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PSO-BELBIC scheme for two-coupled distillation column process

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Journal of Advanced Research (2011) 2, 73–83

Cairo University

Journal of Advanced Research

ORIGINAL ARTICLE

PSO-BELBIC scheme for two-coupled distillation
column process
Hassen T. Dorrah a, Ahmed M. El-Garhy
a
b
c

b,*

, Mohamed E. El-Shimy

c

Department of Electrical Power and Machines, Faculty of Engineering, Cairo University, Giza, Egypt
Department of Electronics, Communications and Computers, Faculty of Engineering, Helwan University, Helwan, Egypt
King Saud University, Riyadh, Saudi Arabia

Received 31 March 2010; revised 2 July 2010; accepted 3 July 2010
Available online 27 November 2010

KEYWORDS
Particle Swarm Optimization
(PSO);


Two-coupled distillation
column;
Brain Emotional Learning
Based Intelligent Controller
(BELBIC);
PID controller

Abstract In the two-coupled distillation column process, keeping the tray temperatures within a
specified range around their steady state values assures the specifications for top and bottom product purity. The two-coupled distillation column is a 4 Input/4 Output process. Normally, control
engineers decouple the process into four independent loops. They assign a PID controller to control
each loop. Tuning of conventional PID controllers is very difficult when the process is subject to
external unknown factors. The paper proposes a Brain Emotional Learning Based Intelligent Controller (BELBIC) to replace conventional PID controllers. Moreover, the values of BELBIC and
PID gains are optimized using a particle swarm optimization (PSO) technique with minimization
of Integral Square Error (ISE) for all loops. The paper compares the performance of the proposed
PSO-BELBICs with that of conventional PSO-PID controllers. PSO-BELBICs prove their usefulness in improving time domain behavior with keeping robustness for all loops.
ª 2010 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

Introduction
Keeping the temperatures of the different trays constant in the
two-coupled distillation columns process is one of the most
* Corresponding author. Tel.: +966 594257945; fax: +966 14696800.
E-mail address: agarhy2003@yahoo.co.in (A.M. El-Garhy).
2090-1232 ª 2010 Cairo University. Production and hosting by
Elsevier B.V. All rights reserved.
Peer review under responsibility of Cairo University.
doi:10.1016/j.jare.2010.08.004

Production and hosting by Elsevier

important control actions in the chemical industries. Recently,

many researchers have devoted much effort in this area. The
designers of control systems decouple the process of the twocoupled distillation columns into a group of independent loops
[1]. They control the temperature for each loop via conventional PID control law [2] and adjust its gains appropriately
according to the process dynamics. The conventional PID controller is hardly efficient to control the disturbed system. Several methods for parameter tuning of non-fixed PID controller
were proposed [3–5].
Particle swarm optimization (PSO) is a population-based
stochastic optimization technique developed by Dr. Eberhart
and Dr. Kennedy in 1995, inspired by social behavior of bird
flocking or fish schooling [6,7]. PSO shares many similarities
with other evolutionary computation techniques such as Genetic
Algorithms (GA) [8,9]. Compared to GA, PSO is easy to


74

H.T. Dorrah et al.

Nomenclature
QE
SAB
RLA
RLB
T11
T30
T34
T48
Yout
H
K
R

U
Gc
Gc11
Gc22
Gc33
Gc44
d

heat added
steam goes from column A to column B
reflux produced from column A
reflux produced from column B
temperature measured for tray 11
temperature measured for tray 30
temperature measured for tray 34
temperature measured for tray 48
the actual outputs of the process
process transfer function matrix
steady state decoupling compensation matrix
the set values of the process inputs
output signals from controllers
the controller transfer function matrix
the controller transfer function for the decoupled
loop (QE; T30 )
the controller transfer function for the decoupled
loop (SAB; T11 )
the controller transfer function for the decoupled
loop (RLA; T34 )
the controller transfer function for the decoupled
loop (RLB; T48 )

