Testing Laguerre-Volterra Model in a Batch Reactor

Carlos Medina-Ramos

Centro de Tecnologıas de Informacion

y Comunicaciones CTIC-UNI

Universidad Nacional de Ingenierıa

Lima - Peru

Email: ccmedina@uni.edu.pe

Huber Nieto-Chaupis

Facultad de Ingenierıa Electrica y Electronica

Seccion de Postgrado y Segunda Especializacion

Universidad Nacional de Ingenierıa

Lima - Peru

Email: huber nieto@hotmail.es

Abstract—We report an application of the formalism ofLaguerre-Volterra to model the temperature of a batch reactorin a chemical plant. To reduce substantially costs during thepreparation of melamine resin, this study focuses on the initialphase where an efficient controllability of the heat providedto reactor is needed. This analysis have shown an acceptableagreement between model and data supporting the idea that thebatch reactor presents interesting prospects to be automated byusing mathematical models containing orthogonal polynomials.

I. INTRODUCTION

Mathematical methodologies in advanced control have

played a crucial role to understand complex nonlinear systems

realized at industrial processes along the last decades [1].

Often one needs to know the internal dynamics in real systems

from the I/O data. Mathematically speaking, the description

of systems can be possible by using the master equation

y(t) = Ox(t), (1)

where O contains information about properties of system. In

those cases what are characterized by delays, sometimes a

reasonable choice is that of the Volterra series expansion,

which in concrete assumes that the input is operated in the

following way [2],

y(t) =

∞∑j=1

Hj [x(t)], (2)

where Hj denotes the Volterra operator of order j which is

evaluated to time invariant nonlinear systems. Recent reports

have indicated some advantages in applying this formalism to

identify nonlinear systems with memory such as behavioral

modelling for radio frequency power amplifier analysis [3].

Other studies have considered to expand the Volterra kernels in

discrete-time Laguerre polynomials to reduce the large number

of parameters which is one of the disadvantages of (2) [4]. In

fact, whereas the usage of Volterra series leads to define a

strategy to identify nonlinear processes, the projection of the

kernel onto Laguerre polynomial basis allows to formulate

a scheme containing a reduced number of parameters to be

identified. In particular, in a chemical reactor whose temper-

ature should be efficiently controlled, a suitable identification

of process is desirable, specifically from the initial phase

which is of enormous importance. Our case focuses in a batch

reactor aimed to produce melamine resin in PISOPAK PERU

S.A.C, a chemical facility located at Lima. Due to reasons

of costs of production and quality, the internal temperature in

this reactor should be monitored. In other words an excellent

controllability of internal temperature is needed. In this report

we have tested the truncated Laguerre-Volterra expansion

for identification of a nonlinear system. The main goal is

to investigate the feasibility for modelling and therefore to

evaluate the automatization of system. It is realized with data

acquired from the plant and confronted to the proposed model.

The results have yielded interesting prospects to automate

the controllability of heat provided to reactor accurately. It

constitutes a positive fact since the necessity in to optimize

the productivity in this kind of facilities turns out to be of

great importance [5]. This paper is structured as follows:

second section gives the main mathematical relations which

were used to identify the process under study. We started

with the Volterra expansion in the discrete case. Thus, the

Laguerre vectors are introduced leading to express the output

as functions of Laguerre polynomials multiplied by real coef-

ficients. The case when a multilevel signal as input function is

considered. A point of importance in our hypothesis is that of

the identification of nonlinear systems through the Laguerre-

Volterra approach can be achieved in terms of their diagonal

entries. Hence, the extraction of the coefficients through a

computational procedure is performed. It is verified that the

modelling adjusts the observed thermodynamics properly. In

third section, data and model are analized regarding their

behavior whereas in a fourth section conclusions are drawn.

II. THE USAGE OF THE LAGUERRE-VOLTERRA MODEL

The starting point is the description of model by written

the original definition of a Volterra series in its fundamental

definition,

y(t) =

∫h(τ1)x(τ1 − t)dτ1 +

∫ ∫h(τ1, τ2)x(τ1 − t)x(τ2 − t)dτ1dτ2 +

∫ ∫ ∫h(τ1, τ2, τ3)x(τ1 − t)x(τ2 − t)x(τ3 − t)dτ1dτ2dτ3

+O(> 3) (3)

that is in essence the definition (2) [6]. The functions h(τi) and

x(τi−t) describe the kernel and delayed function respectively.

978-1-4244-5697-0/10/$25.00 ©2010 IEEE 141

Fig. 1. Illustration of a truncated Laguerre-Volterra model up to third order.The φi functions denote the orthogonal Laguerre polynomials.

