Tests of Hybrid Volterra-Type Models Through

NMPC Schemes at a Batch Reactor

Carlos Medina-Ramos and Huber Nieto-Chaupis

Centro de Tecnologıas de Informacion y Comunicaciones

Universidad Nacional de Ingenierıa

Lima - Peru

ccmedina@uni.edu.pe - hubernieto@uni.edu.pe

Abstract—A simulation of the control of temperature in abatch reactor aimed to preparate melamine resin, is presented.In contrast to traditional techniques like PID by which the setpoint turns out to be of importance, a Nonlinear Model PredictiveControl (NMPC) by using the Volterra model together to theorthogonal polynomials might permit to control the referencetrajectory tracking of the internal temperature of the reactor.This work proposes a NMPC based on the second order hybridVolterra-type model containing only 6 parameters. It allowsto evaluate both advantages and disadvantages of the modelas to automate the temperature trajectory tracking from theinitial phase of the heating. From the results, interesting andfavorable prospects to automate the reactor are expected. It mightsuggest that the hybrid Volterra models are a key step towardscompact technologies containing newest NMPC schemes of highperformance.

I. INTRODUCTION

The searching of new schemes for manufacturing high

quality products has triggered diverse strategies of control

aimed to produce excellent quality products in a full according

to the standards and others global requirements. The control

engineering has proposed and developed solid mathematical

methodologies engaging the formalism to the phenomenology

of systems efficiently [1] . In fact, the robustness of the

mathematical model can be crucial for implementations in

control systems and to understand the phenomenology as well.

Let us to mention the Volterra formalism which has widely

been used for modeling nonlinear systems. Based on the clear

fact that for any system holds the I/O universal relation,

the associated master equation describing such system can be

written by the Volterra model as follows,

y(t) = V[χ(t)] =∞∑

m=1

Vm [χ(t)], (1)

where V is called the Volterra operator by acting on the input

signal given by χ(t). This operator is actually an infinite

composition of terms by which the nonlinear system can be

adequately described. In the practice, the order is truncated by

the integer number N . In reality, the effect of the truncated

Volterra operator on the input signal results in an adequate

model of approximation for [1],

V[χ(t)] =N∑

m=1

∫ b

a

...

∫ b

a

h(τ1, ..., τm)

τm∏τi=τ1

χ(t−τi)dτi, (2)

where the kernels h(τ1, ..., τm) are intrinsically ascribed to

the Volterra operator, whereas a and b denote the limits of

integration corresponding to the range of operation of the

system. Thus, (2) is expressed as a finite sum of processes

of convolution. In this report, we focused on the expansion de

Eq. 2 up to the second order which means that (2) is rewritten

as

V[χ(t)]ForN=2 =

∫ b

a

h(τ1)χ(t− τ1)dτ1

+

∫ b

a

∫ b

a

h(τ1, τ2)χ(t− τ1)χ(t− τ2)dτ1dτ2, (3)

that is called the second order truncated Volterra series. As

observed in previous works [2], the modeling by using pure

Volterra models becomes intractable because the huge number

of kernels. This point has been a topic of exhaust discussions

in literature [3]. Indeed, it has also been the reason of

reconfigurate the Eq. (1-2) to a one much more reasonable for

practical ends. Such reconfiguration has evaluated to use the

orthogonal polynomials as recently reported in order to reduce

the number of kernels leading to establish fluid algorithms to

be used in control system.

The goal of this report is that of applying the hybrid Volterra-

type model in control systems for preparation of melamine

resin of excellent quality. In this way, a simulation of the

control of trajectory (of heating) based on the hybrid Volterra

models containing orthogonal polynomials like Laguerre,

Chebyshev and Hermite is presented. The results have shown

that the scheme of hybrid models adjusts to solve the problem

of temperature trajectory tracking in a 3.5 tons batch reactor.

The reactor under study aims to produce melamine resin

of high quality, however a carefully monitoring of internal

temperature in reactor should be applied to defeat instabilities,

overshoots and random disturbances which might emerge as

consequence of the complex mixing of chemical compounds

and all those fluctuations coming from the heating source.

This paper is organized as follows: in second part the basics

of identification is presented, with a brief description of

the orthogonal polynomials. Third part describes the NMPC

scheme associated to the technology of Nonlinear Dynamics

Matrix Control (NDMC) introduced as a prospective proposal

in optimizing the control of the reference trajectory tracking

of the temperature. Finally, some conclusions regarding the

978-1-4244-6742-6/10/$26.00 ©2010 IEEE

results of this work are listed.