¼ 1; 2; . . . ; D and D is the size of dimensional
vector

¼ 1; 2; . . . ; M and M is the size of the swarm
(i.e. number of particles in the swarm)
c1, c2
positive values, called acceleration constants
r1, r2
random numbers uniformly distributed in [0, 1]
z
¼ 1; 2; . . . ; Z and Z is the maximal times of iteration
w
the inertia weight function
a
the learning rate in amygdala
REW
the reinforcing signal
GAi
the weight of the plastic connection in amygdala
GOi
the weight of orbitofrontal connection
b
the orbitofrontal learning rate
yp
the plant output
e
the error signal
Kp ; Ki ; Kd the gains the designers must tune for satisfactory performance
KpðQE; T30 Þ ; KdðQE; T30 Þ ; KiðQE; T30 Þ the controller’s gains for
loop (QE; T30 )

KpðSAB; T11 Þ ; KdðSAB; T11 Þ ; KiðSAB; T11 Þ the controller’s gains
for loop (SAB; T11 )
KpðRLA; T34 Þ ; KdðRLA; T34 Þ ; KiðRLA; T34 Þ the controller’s gains
for loop ðRLA; T34 Þ
KpðRLB; T48 Þ ; KdðRLB; T48 Þ ; KiðRLB; T48 Þ the controller’s gains
for loop (RLB; T48 )
i

implement with few adjustable gains. PSO has been successfully applied in many areas such as function optimization, artificial neural network training and fuzzy system control. PSO is
already a new and fast-developing research topic [10–13].
Intelligent control designs have received great attentions in
recent years. Control techniques based on artificial neural networks [14], fuzzy control [15] and GA [16] are among them.
Recently, researchers have developed a computational model
of emotional learning in mammalian brain [17,18]. A Brain
Emotional Learning Based Intelligent Controller (BELBIC)
[19–22] has been successfully employed for making decisions
and controlling simple linear systems as in [23], as well as in
non-linear systems such as control of a power system, speed
control of a magnet synchronous motor and automatic voltage
regulator (AVR) system [24–28], micro-heat exchanger [29],
flight control [30], and positioning control of SIMO Overhead
Traveling Crane [31]. The results indicate the ability of BELBIC to control unknown non-linear dynamic systems. In addition, software developers have used the BELBIC toolbox to
control a community as a pattern [32].
Flexibility is one of BELBIC’s characteristics and it has the
capacity to choose the most-favoured response [33,34]. The
utilization of PSO to estimate the optimal BELBIC gains with
minimization of ISE is the goal of this research.
The control scheme for the two-coupled distillation column
process
The two-coupled distillation column [35] shown in Fig. 1 is a 4

Input/4 Output process.

Fig. 1

The two-coupled distillation columns process.


BELBIC scheme for two-coupled distillation column

75

The inputs of the process are QE; SAB; RLA and RLB,
while the outputs of the process are T11 ; T30 ; T34 and T48 .
The following transfer function matrix describes the process:
2 2:6
À6:098
...
1:69sþ1
3:5sþ1
6 7:32ð1:05sþ1Þ
À1:45
6 ð10:4sþ1Þð0:14sþ1Þ 0:4sþ1
...
6
6 4:6ð0:53sþ1Þ
À2:37ð0:23sþ1Þ
6 ð2:78sþ1Þð0:09sþ1Þ ð2sþ1Þð0:3sþ1Þ
...
4
À2:11ð0:06sþ1Þ

2:11
...
ð2:38sþ1Þð0:05sþ1Þ
0:92sþ1
3
ð1Þ
HðsÞ ¼
À4:99ð0:2sþ1Þ
0:071
. . . ð4:5sþ1Þð0:06sþ1Þ 3:5sþ1
7
À1:57ð0:23sþ1Þ
À0:14
7
. . . ð1:34sþ1Þð0:2sþ1Þ
1:92sþ1
7
À0:36ð0:02sþ1Þ 7
À2:7
7
. . . 1:75sþ1
ð2:47sþ1Þð0:04sþ1Þ 5
...

