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An experimental validation of a novel clustering approachto PWARX identication

Zeineb Lassoued, Kamel Abderrahim

Numerical Control of Industrial Processes, National School of Engineers of Gabes, University of Gabes, St Omar Ibn-Khattab, 6029 Gabes, Tunisia

a r t i c l e i n f o

Article history:

Received 14 March 2013

Received in revised form2 September 2013

Accepted 16 October 2013Available online 2 December 2013

Keywords:

Hybrid systems

PWARX models

Clustering

Identication

Chiu's clustering technique

Experimental validation

a b s t r a c t

In this paper, the problem of clustering based procedure for the identication of PieceWise Auto-

Regressive eXogenous (PWARX) models is addressed. This problem involves both the estimation of the

parameters of the afne sub-models and the hyperplanes dening the partitions of the state-inputregression. In fact, we propose the use of the Chiu's clustering algorithm in order to overcome the main

drawbacks of the existing methods which are the poor initialization and the presence of outliers.

In addition, our approach is able to generate automatically the number of sub-models. Simulation results

are presented to illustrate the performance of the proposed method. An application of the developed

approach to an olive oil esterication reactor is also suggested in order to validate simulation results.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Hybrid systems are dynamical systems that involve the interac-

tion of continuous and discrete dynamics. The continuous behavior is

the fact of the natural evolution of the physical process whereas the

discrete behavior can be due to the presence of switches, operating

phases, transitions, computer program codes, etc.

Hybrid systems allow to describe numerous physical processes

such as chemical process control where the continuous evolution

can be changed by the actuators states (valves and pumps),

embedded systems where the discrete part represented by the

program interacts with the physical part represented by the

system, industrial production lines, telecommunication networks,

air trafc management systems, etc. Moreover, the notion of

hybrid system can be used to represent complex nonlinear con-

tinuous systems. In fact, we can exploit the

divide and conquer

strategy which consists in decomposing the domain range of the

nonlinear system into a set of operating regions. For each opera-

tion region, a linear or afne model is associated. So, the con-

sidered complex nonlinear system becomes by modeling as an

hybrid system switching between linear or afne sub-models.

The analysis and control of hybrid system require that the

system's dynamic behavior must be dened by a mathematical

model. This model can be developed using the physical laws that

govern the functioning of the system. But, this approach fails when

the physical laws describing the behavior of the system are notsufciently well known or the system involves not easily measur-

able parameters. In addition, it can lead to very complicated

models that cause problems of exploitation and implementation.

However, for engineering, a mathematical model must provide a

compromise between accuracy and simplicity of implementation.

A solution to this problem consists in using the identication

approach which allows building a mathematical model from

inputoutput data. For this reason, only the problem of identica-

tion of hybrid systems represented by Piecewise AutoRegressive

systems with eXogenous input (PWARX) is considered in this

paper. This problem has received a great attention in the last years,

since the PWARX models can approximate any nonlinear system

with arbitrary accuracy (Lin and Unbehauen, 1992). Moreover, the

properties of equivalence (Heemels et al., 2001) between PWARXmodels and other representations of hybrid systems allow to

transfer the results of PWA models to other classes of hybrid

systems, such as Jump Linear models (JL models) (Feng et al.,

1992), Markov Jump Linear models (MJL models) (Doucet et al.,

2001), Mixed Logic Dynamical models (MLD models) (Bemporad

and Morari, 1999; Bemporad et al., 2000), Max-Min-Plus-Scaling

systems (MMPS models) (De Schutter and Van den Boom, 2000),

Linear Complementarity models (LC models) (van der Schaft and

Schumacher, 1998), Extended Linear Complementarity models

(ELC models) (DeSchutter, 1999).

The PWARX systems are obtained by decomposing the state-

input domain into a nite number of nonoverlapping convex

Contents lists available at ScienceDirect

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Engineering Applications of Articial Intelligence

0952-1976/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.engappai.2013.10.007

E-mail addresses:[email protected] (Z. Lassoued),

[email protected] (K. Abderrahim).

