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Adaptive fuzzy model based inverse controller design using BB-BC
optimization algorithm
Tufan Kumbasar, Ibrahim Eksin
⇑
, Mujde Guzelkaya, Engin Yesil
Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department, Maslak, TR-34469 Istanbul, Turkey
article info
Keywords:
Inverse model based control
Fuzzy model inversion
Fuzzy logic controller
Big Bang-Big Crunch optimization
Heat transfer process
pH process
abstract
The use of inverse system model as a controller might be an efficient way in controlling non-linear
systems. It is also a known fact that fuzzy logic modeling is a powerful tool in representing nonlinear
systems. Therefore, inverse fuzzy model can be used as a controller for controlling nonlinear plants. In
this context, firstly, a new fuzzy model based inverse controller design methodology is presented in this
study. The design methodology introduced here is based on a recursive optimization procedure that
searches for an optimal inverse model control signal at every sampling time. Since the task of optimiza-
tion should be accomplished in between two sampling periods the use of a fast optimization algorithm
becomes essential. For this reason, Big Bang-Big Crunch (BB-BC) optimization algorithm is used due to its
low computational time and high global convergence properties. Even though, inverse model controllers
may produce perfect control while operating in an open loop fashion, this open loop control would not be
sufficient in the case of modeling mismatches or disturbances that might occur over the system. In order
to overcome this problem, secondly, an on-line adaptation mechanism via BB-BC optimization algorithm
is introduced in addition to BB-BC optimization based fuzzy model inverse controller. The adaptation
mechanism is used to update the related parameters of the model while minimizing the absolute value
of the instantaneous error between the system and model outputs. In this manner, the system output is
somehow fed back, the overall control form can be considered as a closed-loop system. The new fuzzy
model based inverse control scheme with the new online adaptation mechanism has been implemented
and tested on the two real time processes; namely, heat transfer and pH processes and very satisfactory
results has been reported.
Ó2011 Elsevier Ltd. All rights reserved.
1. Introduction
The use of the inverse system model might be an efficient way
in the control of non-linear systems. Even though necessary and
sufficient existence conditions of the inverse of a non-linear
system cannot been guaranteed, some numerical methods have
been proposed which generates the inverse of non-linear systems
(Economou, Morari, & Palsson, 1986). It is also a known fact that
fuzzy modeling can be used in the representation of highly non-
linear processes. In this context, describing a non-linear system
by a fuzzy model may provide an opportunity to design an effective
controller for the process using its inverse (Babuska, 1998).
There exist various fuzzy inversion methods developed in liter-
ature. Recently different fuzzy inversion techniques for certain
fuzzy models have been suggested that can only be applied under
certain limitations. Babuska, Sousa, and Verbruggen (1995)
claimed that under some specific invertibility conditions the exact
inverse of a fuzzy model can be attained when the output of fuzzy
model of the system is chosen as singleton type. When these
conditions are not satisfied the fuzzy model must be decomposed
into invertible parts (Baranyi, Korondi, Hashimoto, & Wada, 1997).
The fuzzy inversion method given in Varkonyi-Koczy, Amos, and
Kovicshizy (1999), and Varkonyi-Koczy, Péceli, Dobrowiecki, and
Kovácsházy (1998) does not require any invertibility conditions
to be satisfied. In this method, the role of the output and one of
the inputs of a multi-input single-output fuzzy model is inter-
changed and the arising non-linear equation is solved iteratively.
Furthermore, another method is proposed in Boukezzoula, Gali-
chet, and Folloy (2003, 2007), in which the fuzzy model is decom-
posed into fuzzy meshes and the inverse of the global fuzzy system
is obtained through inversion of each fuzzy mesh. In this inversion
method, there is again no need to check any invertibility condition.
Moreover, in Abonyi, Andersen, Nagy, and Szeifert (1999), Youssef,
Yousef, Sebakhy, and Wahba (2009), Chen and Yu (2009), and You-
sef, Elkhatib, and Sebakhy (2010), the inverse model is obtained
directly mapping the output and input data of the process via neu-
ral networks or fuzzy logic approach.
