Intelligent Control and Automation, 2011, 2, 226-232
doi:10.4236/ica.2011.23027 Published Online August 2011 (http://www.SciRP.org/journal/ica)
Nonlinear Multiple Model Predictive Control of Solution
Polymerization of Methyl Methacrylate
Masoud Abbaszadeh
Department of El ectri cal and C om puter Engineerin g, University of Alberta, Edmonton, Canada
Received December 29, 2010; revised June 10, 2011; accepted June 17, 2011
Abstract
A sequential linearized model based predictive controller is designed using the DMC algorithm to control the
temperature of a batch MMA polymerization process. Using the mechanistic model of the polymerization, a
parametric transfer function is derived to relate the reactor temperature to the power of the heaters. Then, a
multiple model predictive control approach is taken in to track a desired temperature trajectory. The coeffi-
cients of the multiple transfer functions are calculated along the selected temperature trajectory by sequential
linearization and the model is validated experimentally. The controller performance is studied on a small
scale batch reactor.
Keywords: Model Predictive Control, Methyl Methacrylate, Nonlinear Multiple Model Control,
Polymerization
1. Introduction
The importance of effective polymer reactor control has
been emphasized in recent decades. Kinetic studies are
usually complex because of the nonlinearity of the proc-
ess. Hence, the control of the polymerization reactor has
always been a challenging task. Due to its great flexibil-
ity, a batch reactor is suitable to produce small amounts
of special polymers and copolymers. The batch reactor is
always dynamic by its nature. A good dynamic response
over the entire process is necessary to reach an effective
controller performance. To do so, it is essential to have a
suitable dynamic model of the process. Louie et al. [1]
reviewed the gel effect models and their theoretical
foundations. These researchers then modeled the solution
polymerization of methyl methacrylate (MMA) and
validated their model.
Control of MMA polymerization processes has be-
come popular as a benchmark for advaced process con-
trol methods, since the dynamics of the methyl
methacrylate polymerization process is well studied and
several physical models of high fedility are readily
avaliable. Methyl methacrylate is normally produced by
a free radical, chain addition polymerization. Free radical
polymerization consists of three main reactions: initia-
tion, propagation and termination. Free radicals are
formed by the decomposition of initiators. Once formed,
these radicals propagate by reacting with surrounding
monomers to produce long polymer chains; the active
site being shifted to the end of the chain when a new
monomer is added. Rafizadeh [2] presented a review on
the proposed models and suggested an on-line estimation
of some parameters, such as heat transfer coefficients.
The model consists of the oil bath, electrical heaters,
cooling water coil, and reactor. Mendoza-Bustos et al. [3]
derived a first order plus dead time transfer function for
polymerization. Then, they designed PID, Smith predic-
tor, and Dahlin controllers for temperature control. Pe-
terson et al [4] presented a non-linear predictive strategy
for semi batch polymerization of MMA. Penlidis et al. [5]
presented an excellent paper, in which they reviewed a
mechanistic model for bulk and solution free radical po-
lymerization for control purposes. Soroush and Kravaris
[6] applied a Global Linearizing Control (GLC) method
to control the reactor temperature. They compared the
result of GLC and PID controllers. Performance of the
GLC for tracking an optimum temperature trajectory was
found to be suitable. DeSouza Jr. et al. [7] studied an
expert neural network as an internal model in control of
solution polymerization of vinyl estate. The architecture
of their model predicts one step ahead. In their study,
they compared their neural network control with a classic
PID controller. Clarke-Pringle and MacGregor [8] stu-
died the temperature control of a semi-batch industrial