Engineering Applications of Articial Intelligence 28 (2014) 201209
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polyhedral regions, and by associating a simple linear or afne model

to each region. Consequently, the PWARX identication problem

involves both the estimation of the parameters of the afne sub-

models and the hyperplanes dening the partitions of the state-input

regression. This is one of the most difcult problems that represents

an area of research where considerable work has been done in

the last decade (Bako et al., 2012; Ferrari-Trecate et al., 2003). The

proposed methods in the literature for the identication of PWARX

systems can be classied in numerous categories of solutions such asthe algebraic solution (Tian et al., 2009), the clustering-based

solution (Ferrari-Trecate et al., 2003), the Bayesian solution(Juloski

et al., 2005), the bounded-error solution (Bemporad et al., 2005), the

sparse optimization solution (Bako, 2011;Bako and Lecoeuche, 2013)

etc. Only the clustering-based procedure is considered in this paper.

This approach is based on three main steps: data classication,

parameter estimation and region reconstruction. It is easy to remark

that the classication of data represents the corner stone of this

approach toward the objective of PWARX system identication

because a successful identication of models' parameters and hyper-

plans depends on the correct data classication. The early approaches

use classical clustering algorithms for the data classication such as

k-means algorithms (Ferrari-Trecate et al., 2003; Ferrari-Trecate and

Muselli, 2003; Nakada et al., 2005; Boukharouba, 2011). These

algorithms may converge to local minima in the case of poor

initializations because they are based on the minimization of non-

linear criterion. Furthermore, their performances degrade in the case

of the presence of outliers in the data to be classied. In addition,

most of them assume that the number of sub-models is a

priori known.

In this paper, we improve the performance of the existing

methods based on clustering procedure thanks to the use of Chiu's

algorithm for data classication (Chiu, 1994). This algorithm allows

us to reduce the effect of outliers. Moreover, it does not need any

initialization. In addition, it is able to automatically generate the

number of sub-models. The performance of this method is

illustrated by a simulation example. An experimental validation

with an olive oil esterication reactor is also considered in order to

show the efciency of the developed approach.This paper is organized as follows.Section 2presents the model

and its assumptions.Section 3recalls the clustering-based procedure

for the identication of PWARX systems. Moreover, it presents the

main drawbacks of the existing methods for the identication of

PWARX based on clustering procedure. In Section 4, we propose a

solution to overcome the main problems of the existing methods.

This solution consists in using the Chiu's algorithm for data classi-

cation. The performance of the proposed approach is evaluated and

compared through simulation results in Section 5. In Section 6, an

application of the developed approach to olive oil esterication

reactor is presented.

2. Model and assumption

In this paper, we address the problem of identifying PieceWise

Auto-Regressive eXogenous (PWARX) systems described by

yk fk ek; 1

whereykARis the system output, e(k) is the noise, k is the now

time index, k is the vector of regressors which belongs to a

bounded polyhedronHin Rd:

k yk1;;yk nauk1;; uk nbT; 2

whereukARnu is the system inputs (nuis the length ofu),naand

nb are the system orders and d na nunb 1. f is a piecewise

afne function dened by

f

T1 if AH1

Ts if AHs

8>: 3

where T 1T, s is the number of sub-models, fHigsi 1 are

polyhedral partitions of the bounded domainHandiARd 1 is the

parameter vector.

The following assumptions are assumed to be veried:

A1 The number of sub-modelss is unknown.

A2 The orders na and nb are known.

A3 The noise e(k) is Gaussian sequence independent and identi-

cally distributed with zero mean and nite variance s2.

A4 The regions fHigsi 1 are the polyhedral partitions of a bounded

domain H Rd such as

s

i 1

Hi H

Hi \Hj ; 8 iaj

8>: 4

Problem statement: Identify the number of sub-models s,the partitions fHig

si 1 and the parameter vectors fig

si 1 of the

PWARX model using a data set fyk;kgNk 1 where N is the

samples' number.

3. Identication of PWARX models based on clustering

approach

In this section, we recall the main steps of the clustering-based

approach for the identication of PWARX models. These steps can

be summarized as follows: constructing small data set from the

initial data set, estimating a parameter vector for each small data

set, classifying the parameter vectors in s clusters and identifying

the value ofs at the same time, classifying the initial data set andestimating the s sub-models with their partitions.