In this study, firstly, a new on-line fuzzy inverse controller
design methodology is presented which does not require any
invertibility conditions or any limitations on the form of the fuzzy
process model. The design methodology introduced here is based
0957-4174/$ - see front matter Ó2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2011.04.015
⇑
Corresponding author. Tel.: +90 212 2853500; fax: +90 212 2852920.
E-mail address: eksin@itu.edu.tr (I. Eksin).
Expert Systems with Applications 38 (2011) 12356–12364
Contents lists available at ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa
on a recursive optimization procedure that searches for an optimal
inverse model control signal at every sampling time. Therefore, the
task of optimization should be accomplished in between two sam-
pling periods which in return makes the use of a fast optimization
algorithm essential.
Genetic Algorithms (GA) are the most commonly used global
search methods. They are considered to be very effective non-linear
optimization tools generally for off-line purposes. Even though, it
has been pointed out in Lennon and Passino (1999), and Moore,
Musacchio, and Passino (2001) that GA can also be used within an
adaptive control loop in an online fashion, it is still underlined that
GA based controllers are not feasible for real-time control purposes
because of the long computation time required (Wang, Cheng, &
Leu, 2004). Recently, a new evolutionary computation algorithm
named as the Big Bang-Big Crunch (BB-BC) optimization algorithm
has been developed in Erol and Eksin (2006) and the main advan-
tage of BB-BC is its high convergence speed and as a consequence
low computation time. In that respect, the BB-BC optimization algo-
rithm is both appropriate and feasible for online implementations.
In this study, the BB-BC optimization algorithm is used to generate
the optimal control signal so as to minimize the error between the
reference input signal and the output of the model of the system
within every sampling period of time.
In addition to the BB-BC optimization based fuzzy model
inverse controller developed in this study, an on-line adaptation
of the model parameters is also accomplished via BB-BC optimiza-
tion algorithm in the case of process parameter variations and/or
disturbances. It is a very well-known fact that when a perfect
inverse model of a process is obtained, a perfect control in an open
loop form could be accomplished using the inverse model as the
controller of the process as long as there does not exist any param-
eter variations and/or disturbances acting on the system. Here, the
optimization algorithm would try to update the related parameters
of the model while minimizing the absolute value of the instanta-
neous error between the system and model outputs. In this man-
ner, since the system output is somehow fed back, the overall
control form can be considered as a closed-loop system and a ‘‘bet-
ter’’ system performance would be obtained. As it is mentioned
above, fuzzy models can represent nonlinearities arising in
systems; therefore, in this study, the models for the processes
taken under considerations are taken as fuzzy ones. Therefore,
the inverse model of the process obtained via BB-BC algorithm
can be seen as a fuzzy inversion operation.
This paper is organized in six sections excluding the conclusion
section. The BB-BC optimization algorithm is explained in Section 2.
Next, the basics of inverse controller structures are discussed in
Section 3. Then, in Section 4, the newly proposed BB-BC optimiza-
tion based fuzzy model inverse controller (BBBC-FMIC) scheme is
introduced with a simulation example. In Section 5, adaptive fuzzy
model inverse control structure is elaborated for the case of model
mismatches or disturbances in process where the deviations are
also optimized using BB-BC optimization algorithm. The method
has also been tested on a simulation example as in the previous
case described in Section 4. Finally, a heat transfer process (Heat
Transfer Process Trainer PT-326) and pH process (pH Value Control
Trainer RT-552) are introduced and the proposed BB-BC based
adaptive and nonadaptive fuzzy model inverse control schemes
have been implemented and tested on these processes in Section
6. The results have been evaluated and discussed in the conclusion
section.