M. ABBASZADEH227
reactor. They suggested a coupled non-linear strategy
and extended Kalman filter method. They used energy
balance approach for the reactor and jacket to estimate
process parameters. Mutha et al. [9] suggested a non-
linear model based control strategy, which includes a
new estimator as well as Kalman filter. They conducted
experiments in a small reactor for solution polymeriza-
tion of MMA. Rho et al. [10] reviewed the batch polym-
erization modeling and estimated the model parameters
based on the experimental data in the literature. For con-
trol purposes, they assumed a model to pursue the con-
trol studies and estimated the parameters of this model
by on line ARMAX model.
Model predictive control (MPC), on the other hand, is
a model based advanced control technique that have been
proved to be very sussefull in controlling highly complex
dynamic systems. It naturally supports design for MIMO
and time-delayed systems as well as state/input/output
constiants. MPC is generally based on online optimiza-
tion but in the case of unconstrianed linear plants, closed
form solutions can be derived analytically. MPC usally
requires a high computaional power; however, since
chemical processes are typically of slow dynamics, they
have been designed and implemented on various chemi-
cal plnat with great success. Therefore, MPC seems to be
good candicate for controlling MMA polymerization
based on physical (first-principle) modeling.
This paper presents a mechanistic model of batch po-
lymerization. Sequential linearization, along a selected
temperature trajectory, is conducted. Consequently, us-
ing a nonlinear model predictive approach, a controller is
designed. A multiple model adaptive MPC controller is
desined for the trajectory lineairzed model. Results show
the better performance than the performance of adaptive
PI controller [11].
2. Polymerization Mechanism
Methyl methacrylate normally is produced by a free
radical, chain addition polymerization. Free radical po-
lymerization consists of three main reactions: initiation,
propagation and termination. Free radicals are formed by
the decomposition of initiators. Once formed, these radi-
cals propagate by reacting with surrounding monomers
to produce long polymer chains; the active site being
shifted to the end of the chain when a new monomer is
added. During the propagation, millions of monomers are
added to 1 radicals. During termination, due to reac-
tions among free radicals, the concentration of radicals
decreases. Termination is by combination or dispropor-
tionation reactions. With chain transfer reactions to
monomer, initiator, solvent, or even polymer, the active
free radicals are converted to dead polymer [1]. Table 1
o
P
Table 1. Polymerization mechanism.



1
2
2
2
d
i
ti
o k
dd
oo o k
ii
oo k
ti ti
IRG RkI
R
MP RkRM
RI RkR

 
 


 

Initiation


1
d
oo k
nnpp
o
n
P
MP RkMP






Propagation
tc
tc
oo oo k
nmn mtctcnm
oo oo
k
nmnm tdtdnm
PP DRkPP
PPDDRkPP

 

 

Termination



1
f
s
k oo
nnff
oo k
nnss
PMPD RkMP
PMsD RkMP
 o
n
o
n









Transfer
gives the basic free radical polymerization mechanism.
The free radical polymerization rate decreases due to
reduction of monomer and initiator concentration. How-
ever, due to viscosity increase beyond a certain conver-
sion there is a sudden increase in the polymerization rate.
This effect is called Trommsdorff, gel, or auto-accelera-
tion effect. For bulk polymerization of methyl methacry-
late beyond the conversion, reaction rate and mo-
lecular weight suddenly increase. In high conversion,
because of viscosity increase there is a reduction in ter-
mination reaction rate.
20%
3. Mathematical Modeling of Polymerization
Table 2 shows the mass and energy balances of reactor.
The polymer production is accomplished by a reduction
in volume of the mixture. The volumetric reduction fac-
tor is given by:
p
m
p




 (1)
The instantaneous volume of mixture is given by:
Table 2. Mass and energy balances.
 


0
0
0
d2 1
d
dpf
xfk
I
kk x
tMV







dd
dd
d
II
V
kI
tV
 t





0
dd
dd
s
SS V
kS
tV
  t
0
dd
dd
m
VM x
x
tt
d
dt





 








0
d
d
ppp j
r
T
mCH kMV UAT TUAT T
t



 
 
d
d
o
j
opj j
ro
T
mCPP UATTUATT
t



 

0
2d
t
f
kI
k
 

M. ABBASZADEH 228

01
m
M
Vx




 (2)

1
exp 1
p
p
DAB






 




(5)
The parameter

is defined as:
Similarly, termination rate constant, , is given by:
t
k
1
s
s
f
f
  (3)
0
0
11
t
tt
kk D

  (6)
During the free radical polymerization, the cage, glass,
and gel effects occur. For the cage effect, the initiator
efficiency factor is used. The CCS (Chiu, Carrat, and
Soong) model is used in this study to take into considera-
tion the glass and the gel effects. Therefore, propagation
rate constant,
p
k, is changing according to:
0
t is changing as Arrhenius function.
k
p

and t

are adjustable parameters related to propagation and ter-
mination rate constants, respectively. All the other nec-
essary parameters and constants for this model are given
in the literature [1]. The Equations (7)-(10) are essential
for dynamic studies.
0
0
11
p
pp
kk D

  (4)



 
1
2
d1
d
,,
d
pf t
f
kI
xkk x
tk
fxIT
 

(7)
0
p
k is changing as Arrhenius function, and is
given by equation:
D


  



 
2
d2
1
d1
d
dpf
t
If
kIk k IxfxIT
tx k


 


, ,
kI
(8)
 




 
00
3
2
1
d,,,
d
d
pp j
r
t

j
p
fk I
HkVMxUATT UATT
k
T
f
xITT
tmC


 

(9)