3.1. Data classication

For every pair of datafk;ykgNk 1 , we construct a local set Ckcollectingfk;ykg and its n 1 nearest neighbors satisfying:

8

;y

ACk; Jk

J2r Jk J2; 8 ; ^y=2 Ck: 5

The parametern is chosen randomly asn4d 1. This parameter

decisively inuences on the performance of the algorithm.

The optimal value of n is always a compromise between two

phenomena: accuracy and complexity. The bigger this parameter

is, the better the estimation of the parameters is.

We also dene k as follows: k t1k ;; tnk

. It contains, in

ascending order, the indexes of the elements belonging to Ck.

Among the obtained local sets Ck, some may contain only data

from the same model. They are called pure local sets. Others can

collect data from multiple sub-models that are called mixed sets.

For each local set Ck, we identify an afne model using least

square method to determine the local parameters k:

k Tkk

1Tk Yk; 6

where

k t1k t

nk

T;

and

Yk yt

1

k

yt

n

k

T

:

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The obtained parameter vectors fkgNk 1 are then classied into

s disjoint clusters using classical clustering algorithms derived

from the k-means approach.

These algorithms are interesting from a theoretical point of

view because they show the possibility of clustering the obtained

parameter vectors. Moreover, they are characterized by the sim-

plicity of implementation. However, these algorithms poorly per-

form in several cases because they do not remove effectively the

outliers. In addition, they can be trapped in local minima in thecase of poor initializations.

3.2. Parameters estimation

As the obtained data are classied, it is possible to determine

the s ARX sub-models dened by the parameter vectors

i; i 1;; susing again the least square method. More precisely,

the ith sub-model is obtained from the data of cluster Di. Then,

the points in each nal cluster can be used for estimating the

parameter vector of each sub-model according to the rule:

k;ykADi3kADi: 7

3.3. Regions estimation

The nal step consists in determining the regions Hi . The

methods of statistical learning such as the Support Vector

Machines (SVM) offer an interesting solution to accomplish this

task (Wang, 2005; Duda et al., 2001). The SVM are a popular

learning method for classication, regression and other learning

tasks. This study is still an ongoing research issue ( Hsu and Lin,

2002;Weston and Watkins, 1999).

In our case, it is a matter of nding for every iaj the hyper-

plane that separates points existing in Hiand inHj. Given two sets

Hi and Hj,iaj, the linear separation problem is to ndwARd and

bAR such that:

wTk b40 8kAHi;

wTk bo0 8kAHj: 8

This problem can be easily rewritten as a feasibility problem

with linear inequality constraints. The estimated hyperplane

separatingHi from Hj is denoted with Mi;j mi;j where Mi;j and

mi;jare matrices of suitable dimensions. Moreover, we assume that

the points inHi belong to the half-space Mi;jrmi;j.

The regions fHigsi 1 are obtained by solving these linear

inequalities (Ferrari-Trecate et al., 2003):

Mi;1M

i;sMrm

i;1m

i;sm; 9

whereMxrm are the linear inequalities describing the bounded

domain H.

4. Chiu's clustering technique

The classication is an important problem to achieve the

objective of PWARX model identication because successful iden-

tication of both sub-models and partitions depends on the

performance of the used clustering approach. In fact, this problem

presents an area of research in which few results have been

devoted in the past (Bemporad et al., 2000; Ferrari-Trecate and

Muselli, 2003; Ferrari-Trecate et al., 2003). Most of the existing

methods for the identication of PWARX models are based on

classical clustering algorithms such as k-means methods (Ferrari-

Trecate et al., 2003). These methods are based on the minimization

of nonlinear criteria. Consequently, they may converge to local

minima in the case of poor initialization. Moreover, these methods

are not able to effectively remove the outliers which degrade their

performance. Also, most of these algorithms assume that the

number of sub-models is a priori known. To overcome these

problems,we propose, in this section, an alternative solution to

these methods based on the Chui's clustering algorithm.

4.1. Principle

Chiu's classication method consists in computing a potential value

for each point of the data set based on its distances to the other datapoints and consider each data point as a potential cluster center. The

point having the highest potential value is chosen as the rst cluster

center. The key idea in this method is that once the rst cluster center

is chosen, the potential of all the other points is reduced according to

their distance from the cluster center. All the points which are close to

therst cluster center will have greatly reduced potentials. The next

cluster center take then the highest remaining potential value. The

procedure for determining a new center and updating other potentials

is executed until a predened condition is reached. This condition

depends on the minimum value of the potentials or the required

number of clusters which are reached. This clustering method is based

on three steps: potential computing, outlier processing and clusters

number and centers computing.

