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Intelligent Control and Automation, 2011, 2, 383-387
doi:10.4236/ica.2011.24043 Published Online November 2011 (http://www.SciRP.org/journal/ica)
Copyright © 2011 SciRes. ICA
Dynamic Neural Network Based Nonlinear
Control of a Distillation Column
Feng Li1,2
1Department of Mathem at i c s an d St at i st i cs , University of Maryland, Baltimore, USA
2Office of Biostatistics, Center for Drug Evaluation and Research,
Food and Drug Administration, Silver Spring, USA
E-mail: [email protected]
Received July 27, 2011; revised August 19, 2011; accepted August 26, 2011
Abstract
Taking advantage of the knowledge of top and bottom compositions of a distillation column, a dynamic neu-
ral network (DNN) is designed to identify the input-output relationship of the column. The weight-training
algorithm is derived from a Lyapunov function. Based on this empirical model, a nonlinear H controller is
synthesized. The effectiveness of the control strategy is demonstrated using simulation results.
Keywords: Distillation Column, Dynamic Neural Network, Nonlinear H Control
1. Introduction
A distillation column is a strongly nonlinear process with
multivariate interactions among outputs and some uncer-
tainty often exists in the system, which renders the
analysis and control of a distillation column very difficult
[1,2]. In practice, the single-point control is commonly
used which is sample and easy to tune. However, the con-
sumption of energy in single-point control is large and the
product of the other end is not guaranteed. For high purity
distillation column, the two-point control or other strate-
gies have been investigated [1,2]. Considering the intrinsic
nonlinearity of a distillation column, nonlinear controller
has been designed based on the rigorous mathematical
model [3,4]. Although a nonlinear controller may be
effective in simulation, its implementation in practical
plants is complex because of the lack of measurement of
some key variables. Furthermore, a controller based on the
accurate mathematical model may lead to poor perfor-
mance in case of large perturbation. For this reason, most
controllers for distillation columns are synthesized based
on the input-output relations such as transfer function or a
model obtained by system identification [1,5-8].
Because of its capability to approximate arbitrary non-
linear mapping, neural network has been actively used in
nonlinear system identification and control [9-12]. The
multilayer feed-forward neural network (MFNN) is one
of the most widely used neural networks as a system
model in the design of a model-based controller. A dy-
namic neural network (DNN) is more suitable to ap-
proximate a nonlinear dynamic process. Therefore, dy-
namic recurrent neural network has been used to learn the
input-output relationship of a column and a local optimal
controller based on the neural network model was given in
[8]. Non-linear adaptive controllers based on MFNN and
RBFNN have also been studied [13,14]. A recent review
[15] for the past 28 years showed that most of the
implementations of advanced control like internal model
control were based on linear models. Many recently
published papers [16,17] on neural control for distillation
columns are some extensions of previous research and
supported by simulations. Although a neural network can
be trained in simulation to approximate any nonlinear
process, the control law must be designed to be robust for
the modeling error. H control has been a very popular for
robust control of nonlinear system [18-20]. The applica-
tion of H control to distillation columns has not been
widely discussed.
In this paper, a dynamic neural network is designed to
identify the input-output relation of a distillation column
online. The training algorithm of the network is derived
from a Lyapunov function so that the convergence of
weights and the finite bound of the identification error are
guaranteed. To cope with model error, a nonlinear H
controller based on the trained network model is em-
ployed to enhance the robustness of the control system.
Simulation results demonstrate the effectiveness of the
online identifier and the control algorithm.
F. LI
384

1
12
2
()
()()() ()
ut
YfY gYgYut

 

21
(), ()
2. DNN Based On-Line Identifier of a Binary
Distillation Column
Assume the distillation column is with (L, V) control
structure where L and V are reflux flow in condenser and
boil up flow in reboiler (kmol/min) respectively. Then, let
u1, u2 be L, V respectively. The outputs are the light
compositions of the top and bottom products of a distilla-
tion column.
In some set points, assume the nonlinear input-output
relationship of the distillation column can be expressed
approximately as below:
(1)
where i
f
YgYR

()() ()
T
gYg YgY
0,g
22 0gi

12
1
12
()
ˆˆ
(,)(,)
gg
ut
gY
W gYW

ˆ
,12ii
i
, and
according to some mechanistic knowledge, 11
, for any , = 1, 2.
12 21
Assume the outputs1
Y, 2are known, a dynamic neu-
ral network based identifier is designed to approximate
(1) as below:
0, 0gg [0,1]
i
Y
Y
2()ut

ˆ
Y
ˆ
ˆˆ
(,YAYfY
)
f
W AY
(2)
where A is any stable matrix. Y and are the output
vectors of the column and the identifier respectively.
f
and ˆi
g
(i = 1,2) are all MFNN, which are the
estimates of
f
and i
g
in (1) respectively. The active
function of the output-layer of MFNN ˆ
f
is linear. The
active function of 1
ˆ
g
is a sigmoid function
(1 exp(k))xˆ
, and the active function of 2
g
is
(1 xk
exp(k
k)), where is a tunable parameter sat-
isfying .
0
f
W and i
g
W are the weight matrixes.
From [9], for 0
 and any continuous function
()
x there exist a three-layer feedforward network sat-
isfing:
ˆ
F
sup
x( )
Fx ( ,)
xW

W
(, )
where denotes the optimal weight matrix and
ˆ
F
xW denotes the outputs of the network.
Let
f
W and i
g
W be the corresponding optimal
weight matrix, system (1) can be rewritten as：
2
1
ˆˆ
(,)(,) ()()
i
figi
i
fYWAYgYWutY