À1:75
2:16sþ1

À0:3ð1:89sþ1Þ
ð4:35sþ1Þð0:16sþ1Þ


Keeping the tray temperatures T11 ; T30 ; T34 and T48 within a specified range around their steady state values is essential

for specifying top and bottom product purity. The transfer
function matrix demonstrates strong interactions between
process inputs and outputs. For proper control of the process,
decoupling it into four loops is necessary. Some researchers
propose a PSO-based decoupling technique [1]. Such a technique estimates the optimum values of a steady state decoupling compensation matrix that minimizes the interactions
between each input and its unpaired outputs. The decoupling
technique yields to four independent decoupled loops; namely
loop ðQE; T30 Þ, loop ðSAB; T11 Þ, loop ðRLA; T34 Þ and loop
ðRLB; T48 Þ. Fig. 2 depicts the decoupling scheme for the
two-coupled distillation column process.
Based on the decoupling scheme, the following relations are
satisfied in matrix form:
Yout ¼ HKR
3
2 3 2
T11
Y1
6Y 7 6T 7
6 2 7 6 30 7
Yout ¼ 6 7 ¼ 6
7
4 Y3 5 4 T34 5
Y4
T48
2
1
0:1788
6 À1:9273

1
6
K¼6
4 2:9263
0:8865
3:5183 À10:9464
3
2 3 2
R1
QE
6 R 7 6 SAB 7
7
6 27 6
R¼6 7¼6
7
4 R3 5 4 RLA 5
R4

Fig. 2 The decoupling scheme for the two-coupled distillation
column process.

Fig. 3

ð2Þ

ð3Þ
3
0:0608 À0:0078
À0:7906 0:4555 7
7

7
1
À0:5466 5
0:1548
1

ð4Þ

ð5Þ

RLB

Fig. 3 illustrates the step changes of process inputs at different times to check the behavior of the decoupled loops. Fig. 4
illustrates the outputs of different decoupled loops in the case
of no controllers.

Step changes in system inputs.


76

H.T. Dorrah et al.

Fig. 4

The outputs of different decoupled loops in case of no controllers.

Step changes in a specific input cause some small and narrow perturbations (spikes) in its unpaired outputs, while causing a direct step response in its own-paired output. From this
point of view, the decoupling scheme proves its suitability to
control the four decoupled loops using four individual controllers. Fig. 5 presents the control scheme of the two-coupled distillation column process.

The following matrix form fulfils the relations of the control scheme:

Fig. 5

Yout ¼ HKGc ½R À Yout Š

ð6Þ

U ¼ Gc ½R À Yout Š
2 3
U1
6 7
6 U2 7
6 7
U¼6 7
6 U3 7
4 5

ð7Þ

U4

The control scheme of the two-coupled distillation column process.

ð8Þ


BELBIC scheme for two-coupled distillation column
2


Gc11
6
6 0:0
Gc ¼ 6
6 0:0
4
0:0

0:0

0:0

Gc22

0:0

0:0

Gc33

0:0

0:0

0:0

77

3


7
0:0 7
7
0:0 7
5
Gc44

ð9Þ

Particle swarm optimization (PSO)
PSO [6–12] is a population-based search algorithm initialized
with a population of random solutions, called particles. Each
particle in PSO has its associated velocity. Particles fly through
the search space with dynamic adjustable velocities according
to their historical behaviours. Remarkably, in PSO, each individual in the population has an adaptable velocity (position
change), according to which it moves in the search space.
Suppose that the search space is D-dimensional, and then a
D-dimensional vector can represent the ith particle of the
swarm Xi ¼ ½xi1 xi2 . . . xiD ŠT . Another D-dimensional vector
can represent the velocity of the particle Vi ¼ ½vi1 vi2 . . . viD ŠT .
The best previously visited position of the ith particle denoted