In (3) the output is composed by an infinite amount of terms

where each one is accompanied of kernels. In this work the

truncated discrete case is used, so that∫

→ Σ up to third

order. As it is well-known the discrete case is in praxis the

most convenient one. Thus, an alternative way to write the

Volterra expansion reads

y(t) = Σ∞

τ1h(τ1)x(τ1 − t) +

Σ∞

τ1Σ∞

τ2h(τ1, τ2)x(τ1 − t)x(τ2 − t) +

Σ∞

τ1Σ∞

τ2Σ∞

τ3h(τ1, τ2, τ3)x(τ1 − t)x(τ2 − t)x(τ3 − t),

(4)

where τj covers all values. It is clear that the inclusion of

higher orders will make a precise prediction but on the other

hand, the calculation of the Volterra kernels turns out to

be complicated. Although (4) is quite efficient for system

identification as reported by various authors, it is known from

several authors that a relative precision might be acquired for

certain kinds of systems if the Volterra kernels are projected

onto a orthogonal finite Laguerre basis [7] [8]. Translated in

a formal language, we have

h(k1) = ΣLj cjφj(k1) (5)

h2(k1, k2) = ΣLj ΣL

l cjlφj(k1)φl(k2) (6)

h3(k1, k2, k3) = ΣLj ΣL

l ΣLncjlnφj(k1)φl(k2)φn(k3), (7)

where φi denote a L-dimensional Laguerre orthogonal basis.

The ci are real numbers and emerge as result of the projection

of the Volterra kernel onto the Laguerre space. Roughtly

speaking, the usage of the Laguerre expansion (5-7) allows

that the Volterra kernels as defined in (4) become reduced

considerably. It is because the Laguerre polynomials transfer

their orthogonal nature to the Volterra kernels. In other words,

most of them are canceled in according to the orthogonality

in the sense of the inner product.

We note that the coefficients are unknown variables and the

main goal is now to figure out a scheme to extract them. To

be more specific, the expansion (4) used in conjunction to the

kernels expanded in Laguerre polynomials should be truncated

for computational ends. Thus in our calculations up to third

order have been considered. The number of points is truncated

in both N=87 (T=30 s) and N=130 (T=20 s), where N denotes

the upper limit to be used in (4).

Our hypothesis consists in to keep only those diagonal terms

which might be enough to system identification. Therefore our

approach reads

y(k) ≈ ΣNk1=1

c1φ1(k1)x(k − k1) + c2φ2(k1)x(k − k1)

+c3φ3(k1)x(k − k1)

+ΣNk1=1

c11φ2

1(k1)x

2(k − k1) + c22φ2

2(k1)x

2(k − k1)

+c33φ2

3(k1)x

2(k − k1)

+ΣNk1=1

c111φ3

1(k1)x

3(k − k1) + c222φ3

2(k1)x

3(k − k1),

+c333φ3

3(k1)x

3(k − k1), (8)

where the φi are the Z−1-transformed of the orthogonal

Laguerre polynomials. Throughout this study the Laguerre

pole varies between 0.974 and 0.981. This range is fixed

by considerations purely of analicity. In fig. 1 is drawn the

diagram of blocks corresponding to y(k) as expressed in (8).

For the sake of the simplicity first and second order were not

displayed, instead only third order Laguerre-Volterra model is

depicted. In the following, the main goal is to determine the

values of 3, 6, and 9 unknown coefficients corresponding to

first, second and third order respectively. Of course, another

manner is that of keeping all contributions when (5-7) are

replaced in (4), nevertheless in our approximation we try to

demonstrate that only a small amount of elements of the multi-

dimensional matrix is enough to identify the phenomenology

in reactor.

A. Extraction of the Kernels

The output y(k) what are measured as effect of the multi-

level signal χ(k) for the first order system, can be written in

the following way

y(1) = c1φ1(1)χ(1)+ c2φ2(1)χ(1)+ c3φ3(1)χ(1)y(2) = c1φ1(2)χ(2)+ c2φ2(2)χ(2)+ c3φ3(2)χ(2)y(3) = c1φ1(3)χ(3)+ c2φ2(3)χ(3)+ c3φ3(3)χ(3)

. . . .

. . . .y(k) = c1φ1(k)χ(k)+ c2φ2(k)χ(k)+ c3φ3(k)χ(k),

(9)

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Fig. 2. Plotting of third order Laguerre-Volterra kernels. (Top) First orderkernels c1, c2 and c3. (Middle) Second order kernels c11, c22 and c33.(Bottom) Third order kernels c111, c222 and c333. Dark green denotesc1, c11 and c111, blue c2, c22 and c222 and light green c3, c33 and c333.Computational analysis is done with the support of MATLAB.

where only the first order is used for this exercise, thus ci

should be extracted. Note that the kernels can be factorized

by writting (9) in a compact form as follows:

y(1)y(2)y(3)

.

.y(k)

=

φ1(1)χ(1) φ2(1)χ(1) φ3(1)χ(1)φ1(2)χ(2) φ2(2)χ(2) φ3(2)χ(2)φ1(3)χ(3) φ2(3)χ(3) φ3(3)χ(3)

. . .