2. Big Bang-Big Crunch optimization
The Big Bang-Big Crunch (BB-BC) algorithm is a global optimiza-
tion method that has been built on two main steps: the first step is
the ‘‘Big Bang’’ phase where candidate solutions are randomly dis-
tributed over the search space and the next step is the ‘‘Big Crunch’’
phase where a contraction procedure calculates the center of mass
for the population (Erol & Eksin, 2006). The initial Big Bang popu-
lation is randomly generated over the entire search space just like
the other evolutionary search algorithms. All subsequent ‘‘Big
Bang’’ phases are randomly distributed about the center of mass
or the best fit individual in a similar fashion.
After the ‘‘Big Bang’’ phase, a contraction procedure is applied as
the ‘‘Big Crunch’’ phase to form a center or a representative point
for further ‘‘Big Bang’’ operations. In this phase, the contraction
operator takes the current positions of each candidate solution in
the population and its associated cost function value and computes
a center of mass. The center of mass can be computed as:
x
c
¼P
N
i¼11
f
i
x
i
P
N
i¼11
f
i
ð1Þ
where x
c
is the position of the center of mass, x
i
is the position of the
candidate, f
i
is the cost function value of the ith candidate, and Nis
the population size. Instead of the center of mass, the best fit indi-
vidual can also be chosen as the starting point in the ‘‘Big Bang’’
phase. The new generation for the next iteration ‘‘Big Bang’’ phase
is normally distributed around x
c
. The new candidates around the
center of mass are calculated by adding or subtracting a normal ran-
dom number whose value decreases as the iterations elapse. This
can be formalized as
x
new
¼x
c
þr
a
ðx
max
x
min
Þ
kð2Þ
where ris random number;
a
is a parameter limiting the size of the
search space, x
max
and x
min
are the upper and lower limits; and kis
the iteration step. In Erol and Eksin (2006), the working principle of
this evolutionary method is explained as to transform a convergent
solution to a chaotic state which is a new set of solutions.
3. Inverse model based controllers
The simplest way to design a model based controller for pro-
cesses is to set up the inverse model of the system as the controller
(Babuska, 1998). Let us consider a Single Input-Single Output
(SISO) system model without transport delays. The model can be
expressed as
yðkÞ¼fðxðk1Þ;uðk1ÞÞ ð3Þ
where x(k1) are the inputs of the previous state, u(k1) is the
previous input, y(k) is the current output and frepresents the lin-
ear/non-linear relationship of the mapping. The objective of inverse
control is to calculate the control input such that the systems out-
put at the next sampling time will eventually become equal to
the desired reference signal that can be formulated as:
uðkÞ¼f
1
ðxðk1Þ;rðk1ÞÞ ð4Þ
Since it is desired that the system output will converge to the refer-
ence signal, y(k) could naturally be substituted by r(k) in Eq. (4).In
this fashion, the inverse model has been used as an open loop feed
forward controller since state x(k) is updated using the output of the
model. If x(k) is updated via output of the process, the inverse con-
troller is then called as an open loop feedback controller (Babuska,
1998). Thus, as illustrated in Fig. 1, the difference between these
two methods is in the way of how the states are updated.
It is usually difficult to find the inverse function analytically,
thus numerical optimization methods have been utilized to form
this function in a recursive manner. In this recursive numerical
optimization procedure the objective function to be minimized
can be defined as follows:
J¼jrðkÞyðkÞj ð5Þ
T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364 12357
Then, the solution of this optimization problem provides one the
optimal inverse model signal of the inverse function or the best
approximation. Therefore, the control algorithm introduced here
is based on an optimization procedure that has been searching for
an optimal inverse model control signal at every sampling time
(Kumbasar, Eksin, Guzelkaya, & Yesil, 2009). For this reason, the
optimization algorithm must have a high convergence speed and
low computation time. The fitness function has been chosen so that
the error between the model and reference signal minimized. How-
ever, if the model does not match the process perfectly or an input/
output disturbance is applied, a steady state error will be inevitable.