Potential computing: We consider the local parameters obtained

by applying the least square method to the grouping obtained by

associating to each its n 1 nearest neighbors. These Nlocal

parameters (i; i 1;; N) picked out by applying (6) are the

objective of our proposed classication technique. Thus, we

compute a potential value for each local parameter i using the

following expression:

Pi N

j 1

e 4=r2a Ji jJ

2

: 10

The potential of each local parameter is a function of the distance

from this parameter to all the other local parameters. Thus, a local

parameter with many neighboring local parameters will have the

highest potential value. The constant ra is the radius dening the

neighborhood which can be determined by the following expres-

sion:

ra

N

N

i 1

1

nn

j 1

Ji j J ; 11

wherecan be chosen as follows 0oo1.

Outlier processing: From the set of local parameters (i; i

1;; N), there are some parameters obtained from mixed local

sets, it is clear that we have interest in eliminating them. Eq. (10)

can be exploited to eliminate the misclassied points. As this

equation attribute to the outliers a low potential, we can x a

thresholdunder which the local parameters are not accepted and

then removed from the data set. This threshold is described by the

following equation:

minP maxP minP; 12where P is the vector containing the potentials Pi such that

P P1;; PN and is a parameter chosen as 0oo1.

Cluster number and centers computing: After this treatment, the

set of local parameters is ltered and reduced to (i; i 1;; N)

NoN. Then, from this new data set, we select the data point

with the highest potential value as the rst cluster center.

Let n1 be this rst center and Pn

1 be its potential. The other

potentials Pj, j 1;; Nare then updated using this expression:

Pi ( Pi Pn

1e 4=r2

b Ji

n

1J2

: 13

Expression (13) allows to associate lower potentials to the local

parameters close to the rst center. Consequently, this choice

guaranties that these parameters are not selected as cluster

centers in the next step. The parameter rb is a positive constant

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that must be chosen larger than ra to avoid obtaining cluster

centers which are too close to each other. The constant rb is

computed using this formula:

rb

N

N

i 1

maxj 1:n

Ji j J: 14

In general after obtaining the kth cluster center, the potential of

every local parameter is updated by the following formula:

Pi ( Pi Pn

ke 4=r2b J i

nk J 2 ; 15

where Pnk and n

k are respectively the potential and the center of

the kth local parameter.

The number of sub-models s is a parameter that we would like

to determine. Therefore, we have developed some criteria for

accepting or rejecting the cluster centers as it is explained in the

algorithm of the next section.

To search the elements belonging to each cluster, we compute

the distance between the estimated output and the real one and

classifyk within the cluster which has the minimum distance.

arg minTi k yk; i 1;; s: 16

4.2. Algorithm

Algorithm 1. Identication algorithm

Data: Dispose offigNi 1 from a given data set i;yi

Main steps:

- Compute Pi for every figNi 1 according to(10)

- Determine the ltered local parameters figNi 1 , NoN

- Compute the rst cluster center n1 from(10)

repeat

Compute the other cluster centers according to the

updated potential formula15

IfPnk4then

Compute Vcsuch as :

Vc J nk n

cJ ; c 1;; k 1: 17

wherenk is the current cluster center and n

c; c 1;; k 1

are the last selected ones:

if Vc4; c 1;; k 1then

j accept nk as a cluster center and continue

else

j reject nk and compute a new potential

end

else

j reject nk and break

end

until Vcr; c 1;k 1

Result:Determination of the number of clusters s and theparametersfig

si 1

While is a small parameter characterizing the minimum distance

between the new cluster center and the existing ones.

4.3. Properties

The new clustering technique has several interesting properties

which can be summarized as follows:

This method automatically generates the number of sub-models. This method does not require the initialization of centers.

Therefore, the problem of convergence towards local minima

does not appear.

This method removes the outliers by associating small values totheir potentials. This operation improves the performance of

the clustering step.

5. Simulation example

We now present an example to illustrate the performance of

the proposed method.