II. SYSTEM IDENTIFICATION

We argued that the Volterra series as expressed in Eq. 2, can

be applied to identify the heating of the batch reactor under

study. Thus we propose the following:

• rewrite Eq. 2 in its discrete form;

• to expand the kernels onto orthogonal polynomials;

• to keep the diagonal terms of the full algorithm;

• to choose the family of polynomials to be used.

Respect to the former, this report has used up to three different

families of orthogonal polynomials: Laguerre, Chebishev and

Hermite. So that, we shall establish up to three algorithms

which might be incorporated in the control system.

A. The Second Order Discrete Volterra Algorithm

In praxis, one needs to digitalize Eq. 2 so that it is necessary

to rewrite it in a much more appropriated manner,

y(k) =

q∑p=1

h(p)χ(k − p) +

q∑p,m=1

h(p, m)χ(k − p)χ(k −m),

(4)

where it was assumed N=2. The terms denote the first and

second order Volterra model, respectively. For the sake of the

simplicity, q is the same in both first and second order. Indeed,

it is observed that the integer numbers p and m run over the

values allowed by the horizon determined by the range [1,q].

B. The Hybrid Volterra-type Model

An interesting scheme to reduce the number of kernels

in Eq. 4 is that of expanding the kernels onto orthogonal

polynomials as follows:

h(p) =

r∑j=1

CjΨj(p), (5)

h(p, m) =

r∑j=1

s∑l=1

Cj,lΨj(p)Ψl(m), (6)

where r and s denote the highest value of order of the

polynomials to be considered (it is helpful to mention that

the projections (5) and (6) are established in virtue of the

Rietz-Fisher theorem.

When Eq. 5 and Eq. 6 are inserted in Eq. 4 the resulting

algorithm reads

y(k) =

q∑p=1

r∑j=1

CjΨj(p)χ(k − p)

+

q∑p,m=1

r∑j=1

s∑l=1

Cj,lΨj(p)Ψl(m)χ(k − p)χ(k −m). (7)

The hypothesis in this work consists in keeping only those

diagonal elements of Eq. 7, which would be enough for

system identification. This assumption reduces drastically the

number of parameters. Finally, the model algorithm to be used

throughout this report can be written as follows:

y(k) =

q∑p=1

r∑j=1

CjΨj(p)χ(k − p)

+

q∑p=1

r∑j=1

CjjΨ2l (p)χ2(k − p). (8)

which is denominated the second order truncated hybrid

Volterra-type model.

C. The Candidates Orthogonal Polynomials

Below are listed those orthogonal polynomials to be used

in the algorithm of the truncated hybrid Volterra-model. To

this end, it is required that any orthogonal polynomial should

satisfy the orthogonality relation∫ W(t)PQ(t)PS(t)dt = δQS

where W (t) is called the weight function associated to the

orthogonal polynomial PQ.

1) The Generalized Laguerre Polynomials: As mentioned

in several references, the Laguerre polynomials have been used

in the identification of nonlinear systems due to the fact that

they adjust well to those special system which are free of

strong oscillations [5]. They are generated from

LQ(t) =√

2aeat

(Q− 1)!

dQ−1

dtQ−1(tQ−1e−2at) (9)

for 0 ≤ t < ∞. To note that Q an integer number by denoting

the order of the polynomials and a the Laguerre pole. The

Laguerre polynomials are orthogonal respect to the weight

function tae−t.

2) The Chebishev Polynomials: The first kind Chebyshev

polynomials can be expressed as

CQ(t) = cos(Qcos−1(t)) (10)

for −1 ≤ t ≤ 1. The Chebyshev functions are orthogonal

respect to the weight function (1 − t2)−1/2 by assuring the

orthogonality relation inside the range of validity.

3) The Hermite Polynomials: The generation of orthogonal

functions is given by the following relation

HQ(t) = (−1)Qet2 dQ

dtQ

[e−t2

], (11)

for −∞ < t < ∞. The H functions are orthogonal respect to

the weight function defined by e−t2 along the all values of t.

D. Model Parameters Identification

So far it has been explained the following: (a) the iden-

tification algorithm given by Eq. 8, and (b) the orthogonal

polynomials Eq. (9-11) by which one should evaluate their

performance in (8) for the temperature identification.