In order to compensate this error, an optimization based online
adaptation mechanism is proposed. As the parameters of the pro-
cess changes, the online adaptation mechanism will update the
model parameters via an optimization procedure. As the new
parameters are transferred to the inverse model controller, the con-
troller generates the new optimal control signal for the changed
process. Thus, the inverse model controller that can be considered
as an open loop control scheme has been transformed into some
kind of a closed loop control scheme which is robust to disturbances
and parameter changes in the process. An adaptive inverse control-
ler structure given in Fig. 2 has been obtained when the proposed
optimization based techniques are combined together.
4. BB-BC based fuzzy model inverse control (BBBC-FMIC)
scheme
In this proposed control structure, the inverse model control
signal generation is handled as an optimization problem. A fuzzy
model has been used to represent process dynamics, which is usu-
ally non-linear, and the optimization problem has been defined as
minimization of the error between the fuzzy model output and the
reference signal (Kumbasar, Eksin, Guzelkaya, & Yesil, 2008). If the
fuzzy model matches the process dynamics perfectly, then one can
assume that the process output will converge to the set point.
Here, the optimization problem that will provide the required con-
trol signal is handled using the nature inspired global BB-BC Algo-
rithm. An online implementation of this evolutionary algorithm
has been feasible at each sampling time due to the simplicity
and high convergence speed of the algorithm (Erol & Eksin, 2006).
The cost function J
1
which is minimized at every sampling time
is chosen as:
J
1
¼jrðkÞy
m
ðkÞj ð6Þ
This optimization problem of J
1
gives the optimal inverse model sig-
nal or the best approximation defined as u(k). Then, that optimal
control signal is applied to the process. The proposed BBBC-FMIC
scheme is given in Fig. 3.
In the case of existence of a transport delay in the process, a pre-
dicted fuzzy model should be used to eliminate effect of the time
delay. If the process has no time delay, then the normal or unpre-
dicted fuzzy model would be used in the optimization procedure.
4.1. Simulation example for BBBC-FMIC scheme
A simulation study has been performed initially in order to
show the effectiveness of the basic BB-BC based fuzzy model
inverse control (BBBC-FMIC) scheme which is an open loop
controller. In this simulation, a first order system plus dead time
(FOPDT) system with the following transfer function is chosen:
GðsÞ¼ 0:87
0:52sþ1e
0:2s
ð7Þ
Then, in order to apply the BBBC-FMIC scheme a fuzzy model of the
process is attained using the MATLAB/ANFIS toolbox. For this simu-
lation, the sampling time is chosen as 100 ms.
The population size and the number of iterations for the BB-BC
optimization method have been chosen as 20 which is a small and
limited number. Even for such a choice satisfactory results have
been obtained within each sampling time interval; and that clearly
shows the high convergence speed of this algorithm. Increasing the
population size and number of iterations may provide much more
accurate results; however, because of the hardware limitations, the
processor may fail to finish the calculations within the sampling
time in real time applications.
Since the BB-BC optimization method is a stochastic random
search technique each run might provide a different result. To
show this characteristic, three different system responses have
been obtained for three different trial runs and they are shown
in Fig. 4. Moreover, as it is expected and discussed above, in the
case of a disturbance or model mismatch cases, the system output
will naturally not converge to the reference signal as illustrated in
Fig. 1. (a) Open loop feed forward inverse control scheme and (b) open loop
feedback inverse control scheme.
Fig. 2. Inverse control scheme with an adaptation mechanism.
Fig. 3. BB-BC based fuzzy model inverse control (BBBC-FMIC) scheme.
12358 T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364
Fig. 5, since the proposed controller has an open loop
characteristic.
5. BB-BC based adaptive fuzzy model inverse control structure
In order to cope with disturbances and model mismatches
BBBC-FMIC structure should have a feedback or an adaptation
mechanism. For this purpose, firstly the adaptation mechanism
described in this section is proposed and then the proposed adap-
tation mechanism is fused with the basic fuzzy model inverse con-
trol (BBBC-FMIC) scheme discussed in Section 4.