Fig. 6

as Pi ¼ ½pi1 pi2 . . . piD ŠT . Defining ‘‘g’’ as the index of the best
particle in the swarm, where the gth particle is the best, and
let the superscripts denote the iteration number, then the following two equations manipulate the swarm as follows:
zþ1 n
vzþ1
vid þ c1 rz1 ðpzid À xzid Þ þ c2 rz2 ðpzgd À xzid Þ

id ¼ wi

ð10Þ

xzþ1
id

ð11Þ

xzid

vzþ1
id

¼
þ
0:5z
0:4 À 0:9Z
þ
wz ¼
1ÀZ
1ÀZ

ð12Þ

The inertia weight decreases from 0.9 to 0.4 through the run
to adjust the global and local searching capability. The large
inertia weight facilitates global search abilities while the small
inertia weight facilitates local search abilities.
Fig. 6 displays the flow chart of the PSO algorithm.

Brain Emotional Learning Based Intelligent Controller
(BELBIC) model
Brain Emotional Learning (BEL) is divided into two parts [26],
very roughly corresponding to the amygdala and the orbito-

Flow chart of the PSO algorithm.


78

H.T. Dorrah et al.

Fig. 7

Scheme of BEL strucure.

frontal cortex, respectively. The amygdaloid part receives inputs from the thalamus and from cortical areas, while the orbital part receives inputs from the cortical areas and the
amygdala only. The system also receives reinforcing (REW)
signal. There is one A node for every stimulus S (including
one for the thalamic stimulus). There is also one O node for
each of the stimuli (except for the thalamic node). There is
one output node in common for all outputs of the model called
MO. Fig. 7 reveals the scheme of BEL structure.
The MO node simply sums the outputs from the A nodes,
and then subtracts the inhibitory outputs from the O nodes.
The result is the output from the model.
X
X
MO ¼
Ai À

Oi
ð13Þ
i

i

Unlike other inputs to the amygdala, the orbitofrontal part
does not project or inhabit with the thalamic input. Eq. (14)
represents that emotional learning occurs mainly in the
amygdala:
!!
X
DGAi ¼ a Á Si Á max 0; REW À
ð14Þ
Ai
i

Equations (15) and (16) give the learning rule in the orbitofrontal cortex as follow:

Fig. 8

DGOi ¼ b Á Si Á Ro
ð15Þ
8


P
P
>
>

>
< max 0; Ai À REW À Oi 8REW – 0
i
i


where Ro ¼
ð16Þ
P
P
>
>
>
8REW ¼ 0
: max 0; Ai À Oi
i

i

As is evident, the orbitofrontal learning rule is very similar to
the amygdaloid rule. The only difference is that the weight of
orbitofrontal connection can both increase and decrease as
needed to track the required inhibition.
Eqs. (17) and (18) calculate the values of nodes as:
Ai ¼ GAi Á Si
Oi ¼ GOi Á Si

ð17Þ
ð18Þ


Note that this system works at two levels: the amygdaloid
part learns to predict and react to a given reinforcer. The
orbitofrontal system tracks mismatches between the base system’s predictions and the actual received reinforcer and learns
to inhibit the system output in proportion to the mismatch.
The reinforcing signal REW comes as a function of the
other signals, which can represent a cost function validation
i.e. reward and punishment are applied on the basis of the previously defined cost function.
REW ¼ JðSi ; e; yp Þ

Control system configuration using BELBIC.

ð19Þ


BELBIC scheme for two-coupled distillation column
Table 1

79
REW ¼ Kp Á e þ Ki Á

Gains of PSO.

