a methodology for modeling based on fuzzy clustering

7/31/2019 A Methodology for Modeling Based on Fuzzy Clustering...

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5TH INTERNATIONAL SYMPOSIUM ON ROBOTICS AND AUTOMATION 2006, SAN MIGUEL REGLA HIDALGO, MEXICO, AUGUST 25-28, 2006. 1

A methodology for modeling based on fuzzy

clustering - cubic splines for nonlinear systems

Julio Cesar Ramos Fernandez1, Virgilio Lopez Morales2, Gilles Enea3, Jean Duplaix4.

Abstract This paper proposes a novel methodology based onfuzzy clustering and cubic splines for modeling nonlinear systems.The Gustafson-Kessel (G-K) algorithm is used in order to classifyclusters with linear domains. At least in each three different andordered consecutive clusters (with linear trend), there are twopoints of inflexion, one maximum and one minimum. Then, acubic spline can be synthesized, and the intersection with thenext spline is smoothing via fuzzy sub-models. An advantage ofthis approach is a reduction on rules number, and consequentlyan important reduction of Takagi-Sugeno (T-S) models. We obtainmodels with the rule-base, which have trapezoidal membershipfunctions for input and cubic spline for output. An example is

proposed to illustrate our modeling approach.Keywords models reduction, fuzzy modeling, local models,

cubic spline.

I. INTRODUCTION

In control engineering, modeling and identification are

important steps in the design of control laws, supervision and

fault-detection systems. Traditionally modeling is seen as a

conjunction of a thorough understanding of the systems nature

and behavior, and of a suitable mathematical treatment that

leads to an usable model. This approach is usually termed

white box (physical, mechanistic, first-principle) modeling

[Babuska, 1998].

The identification problem consists of estimating the param-

eters in the model. If representative process data is available,

black-box usually models can be developed easily, without

requiring process specific knowledge. A severe drawback of

this approach is that the structure and parameters of these

models usually do not have any physical significance. There

is a range of modeling techniques that attempt to combine the

advantages of the white-box and black-box, such that the know

parts of the systems are modelled using physical knowledge,

and the unknown or less certain parts are approximated in a

black-box approach, using process data and black-box model-

ing structures with suitable approximation properties. These

methods are often denoted as hybrid, semi-mechanistic or

The work of Julio Cesar Ramos Fernandez is partially supported by thePROMEP grant and ANUIES-SEP-CONACyT/ECOS NORD, M02:M03. Thework of Virgilio Lopez Morales is partially supported by the SEP-SESIC-PROMEP research project grant PROMEP - UAEH-PTC-1004 and ANUIES-SEP-CONACyT/ECOS-NORD, M02:M03.

1 Author is at the Universidad Tecnologica de Tula-Tepeji, Col. El33 El Carmen Tula de Allende Hidalgo, Mexico; 2 Author is at theUniversidad Autonoma del Estado de Hidalgo, Centro de Investigacionen Tecnologas de Informacion y Sistemas, Carr. Pachuca-TulancingoKm. 4.5, C.P. 42090, Pachuca, HGO, Mexico; 3,4 Authors are atthe Laboratoire des Sciences de lInformation et des Systemes UMR-CNRS 6168 Equipe COSI, Universite du Sud-Toulon Var, La [email protected], [email protected],[enea,duplaix]@univ-tln.fr

gray modeling [Lafont and Balmat, 2003], [Babuska, 1998].

The modeling approach proposed in this paper is based on

fuzzy models, which describe relationships between variables

by means of If-Then rules, such as:

Ri : If x is region A Then y = f(x) (1)

where Ri is the number of rules or models. Among the differ-

ent fuzzy models methods, the T-S has attracted most attention

[Tanaka and Wang, 2001]. In fact this model consist of rules

like [1], with fuzzy antecedents (region A) and mathematical

functions in the consequent part (y = f(x)). In this paperwe obtain fuzzy models by using G-K algorithm to detectconsecutive linear regions. A cubic spline is calculated by each

three linear regions. An important issue in fuzzy-rule-based

modeling is how to select a set of important fuzzy rules from

the rule base can result in a compact fuzzy model with better

generalizing ability. In [Yen and Wang, 1999] it is introduced

several orthogonal transformation-based methods that provide

new or alternative tools for parameters and rule selection.

Orthogonal Least Squares (OLS) method is used to remove

redundant or less important clusters during the clustering

process. The drawbacks on these approaches is the initializa-

tion of clustering process with an overestimated number of

clusters. The big question is how many number of clustersin the partition are overestimated?. In [Setnes et al., 1998]

it is shown some results to reduce the number of initials

models by using a measure of similarity between the initial

fuzzy sets. Therefore the fuzzy sets with redundant information

are merged to create a common fuzzy set to replace them

in the rule base. In nonlinear systems identification, both

the amplitude and frequency contents of the input signals

are of major importance [Johansen and Murray-Smith, 2000].

The principal components analysis (PCA) allows to reduce

a complex correlation systems into a smaller number of

dimensions, in [Pessel and Balmat, 2005], a reduction in the

matrix of variables input/output is achieved, an application on

a real system (an experimental greenhouse) with a neuronalmodeling. A drawback is that reduction on input variables or

excitations of the modeling, avoid the possibility to design

control laws. Several works with the B-spline basis functions

of different orders are regarded as a class of a membership

functions. Recently, splines have also been proposed for neu-

ronal network modelling [Brown and Harris, 1987] and design

of fuzzy controllers using periodical non uniform B-spline

In [Shimojima et al., 1994] it is proposed a self-tuning fuzzy

inference neuronal networks of membership functions that

are represented by spline function. A key problem is on the

obtention of the region of the spline in order compute the knots


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position of the spline, since the shape of the spline function

depends of the knots.