5.1. On-line model adaptation mechanism
An on-line adaptation should be applied to the BB-BC based fuz-
zy model inverse control (BBBC-FMIC) scheme described above in
the case of a disturbance or in the case of a model mismatch that
might occur when the processes parameters changes permanently
or temporarily (Kumbasar, Yesil, Eksin, & Guzelkaya, 2008). In this
manner, the control scheme that has an open loop nature would
acquire a closed loop property.
The proposed BB-BC based online adaptation scheme is illus-
trated in Fig. 6. Here, it can easily be deduced that the model is
adapted when a difference between the real system and model
outputs occurs. This model adaptation is accomplished via the
Big Bang-Big Crunch optimization algorithm. This optimization
algorithm is again feasible for on-line use since the convergence
speed is fast and the computation time is low (Erol & Eksin,
2006). In order to reduce computation time further, only the con-
sequent parameters that are fired in rule base of the fuzzy model
have been chosen as the parameters to be optimized. The cost
function J
2
which has to be minimized in every sampling time is
chosen as:
J
2
¼jyðkÞy
m
ðkÞj ð8Þ
Fig. 4. System responses and control signals for three different trials.
Fig. 5. The system outputs for the three trial runs (a) in the case of input
disturbance and (b) in the case of parameter variation.
Fig. 6. The BB-BC based online adaptation scheme.
T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364 12359
5.2. Adaptive fuzzy model inverse control (BBBC-AFMIC) structure
Combining the BBBC-FMIC structure and the BB-BC based
online adaptation scheme described above, a closed loop control
structure, named as BB-BC based adaptive fuzzy model inverse
control (BBBC-AFMIC) scheme, could be obtained as illustrated in
Fig. 7. If there is a need for adaptation, the consequent parameters
of the fuzzy or predicted fuzzy models will be optimized and
updated via BB-BC optimization algorithm using the cost function
in Eq. (8). Afterwards, the BBBC-FMIC part will calculate new opti-
mal control signals using the cost function given in Eq. (6) for the
process with its updated and optimized cost function which trans-
forms the open loop control structure into a closed loop control
structure.
5.2.1. Simulation example for the BBBC-AFMIC scheme
A simulation study has been done on the same process model
given in Eq. (7) for this proposed control structure. First of all,
the case of a disturbance rejection performance of the proposed
BBBC-FMIC structure has been examined. For this reason, a step in-
put disturbance and a step output disturbance with the magnitude
of 0.1 applied in 3rd and 7th seconds, respectively. Then, in order
to examine the performance of the proposed scheme for model
mismatch case, the gain of the model has been decreased from
0.87 to 0.77 in the 10th second. As it can be seen from Fig. 8, the
system response is quite satisfactory for both cases. This fast evo-
lutionary algorithm made it easier with the proposed control struc-
ture to compensate very effectively the model mismatch and
disturbances in a short period of time.
6. Applications
In order to show the effectiveness of BB-BC based adaptive and
nonadaptive fuzzy model inverse control structures, two real time
control system applications have been performed. In the first appli-
cation, the real time Heat Transfer Process Trainer (PT326 – Feed-
back) is used. Since the time constant of this system is relatively
small, this real time application case is designed to demonstrate
that the inverse fuzzy model control signal can be generated with-
in a short sampling time and still the system output converges to
the desired reference successfully. Therefore, the applicability of
proposed methods to a fast system with a small nonlinearity has
been underlined.
Secondly, the proposed structures have also been applied to a
highly non-linear pH control process. In this case, the advantage
of using the fuzzy model for the system to be controlled has been
elaborated. It has been demonstrated that describing a highly non-
linear system by a fuzzy model provided an opportunity to make
an effective control process using its inverse model.
