaiate tiangula fuzzy patition {B j } (spline of ode two) into a, Figue 4: Nonunifom fuzzification tansfom

4 The following popety justifies the piecewise constant natue of nonunifom fuzzification tansfom consideed. Affine invaiance popety. Affine functions ae fixed points of nonunifom fuzzification tansfom: (a v + b) / w N (( u v) / w) dv = a u + U Poof. It is a consequence of the even and unit-aea popety of the nonunifom fuzzification function N w(u)=/w N (u/w) used. In each position of the coesponding ectangula window filte, the coesponding aveage value, fo two symmetical points of this window, is equal to the coesponding cental value. Next section analyzes the main chaacteistics of the obtained univaiate ϕ w -spline patitions. 4 Quadatic j w -spline Patitions Univaiant quadatic ϕ w -spline patitions can be consideed as a genealization of quadatic B-splines [][4][9][] and ae defined in this section by means of the efeed nonunifom even fist-ode B-spline filte N w applied to a nomal tiangula fuzzy patition {B j }: N w {B j } = {B j w } An example of a quadatic ϕ w -spline patition defined by the knots sequence T =(, 6,, 8,, 5.4) is shown in Figue 5. The nonunifom even B-spline convolution filte used N w detemines the smoothness and tension of the,,8,6,4, Figue 5: A nonunifom quadatic ϕ w -spline fuzzy patition {B j w } b elated output function. This way the designe can initially deive a nomal tiangula fuzzy patition, which gives a non-smooth piecewise multilinea output function. Futhemoe, it is possible to add a nonunifom fist-ode even B-spline filte to adjust the smoothness of the coesponding output function. Popeties. Quadatic ϕ w -splines {B j } of degee and thid ode consideed have the following popeties:. Suppot width: supp(b j ) = supp(b j ) + w j- +w j+ whee w j- and w j+ ae espectively the left and ight fuzzification window width of knots x j- and x j+.. Coe width: if w j > then coe(b j ) = Ø;. Continuity class: {B j } C 4. Polynomial fom: {B j } ae fomed by piecewise affine and quadatic functions. 5. Lineaity pesevation: {B j } ae affine functions outside the fuzzification intevals, whee the window width function w is zeo (Figue 4 and 5). 6. Equivalence between tiangula splines and ϕ w - splines: B j (u)=b j (u) and between thei deivatives: B j (u)= B j (u) outside the fuzzification intevals. 7. ϕ w -splines {B j } ae vaiation diminishing, meaning that thee ae no moe local exteme points in the linea combination of ϕ w -splines than thee ae in the data set of the coesponding weighting points. 8. ϕ w -splines {B j } ae shape-peseving and conseve the positivity, monotony o convexity of the coesponding data set. 9. Univaiate ϕ w -splines ae easily 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 the ectangula window convolution (low-pass filte) popeties applied to tiangula fuzzy patitions. To compute the ovelapping thid ode ϕ w -splines: B j-, B j, B j+, we calculate fo k = j-, j and j+: B k,w u wj j + u wj k (u) = (/ w ) B ( v) dv in each fuzzification inteval Iw j = [x j -w j, x j +w j ] elative to each intemediate knot x j. Outside the fuzzification

5 intevals, the esultant quadatic splines B j ovelap the oiginal tiangula splines B j. Fo each fuzzification inteval Iw j = [x j -w j, x j +w j ] of an intemediate knot x j, the concened quadatic splines (B j-, B j, B j+ ) have the following expessions: B j- (u)= (u-(x j +w j )) /(4w j (x j -x j- )) B j+ (u) = (u-(x j -w j )) /(4w j (x j+ -x j )) B j (u)= - B j- - B j+ It is impotant to note that these fomulas only give a single quadatic segment of the coesponding ϕ w - splines ove the fuzzification inteval [x j -w j, x j +w j ]. Notice that fo k Wj =/ the coesponding intemediate affine appoximation inteval degeneates into a point. 5 Example of a SISO System The paticula univaiate nonlinea ATS system consideed, is specified on a tapezoidal fuzzy patition {A, A, A } defined by the knot sequence: T=(x,x,...,x 5 ) = (, 6,, 8,, 5.4) The coesponding univaiate affine fuzzy ules ae: R: IF x IS A then z = (.8/6) (x-) +. R: IF x IS A then z = (.5/7) (x-) +.5 R: IF x IS A then z = - (/4.4) (x-) +.5 Using the same knot sequence, an equivalent tiangula fuzzy patition {B, B, B, B, B 4, B 5 } is deived. The coesponding univaiate zeo-ode TS fuzzy ules ae: R: If x is B then z =. R: If x is B then z = R: If x is B then z =.5 R: If x is B then z = 4 R4: If x is B then z =.5 R5: If x is B then z =.5 The additional nonunifom fuzzification filte applied is a fist-ode B-spline N w(u)=/w N (u/w) defined by the global tension paamete k W =.4. In Figue 6 it is also shown the coesponding piecewise constant window width function w= w(u). Figue 6 also epesents the piecewise affine function C obtained using the intemediate zeo-ode TS model and the fuzzified 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 fuzzification intevals. 4,5 4,5,5,5, Figue 6: A nonunifom quadatic ϕ w -spline fuzzy patition {B j w } 6 Example of a MISO system 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 = (x, x, x, x ) = (,, 5, 7) T = (x, x, x, x ) = (,, 5, 7) The coesponding bivaiate ATS model is defined by the single ule: R: If x is A and x is A then z = +4x +x /8 else z=; The additional nonunifom fuzzification filte applied is a bivaiate fist-ode even B-spline tensopoduct N w(u ) N w(u ) espectively defined by the global tension paametes: k W =.4 and k W =.4. The obtained shape-peseving contol suface is depicted in Figue 7.a, whee it is possible to appeciate that the lineaity of the oiginal suface is also peseved excluding the coesponding bivaiate fuzzification intevals. This fuzzified SPATS contolle also poduces a suitable smooth output. Figue 7.b shows the coelative contol suface of the standad tapezoidal ATS model, whee it is possible to appeciate the efeed shape failues.

6 (a) (b) Figue 7. Bivaiate contol sufaces: (a) SPATS model. (b) Standad ATS model 7 Conclusions In this pape a new type of multiquadatic shapepeseving affine Takagi-Sugeno system has been intoduced based on the tansfomation of the oiginal univaiate tapezoidal patitions into equivalent tiangula one. The pesented SPATS model integates an additional nonunifom fuzzification function that is adapted to the specified tapezoidal patition. The stability analysis of the coesponding contolle can be facilitated taking into account the shape-peseving natue of the descibed model. The descibed model also allows specifying diffeent fuzziness of a fuzzy system by means of the nonunifom fist-ode even B-spline fuzzification consideed. The deived smooth C SPATS model povides a suitable appoximation method that can be applied to the poblems of system identification, signal pocessing and contol. 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] G. Fain, Cuves and Sufaces fo Compute- Aided Geometic design, Fouth edition, Academic Pess, 998. [5] D. Diankov, H. Hellendoon, M. Reinfank, An Intoduction to Fuzzy Contol, Spinge-Velag, 99. [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 [7] G. Kli, B. Yuan, Fuzzy Sets and Fuzzy Logic, Pentice Hall, 995. [8] B. I. Kvasov, Methods of Shape-Peseving spline appoximation, Wod Scientific,. [9] J. Zhang and A. Knoll, Constucting Fuzzy Contolles with B-spline Models, IEEE Intenational 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