Parameter

Value

Number of particles
Maximum number of iterations
Inertia weight


50
1000
Linearly decreasing
from 0.9 to 0.4
2
0.01
100,001

Acceleration constants
Sampling time
Number of samples in each iteration

e Á dt þ Kd Á

de
dt

ð20Þ

ð24Þ

Although BELBIC demonstrated effective control performance in many applications, its gains were adjusted using trial
and error rather than an optimal approach. To deal with this
drawback, this paper employs the PSO for tuning BELBIC
gains for all loops. PSO uses the summation of the Integral
Square Errors (ISE) of different decoupled loops as a fitness
function, such that:
Z 1
Z 1
Fitness function ¼

ðQE À T30 Þ2 þ
ðSAB À T11 Þ2
0
0
Z 1
Z 1
þ
ðRLA À T34 Þ2 þ
ðRLB À T48 Þ2
0

As illustrated in Eqs. (19) and (20), reward signal and sensory input can be an arbitrary function of reference input, r,
controller output, u, error (e) signal, and the plant output yp.
It is for the designer to find a proper function for control.
This paper has used the continuous form of BEL. In continuous form, the updating of weights for both the plastic connection in amygdala and the orbitofrontal connection do not
follow a discrete relation but a continuous one. These continuous relations are:
dGAi
¼ a Á Si Á ðREW À Ai Þ
dt
dGOi
¼ b Á Si Á ðAi À Oi À REWÞ
dt

ð21Þ
ð22Þ

Fig. 8 demonstrates the control system configuration using
the Brain Emotional Learning Based Intelligent Controller
(BELBIC).
Methodology of the proposed PSO-BELBIC scheme

As explained in section ‘Particle swarm optimization (PSO)’,
the four decoupled loops constitute the two-coupled distillation column process. The methodology of the proposed
PSO-BELBIC scheme assigns one BELBIC for each loop.
The following relations yield the functions used in reward signal and sensory input blocks for each control loop:

Fig. 9

ð23Þ

S¼KÁe

Similarly, the sensory inputs must be a function of plant
outputs and controller outputs as follows:
Si ¼ fðu; e; yp ; rÞ

Z

0

ð25Þ
QE; T30 ; . . . ; RLB; T48 are in the form of electric signals.
Twenty-four gains should be tuned simultaneously (six for
each loop) with the objective of minimizing the fitness
function.
Table 1 gives the gains of PSO.
Fig. 9 evolutes the fitness function in all iterations for the
BELBIC scheme.
Table 2 gives, for different BELBICs, the resulting best
gains values that minimize the fitness function.
Hence, in order to evaluate the control capability of the

proposed PSO-BELBIC scheme, simulation with step changes
in system inputs investigates the performance of the two-coupled distillation column process. Fig. 10 simulates the response
of the four decoupled loops.
The values of steady state errors (ess ) and integral square
errors (ISE) for PSO-BELBIC scheme are summarized in
Table 3.

Comparing PSO-BELBIC with PSO-PID
Evaluation and validation of the proposed PSO-BELBIC
scheme requires it to be compared to the PID scheme. The
PID control scheme for the two-coupled distillation column

The evolution of fitness function in all iterations for BELBIC scheme.


80

H.T. Dorrah et al.

Table 2

The best gains of the BELBIC optimized by PSO for different loops.

Loop

ðQE; T30 Þ
ðSAB; T11 Þ
ðRLA; T34 Þ
ðRLB; T48 Þ


Gains
a

b

K

Kp

Ki

Kd

5.89eÀ09
2.78eÀ09
6.13eÀ09
2.84eÀ09

7.79eÀ08
6.81eÀ08
7.89eÀ08
8.01eÀ08

149.77
275.12
274.88
69.950

59.9800
À743.260

À2.95e+03
À891.460

5.10eÀ04
2.91eÀ03
0.82640
0.68420

124.99
52.365
59.564
364.45

Fig. 10

The response of the four decoupled loops using PSO-BELBIC scheme.

process includes four PID controllers, one for each decoupled
loop, given by:
KiðQE; T30 Þ
þ KdðQE; T30 Þ s
s
KiðSAB; T11 Þ
¼ KpðSAB; T11 Þ þ
þ KdðSAB; T11 Þ s
s
KiðRLA; T34 Þ
þ KdðRLA; T34 Þ s
¼ KpðRLA; T34 Þ þ
s

