Stability of interpolative fuzzy KHcontrollers 1

Domonkos Tikk a,2,3 Istvan Joo a Laszlo T. Koczy a

Peter Varlaki b Bernhard Moser c Tamas D. Gedeon d,3

aDepartment of Telecommunications & Telematics, Budapest University ofTechnology and Economics, H-1117 Pazmany setany 1/d, Budapest, Hungary

bDepartment of Road Vehicles, Budapest University of Technology and Economics,H-1111 Sztoczek u. 6., Budapest, Hungary

cFuzzy Logic Laboratory Linz-Hagenberg, Johannes Kepler University, A-4040,Linz, Austria

dSchool of Information Technology, Murdoch University, South Street, Murdoch,6150 W.A., Australia

Abstract

The classical approaches in fuzzy control (Zadeh and Mamdani) deal with denserule bases. When this is not the case, i.e. in sparse rule bases, one has to chooseanother method. Fuzzy rule interpolation (proposed first by Koczy and Hirota [15])offers a possibility to construct fuzzy controllers (KH controllers) under such con-ditions. The main result of this paper shows that the KH interpolation method isstable. It also contributes to the application oriented use of Balazs–Shepard inter-polation operators investigated extensively by researchers in approximation theory.The numerical analysis aspect of the result contributes to the well-known problemof finding a stable interpolation method in the following sense.

We would like to approximate a function that is only known at distinct points(e.g. where it can be measured). Since measurement errors cannot be eliminated inpractice, it is a natural requirement that the interpolation method should be sta-ble independently from the selection of measurement points. The classical methodsgenerally do not fulfil this condition, only with some strong restrictions concerningthe measured points. If we neglect the classical form of the approximating function(polynomial and trigonometrical), we can produce better behaving approximation.The KH controller approximating function, being a simple fractional function, fulfilsthe stability condition. This can be interpreted also as KH controllers are univer-sal approximators in the space of continuous functions (with respect to, e.g., thesupremum norm).

Key words: Fuzzy inference systems; Fuzzy control; Fuzzy rule interpolation;Stability; Universal approximation

Preprint submitted to Elsevier Preprint 17 August 2000

1 Introduction

The classical approaches of fuzzy control deal with dense rule bases wherethe input universe of discourse is fully α-covered by the antecedent fuzzysets of the rule base in each dimension. (Fully α-covering means that for anyx = (x1, x2, . . . , xN) ∈ X = X1×X2×· · ·×XN in each Xi (i = 1, . . . , N) thereis a fuzzy set A ∈ F(Xi) for that A(xi) ≥ α holds, where α ∈ (0, 1] usuallyα ≥ 0.5. Here, F(Z) denotes the set of fuzzy sets defined on Z.) However, thisis not always the case, because several reasons can lead to α-sparse or sparserule bases (i.e. where at least one of the input universes is not fully (α-)coveredby the rule antecedents). A good example is when the starting term set is ofα-cover type. Nevertheless, by tuning the rules, the antecedents are partiallyshifted and shrunk so the tuned model will contain gaps [6]. Another causecan be the omission of rules from a complete but redundant rule base in orderto reduce complexity, which may also easily result in a sparse rule base [17].In such a sparse rule base no rule can be applied, and hence, no consequentcan be constructed by means of the classical methods if the observation iswithin a gap. The fuzzy rule interpolation method (proposed first by Koczyand Hirota [15]) provides a tool to construct fuzzy controllers handling sparserule bases.

On the other hand there is a great demand for finding a stable interpolatingmethod among researchers in the field of mathematical analysis. In general,the interpolation problem aims at the exact implementation (or approxima-tion) of a continuous function that is known only at certain points (i.e. whereit can be measured). It is a natural requirement of an interpolation methodthat it should be stable independently from the measurement points, e.g. itshould converge to the approximated function regardless of the measurementpoints if their number converges to infinity. The classical mathematical in-terpolation methods generally do not fulfil this condition. These approachesrequire that the approximated function should be measured at some specialexact points. In practice, it can hardly be solved because of the uncertaintyof the measuring process. If we consider the input–output function of the KHinterpolation method as a function approximator it fulfils the stability condi-

1 Research supported by the National Scientific Research Foundation (OTKA)Grants No. T019671, T030655, by the Hungarian Ministry of Culture and Edu-cation (MKM) Grant No. FKFP 042/1997.2 Corresponding author, email: tikk@david.ttt.bme.hu3 Partially funded by the Australian Research Council large grants scheme

2

tion, that is it converges to the approximated function independently from thepoints where the latter have been measured. Further, a class of stable interpo-lating functions can be derived from the original KH function. We also presentthe so-called (Balazs–)Shepard interpolation which has a strong relation withthe KH interpolation for every fixed α level. Some corresponding results forShepard interpolation can be exploited to characterize the approximation byKH interpolation.