4
2
d,,
do
jj
jro j
op
PUATTUATT
T

f
TT P
tmC




 (10)
Equations (7) and (8) are mass balances for monomer
and initiator, respectively. Long Chain Approximation
(LCA) and Quasi Steady State Approximation (QSSA)
are used in this study. Equations (9) and (10) show en-
ergy balances for the reactant mixture and oil, respec-
tively. In this study, heat transfer coefficients are esti-
mated experimentally [2]. Equations (7)-(10) are highly
nonlinear and, using Taylor expansion series, these equa-
tions were converted to linearized form. The linearized
state space form is given by:



11 1
22 21
33
0
d
d0
d00
d0
d2
d
d
d00
ss
s
ss
s
rr
j
spp
sopo
j
ro r
opo opo
ff f
XxI T
tX
ff f
ixI Ti
tT
TUA UAUA
ff
tT
x ImCmCmC
TUA UA
UA
tmC mC




 

 






 

 
 


 
 
 


 

 

 

 


 




0010
j
P
X
i
TT
T























(11)

M. ABBASZADEH 229
where
 






,,
,
s
s
jjjss
,
s
X
xxiI IT TT
TTT PPP

 

  (12)
Equation (11) and is converted to the transfer function
form:
2
345
432
1234
()
()
nsns n
Ts
Psdsdsds dsd


5
(13)
4. The Experimental Setup
A schematic representation of the experimental batch
reactor setup is shown in Figure 1. The reactor is a Bu-
chi type jacketed, cylindrical glass vessel. A multi-pad-
dle agitator mixes the content. A Pentium II 500 MHz
computer is connected to the reactor via an ADCPWM-
01 analog/digital Input/Output data acquisition card. The
data acquisition software was developed in-house. The
heating oil was circulated by a gear pump and its flow
rate was about 15 minlit . The heating/cooling system
of the oil consisted of two 1500W electrical heaters and a
coolant water coil, which was operated by an On/Off
Acco brand solenoid valve. Two Resistance Temperature
Detectors (RTDs), were used with accuracy of .
Methyl methacrylate and toluene were used as monomer
and solvent, respectively. Benzoyl peroxide (BPO) was
used as the initiator. The molecular weight of the pro-
duced polymer was measured using an Ubbelohde vis-
cometer.
o
0.2 C
5. An Overview of MPC
Due to its high performance, Model predictive control
method has recieved a great deal of attention to control
chemical processes, in last few years. This approach is
applicable to multivariable systems and canstrained sys-
tems. Monuverability in design, noise and disturbance
rejection and robustness under model mismatch are the
most important ability of this method. Cumbersome
Motor
Data acquisition
card
T
T
Polymerization
reactorOil bath
Coolant water
Oil circulating pump
Control
Figure 1. The experimental setup.
computation, lack of systematic rules for controller tun-
ing are some drawback of this method. Model predictive
control is based on a process model. Although impulse or
step responses have some limitation for nonlinear proc-
ess, they may be used to develop a model. During the the
model predictive control following steps should be con-
ducted:
 Explicit prediction of future output (prediction hori-
zon).
 Calculation of a control sequence based on the mini-
mized cost function (control horizon).
 Receding strategy.
The Dynamic Matrix Control (DMC) is used in this
research. Its cost function is:
 
1
1
22
11
1
NP M
QR
iN i
Jeti uti

 

 (14)
where P, M and N1 are prediction horizon, control hori-
zon and pure time delay, respectively, ,
M
MP
RQ
P
are
whigthing martices. The prediction horizon must be at
least equal to the pure time delay.




 
dp
dm
et iyt iyt i
ytiytidti
 


 (15)
where yp is the process output, ym is the model output and
d is the process and model outputs diffrence, including
noise, disturbance and model mismatch. yd(t) is the de-
sired output based on the refrence input. If ysp(t) is the
refrence input, the following filtered form is used as the
tracking trajectory:






11(); 0
dd sp
yt ytyt  1 

  (16)

changes the first order smoothing filter pole place.
The smaller

the faster output. It has been shown that
system robustness can be decreased by the reduction of

and increment of the manipulated signal [12]. Figure
2 shows the block diagram of DMC.
The cost function in equation 10 can be rearrenged to:

TT
mDmD
J
YDYQYDY URU

  (17)
without loss of generality, if N1 is assumed zero, then:
Model -
ym
ysp
M , P
System
d
R , Q α
yp
+
u(i-1)
Δu(i-1)
Optimization
+
u(i)
Figure 2. Block diagram of DMC.