6.1. Application 1: Process Trainer PT326
The widely used heat transfer control experiment set Process
Trainer (PT 326) is also used here to test the new control strategy
and present its advantages (Dias & Dourado, 1999; Genc, Yesil,
Eksin, Guzelkaya, & Tekin, 2009; Ng & Cook, 1998; Pereira,
Henriques, & Dourado, 2000; Yesil, Guzelkaya, Eksin, & Tekin,
2007). Process Trainer PT 326 involves a tube through which air
is drawn from atmosphere by a centrifugal blower and the air is
heated as it passes over a heater grid before being released into
the atmosphere. Temperature control is achieved varying the elec-
trical power supplied to the heater grid. The mass flow of air
through the duct can be adjusted by setting the opening of the
throttle. The pure time delay depends on the position of the tem-
perature sensor element that can be inserted into the air stream
at any one of the three points spaced at 28, 140 and 280 mm away
from the heating point. In this study, the sensor element is placed
at the third place and therefore the allowable longest dead time is
achieved. The damper position is set to 40°. The system input u(k)
is the voltage applied to the power electronic circuit and the out-
put y(k) is the outlet air temperature.
6.1.1. Identification and modeling of Process Trainer PT-326
An identification experiment is applied to the heat transfer
plant PT-326 in order to find the fuzzy model of the process. For
that purpose, an input of multi-sinusoidal signal (signal with three
different frequencies and amplitudes) has been applied to the pro-
cess and the data has been collected via the experimental setup
Fig. 7. BB-BC based adaptive fuzzy model inverse control (BBBC-AFMIC) scheme.
Fig. 8. System response of the BBBC-AFMIC scheme in case of disturbance and
model mismatches.
12360 T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364
shown in Fig. 9. The regression vector is constructed by considering
u(k3) and y(k1) values and the collected input and output
data has then been trained by the MATLAB/ANFIS toolbox to obtain
the fuzzy model with sampling period of 100 ms. Thus, the output
vector y(k) can be expressed as follows:
yðkÞ¼fðuðk3Þ;yðk1ÞÞ ð9Þ
The consequent singleton parameters of the obtained fuzzy rule
base are given in Table 1. The membership functions of y(k1)
and u(k3) are designed as %50 overlapped triangular functions.
6.1.2. The results for PT-326 with the BB-BC basedfuzzy model inverse
controller (BBBC-FMIC)
When the BB-BC based fuzzy model inverse controller has been
applied to the PT-326 with the reference signal set to a tempera-
ture of 36 °C, and the results given in Fig. 10 have been obtained.
It should be noted again that since the proposed fuzzy model in-
verse controller is based on a stochastic search algorithm, the tran-
sient response of process response might vary for each trial. Fig. 10
demonstrates only one of the results attained by applying the open
loop version of the proposed method. The effect of disturbance has
not been considered in this case since it is obvious that the BB-BC
based inverse model fuzzy controller could not cope with distur-
bances due to its open loop nature.
6.1.3. The results for PT-326 with the BB-BC based adaptive fuzzy
model inverse controller (BBBC-AFMIC)
Firstly, an input step disturbance and an output disturbance
with the magnitude of 1.5 V have been applied at 5th and 9th sec-
ond, respectively. As it is illustrated in Fig 11, the proposed BBBC-
AFMIC structure has compensated these disturbances in a very
short period of time as expected. Secondly, the damper position
of PT-326 set has been changed from 40°to 60°in order to examine
a model-mismatch case. Changing the damper position in such
way at 14th second has caused a sudden temperature decrease
in system output; however, this has also been compensated and
the system output has converged to the set point value very
quickly as seen in Fig. 11.
6.2. Application 2: The pH Value Control Trainer GUNT RT-552
The control of the neutralization process has been studied for
several years but still remains to be a challenging problem (Wright
& Kravaris, 1991). The pH process inherits nonlinearity and high
sensitivity near the neutralization point (Pishvaie & Shahrokhi,
2006). Several control techniques have been proposed lately rang-
ing from linear to non-linear controllers. Most of the successful lin-
ear controllers use multiple linear models (Nystrom, Sandstrom,
Gustafsson, & Toivonen, 1998). In these approaches, the controller
design is based on solving linear quadratic (LQ) problem, and then
combining the controllers via gain scheduling methods. Even
though the linear controllers are easy to implement, the best
performances are obtained via non-linear controllers because of
the high nonlinearity inherited by the neutralization process. An
Fig. 9. (a) A view of the Process Trainer PT-326 and real time control equipment
and (b) schematic working principle of the PT326 Process.