This paper is organized as follows. Section 2 gives a short overview of theKH rule interpolation method. Section 3 recalls some problems from the fieldof mathematical (numerical) analysis relating to the selection of the measure-ment points. Section 4 reviews the relating results about Shepard interpolationin approximation theory. Section 5 introduces the main results concerning thestability property of the KH interpolation. At the end of this Section, a coun-terexample is introduced to illustrate that the second statement of the maintheorem is sensitive to the value of its parameters. Finally, section 6 presentsour conclusions.

2 Rule interpolation

Rule bases containing gaps require completely new techniques of reasoningand control. This family of methods works well only if the system has some“nice” properties: it is not allowed to behave too unexpectedly in the areaswhere the model does not cover. Luckily, in practice such a nice behaviourmight be expected in most cases. The term for the class of systems where thefollowing algorithms are applicable is interpolative system. (We remark herethat interpolativity is connected with the Shannon sampling theorem).

Let us introduce the extended concept of rule interpolation. The starting ideasare the Extension and Resolution Principles. The former states that the solu-tion of a problem for fuzzy sets can be obtained by solving it first for arbitraryα-cuts (N.B. these are crisp sets) and then extending the solution to the fuzzycase. The latter describes the decomposition of fuzzy sets to α-cuts:

A =⋃

α∈[0,1]

αAα, (1)

(here the union means maximum).

Every fuzzy set can be approximated by the family of approximations of itscuts. Although theoretically all infinite cuts should be treated separately, inmost practical cases, if the membership functions are piecewise linear, it isoften sufficient to calculate for only a few important or typical cuts [18,25–27], e.g. α = 0 and α = 1.

3

Some conditions must be fulfilled for the applicability of the interpolationmethod: the fuzzy sets must be normal and convex, (briefly: CNF sets) andthe state variables, including Xi and Y as well, must be bounded and gradualwhich guarantees that a full ordering on each of them exists. This impliesthe existence of a partial ordering on the whole input universe, so a partialordering can be introduced among the elements of X (i.e. among CNF sets)with the help of their α-cuts. If

∀α ∈ [0, 1] : infAα ≤ infBα and supAα ≤ supBα

then A and B are comparable, i.e. A ≺ B.

Among comparable fuzzy sets the new concept of distance can be defined. Forall α ∈ [0, 1], in practice, only for each significant α (e.g. α = 0 and α = 1),the two extremes of the α-cuts are considered and their pairwise distances willbe named the “lower” and the “upper fuzzy distance” of the two cuts. Thedefinition of these distances results in two families of distances as follows:

µdL(A,B)(zi) =∑

α∈[0,1]

α/| infAα − infBα|;

µdU (A,B)(zi) =∑

α∈[0,1]

α/| supAα − supBα|

where zi ∈ Z is the variable representing the possible values of distances in Xi

and∑

α/xi denotes µ(xi) = α. (For more details on these distances see [16]).Using the concept of fuzzy distance, the closeness of two comparable fuzzysets can be determined even if their supports are disjoint.

With the help of fuzzy distance, the classical methods of function approxima-tion can be applied to rule bases, even if they are sparse. Using (1), a rulebase can be represented by a family of hyperintervals in X × Y , that is, forevery α, Aiα, Biα form a hyperinterval for CNF sets. By the Extension Prin-ciple, instead of approximating the original fuzzy mapping, only its importantcuts (breakpoint levels or sets) will be approximated. Hence the problem isreduced to a family of non-fuzzy approximation problems which can be solvedwith any of the classical function approximation methods like interpolation orextrapolation, etc.

The simplest of these methods is the linear interpolation of two rules forthe area between their antecedents. This can be applied if the observation islocated so that

Ai1 ≺ A∗ ≺ Ai2 and Bi1 ≺ Bi2.