KiðRLB; T48 Þ
þ KdðRLB; T48 Þ s
¼ KpðRLB; T48 Þ þ
s

Gc11 ¼ KpðQE; T30 Þ þ

ð26aÞ

Gc22

ð26bÞ

Gc33
Gc44

ð26cÞ

Table 3 The steady state and integral square errors for PSOBELBIC scheme.
Loop

ðQE; T30 Þ
ðSAB; T11 Þ
ðRLA; T34 Þ
ðRLB; T48 Þ

Parameter
ess

ISE


0.087510
0.019915
0.036152
0.189710

18.0900
1.07800
2.45940
45.3900

ð26dÞ

For the fairness of comparison, the gains of different PID
controllers are also subjected to the PSO-based tuning method
with the same fitness function as defined in Eq. (25) and the
same gains given in Table 1, then twelve gains should be tuned
simultaneously (three for each loop). Fig. 11 evolutes the fitness function in all iterations for the PSO-PID scheme.
Table 4 gives, for different PID controllers, the resulting
best gains values that minimize the fitness function.

Simulation with step changes in system inputs investigates
the performance of the two-coupled distillation column process using the PSO-PID scheme. Fig. 12 exhibits the simulated
response of the four decoupled loops.
For detailed comparison, Fig. 13 scrutinizes the outputs
subjected to step changes in inputs at different times for both
schemes.


BELBIC scheme for two-coupled distillation column


Fig. 11

81

The evolution of fitness function in all iterations for PID scheme.

Table 4 The best gains of the PID controller optimized by
PSO for different loops.
Loop

ðQE; T30 Þ
ðSAB; T11 Þ
ðRLA; T34 Þ
ðRLB; T48 Þ

Gains

The steady state errors (ess ) and the integral square errors
(ISE) for the PSO-PID scheme are summarized in Table 5.
Discussion and conclusion

Kp

Ki

Kd

0.656740
À1.78600

À24.6470
À7.96700

5.137eÀ05
6.034eÀ05
6.985eÀ05
4.967eÀ05

0.08574
0.06850
0.04560
0.00500

Fig. 12

In spite of overdamped responses noticed in all loops, the PSOBELBIC scheme proves its usefulness over the PSO-PID
scheme. Figs. 10 and 12 prove that the performance of the
PSO-BELBIC scheme is much better than that of the PSOPID. Although it gave a slower response compared with

The response of the four decoupled loops using PSO-PID scheme.


82

H.T. Dorrah et al.

Fig. 13

The detailed outputs at instants of step input changes for PSO-PID (- - -) and PSO-BELBIC (


Table 5 The steady state and integral square errors for PSOPID scheme.
Loop

ðQE; T30 Þ
ðSAB; T11 Þ
ðRLA; T34 Þ
ðRLB; T48 Þ

Parameter
ess

ISE

0.215160
0.049978
0.062051
0.279180

42.9560
2.32600
2.61740
47.3280

PSO-PID due to its learning capability, Tables 3 and 5 present
a remarkable reduction in both ess and ISE for all loops in case
of the PSO-BELBIC scheme. In addition, using a PID controller as a reward signal builder with availability of reinforcement
or punishment by BELBIC can have some advantages of the
PID scheme, such as robustness. Fig. 13 confirms clearly that
the perturbations (spikes) that occurred in unpaired outputs at
the instant of change of specific input are remarkably reduced

in the case of PSO-BELBIC and smoother performances are
achieved. Simulation implementation for the two-coupled distillation column process demonstrates the effectiveness of the
proposed scheme. PSO-BELBIC improves the time domain
parameters of all loops of the process. Previous researchers
have used PSO-BELBIC with a single input/single output process to produce therefore a limited number of adjustable gains.
The main contribution of this paper is to use PSO-BELBIC
with a more complex, multi input/multi output process, which
produced 24 adjustable gains.

) schemes.

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