Table 1
Fuzzy rule base for PT-326 Heat Transfer Process.
u(k3)/y(k1) NB NS PS PB
NB 0.0017 0.0532 0.0532 0
NS 0.1600 0.2170 0.2743 0
PS 0.3085 0.3758 0.4246 0.4194
PB 0.6009 0.5366 0.5790 0.5864
Fig. 10. Results for PT-326 with the BBBC-IMFC: (a) the system output and (b) the
control signal.
T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364 12361
implementable version of the indirect adaptive non-linear control
strategy is proposed by Henson and Seborg (1994). Since the pro-
cess is highly non-linear, fuzzy control structures have also been
examined by the researchers. Babuska, Oosterhoff, Oudshoorn,
and Bruijn (2002) have proposed a fuzzy self-tuning PI controller
for pH control in a fermentation system where PI parameters are
tuned in an on-line fashion. Fuente, Robles, Casado, Syafiie, and Ta-
deo (2006) suggested a fuzzy PI controller which is designed based
on titration curve. Beside this, non-linear model predictive control
structures have also been applied by Mahmoodi, Poshtan, Jahed-
Motlagh, and Montazeri (2009).
The general view of the pH Value Control Trainer GUNT RT-552
and the schematic working principles are illustrated in Fig. 12. The
RT-552 provides a comprehensive experimental introduction to
the fundamentals of pH process control. A caustic solution is added
to fresh water by way of a metering pump and the value of this
solution is measured as pH
1
. Then, a basic solution is then added
as a neutralizing agent via a second metering pump and the pH va-
lue is measured as pH
2
after the chemical reaction occurred in the
pipeline. A digital industrial controller tries to adjust this pH
2
value
to a set point value by controlling flow rate of the second metering
pump. Here, the base flow rate is considered as the manipulated
variable (F
b
). The process reaction consists of a strong acid (HCl)
and strong base (NaOH) as a titrating stream. The operating condi-
tions and the parameters of the neutralization process are given in
Table 2.
6.2.1. Identification and modeling of RT-552
In the beginning, an identification experiment is applied to RT-
552 in order to find a fuzzy model of this process. For that purpose,
a multi-signal (three random number generators with different fre-
quencies and amplitudes and a sinusoidal signal) is used as input
signal of the process and data set of 3000 samples with the sam-
pling time of 4 s has been collected. The regression vector is con-
structed by considering pH(k1) and F
b
(k1) values and the
process is then approximated as
pHðkÞ¼fðpHðk1Þ;F
b
ðk1ÞÞ ð10Þ
The collected input and output data has then been trained by
fuzzy identification toolbox (Babuska, 1998) of MATLAB to obtain
the fuzzy model. In order to represent the nonlinearities of pH pro-
cess better, the membership functions of each input in the fuzzy
rule base are defined with six Gaussian membership functions
and the rule structure is defined in the following format:
Rule i: If pH(k1) is A
i
and F
b
(k1) is B
i
THEN pH(k)=a
i
pH(k1) + b
i
F
b
(k1) + c
i
The consequent parameters of obtained fuzzy rules are tabu-
lated in Table 3.
6.2.2. The results for RT-552 with BB-BC based fuzzy model inverse
controller (BBBC-FMIC)
In this real time application, the system to be controlled does
possess high nonlinearity related to the molarities of [H] (or
[OH]); thus the controller has to be tested under varying reference
values. The desired reference trajectories have been chosen as 6, 9,
Fig. 11. Results for PT-326 with the BBBC-AIMFC under disturbance and model
mismatch: (a) the system output and (b) the control signal.
Fig. 12. (a) A general view of RT552-pH Process Set and (b) schematic working
principle of the GUNT RT 552-pH Process Set.