Using the concept of fuzzy distance, the following fundamental equation oflinear interpolation can be written, in accordance with the gradual semantic

4

interpretation of fuzzy rules by [13]:

d(A∗, Ai1) : d(A∗, Ai2) = d(B∗, Bi1) : d(B∗, Bi2). (2)

After decomposing (2) to every α ∈ [0, 1] or in breakpoint sets it can be solvedfor B∗

α. The following formulae are the solution for linear KH controllers:

minB∗α=

infBi1αdL(A∗

α, Ai1α)+

infBi2αdL(A∗

α, Ai2α)1

dL(A∗α, Ai1α)

+1

dL(A∗α, Ai2α)

;

(3)

maxB∗α=

supBi1αdU(A∗

α, Ai1α)+

supBi2αdU(A∗

α, Ai2α)1

dU(A∗α, Ai1α)

+1

dU(A∗α, Ai2α)

,

For the two families of solutions to determine a fuzzy set B∗, they should sat-isfy minB∗

α ≤ maxB∗α for every α, cf. [16]. In certain cases this condition is

not satisfied, and hence we obtain a conclusion not directly interpretable as afuzzy set, though there exist methods which provide a fuzzy set as a conclusionregardless of the shape of the membership functions involved [2,31].

The principle of interpolating two rules can be extended in many differentways. The most obvious extension of the linear interpolation is the interpo-lation of more rules flanking the observation (with respect to the orderingon the input space) from both sides. Intuitively, the further a rule from theobservation is located, the less weight the respective consequent plays in theconstruction of the conclusion. The general KH interpolation (also termed asextended) for this type of interpolation are obtained from the solution of (3)repeatedly for the points taken into consideration and then normalizing thesolution as:

min B∗α =

m∑

i=1

infBiαdL(A∗

α, Aiα)m

∑

i=1

1dL(A∗

α, Aiα)

; max B∗α =

m∑

i=1

supBiαdU(A∗

α, Aiα)m

∑

i=1

1dU(A∗

α, Aiα)

, (4)

where m is the number of rules taken into account. More details on this methodcan be found in [15].

In the main part of this work (Section 5) we will use a slightly different versionof the general KH interpolation. So, we change (4) by taking the Nth power

5

dN of the distance function, where N denotes the dimension of the domain X:

min B∗α =

m∑

i=1

infBiαdn

L(A∗α, Aiα)

m∑

i=1

1dn

L(A∗α, Aiα)

; max B∗α =

m∑

i=1

supBiαdn

U(A∗α, Aiα)

m∑

i=1

1dn

U(A∗α, Aiα)

, (5)

We will refer to formula (5) as stabilized (general) KH interpolation. Its sta-bility will be proved in Section 5.

3 Problems of measurement points selection

Interpolation is a method where a continuous function is approximated bysome analytical (e.g. algebraic) function, based on a finite number of knownpoints of the original function. The essential idea is to construct a polynomialwhich coincides at certain given points (so-called measurement points) withthe function to be approximated, f(x).

The selection of the measurement points plays an important role in the theoryof interpolation. This section briefly reviews the related results (cf. [21]).

The first result in this direction was given by Gauss. The quadrature formula∫ b

af(x) dx =

n∑

k=1

Akf(xk) (6)

is exact if the degree of polynomial f(x) is at most n − 1 with arbitraryselection of the measurement points xi, because in this case, f(x) is equal toits own Lagrange interpolation polynomial. (Here we approximate the valueof the “determined integral”.) Gauss has shown that if measurement pointsare selected specially, then (6) is exact for any polynomial of degree at most2n− 1, which holds for the following more general case, as well.

Let p(x) be a weighting function on [a, b]. Let us construct the Lagrangepolynomial L(x) =

∑nk=1f(xk)lk(x) for the continuous f(x) on [a, b] where

x1, . . . , xn are the measurement points and lk(x), 1 ≤ k ≤ n are the Lagrangecoefficients. Multiplying L(x) by p(x) and then integrating we get

∫ b

ap(x)L(x) dx =

n∑

k=1

(

∫ b

ap(x)lk(x) dx

)

f(xk) =n

∑

k=1

Akf(xk)

and if L(x) is considered as the approximation of f(x) then we obtain the∫ b

ap(x)f(x) dx =

n∑

k=1

Akf(xk) (7)

6

approximating quadrature formula, this is exact for any polynomial f(x) ofdegree at most 2n − 1 with the special selection of measurement points de-scribed in the next paragraph. (When p(x) = 1, the formulae (6) and (7) areequivalent.)

According to Gauss, to achieve the exact interpolation, the measurementpoints have to satisfy the following condition. If the measurement points arechosen as the zeros of a polynomial of degree n ωn(x) which is orthogonal withrespect to p(x) for every polynomial of degree at most n−1, then the formula(7) is exact.

Example 1 If a = −1, b = 1 and p(x) = 1, then the solution of the Gauss’problem is formed by the zeros of the Legendre polynomial Xn(x).