M. ABBASZADEH 230

For LTI system, without any constraints on output or
co
where:

1T YytytP   
mm m

 
1T
Dd d
Yyt ytP  


 

1,
1
T
T
UututM
Ddt dtP


 



ntrol signal, optimization has the following closed form:

1
TT
UG QGRG QE
 (18)
  
 
P
D
EYY

 (19)


s are the step response samples and G+ is a Toeplitz
m
N


(20)
where N is the number of system step response samples
 
1
1
,
1
P
T
g
gg
G
g
UUutut M

















i
g
atrix consisting the step response samples. The model
output has calculated by:
YGUG
 
12 1
21
,
ut 1ut1
mN
N
N
p
T
UgU
gg g
gg
G
g
UN
 












 




 

reaching to steady state or equvalent impulse response
steps which lead to zero; and gN is the system dc gain.
N
g
dcgain
 
11
T
N
U utNutNutNP  


6. The Modified DMC
If there is any pole close to origin, the step response will
-
co
be very slow and the required N is very large. Then, a
system including integrator never reaches to the steady
state (this case exists in the set of linearized models of
the MMA reactor) and N lead to infinity. Hence, unsta-
bility occurs. This is one of the DMC limitations [13].
Researchers have suggested some methods to over
me this problem, for example formulating DMC in the
state space an then using an state observer [14]. Because
of model mismatch this method doesn’t have proper per-
formance in real time applications. The alternative is:
mNN
P
astNNmPast
YG
UgU YGUY
 

 
  
YGUGUgU
(21)
where
P
ast
Y
utp
is “the effect of past input to the future
system outs without considering the effect of present
and future inputs”. Consequently,
P
ast
Y can be calcu-
lated by setting the future “Δu”s equ zero and solv-
ing the model P steps ahead.
UY
al to
mPas
Y


t  (22)
As seen in equations 14 and 15, G+ and U are inde-

pendent of N. G

dimension is determined by N. There-
fore, the DMC culation is independent than N. YD is:
 
1T
cal
Dd d
Yyt ytp




 (23)
7. Results and Discussion
Figure 3 shows the model validation results. The simu-
lation follows the experimental data very well. The DMC
algorithm was applied to control a MMA polymerization
reactor. The reactor temperature trajectory is known,
hence, the refrence input is known for all times so the
programmed MPC is used. The DMC controller gain
defined as:


1
TT
DMC
K
GQGR GQ

 
 (24)
DMC
UkE



The
(25)
G

ch
of present model is useDMC
kD
d to calculate its k.
MC is anged in the appropriate model switching in-
stant. Therefore, a multiple model strategy is used im-
plicitly. However the valid model is known before.
020406080 100 120 140 160
50
55
60
65
70
75
80
85
90
95
100
Ti me,min
Reac tor temperature
E x perim ent al dat a
S im ulat ion res ult
Figure 3. Model validation.

M. ABBASZADEH231
Figures 4 an ity to track the
te
d 5 show the controller abil
mperature trajectory. The average error is 0.3˚C. Due
to the controller robustness, switching between models
causes no unstability in closed loop system. Furthermore,
appropriate selection of controller parameters could pre-
vent the unstability. The selected sampling period is T =
10s. Other parameters are 5P, 2
M
, 0.05


,
5*5
QI, 3*3 .05*RI. Thetiipl
nsures the reactor temperature
tracking error to with in 0.3˚C while the adaptive PI con-
trol in [11] has a 2˚C average error and the Generalized
Takagi-Sugeno-Kang fuzzy controller proposed in [16]
has a 1˚C average error; demonstrating the superior per-
formance of the MPC.
adapve multle mode
MPC designed here e
0500010000 15000
75
80
85
90
95
100 output
0500010000 15000
0
500
1000
1500 c ont rol s ignal
Ti me
(
sec
)
Figure 4. Controller performance in the absence of distur-
bance and noise.
0500010000 15000
75
80
85
90
95
100 output
0500010000 15000
0
100
200
300
c ont rol si gnal
Time(sec)
Figure 5. Controller performance in the presense of ste
zed model based predictive controller
based on the DMC algorithm was designed to control the
. C. Carratt and D. S. Soong, “Modeling
the Free Radical Solution and Bulk Polymerization of
p
disturbance (dashed line) and Guassian measurem e nt noise.
8. Conclusions
A sequential lineari
temperature of a batch MMA polymerization reactor.
Using the mechanistic model of the polymerization, a
transfer function was derived to relate the reactor tem-
perature to the power of the heaters. The coefficients of
the transfer function were calculated along the selected
temperature trajectory by sequential linearization. The
controller performance was studied experimentally on a
small scale batch reactor.
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