Hence, the objective of the simulation is to compare theefciency of the proposed approach with that of the modied k-

means method (Ferrari-Trecate et al., 2003), the following quality

measures are used (Juloski et al., 2006):

The maximum of relative error of parameter vectors is denedby

maxi 1;;s

Ji i J2

Ji J2; 18

where i and i are the true and the estimated parameter

vectors for the sub-model i. The identied model is deemed

acceptable if is small or close to zero. The averaged sum of the squared residual errors is dened by:

s2e 1s

si 1

SSRijDij

; 19

where SSRi yk;kADi yk k1i2 and jDij is the car-

dinality of the cluster Di.The identied model is considered

acceptable if se2 is small and/or close to the expected noise

variance of the true system. The percentage of the output variation explained by the model

is dened by:

FIT 100 1J ^y yJ2Jy y J2

; 20

where ^y and y are the estimated and the real outputs, and y is

the mean value ofy .

The identied model is considered acceptable if the FIT is close

to 100.

Consider the following PWARX model (Boukharouba, 2011):

yk

0:4 0:5 0:3k ek if kAH1;

0:7 0:6 0:5k ek if kAH2;

0:4 0:2 0:2k ek if kAH3;

8>: 21

where s 3, na1, nb1 and k yk 1 uk 1T is the

regressor vector and the partitions fHigsi 1 are dened by

H1 fAR2

:1 0:3 0Z0 and 0 0:5 040g;

H2 fAR2

:1 0:3 0r0 and 1 0:3 0o0g;

H3 fAR2

:1 0:3 0Z0 and 0 0:5 0r0g: 22

The input signal u(k) and the noise signal e(k) are zero mean,

Gaussian distributed with variances respectively 0.5 and 0.05. The

output y(k) is illustrated inFig. 1.

Firstly, we determine the local parameters using the least

square method with n 20. The obtained parameter vectors are

presented inFig. 2.

Secondly, we use the Chui's clustering algorithm in order to

separate the obtained local parameters. This algorithm consists in

computing the potential of each local parameter.The large poten-

tials represents the cluster centers. However, the lower potentials

represent the outliers which must be removed. It is easy to deduce

that the proposed method is able to remove the outliers. More-

over, it generates automatically the number of submodels. This is

shown inFig. 3that illustrates the local parameters separated into


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s3 sets where the clusters centers are depicted by the star

symbols.

Table 1 presents the estimated parameter vectors obtained

with the proposed method and the k-means one.

The regionsHiare determined using the SVM algorithm. For the

proposed method they are dened by the following inequalities:

H1 AR2

:0:2161 1:9693 0:1850

1:7292 0:4728 0:1822 Z0 ; 23

H2 AR2 :0:2166 1:9704 0:1853

2:1873 0:7128 0:0906

Z0

; 24

H3 AR2

:1:7292 0:4728 0:1822

2:1873 0:7128 0:0906

Z0

: 25

For the k-means method, the following inequalities are obtained:

H1 AR2

:0:3255 3:1594 0:1354

3:1094 1:4634 0:1090

Z0

; 26

H2 AR2

:2:9235 1:4932 0:1469

2:4998 1:6640 0:1243

Z0

; 27

H3 AR2

:0:3255 3:1594 0:1354

2:4998 1:6640 0:1243

Z0

: 28

Fig. 2. Local parameters.

Fig. 3. Local parameters separated into 3 sets.

Table 1

Estimated parameters.

True values Proposed method k-means

1 0:4

0:5

0:3

264

375

0:3903

0:4653

0:3075

264

375

0:4064

0:5464

0:2598

264

375

2 0:7

0:6 0:5

264 3750:7406

0:60350:5027

264 375 0:6955

0:5903 0:4939

264 375

3 0:4

0:2

0:2

264

375

0:4023

0:1601

0:1847

264

375

0:4792

0:2101

0:2406

264

375

Fig. 1. The real output of the system.

1

0.5

0

0.5

1

1.5

y(k)

0 50 100 150 200 250

k

Fig. 4. Real and estimated outputs: proposed method.

1

0.5

0

0.5

1

1.5

y(k)

0 50 100 150 200 250

k

Fig. 5. Real and estimated outputs: k-means method.