Let us consider a more general interpolational problem, the so-called Hermiteinterpolation. The aim is to construct the polynomial H(x) of the minimaldegree which satisfies the conditions

H(r)(xi) = y(r)i (i = 1, 2, . . . , n; r = 0, 1, . . . , ai − 1) (8)

where y(r)1 and xi are given and r is the highest rank of derivatives taken into

account. We consider only the case when a1 = a2 = · · · = an = 2. The solutionof (8) is given by the formula

H(x) =∑

k=1

[

1− ω′′(xk)ω′(xk)

(x− xk)]

l2k(x) +n

∑

k=1

y′k(x− xk)l2k(x). (9)

using the ordinary notation. We note that the degree of H(x) is 2n− 1.

Usually the continuity of the function and its derivatives does not give suffi-cient conditions for the convergence of the interpolation method. Numerouscounterexamples are known e.g. from Bernstein or Marcinkiewicz, cf. [21]. Nev-ertheless, Fejer has shown if xk are the zeros of the Tchebysheff polynomial,the convergence can be proved [21].

As we have shown in this section the choice of the measurement points playsa very important role in many areas concerning numerical interpolation. Fromthe practical point of view, if we would approximate a function by interpolationmeasuring its value at distinct points it is almost impossible to choose themexactly as, e.g., the zeros of the Tchebysheff polynomial.

The interpolation method introduced by Koczy and Hirota [15] for fuzzy con-trol satisfies the property of convergence independently from the values of xk

as will be shown in Section 5.

7

4 Shepard interpolation and KH interpolation

The Shepard interpolation method was first introduced in [24] for arbitrarilyplaced bivariate data as

S0(f, x, y) =

fk, if (x = y) = (xk, yk) for some k( n

∑

k=0f(xk, yk)/dλ

i

) / ( n∑

k=01/dλ

i

)

, otherwise

(10)

where measurement points xk, yk (k ∈ [0, n]) are irregularly spaced on thedomain of f ∈ R2 → R, λ > 0, and di = [(x − xk)2 + (y − yk)2]1/2. Thisfunction can be used typically when a surface model is required to interpolatescattered spatial measurements. This problem is encountered in such areas aspattern recognition, geology, cartography, earth sciences, fluid dynamics andmany others.

Because this method has some unsatisfactory properties (mostly for smallλ ∈ (0, 1]) in [4] an extension of Shepard’s method was introduced whicheliminated these difficulties.

From a practical point of view, formula (10) also has the defect that if an-other interpolation point is added, then all weights must be reformulated. Thisdeficiency can be eliminated by the recursive “expandable” Shepard methodproposed also in [4]. This modification is based on the similar idea of how onecan derive the Newton form of linear interpolation from the Lagrange typeinterpolation.

Beside the application oriented investigation of Shepard’s method such as [4](see also [3,14,20,23,22,5]), an increasing interest has arisen from mathematicalresearchers to examine the approximation property of formula (10).

Formula (10) was generalized as follows

Sn,λ(f, x) =∑n

k=0 f(xk)(x− xk)−λ

∑nk=0(x− xk)−λ , λ > 0, n = 1, 2 . . . (11)

for an arbitrary f ∈ C[0, 1], where xk (k = 0, . . . , n), in general, denotes thenodes of the equidistant distribution of the domain [0, 1]. We note that fixingthe domain to the interval [0, 1] does not mean any restriction, because byproper linear transformation any finite interval can be mapped into anotherone.

The use of (11) type of rational functions as approximating means was first

8

discovered by J. Balazs [1]. (After his name this operator is often termed in theliterature of approximation theory as Balazs–Shepard operator.) The proper-ties of the (11) operator was widely investigated by J. Szabados [29,30], G.Somorjai [28], by several Italian mathematicians B. Della Vecchia, G. Mas-troianni, G. Criscuolo etc. see e.g. [9,12,8,10,11].

The first result on the approximation property of the Shepard operator waspublished by Szabados [29], where the author showed that for λ = 4 the (11)realized the so-called Jackson order of approximation, i.e.,

‖f − Sn(f)‖ = O(

ω(f, n−1))

where ω(f, n−1) was the modulus of smoothness (or continuity) of function f ,that is,

ω(f, n−1) = maxx,y∈Ω

|x−y|≤n−1

|f(x)− f(y)|. (12)

The saturation problem, i.e., the determination of the optimal order of conver-gence was also investigated by Somorjai [28]. He gave a complete descriptionof the order and class of saturation when the weights (x−xk)−λ were replaced

by∣

∣

∣x− kn

∣

∣

∣

−λ, k = 0, . . . , n for λ > 2. In [28] it was shown that for λ > 2

‖f − Sn(f, x)‖= o(1/n) iff f = const‖f − Sn(f, x)‖= O(1/n) iff f ∈ Lip1.

