On soft-fuzzy computing and modelling

Witold Kosinski1,2

1 Polish-Japanese Institute of Information Technology, PJWSTK, Research Center, ul. Koszykowa 86, 02-008 Warszawa,Poland.

2 Kazimierz Wielki University, Institute of Environmental Mechanics and Applied Computer Science, ul. Chodkiewicza 30,85-072 Bydgoszcz, Poland.

In real-life problems both parameters and data used in mathematical modelling are vague. Pattern recognition, system mod-elling, diagnosis, image analysis, fault detection and others are fields where soft calculation with unprecise, fuzzy, objectsplays an important role. In the presentation recent results in the theory of ordered fuzzy numbers and their normed algebra areshortly reviewed. Then possible applications in modelling dynamical systems and mechanics are presented. From the classicalframework known algebraic and evolution equations describing such systems are transformed to their fuzzy versions.

1 Motivation

One can state the following questions: 1) How to deal with imprecise concepts such as small , big, warm, hot, slow, fast oraround 2, around 5 and around 11 ? 2) How to add or multiply around 2 and around 11? 3) How to include in constitutivemodelling and governing field equations vagueness in experimental data and unprecise initial or boundary conditions in PDEs?

For example: hot weather can be characterised by the temperature range [24C, 36C]. If we take any temperature from thisrange we say it is hot, while temperature equal 21 C means it is not hot. Saying this we stay within two-valued logic. Noticeit is a jump from 23.9 to 24 C. Could we smooth it out?

To make calculation on such objects we need a model for them and then an appropriate arithmetic or algebra. Two possibleapproaches: intervals and fuzzy numbers or fuzzy sets, are possible

2 Multivalued logic and fuzzy sets

To solve the problem one can make a generalisation due to J. Łukasiewicz (1929) [7] and L. Zadeh (1965) [9] who introducedrespectively: the three–valued logic and the continuous–valued logic. If 0 stays for the complete false and 1 for the completetruth, then in Zadeh’s logic any partial truth, i.e. value from [0, 1], can be attained. It is a kind a level of truth, or better to say,a level of membership to a given concept (in our case to an interval, representing the hot weather).

In this way we arrive at fuzzy sets. Classically in [9] in the definition of a fuzzy set A on R one relates to A the so-calledmembership function µA : R → [0, 1] with values between 1 and 0. The number µA(x) denotes the level at which x belongsto the fuzzy set A.

Fuzzy number is a particular case of fuzzy sets. To make calculation easier one restricts the shape of membership functionsrequiring a kind of convexity, then we get so–called convex fuzzy numbers [1–3, 8]. Examples of them are triangular andtrapezoidal fuzzy numbers. Arithmetic operations on fuzzy numbers have been developed with both the Zadeh’s extensionprinciple [9, 10] and the α-cut with interval arithmetic method. However, they do not lead to algebraic structures.

For a convex fuzzy number A with its membership function µA and for its α-cut, i.e. the set A[α] = {x ∈ R : µA(x) =y ≥ α}, with α ∈ [0, 1] we define two functions a1, a2 on [0, 1] that give lower and upper bounds of each

A[α] = {x : µA(x) ≥ α} = [a1(α), a2(α)]. (1)

Notice, that if the membership function µA is partially invertible on increasing µA|incr and decreasing µA|decr parts, thena1(α) = µA|

−1

incr(α) and a2(α) = µA|−1

decr(α) . Now we are ready to make our generalisation.

3 Membership relation and ordered fuzzy numbers

In the series of papers [4–6] we have introduced and then developed main concepts of the space of ordered fuzzy numbers.In our approach the concept of membership functions has been weakened by requiring a mere membership relation. In ourapproach a ’new’ fuzzy number A is identified with the pair of functions a1 and a2 defined on the interval [0, 1], i.e.

Definition 1. By an ordered fuzzy number A we mean an ordered pair (f, g) of functions such that f, g : [0, 1] → R arecontinuous.

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© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We do not require that two continuous functions f and g are inverse functions of some membership function, in generale itneeds not to exist. If it exists then f(y) = µA|

−1

incr(y) and g(y) = µA|−1

decr(y) or vice versa. To be in agreement with classicaldenotations of fuzzy sets (numbers) the independent variable of f and g is denoted by y, and the values of them by x.

According to Kosinski et al. 2002-2003 we perform arithmetic operations componentwise for A = (fA, gA), B = (fB, gB)and C = (fC , gC) .