To identify completely the system it is needed to know so

well the system and therefore to design the input function

χ(k). In the present simulation we shall use a Pseudo Random

Multi Level (PRMS)[4] input which must have the property

of exciting permanently the system during a sampling time

of order of 6000 s. Even though such excitation should

be permanent in order to guarantee a reasonable relation

signal/noise.

The next step is to provide a mechanism which should be

capable of extracting the Cj and Cjj parameters together to

their associated identification error e(k) = y(k)−y(k), where

y(k) becomes output data acquired from the plant, by effect

of the PRMS.

The Procedure: It is possible to write Eq. 8 in its matrix form.

In this way, [Y]T = [y(1), y(2), y(3), ..., y(r−2), y(r−1), y(r)]and [C]T = [C1, C2, C3, C11, C22, C33]. Thus one can express

Eq. 8 as

[Y(r)] = [Ψ(r)χ(r)][C(r)] = [A(r)][C(r)]. (12)

To be concrete, the extraction of the C parameters, is achieved

from the identification error e(r) = [Y(r)]−[Y(r)] = [Y(r)]−[A(r)][C(r)]. To note that r ∈ [6, P ], for the second order (the

six parameter model are derived from Eq. 8).

Therefore, the resulting matrix equation Eq. 12 can be written

in a much more compact form Y = Y(C) = A · C, so

to identify the entries of matrix C one should consider a cost

function whose purpose is that of minimizing the identification

error,

J(C) =(Y −Y

)2

= (Y −Y)T(Y −Y), (13)

through the operation dJdC=0. It is easy to demonstrate that

after a straightforward algebra C = (ATA)−1

ATY resulting

in the Moore-Penrose pseudo-inverse matrix. Consequently,

the C matrix is written in a much more explicit manner,

C(r) =[[A(Ψ(r)χ(r))]

TA [Ψ(r)χ(r)]

]−1

×A

T(Ψ(r)χ(r))Y(r). (14)

One can to note that the suitable applicability of the Moore-

Penrose equation C = (ATA)−1

ATY is manifested in Eq.

14.

E. Functionality of the Heating Mechanism

In Fig. 1 is depicted the functionality of the preparation

of melamine resin at the R01 reactor of PISOPAK PERU

S.A.C. Firstly, the mixer is set on in order to uniformize the

chemical ingredients. Consequently, a signal which is denoted

by χ(k) is applied. This input is applied inside the range 4/20

mA on the transducer I/P converting the current to a signal

3/15 psi. This pressure governs a three way pneumatic valve

whose role is that of controlling the flux of the diathermic oil

( 220 ±5 0C ) coming from a heater oil, as shown on the left

part of Fig. 1. The oil goes to the batch reactor and it is the

element responsible for heating the reactor. A PT100 is used

as a transducer element of the temperature in reactor Y (k) as

observed in right side of Fig. 1.

F. Results of the Identification

Resulting coefficients were obtained from the Moore-

Penrose extraction process including a bias which is the

nominal value of the environment temperature.

� �

IP

�������

���

������

��������

����������

����

������������ ��

�����

!"# "����$

���

%# &'�("#& ��

Fig. 1. Scheme of chemical plant at PISOPAK. To note that the input isdriven by the I/P transducer followed by a three way pneumatic valve whoserole is that of regulating the flow of oil to reactor.

The initial parameters to be used in the algorithm are listed

below

Grade of Model 2

Number of Polynomials 3

Polynomial Type Laguerre, Chebyshev

and Hermite

Sampling Period T 30 s.

Time constant 90 s.

System Horizon 2640 s.

Discrete Horizon 88

With these initial parameters and the full algorithm derived in

Eq. 8, one can to write the one to be used for identification,

y(k) =

88∑p=1

3∑j=1

[CjΨj(p)χ(k − p) + CjjΨ2l (p)χ2(k − p)

](15)

and the results of identifying the system parameters are written

in the next tables,

1) Laguerre (pole=0.978):

C1 C2 C3 C11 C22 C33

1.523 0.770 -0.577 -0.605 0.737 -0.435

2) Hermite:

C1 C2 C3 C11 C22 C33

0.04 0.06 0.01 0.03 -4 10−3 10−5

3) Chebyshev:

C1 C2 C3 C11 C22 C33

-1.40 -1.28 -0.72 -0.06 -0.31 -0.04

In Fig. 2 are displayed the curves of temperature for the three

families of orthogonal polynomials. The best approximation is

given by the Chebyshev polynomials.