Several algorithms for ordinary least squares regression

spline fitting are to exhibit in [Thomas, 2002], an important

issue associated with it is the choice of a theoretical best set

of knots. Typically these best knot sets are defined implicitly

as the optimizers of some objective functions.

The methodology proposed in this paper is on the rule-based

structure proposed by T-S, where it is used linear consequents.

The advantages with our approach is a reduction in the rule-

based, and the consequents are nonlinear (cubic) with constant

parameters. We thought that the cubic consequents may char-

acterizer the transients or nonlinearity of the system better

than the linear consequents (T-S modeling). Our methodology

proposed in this paper for modeling nonlinear systems is based

in fuzzy clustering. Because fuzzy clustering classifies objets

according to similarities among them, and the organizing of

data in groups. The term similarity should be understood as

mathematical similarity, measured in some well-defined sense.

In metric spaces, similarity is often defined by means of a

distance norm [Bezdek et al., 1981].This paper has the following structure, in Section 2, we

describe the algorithm of G-K and the knot selection,

and its ability to detect linear regions, by using its

eigenvalues of the covariance matrix on each cluster

obtained. In Section 3, the interpolation cubic splinealgorithm is explained, and we obtain cubic polynomials, are

obtained for the local interpolation, on the interpretation of

local models. This approach is widely used by different

authors, but they was computed linear consequents,

[Babuska, 1998], [Trabelsi et al., 2004], [Yen et al., 1998],

[Foulloy et al., 2003], [Tanaka and Wang, 2001],

[Sam and Rui, 1993], [Bernal Reza, 2005]. Our approach

proposes cubic consequents with the following:

y(x) = a3 x3 + a2 x2 + a1 x + a0. (2)In Section 4 the fuzzy modeling is obtained. The antecedentsare defined by trapezoidal membership function, and the

consequents by cubic equations. Section 5 discusses the mainresults and gives the concluding remarks.

I I . GUSTAFSON-K ESSEL ALGORITHM AND KNOT

SELECTION

The G-K algorithm is a derived from de basic fuzzy c-

means (FCM), by adaptingdistance norm

, in order to de-tect clusters of different geometrical shapes in one data set

[Babuska, 1998], another algorithm that use the adapting dis-

tance norm is based on fuzzy maximum likelihood estimates

clustering (FMLE).

Each cluster has its own norm-inducing matrix Ai, which

yields the following inner-product norm:

D2ikAi = (zk vi)TAi(zk vi) (3)in the FCM algorithm the matrices Ai are identity matrix

(I), in G-K algorithm the matrices Ai employing an adaptivedistance norm. The cost function of the G-K algorithm is

defined by:

J(Z; U , V, A) =c

i=1

Nk=1

(ik)mD2ikA (4)

where U Mfc, V nc and m > 1.Definition 1. (fuzzy partition space) Let

Z = {z1, z2,...,zN} be a finite set and let 2 c N be an in-teger. The fuzzy partitioning space for data matrix Z is the set

Mfc =

U cn ik [0, 1] , i, k; ci=1

ik = 1, k;

0 1 and the termination tolerance > 0. Initializethe partition matrix randomly, such that U0 Mfc Repeat forl = 1, 2,... Step 1: Compute clusters prototypes, 1 i c:

vli = N

k=1 l1ik

mzk

N

k=1 l1ik

m

1

(12)

Step 2: Compute the cluster covariance matrices, 1 i c:

Fi =N

k=1 (l1ik )

m(zk vli)(zk vli)T

Nk=1 (

l1ik )

m1 (13)

Step 3: Compute the distances, 1 i c, 1 k N:D2ikAi = (zk vli)T

i det(Fi)

1/nF1i

(zk vli) (14)

Step 4: Update the partition matrix, if (DikAi > 0), for(1 i c, 1 k N):

lik =

c

j=1

DikAi/DjkAi

2/(m 1)1 (15)

otherwise

lik = 0 if DikAi > 0, and lik [0, 1] with

ci=1

lik = 1.

UntilUl Ul1 <

III. INTERPOLATION CUBIC SPLINE ALGORITHM, AN D

CUBIC CONSEQUENT

In this paper we have integrated two well known technics:

fuzzy clustering, and cubic spline interpolation. The goal is to


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obtained a structure in the rule base with cubic consequents.

The main steps are: for each rule a submodel is obtained, the

consequents are analytical functions, a reduction in the rule

base compared with the T-S models with linear consequents.

The algorithm that we have used for the cubic spline interpola-

tion is shown in [Madar et al., 2003]. The first version appears

in [Horiuchi et al., 1993]. The cubic splines are defined as

a continuous function and fitted to the available measured

data system input/output (SISO). The input/output vector Y =[y1,...,yN]

T, X = [x1,...,xN]T, which are defined by each

pair of knot: [K11 , K12 ], [K

21 , K

22 ], ..., [K

n11 , K

n2 ].

In order to formulate this algorithm, let us define a cubic

spline for a knot sequence of cubic polynomials: x1 = K11

a methodology for modeling based on fuzzy clustering

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