(The function f : [a, b] → R is called Lipschitz continuous with Lipschitzcoefficient α (notation: f ∈ Lipα) if

|f(x)− f(y)| ≤ α|x− y| for all x, y ∈ [a, b]). (13)

In [12] the authors investigated the saturation problem of (11) for λ = 2. Theyshowed that if

‖Sn,2(f, x)− f(x)‖ = o(n−1)is satisfied for a continuous function f for all x ∈ [0, 1], then f is constant.This statement shows that the λ = 2 case represents additional difficultiescompared to λ > 2 because we do not have strong localization like for λ > 2.This peculiarity can be best illustrated by noting that the linear functionsare not in the saturation class of the operator Sn,2. In [12] it is also shownthat the saturation class of Sn,2 is the set of constant functions. Furthermore,nonconstant linear functions do not belong to the saturation class, and thesaturation class is contained in

⋃

α<1 Lip α. Here it is reasonable to remarkthat classical approximation processes (e.g. Bernstein polynomials) are allsaturated with order n−1, and with trivial classes of functions, either the setof linear or the set of constant functions.

9

In [30] direct and converse estimates were given for the case where the weightsare replaced by their absolute values |x− k/n|−λ (with equidistant nodes) forall λ ≥ 1. Namely, it was shown that

‖Sn,λ(f, x)− f(x)‖ =

O(

ω(

f, 1n

))

, if λ > 2

O(n−1−λ)∫ 11/n t−λω(f, y) dt, if 1 < λ ≤ 2

O(log−1 n)∫ 11/n t−λω(f, y) dt, if λ = 1

(14)

for any f ∈ C[0, 1]. These results do not give the optimal order of approxima-tion for λ < 2, thus in this paper the saturation problem remains unsettled.

In [10] the authors obtained pointwise simultaneous approximation estimatesfor rational operators, being a generalization of the Shepard operator (11),which were not possible by algebraic polynomials. In this result, opposed tothe previous ones, the nodes are placed arbitrarily.

In [8] estimation of the convergence of (11) was given with respect to differentmatrices of knots. For some special knot point matrices, such as zeros of or-thogonal polynomials, the F-stable property (stability in Fejer sense, see also,e.g., [23]) was also proved.

In [11] the characterization of Shepard operator on infinite intervals was given.

Summarizing the characterization of the interpolatory Shepard operator wecan state that one of the main goals, the saturation problem, has been solvedonly for the case λ > 2 and for equispaced nodes. Other node point systemswere also investigated, but many open problems remained unsettled.

It was shown in this section the Shepard operator was extensively investigatedby researchers in the field of approximation theory and by authors interestedin some applications, such as fitting data, curves, and surface, fluid dynamicsproblems. In the next Section, we will return to Shepard interpolation showingits relationship with KH interpolation.

5 Main results

Definition 2 Let RN ⊃ Ω = [a1, b1]× · · · × [aN , bN ], further let Γn∞n=1 be asequence of finite subsets of Ω with #Γn = n. If

∀ε > 0 ∃n0 ∀ω ∈ Ω ∀n ≥ n0 :

∣

∣

∣

∣

∣

#(Γn ∩ ω)#Γn

− |ω||Ω|

∣

∣

∣

∣

∣

< ε (15)

then the set Γn are uniformly distributed on the domain Ω. Here #(Γn ∩ ω)denotes the cardinality of the finite set (Γn∩ω) and |ω| is the Lebesque measure

10

of ω.

Theorem 3 Consider the Lp norm ‖ · ‖p with p ∈ [1,∞], the domain RN ⊃Ω = [a1, b1]× · · · × [aN , bN ] and a continuous function f : Ω → R, then

∀x ∈ Ω : limn→∞

Kn(f, x) := limn→∞

n∑

k=1

f(

x(n)k

)

1∥

∥

∥x− x(n)k

∥

∥

∥

N

pn

∑

j=1

1∥

∥

∥x− x(n)j

∥

∥

∥

N

p

= f(x) (16)

where measurement points x(n)k are uniformly distributed on Ω in the sense of

(15).

PROOF. Let us estimate the following difference

Kn(f, x)− f(x) =n

∑

k=1

[f(xk)− f(x)]1

‖x−xk‖Np

∑nj=1

1‖x−xj‖N

p

.