Definition 2. Let A = (fA, gA), B = (fB, gB) and C = (fC , gC) are mathematical objects called ordered fuzzy numbers.The sum C = A + B, subtraction C = A − B, product C = A · B, and division C = A ÷ B are defined by formula

fC(y) = fA(y) � fB(y) and gA(y) � gB(y) (2)

where ”�” works for ”+”, ”−”, ”·”, and ”÷”, respectively, and where A ÷ B is defined, if the functions |fB| and |gB| arebigger than zero. Notice that the subtraction of B is the same as the addition of the opposite of B, i.e. the number (−1) · B.

In the universe R of all ordered fuzzy numbers (OFN’s) one has the linear structure since R is isomorphic to the linearspace of real 2D-vectors valued functions defined on the unit interval I = [0, 1]. The normed structure is introduced by thenorm

||A|| = max(sups∈I

|fA(s)|, sups∈I

|gA(s)|).

Hence R can be identified with C0([0, 1])×C0([0, 1]) is topologically a Banach space. A class of defuzzyfication operators ofordered fuzzy numbers can be defined [5], as a linear and continuous functionals on the Banach space R. Introducing positiveOFNs as those A = (fA, gA) for which fA ≥ 0 and gA ≥ 0, we may introduce a partial order in R .

4 Modelling

In the literature developed theories of fuzzy sets and numbers have most natural application in the control theory. We may addconstitutive modelling.

Our OFN’s in the space R have extra properties: calculation on them is most natural and compatible with calculations onreal (crisp) numbers. For example for the Cauchy problem of ODE in Rn

x(t) = F (x, t) , with initial condition x(0) = x0 (3)

corresponds to an evolution equation in the algebraR: variable x = (x1, ..., xn) from Rn is substituted by variable (xup, xdown)from Rn, and Eq. (3) is formed in two n-fold products of the Banach space C([0, 1]):

xup = F (xup, t), xdown = F (xdown, t) (4)

with xup(0) = xup0(s), xdown(0) = xdown0(s), s ∈ [0, 1] . Notice the solutions to Eq.(4) are functions of two variablessince the initial conditions are already functions of s ∈ [0, 1].

For example if xup = (x1

up, ..., xnup) and xdown = (x1

down, ..., xndown) and the function F (x) is linear, equal to Ax, with

n × n matrix A = (Axup, Axdown), then solutions to (4) with xup(0) = xup0(s), xdown(0) = xdown0(s), s ∈ [0, 1] are

xup(t, s) = xup0(s) exp(At), xdown(t, s) = xdown0(s) exp(At) with t ≥ 0, s ∈ [0, 1].

To get vector functions with values in reals we need to superposed a defuzzyfication operator (functional). One can go furtherand to use this approach in modelling mechanical or deformable systems, in which known governing PDE’s will be equippedwith fuzzy material coefficients or fuzzy initial or boundary conditions. It will be the subject of further paper.

References[1] J. Buckley James and E. Eslami, An Introduction to Fuzzy Logic and Fuzzy Sets, Physica-Verlag, A Springer-Verlag Company, Hei-

delberg, 2005.[2] G. Chen and T.T. Pham, Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems, CRS Press, Boca Raton, London, New York, Washington,

D.C., 2001.[3] A. Kaufman and M.M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold, New York, 1991.[4] W. Kosinski, On fuzzy number calculus, Int. J. Appl. Math. Comput. Sci., 16 (1), 51–57 (2006).[5] W. Kosinski, On defuzzyfication of ordered fuzzy numbers, in: ICAISC 2004, 7th Int. Conference, Zakopane, Poland, June 2004,

L. Rutkowski, Jorg Siekmann, Ryszard Tadeusiewicz, Lofti A. Zadeh (Eds.) LNAI, vol. 3070, pp. 326–331, Springer-Verlag, Berlin,Heidelberg, 2004.

[6] W. Kosinski, P. Prokopowicz and D. Slezak, Ordered fuzzy numbers, Bulletin of the Polish Academy of Sciences, Ser. Sci. Math., 51(3), 327-338 (2003).

[7] J. Łukasiewicz, Elementy logiki matematycznej, (Elements of mathematical logic, in Polish), Koło Matematyczne-FizyczneSluchaczow Uniwersytetu Warszawskiego,Warszawa, 1929; 2nd ed. Panstwowe Wydawnictwo Naukowe, Warszawa, 1958.

[8] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64, 369-380, (1978).[9] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (3), 338–353 (1965).

[10] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part I, Information Sciences, 8(3),199–249 (1975).

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ICIAM07 Contributed Papers 2010006