. . .φ1(k)χ(k) φ2(k)χ(k) φ3(k)χ(k)

c1

c2

c3

(10)

where k takes the values from 1 to 130 (T = 20 s) and 1 to 87

(T = 20 s). The resulting matricial equation can be written as

Yk = Ak,3C3, (11)

and it is clear that Y is proportional to a non-quadratic matrix.

Multiplying both sides by AT

3,k one gets

AT

3,kYk = AT

3,kAk,3C3. (12)

Fig. 3. Behavior of temperature versus time for both model (dots) andreactor data (blue line) for a multilevel signal. We have used samples of 30 s.First, second and third order Laguerre-Volterra versus data are plotted in top,middle, bottom panels respectively. To note that the usage of second orderimproves notably the adjustability to data.

With this we can arrive to

C3 = (AT

3,kAk,3)−1A

T

3,kYk (13)

which is known as the Moore-Penrose pseudo-inverse matrix.

It is interesting to note that (13) is in agreement with the one

obtained in [9] by which the parameters are computed through

the optimal solution by minimizing the cost function. It should

be clarified that the coefficients (cj ,cjj) of the second order

model as well as those (ci,cii y ciii) of third order model were

obtained with a similar criterion to that used for extracting the

ones of the first order. All calculations to extract the ci, cii and

ciii were performed with the support of MATLAB [10]. In fig.

2 the evolution of ci, cii and ciii with respect to samples is

displayed. It is interesting to note that all panels are featured

by a stability that is reached after 47 samples in average, where

the convergence takes place.

III. RESULTS AND INTERPRETATION

We have collected data in samples of 20 s and 30 s from

20oC up to 80oC during 2500 s, approximately. The input in

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the analysis is defined as the initial manifestation inside the

production chain in plant. Thus a hardware was implemented

and it provides a 4-level pseudorandom signal in order to drive

a I/P transducer (4,20 mA) which controls a 3 way pneumatic

valve. It serves to regulate the flow of diathermic oil necessary

to heat the reactor. The main results are plotted in fig. 3. There

the behavior of Laguerre-Volterra model up to third order

together to reactor data tell us the excellent adjustability of

model with a minor discrepancy above 2000 s.

In top panel the first order model presents minor fluctuations

when is superposed to data as indicated at the range between

1000 and 2000 s. It should be noted the existence of a

discrepancy during the first 500 s. The second order model

and data are plotted in the middle. There a quite acceptable

agreement is observed. However a rough separation of model

to data appears after 2000 s.

The third order model at the bottom of fig. 3 is displayed.

Although there is a clear disagreement ranging between 1800

and 2500 s, the model is still realiable during the first 1000

s. On the other hand, one can see data is clearly featured

by 4 phases: the first one between 0 and 300 s, appears as

consequence of an early stage of condensation intrinsically

ascribed to melamine resin reactor. In a second stage ranging

between 300 and 1700 s, data exhibit solely heating. A third

one between 1700 and 2200 s, reveals exothermic reactions.

Finally, the last stage reaching 840C beyond 2500 s, the heat

provided to reactor is turned off. Although exist there a small

discrepancy in a 5% it is observed that the model in up to third

order have demonstrated an excellent compatibility to data. In

fig. 4 all coefficients for both 20 s and 30 s, are listed. A small

difference for those values computed with different poles is

viewed, but a noteworthy fact is that of the convergence despite

of the fact that only diagonal terms from the multidimensional

Laguerre-Volterra matrix are used.

IV. CONCLUSIONS

In this paper, we have concentrated on the effect in to

incorporate only diagonal terms from a generalized Laguerre-

Volterra model into a reduced scheme of up to third order

capable to model the temperature of a bath reactor which is

featured by several phases.

The model exhibits a notable adjustability to data in its third

order of approximation, whereas in a second order a similar

compatibility is found. The case when diagonal components

are kept in analysis might represent an interesting option in

order to avoid time consuming in those cases where a huge

amount of kernels should be computed.

Although one can observe a small discrepancy of up to 5 %

interesting prospects of the model to automate a batch reactor

is clearly evident. In a future work, Chebyshev polynomials

or similar, will be used in order to find advantages and

disadvantages against our proposed model.

Fig. 4. Values of kernels up to third order obtained for the Laguerre-Volterramodel used in this work.

ACKNOWLEDGMENTS

The authors would like to thank to PISOPAK PERU S.A.C

where the experimental tests were performed. H.N-Ch is

very grateful to the nice atmosphere found in the FIEE-UNI

during the academic year 2008, and therefore thanks to F.

Merchan, D. Carbonel, J. Betetta and A. Rocha, as well as to

postgraduate section staff Mrs. D. Rojas and Mrs. M. Alvarez.

Finally, the authors would like to thank to Mrs. K. Mesıa

whom have help us with the technical details concerning this

manuscript in an early version.

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