For brevity, we have written xk instead of x(n)k . For arbitrary ε > 0, because

of the continuity of f we can choose a sufficiently small real number δ, suchthat for ‖x− xk‖ < δ

|f(xk)− f(x)| < ε (17)

holds. δ can be chosen on Ω since f is uniformly continuous. Then for fixedx the numbers 1, 2, . . . , n are divided into two groups: group I contains thosenumbers k for which ‖x− xk‖p ≤ δ, group II contains the rest.

∑

k∈ I

[f(xk)− f(x)]1

‖x−xk‖Np

∑nj=1

1‖x−xj‖N

p

+∑

k∈ II

[f(xk)− f(x)]1

‖x−xk‖Np

∑nj=1

1‖x−xj‖N

p

=:∑

I+

∑

II. (18)

Because of (17)

∣

∣

∣

∑

I

∣

∣

∣ ≤ ε ·∑

k∈ I

1‖x−xk‖N

p∑n

j=11

‖x−xj‖Np

≤ ε ·n

∑

k=1

1‖x−xk‖N

p∑n

j=11

‖x−xj‖Np

= ε. (19)

Now let us turn to estimating∑

II. The continuous function f is bounded onthe compact domain Ω, thus

|f(xk)− f(x)| ≤ 2 maxΩ|f | = 2‖f‖C =: M

11

and so

∣

∣

∣

∑

II

∣

∣

∣≤∑

k∈ II

|f(xk)− f(x)|1

‖x−xk‖Np

∑nj=1

1‖x−xj‖N

p

≤M ·∑

k∈ II

1‖x−xk‖N

p∑n

j=11

‖x−xj‖Np

≤ MV

V + W, (20)

where V =∑

j ∈ II1

‖x−xj‖Np

, and W =∑

j ∈ I1

‖x−xj‖Np

. In the following, withoutloss of generality, we will apply the maximum norm ‖ · ‖ = ‖ · ‖C to simplifyfurther computation. To estimate the sums V and W we proceed as follows.We define a hypercube covering the domain of f . Then we estimate the sumsby counting the number of grid points on the surface of hypercubes (moreprecisely on the shell of hypercubes) depending on the length of edges. Finally,summing these values on all the shells that forms the covering hypercube weobtain the required estimations.

Assuming uniform distribution (15) of knot points for sufficiently great n, thisresults in an equidistant N -dimensional grid on Ω. Choose a sufficiently greathypercube H(R) with edge R to cover the entire domain of function f . Letη be the grid spacing in the hypercube H(R) (distance between neighbouringnodes)

η =R

N√

n. (21)

Further, let us compute the asymptotic number of the grid points on thesurface of the hypercube H(r)

2N · (r)N−1

ηN−1 = 2N ·(

rη

)N−1

= 2N ·(

r N√

nR

)N−1

= 2N · n1− 1N ·

( rR

)N−1.

First, let us estimate V by summing the reciprocal of the distances on thesurface of hypercubes H(ri) where ri+1 − ri = η, and δ ≤ ri ≤ R:

V =iVmax∑

i=02n · n1− 1

N ·(ri

R

)N−1· 1rNi

, (22)

where

iVmax =R− δ

η=

R− δR

N√

n

from ri = δ + iη = δ + i RN√n ≤ R. Hence

12

V = 2N · n1− 1N

1RN−1

iVmax∑

i=0

1ri

= 2N · n1− 1N

1RN−1

iVmax∑

i=0

1(

RN√

n

) (

δN√

nR

+ i)

= 2N · n 1RN

R−δη

∑

i=0

1(

δ1η

+ i) (23)

We can estimate W analogously, only the boundaries of the sum have to bemodified. More precisely, there are two cases. If x identical to a grid point, Wtakes the value ∞, thus (16) is proved by (18), (19) and (20). Otherwise thereis a closest grid point xj to x such that

δ0 := ‖x− xj‖ = min1≤k≤n

‖x− xk‖ < η.

Thus

W =∑

k,δ0≤‖x−xk‖≤δ

1‖x− xk‖N = 2N · n 1

RN

iWmax∑

i=0

1(

δ01η + i

) , (24)

and here

iWmax =δ − δ0

η=

(

δ − δ0

R

)

N√

n

by ri = δ0 + iη = δ0 + i RN√n ≤ δ.

By (23) and (24) inequality (20) leads to

∣

∣

∣

∑

II

∣

∣

∣≤M ·2N

nRN

iVmax∑

i=0

1αV

n + i

2Nn

RN

iVmax∑

i=0

1αV

n + i+ 2N

nRN

iWmax∑

i=0

1αW

n + i

= M ·

iVmax∑

i=0

1αV

n + iiVmax∑

i=0

1αV

n + i+

iWmax∑

i=0

1αW

n + i

(25)

where αVn and αW

n are δN√nR and δ0

N√nR , respectively.