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Now, we can get the output of the system with the two

methods as it is shown in Figs. 4and 5.

Table 2presents the quality measures(18), (19) and (20) of the

proposed method and the k-means method.

Based on the results presented inTable 2we observe that the

proposed method gives the best results compared with those

obtained by the k-means method ( and s2e are close to zeroand

FIT is close to 100).

6. Experimental example: a semi-batch reactor

This section presents an application of the proposed method to

an olive oil esterication reactor producing ester with a very high

added value which is used in ne chemical industry such as

cosmetic products.

6.1. The semi batch-reactor

The esterication reaction between vegetable olive oil with free

fatty acid and alcohol, producing ester, is given by the following

equation:

AcideAlcohol2Ester Water: 29

The ratio of the alcohol to acid represents the main factor of this

reaction because the esterication reaction is an equilibrium

reaction i.e. the reaction products, water and ester, are formed

when equilibrium is reached. In addition, the yield of ester may be

increased if water is removed from the reaction. The removal of

water is achieved by the vaporization technique while avoiding

the boiling of the alcohol. In fact, we have used an alcohol (1-

butanol), characterized by a boiling temperature of 1181C which is

greater than the boiling temperature of the water (close to 100 1C).

In addition, the boiling temperatures of the fatty acid (oleic acid)

and the ester are close to 300 1C. Therefore, the boiling point of

water may be provided by a temperature slightly greater than

1001C.

The block diagram of the process is shown in Fig. 6. It i s

constituted essentially of:

A reactor with double-jackets: It has a cylindrical shapemanufactured in stainless steel. It is equipped with a bottom

valve for emptying the product, an agitator, an orice introdu-

cing the reactants, a sensor of the reaction mixture tempera-

ture, a pressure sensor and an orice for the condenser. The

double-jackets ensure the circulation of a coolant uid which isintended for heating or for cooling the reactor. A heat exchanger: It allows to heat or to cool the coolantuid

circulating through the reactor jacket. Heating is carried out by

three electrical resistances controlled by a dimmer for varying

the heating power. It is intended to achieve the required

reaction temperature of the esterication. Cooling is provided

by circulating cold water through the heat exchanger. It is used

to cool the reactor when the reaction is completed. A condenser: It allows to condense the steam generated during

the reaction. It plays an important role because it is also used to

indicate the end of the reaction which can be deduced when no

more water is dripping out of the condenser. A data acquisition card between the reactor and the calculator.

The ester production by this reactor is based on three main steps

as illustrated inFig. 7

Heating stage: The reactor's temperature Tr is increased to1051C.

Table 2

Validation results.

Quality measures Proposed method k-means

0.0876 0.1828

se2 0.0023 0.0109

FIT 89.4191 74.195

Fig. 6. Block diagram of the reactor.

0 300 540

100

Time(min)

Reacting stage Cooling stageHeating Stage

Fig. 7. Specic trajectory of the reactor temperature.


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Reaction stage: The temperature should be kept constant at1051C until the end of the reaction which can be detected by

the absence of water at the condenser (when no more water is

dripped out of the condenser). Cooling stage: The reactor's temperature is decreased.

6.2. Existing method for modeling the reactor

An experimental study carried out on the reactor showed that the

reactor can be considered as a Single-Input Single-Output (SISO)

plant. The heating power Pe and the reaction temperature Trrepresent respectively the input and the output of the system.

However, the variation of the quality of the reactants inside the

reactor as well as the external effects can be considered as random

disturbances. In previous work of our laboratory based on some

assumptions, the system is considered as a linear one (Mihoub et al.,

2009). An identication approach is used to estimate the structure

and the parameters of the model of this reactor. In fact, a pseudo-

random binary sequence is applied as input to the real system with

three different amplitude domains in order to excite the three steps

of the operating cycle of the reactor (heating, reaction and cooling).

The obtained measurements, presented in Fig. 8, are then used toestimate the system structure using the instrumental determinants

ratio method (Mihoub et al., 2009) and to identify the system's

parameters with the least square approach.

The obtained system is described by the following discrete

transfer function (Mihoub et al., 2009):

Hz 0:0012

z2 1:5276:z0:5468 30

A validation experiment of this model is then applied. The

obtained results are presented in Fig. 9 which shows that the

obtained model generates an estimated output with large error.

