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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

ð6Þ

U ¼ Gc ½R À Yout

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 T . Another D-dimensional vector

can represent the velocity of the particle Vi ¼ ½vi1 vi2 . . . viD T .

The best previously visited position of the ith particle denoted

Fig. 6

as Pi ¼ ½pi1 pi2 . . . piD T . 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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