We use integral approximation for estimating the sums in (25).

13

iVmax∑

i=0

1αV

n + i≥

∫ αVn +iVmax+1

αVn +1

1x + 1

dx = log(αVn + iVmax + 2)− log(αV

n + 2)

= log

δη + R−δ

η + 2δη + 2

n→∞= log(R

δ

)

, (26)

analogously,

iWmax∑

i=0

1αW

n + i≥ log

δ0η + δ−δ0

η + 2δ0η + 2

n→∞= log(

δδ0

)

. (27)

By (25), (26) and (27)

∣

∣

∣

∑

II

∣

∣

∣ ≤ M · ln(R/δ)ln(R/δ) + ln(δ/δ0)

= const · ln R− ln δln R− ln δ0

< ε (28)

because for sufficiently great n when 0 < δ0 δ < 1 R holds, the rightside of inequality (28) can be less than arbitrary ε > 0. Observe that δ0 isindependent from δ due to its definition. This completes the proof.

We can generalize the above theorem as follows:

Theorem 4 Consider the same conditions as in Theorem 3. If we substituteλ for the exponent N in the expression of the distance, where s ≥ N , theresulting Kλ

n(f, x) converges to the approximated function f , as well:

∀x ∈ Ω : limn→∞

Kλn(f, x) := lim

n→∞

n∑

k=1

f(

x(n)k

)

1∥

∥

∥x− x(n)k

∥

∥

∥

λ

pn

∑

j=1

1∥

∥

∥x− x(n)k

∥

∥

∥

λ

p

= f(x) (29)

where (λ ≥ N), and with uniform distribution of measurement points x(n)k .

PROOF. For proving the convergence of (29) we use similar reasoning asabove. Let t = λ−N > 0. Following the train of thought of the former proofand using its notation we should only modify the estimation of V and W(for clarity, here we denote them by Vλ and Wλ, respectively) as follows. (22)changes as

Vλ =iVmax∑

i=02N · n1− 1

N ·(ri

R

)N−1· 1rλi,

14

and we obtain

Vλ = 2N · nN−1

N−λ−1 · 1Rλ

iVmax∑

i=0

(

δη

+ i)N−λ−1

(30)

in the same way as in (23). Similarly

Wλ = 2N · nN−1

N−λ−1 · 1Rλ

iWmax∑

i=0

(

δ0

η+ i

)N−λ−1

(31)

hence by (30) and (31)

∣

∣

∣

∑

II

∣

∣

∣ ≤ M · Vλ

Vλ + Wλ= M ·

iVmax∑

i=0

(

αVn + i

)N−λ−1

iVmax∑

i=0

(

αVn + i

)N−λ−1+

iWmax∑

i=0

(

αWn + i

)N−λ−1(32)

By integral approximation of the sums in (32)

iVmax∑

i=0

1(αV

n + i)t+1 ≥∫ αV

n +iVmax+1

αVn +1

1(x + 1)t+1 dx

= (t + 1)

1(

δη + 2

)t −1

(

δη + R−δ

η + 2)t

= (t + 1) · 1N√

nt

(

1 + 2N√n

)t−

(

δR + 2

N√n

)t

(

1 + 2N√n

)t·(

δR + 2

N√n

)t

using simplification to obtain the last expression. For n →∞ this results in

(t + 1) · 1N√

nt

(

(Rδ

)t

− 1)

. (33)

In the same way we get

iWmax∑

i=0

1(αW

n + i)t+1 ≥ (t + 1) · 1N√

nt

(δ/R)t − (δ0/R)t

(δδ0/R2)t (34)

Finally, computing the fraction (32) by (33) and (34) leads to

δt0(R

t − δt)δt(Rt − δt

0)<

(

δ0

δ

)t

.

15

This expression can be less than arbitrary ε > 0 using similar arguments asfor (28), which completes the proof.

Remark 5 As we mentioned in Section 2, in both theorems we used the stabi-lized version (5) of the general KH interpolation (4). When N = 1 (16) yields(4) with some apparent substitutions: xk for Ai, f(xk) for inf Bi, sup Bi, x forA∗ and |x− xk| for dL(A∗, Ai), dU(A∗, Ai).

Remark 6 The KH interpolation function is a rational fraction with orderN over N . Hence its computational complexity is not significantly higher thanthose of the classical algorithms.