These poor results are due to the nonlinearity that charac-

terizes the dynamic behavior of the reactor as illustrated in

Fig. 10 that depicts the static characteristic of the system. It is

easy to remark that each step of the operating cycle present a

linear behavior. Consequently, this reactor can be modeled by a

PWARX model.

6.3. PWARX model for the semi-batch reactor

The alternative of considering a PWA map is very interesting

because the characteristic of the system can be considered as

piecewise linear in each operating phase: the heating phase,

the reacting phase and the cooling phase. Previous works have

demonstrated that the adequate estimated orders na and nb of

each sub-model are equal to two (Talmoudi et al., 2008). Thus, we

can adopt the following structure:

yk

a1;1yk 1 a1;2yk 2 b1;1uk 1 b1;2uk 2 if kAH1

as;1yk 1 as;2yk 2 bs;1uk 1

bs;2uk 2 if kAHs

8>>>>>>>>>>>:

31

where the regressor vector is dened by

k yk 1;yk2; uk 1; uk2T

and the parameter vectors is denoted by

ik ai;1; ai;2; bi;1; bi;2; i 1;; s:

We have picked out some inputoutput measurements from thereactor in order to identify a model to this process. We have taken

two measurement les, one for the identication having a length

N220 and another one of length N160 for the validation.

The measurement le used in this identication is presented in

Fig. 8. We apply the proposed identication procedure in order

to represent the reactor by a PWARX model with a number of

neighboring n 70. Our purpose is to estimate the number of

sub-models s, the parameter vectors ik; i 1;; s and the

hyperplanes dening the partitions fHigsi 1 .

The obtained results are as follows:

The number of sub-models is s 3. The parameter vectors ik, i 1; 2 and 3 are illustrated in

Table 3.Fig. 8. The real inputoutput evolution.

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

k

Fig. 9. Model validation: real and estimated system outputs.

65

70

75

80

85

90

95

100

105

110

1000 1200 1400 1600 1800 2000 2200

P (W)

Fig. 10. Static characteristic of the reactor.


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It remains only to determine the partitions fHigsi 1 that are

dened by the following inequalities:

Hi fAR4

:MiZ0g; 32

M1 0:3869 0:3130 0:0001 0:0015 4:8740

0:0164 0:0035 0:0018 0:0026 0:7355

; 33

M20:4548 0:3862 0:0001 0:0014 4:5891

0:2871 0:2519 0:0021 0:0034 0:4359

;

34

M30:0163 0:0034 0:0018 0:0026 0:7335

0:1342 0:1041 0:0019 0:0031 0:2969

: 35

To validate the obtained models, we have considered a new input

output measurement le having a length N160. The obtained results

are presented in Fig.11 which shows that the proposed approach gives

the best results in terms of precision.

7. Conclusions

In this paper, we have considered the problem of clustering

based procedure for the identication of PWARX models. In fact,

we have recalled the main steps of clustering based approach.

This presentation shows that the used clustering algorithms

require a good initial guess in order to converge to global minima.

In addition, these algorithms are sensitive to the presence of

outliers. To overcome this problem, we have suggested the Chiu'salgorithm for data classication. This algorithm presents several

advantages. Firstly, it does not require any initialization and the

problem of convergence towards local minima is overcome.

Secondly, this method is able to remove the outliers by associating

potentials to these points. Finally, our approach generates the

number of sub-models. Numerical simulation results are pre-

sented to demonstrate the performance of the proposed approach.

Also, an experimental validation with an olive oil reactor is

presented to illustrate the efciency of the developed method.

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Table 3

Estimated parameter vectors.

Parameter vectors Estimated parameters

1 1:4256

0:4508

4:9853 10 4

0:0010

26664

37775

2 1:16040:2111

0:0015

0:0014

26664

37775

3 1:0847

0:1490

3:9782 10 4

0:0040

26664

37775

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

k

y(k)

Estimated Output PWARX Model Proposed Method

Real System Output

Estimated Output Validation Existing Method

0 20 40 60 80 100 120 140 160 1800

500

1000

1500

2000

u

(k)

Fig. 11. The input, the real and the estimated validation outputs.

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