20

40

80

60

20

40

60

80

20

40

60

80

T E

M P

E R

A T

U R

E (

º C

)

500 1000 1500 2000 2500TIME (s)

Fig. 2. Curves of temperature (in centigrades) against time (seconds). Redcontinue lines denote the simulation. The dashed lines denote the curvesobtained from the usage of the Laguerre (top), Chebyshev (middle), andHermite (bottom) orthogonal polynomials. It is noteworthy the one derivedfrom the usage of the Chebyshev which adjusts notably to the data along thewhole initial phase between 0 and 2500 s. Curves also shows the effectivityof the usage of Volterra model as discussed in literature.

III. STRATEGY OF THE NMPC PROPOSED AND RESULTS

Normally, the system under study operates through a PID

controller together to a well tuned fuzzy logic scheme for a

determined set point. Actually, it is not coherent since the

system should control the trajectory of heating just over 2500

s. The continue line in Fig. 4 is one example of the resulting

control system under this scheme. In other words, PID seems

to be the one which would not be adequate for the control

strategy of the trajectory tracking. Therefore, a NMPC (or

another) could be an excellent control scheme for this exercise

in to find an efficient mechanism for controlling the reference

trajectory.

For the present study, one should indicate that the matrix

elements are obtained from the hybrid Volterra-type models

instead of I/O samples, as response to the applications of

steps functions.

A. The Control Algorithms

The output y(k) can be expressed in terms of the truncated

hybrid Volterra-types in the following form

y(k) =

P∑j=1

[h1jχ(k − i) + h2jχ

2(k − i)]+ e(k) (16)

where e(k) represent the identification error and P the predic-

tion horizon, χ(k) the PRMS signal input, and hij the second

order kernels.

a.- Kernels of the hybrid Volterra-type system: they are pro-

jected onto an orthogonal basis function as

h1j =

j∑r=1

[C1Ψ1(r) + C2Ψ2(r) + C3Ψ3(r)] ,

h2j =

j∑r=1

[C11Ψ21(r) + C22Ψ2

2(r) + C33Ψ23(r)

](17)

b.- Formulation of NMPC Consider the prediction horizon P ,

the control horizon M and the system output y(k) that can be

written in the predictive form as (to see Eq. 5 and Eq. 6)

y(k + n) =

n∑j=θ

h1jχ(k + n− j) +

P∑j=n+1

h1jχ(k + n− j)+

n∑j=θ

h2jχ2(k + n− j) +

P∑j=n+1

h2jχ2(k + n− j) + e(k) (18)

where θ=1 for n ≤ M and θ = n −M + 1 for M < n ≤ Pand e(k) the prediction error.

c.- Formulation of the Nonlinear Dynamical Matrix Control

(NDMC): NDMC is actually an interesting methodology to

be implemented in our analysis [5][6][8][9]. The elements

of the proposed NDMC of order 2, shall use up to three

lineal and quadratic combination of orthogonal polynomials

for describing h1j and h2j respectively.

It should be mentioned that the system horizon is given by

N=P, with the integer n ∈ [0, N ]. So that it is possible to

group the terms of y(k + n) in its matrix form as YT =[y(k)y(k + 1)...y(k +N )] as well as the χ(k+n), χ2(k+n),χ(k + n− i), and χ2(k + n− i) by

χT+ = [χ(k), χ(k + 1), ...χ(k + M − 1)]

χ2T+ =

[χ2(k − 1), χ2(k + 1), ...χ2(k + M − 1)

](19)

χT− = [χ(k − 1), χ(k − 2), ...χ(k −N )]

χ2T− =

[χ2(k − 1), χ2(k − 2), ...χ2(k −N )

], (20)

with χT+, χT

−, χ2T+ , and χ2T

− , denoting the future, past,

quadratic future, and quadratic past inputs respectively. With

these definitions, the Y can be written as

Y = Gχ+ + G2χ2+ + Hχ− + H2χ

2− + e (21)

where the G, G2, H and H2 are matrix built from the terms

of Eq. 18,

G =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h11 0 . . 0h12 h11 . . 0. . . . .. . . . .

h1M h1,M−1 . . h11

. . . . .