12362 T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364
and 7 pH values, in this successive order. It can be seen from Fig. 13
that the BBBC-FMIC provides satisfactory performances for differ-
ent reference signals. However, it is obvious that a small steady
state error occurs since the fuzzy model could not match the real
pH process exactly. Beside this fact, this controller structure pro-
vides quite satisfactory results for varying reference signals. There-
fore, an on-line adaptation is inevitable in the case of a model
mismatch that might occur when the parameters of the process
changes permanently or temporarily.
6.2.3. The results for RT 552 with BB-BC based adaptive fuzzy model
inverse controller (BBBC-AFMIC)
In this real-time application, the BB-BC fuzzy model inverse
controller with the online adaptation has been tested again for
varying reference signals. Since the fuzzy model is updated via
adaptation mechanism included in this structure, the small steady
state error, which was occurring when BBBC-FMIC structure is
used, has been eliminated as illustrated in Fig. 14.
7. Conclusions
In this study, firstly, a new on-line fuzzy inverse controller
design methodology is presented which does not require any
invertibility conditions or any limitations on the form of the fuzzy
process model. The global evolutionary Big Bang-Big Crunch
(BB-BC) optimization algorithm is used to generate the optimal
fuzzy inverse model output as the control signal at every sampling
time. The low computational time and high convergence properties
of this optimization algorithm make it possible to calculate the
inverse model output signal within two sampling periods.
The proposed BB-BC based fuzzy model inverse control scheme
performs very well when a perfect fuzzy model is attained. The
simulation studies show that the open loop structure of the pro-
posed inverse controller causes a steady state error in case of a
model mismatch, external disturbances or parameter changes that
might occur within the system. In that case, a BB-BC based on-line
Table 2
Parameters of the pH neutralization process.
Symbols Description Value
VVolume of the CSTR 0.8 lt
F
a
Flow rate of the influent stream 1 l/s
F
b
Flow rate of the titrating stream 0–2.1 ml/s (%0–100)
C
a
Concentration of the influent stream 6.3096 10
4
M
C
b
Concentration of the titrating stream 13 10
4
M
Table 3
Fuzzy rule base for RT 552 pH Value Control Trainer.
Rule no. A
i
B
i
a
i
b
i
c
i
1 VVLOW VVLOW 0.98807 0.00045984 0.033246
2 VLOW VLOW 0.97485 0.00087936 0.079457
3 LOW LOW 0.9784 0.0010148 0.087404
4 HIGH HIGH 0.98602 0.0013444 0.011469
5 VHIGH VHIGH 0.98639 0.0017373 0.029662
6 VVHIGH VVHIGH 0.99311 0.0010328 0.0034276
Fig. 13. The performance of the BBBC-IMFC: (a) the system output and (b) the
control signal.
Fig. 14. The performance of the BBBC-AIMFC: (a) the system output and (b) the
control signal.
T. Kumbasar et al. / Expert Systems with Applications 38 (2011) 12356–12364 12363
adaptation mechanism takes the task along with the BB-BC based
fuzzy model inverse controller. Since the adaptation mechanism
updates the related parameters of the model while minimizing
the absolute value of the instantaneous error between the system
and model outputs, the system output is somehow fed back, and
naturally a superior system performance is obtained.
The performances of the proposed non-adaptive and adaptive
BB-BC based fuzzy model inverse controllers are tested on two
experimental setups. In the first application, the proposed control
structures are implemented on a heat transfer process which can
be considered as a fast but practically non-linear system. In the
second application, a pH control process is preferred that is rela-
tively slow but inherits a highly non-linear characteristic. It is ob-
served that especially the proposed adaptive fuzzy model inverse
control structure demonstrates a satisfactory performance for both
relatively slow and fast non-linear systems. The results of the
experiments show that the disturbances and the system parameter
changes are compensated in a very short period of time with the
use of the new online adaptation mechanism. It should be under-
lined that the BB-BC optimization algorithm that has been used
in the proposed fuzzy model inverse control structures is both
appropriate and feasible for online implementations even in the
case of fast system dynamics.
Acknowledgments
This research is supported by the Project (108E047) of Scientific
and Technological Research Council of Turkey. All of these supports
are appreciated.
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