Remark 7 Theorem 4 is sharp in the sense that for exponents less than Nin the expression of distance, the fraction V

V +W converges to a constant, as itwill be shown in the next example.

The stability result of the stabilized KH interpolation is the generalizationof Shepard-interpolation, because the stability is valid for the fuzzy-to-fuzzyfunction approximated by the stabilized KH interpolation, or in other words,for a family of real-valued functions at each α ∈ (0, 1]. Moreover, the stabil-ity theorems 3 and 4 are established for uniformly distributed node points,which relax the condition of the equispaced node point system for Shepardinterpolation. In our future work, we expect to determine the optimal orderof convergence in the approximation of the fuzzy-to-fuzzy function exploitingthe related results for Shepard interpolation.

It is worth noting that the stability of stabilized KH controllers can be viewedfrom the aspect of universal approximation capability. Without going deeplyinto the details, we can state that theorems 3 and 4 imply that the stabilizedKH controllers are capable of approximating any continuous function on acompact domain with respect to the supremum or Lp (p ∈ [1,∞) norm. Thisstatement contributes to the topic of the universal approximation capabilitiesof fuzzy controllers (see e.g. [7,19,32]). It shows that even relaxing the con-dition of full α-covering type rule base fuzzy systems in general preserve itsapproximation capabilities.

Now, we turn to show that the theorems are sensitive to the value of dis-tance’s quotient. Let us consider the bivariate function f(x, y) = (x2 + y2)c,where c > 0 on the unit square. Let us choose equidistant distribution of themeasurement points x(n)

k (see Fig. 1). We will show that the approximatingfunction Kλ

n in (29) does not converges to the approximated function f whenλ < n. For simplicity, we approximate the function f in the point (0, 0). Hencesubstituting this value in (18)

16

-

6

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq

qqqqqqqqq1

1Fig. 1. The distribution of measurement points.

Kλn(f, (0, 0))− f(0, 0) =

=n−1∑

i=1

n−1∑

j=1

(

( in

)2

+( j

n

)2)c

1(

(

in

)2+

(

jn

)2)λ/2

n−1∑

i,j=1

1(

(

in

)2+

(

jn

)2)λ/2

=

∑n−1i,j=1

(

(

in

)2+

(

jn

)2)c−λ

2

∑n−1i,j=1

1(

(

in

)2+

(

jn

)2)λ/2

=

∑n−1i,j=1

(

(

in

)2+

(

jn

)2)c−λ

2

nλ ∑n−1i,j=1

1(i2+j2)λ/2

(35)

Let us investigate the case when λ = 1 < n(= 2) and c = 1/2. Thus (35)becomes

=

∑n−1i,j=1

(

(

in

)2+

(

jn

)2)0

n∑n−1

i,j=11

(i2+j2)1/2

=(n− 1)2

n∑n−1

i,j=11

(i2+j2)1/2

<1

1n

∑n−1i,j=1

1(i2+j2)1/2

Let us further estimate the denominator

1n

n−1∑

i,j=1

1(i2 + j2)1/2 =

n−1∑

i,j=1

1n2

1(

(

in

)2+

(

jn

)2)1/2

n→∞−→∫ ∫

[0,1]2

1√x2 + y2

dx dy

using the fact that(

1n

)2≈ ∆xi∆yi is the distance between the measurement

17

points. Finally, we get

0 <∫ ∫

[0,1]2

1√x2 + y2

dx dy ≤ 2 · π∫

√2

r=0

r4

1rdr =

π2·√

2 = const

thus the difference in (35) converges to a positive constant.

6 Conclusions

The results presented in Theorems 3 and 4 prove the stability of the input–output function of the interpolative KH controllers. This stability is inheritedfrom the α-cuts to the whole fuzzy control algorithm and so, provides a meansfor constructing tunable non-linear controllers, obviously a reason for the suc-cess of fuzzy controllers. Hopefully, similar results can be established for widerclasses of fuzzy controllers.

The stability of the KH controller can be placed in a wider context as hasbeen shown in sections 3 and 4. As it was pointed out in the paper, the well-known interpolation methods have usually searched for an interpolation func-tion in polynomial (or trigonometrical) form. If one neglects this restrictionand takes simple rational functions into consideration a new class of inter-polation methods can be obtained providing stability independently from thechoice of measurement points. We have introduced a class of function derivedfrom the input–output function of the KH interpolation method that fulfilsthis property. We have pointed out the relevance of this result concerning theuniversal approximation capabilities of fuzzy systems, and the relationshipbetween KH and Shepard interpolation (the former being the generalizationof the latter).

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