. . . . .h1N h1,N−1 . . h1c

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦N×M

(22)

and

H2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h22 h23 . h2N 0. . . . .. . . . .. . . . .. . . . .. . . . .

h2N 0 . . .0 0 . . 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦N×N

(23)

,

G2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h21 0 . . 0h22 h21 . . 0. . . . .. . . . .

h2M h2,M−1 . . h21

. . . . .

. . . . .h2N h2,N−1 . . h2c

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦N×M

(24)

and

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h12 h13 . h1N 0. . . . .. . . . .. . . . .. . . . .. . . . .

h1N 0 . . .0 0 . . 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦N×N

(25)

where the relation among the matrix elements is given by

hic =

c−M+1∑j=1

hij , c ∈ [M,N ], i = (1, 2) (26)

B. Expression for the NMPC based on the control increments

on the future

For the NMPC the terms to be taken account on the

increments of control for future are χ(k) = Xχ(k − 1) +

Δχ(k). Its predictive expression for χ(k) reads χ(k + M) =χ(k−1)+

∑Mi=1

Δχ(k +M −1). As consequence the G ·χ+

from Eq. 21 can written as G ·χ+ = G ·VuΔχ+G ·VM ·χ−where Δχ = [Δχ(k)...Δχ(k−M −1)], Vu a lower triangular

matrix with Vu(i, j) = 1 for i ≥ j otherwise is null, and VM a

lower triangular matrix with VM (i, 1) = 1 for all i, otherwise

is null. Consequently, one gets a matrix form for Eq. 21 in

terms of the control increments on the future Δχ,

Y = G · Vu ·Δχ + (H + G · VM )χ−

+H2− · χ2

− + G2 ·X2+ + e (27)

� �

��

��

��

��

���

��

��

��

��

��

��

��

��

��� ��� ����

�� ������

���� ���� ���� ���� ����

���� ���� ����

���� ���� ���� ����

��

����

����

����

����

�

���

����

����

����

����

����

����

��

����

����

����

����

�

��� ���� ���� ���� ����

���

���� ���� ���� ���� �������

���� ���� ����

������ ������

Fig. 3. Simulation of control of the internal temperature in a batch reactor.Plots on the left side depict the control for a reference trajectory of typetanh(t) (blue continue lines), whereas the dashed lines are the results of thesimulation. The ones of the right side depict the input signal actuators inunits of current before of entering to the transducer (see Fig. 1). Top left andright panels describe those when the Laguerre polynomials are used in thealgorithm of Eq. 8 [7]. Those of the middle are obtained for the case of theChebishev polynomials. The ones of the bottom correspond to the case whenthe Hermite polynomials are used.

C. Cost Function and Control Law for the Proposed NMPC

To evaluate the vector Δχ it is necessary to define a cost

function which involves the error between the reference tra-

jectory and the full output, together to their future increments

of the control signal. It is traduced in the following:

[J(Δχ)] =

N∑i=1

||ek+i||2 + λ

M∑i=1

||Δχ(k + i− 1)||2 (28)

which can be expresses in its matrix form as [4]

[J(Δχ)] = [Y − Ref]T [Y − Ref] + λΔχT Δχ (29)

with λ a ponderation factor. In order to carry out the mini-

mization of J a few considerations are required: the variation

of G2 · χ2+ respect to Δχ is both dG2·χ

2

dΔχ << Ref anddG2·χ

2+

dΔχ << Vu.

With this, for the case where is free, one can approximate it to

Yf = Hvuχ−+H2χ2− where Yf the free response of system.

� �

������������ �

�������

��

��

��

��

��

��

��

��

����

��������� ���� ���� ����

Fig. 4. Comparison between PID and NMPC schemes by showing theeffectivity of the proposed control system. Blue shaded line denotes thesimulation achieved in this work by which the Chebyshev-Volterra modelis used , whereas the continue line is derived from the plant which uses aPID controller thereby showing an evident poor performance during the initialphase.

On the other hand, in order to dJdΔX = 0

Δχ = inv(V T

u Vu + λI)V Tu (Ref − Yf ) (30)

where Ref = tanh(t) and Δχ the future increment control

vector on which the action of the control shall use the first

term.

It should be stressed that for the control law only those matrix

built from the hybrid Volterra-type models are considered,

instead of applying step functions. It constitutes the NDMC-

like version of this work.

D. Restrictions Proposed to Δχ

For the case of the reactor under study a few restrictions

are needed in order to achieve an efficient trajectory tracking

between 19 and 88 degrees, in according to the tanh(t) to be

applied to the control system. So that the imposed restrictions

are

• if Δχ < ΔχMin then Δχ = ΔχMin;

• if Δχ > ΔχMax then Δχ = ΔχMax;

• if |Y − YRef | > eMax then Δχ = Δχc;

• if |Y − YRef | < eMin then Δχ = Δχv .

It is important to underline that the Δχc and Δχv are

obtained from the experimental tests. These considerations

were important for the tuning and the achievement of the

tracking trajectory proposed. In Fig. 4, it is displayed a clear

comparison between the NMPC and PID. The evidence of

the robustness of the NMPC is manifested in the excellent

tuning of the control system. It should be noted that the tests

of identification were carried out while the reactor was under

operation of fabrication of melamine resin.

Although statistics of data is not discussed, in a future work

shall be tested the methodology used in this report to find

statistical fluctuations and their impact on the identification.

IV. CONCLUSIONS

In this report we have developed a strategy of control aimed

to monitor the heating provided to a batch reactor based on the

hybrid Volterra-type models which are shown to be sustainable

by the usage of the so-called orthogonal polynomials. In one

hand, the Volterra model has been applied to the concrete case

of system identification, where an excellent performance of the

hybrid Volterra-type models have been observed. On the other

hand, the incorporation of the Volterra models into a MPC

algorithm has been tested in conjunction to a similar DMC

scheme by showing the efficiency of the control strategy in

comparison to classical methodologies like PID (see Fig. 4).

Finally, we argue that the control strategy based on the so-

called orthogonal polynomials get its optimum performance

when the Chebyshev polynomials are used. It might to lead to

establish that those processes containing a well define initial

phase might be controlled through a NMPC based on orthog-

onal families like Chebyshev polynomials featured by only 6

parameters. In future, we shall explore the feasibility in testing

another methodologies containing only a few parameters and

therefore to be able to conclude about the robusteness of

the Volterra formalism to analyze nonlinear systems in batch

reactors.

ACKNOWLEDGMENTS

The authors of the present work would like to kindly

thank to PISOPAK PERU S.A.C where the realizations of

identification were achieved.

C. M-R is really grateful to the support from the Postgraduate

Central Office at Universidad Nacional de Ingenieria.

Finally, we are also indebted to J. Betetta-Gomez and D.

Carbonel-Olazabal whom have provided their suggestions and

valuable comments on the initial versions of the manuscript.

REFERENCES

[1] E. Camacho and C. Bordons, Model Predictive Control. Springer, London(1998).

[2] I. Sandberg Uniform approximation with doubly finite Volterra series ,Signal Processing, IEEE Transaction, 40 (1992) 1438.

[3] A Zhu and T Brazil, RF power Amplifier Behavioral Modeling Using

Volterra Expansion with Laguerre Functions Microwave SymposiumDigest, 2005, IEEE MTT-S International.

[4] R. Nowak and B. Van Veen, Nonlinear System Identification with

Pseudorandom Multilevel Excitation Sequences , Acoustics,Speech andSignal Processing 1993. ICASSP-93 1993 IEEE Int. Conf, 4 , (1993)456.

[5] Pan-yen Ho, Cheng-Ching Yu and T.P Chiang, Two Degree-of-freedom

Dynamic Matrix Control for Distilation Columns Journal of the ChineseInstitute of Eng. 16 (1993) 101.

[6] H. Hapoglu and M. Alpbaz, Experimental Application of Lin-

ear/Nonlinear Dynamic Matrix Control to a Reactor of Limestone Slurry

Titrated with Sulfuric Acid Chemical Engineering Journal, Volume 137,Issue 2, 1 April 2008, Pages 320-327

[7] A. Back and A. Chung Tsoi, Nonlinear System Identification Using

Discrete Laguerre Functions , Journal Of Systems Engineering 6 (1996)6.

[8] Cutler, C.R Ramaker, B.C. (1979), Dynamic Matrix Control-to Computer

Control Algorithm Proceeding of the 86th National Meeting of theAmerican Institute of Chemical Engineering. Houston, TX. WP5-B.

[9] P. Aadaleesa, Nitin Miglan, Rajesh Sharm and Prabirkumar Saha, Nonlin-

ear System Identification Using Wiener Type Laguerre Wavelet Network

Model Chemical Engineering Science, 63 (2008)3932