 
)
YA
Y (3)
where (Y
is the modeling error of the optimal identi-
fer satisfing:
sup( )
Y
YY

，

12
0, 1
YYYY
which can be made as small as possible by adjusting the
neural network structure.
Let
be an extended column vector of all the ele-
ments of a weight matrix W. denotes the corre-
sponding extended vector of the optimal weight matrix.
Then the extended column vector of
f
Wi
and
g
Ware
f
and i
g
. Define identification error as
ˆ
YeY
and parameter estimation error as
j
jj
,i
g (jf
). From (2) and (3) we have
2
1
)
ii
i
eAef gY

()ut (
(4)
where
1
ˆ
ˆˆ
) (,
fY

(,
Y)
ffff
f
f
fO




01
ff
,
,
1
ˆ
ˆˆ i
i
g






(, )(, )
iiii
i
ii gigg gg
g
gg
Y gYO
Define
2
1i
fi
O O

()()()
gi
dtu tY
(5)
Assume:
sup( )sup


fg
dt u
(6)
where
j
(,jfg
) is a finite constant.
The updating algorithms of the parameter vectors
f
and
g
i
are as followings：
ˆ
ff
f
fPe kPe
f





,
ˆ()
ii
i
i
i
ig g
g
gPeu tkPe



 (7)
g

0, ,
where
j
i, and P is the positive defined
solution of the following Lyapunov equation:
kjfg
T
PAA PQ

Q is a positive defined matrix and the minimal eigenvalue
of Q is0
.
Theorem 1: If is bounded and the weight updat-
ing alogrithm is (7), then the identification error and
parameter estimation error
()ut e
j
( are all ulti-
mately uniformly bounded (UUB)
,)
i
jfg
Proof: Consider the Lyapunov function candidate：
2
11
22 ii
ff gg
Vee

1i
P



 

 .
Differentiating it along (4) and (5) yields
2
1
ˆ
()
()
i
ii
1
2
ˆ
1
2i
i
f
ig
ff gg
i
ut
ePdt
i
fg
g
f
VeQeeP eP
 

 




 

 
C
opyright © 2011 SciRes. ICA
F. LI385
Substituting (10) into the above equation leads to
2
1()( ()
2
()
ii ii
1
f
ff f
gg gg
VeQeePdtePk
k
 
i

 
 



2
2
0
22
2
11
(
2
)
iiii i
ff
ggff fgg g
ii
VeePdePk
kk k
 


 


 
We can further have:
22
2*
0
2
22
**
1
22
11
f
ff f
k
VePek

 






 

*
1
4
24
ii ii
f
gg gg
i
kd




Assuming is bounded and combining it with (6),
we have
()ut
()
f
gm
dt u

 .
So m
is bounded.
Define
22
2
1
11
44
ii
fg gm
i
kk
df


 
.
It is easy to verify that
if
0
2
ePd
or: 1
2
f
fdf
k


，
or: 1
i
2
ii
g
gdg
k
0V


then .
For i
L being bounded, when i
ut , we have
d
() L
. According to Barbalat’s theorem [21], it can be
seen that:
,,
i
fg


eL
eL
.
Considering (5), we further have .
3. Nonlinear H Control of the Binary
Distillation Column
3.1. Nonlinear H Control
Consider an affine nonlinear controller system
12
() ()()
x
fxgxg xu
 
() ()yhx kxu
n
(8)
where
x
Rm
uR, ,
s
R
,
p
y
R;()
f
x,1()
g
x,
g
()
2x, h(x) and k(x) are smooth functions; u is control
input
is uncertainty noise and (0)0, (0)0fh
0
.
For system (8), the design objective of nonlinear H
control is to make the following conditions guaranteed
[19,20].
1) The closed-loop system is asymptotically stable;
and,
2) for a given , the following inequality holds
2
22
00
() d() d
 

tt
y0t .
According to [19], defining xV
V
x
n
for
x
R
() T
Rx k
and , we have:
()()xkx
()Rx
Proposition: If is nonsingular, nonlinear H
control law is expressed as
1
2
() () ()
2x
uRxgxx

 

1()Vkxh

(9)
where V(x) (V(0) = 0) satisfies
1
ˆˆˆ ˆ
())()() 0
4
xxx
Vfxhx VRxV

(h x

where
1
ˆ()()()() ()()
2
f
xf xgxRxkxhx

1
ˆ() (()() ())()hxIkxRxkxhx

1
1
ˆ()() ()()()Rxg xgxgxRxg
11 2 2
2
.

12
ˆˆˆ()
YAYfAY ggut
3.2. Nonlinear H Control of the Binary
Distillation Column
Considering the existence of the modeling error, system
(3) can be rewritten as:
 
dd
YAYs
(10)
where
is the uncertainty variable denoting the mode-
ling error.
Assuming the desired tracking trajectory is
s
is a reference variable.
, where
Define .
111d
d
YYxYY YYx


 


 
222d
2
From (10), the tracking error equation can be derived
as below:
1
()
x
axAx GGu
 
()ax
(11)
where
ˆ
fAY s
 10
,11 12
12
21 22
ˆˆ
ˆˆ
01
g
g
,GG
g
g



 

 .
Copyright © 2011 SciRes. ICA
F. LI
386
()GY
u
1
2
For the binary distillation column, 2 is inver-
tible. If the penalty variables are chosen to be
2
() ()()zhx kxuaxG . (12)
Substituting (9) into (12), we have:

2() ()()
()
TT
xGxPxkxhx
xax


1
()uR
GP

 (13)
where P is the positive defined solution of the following
Riccati inequality:
2

110
T
PA AP PP

 
. (14)
Remark: P can be obtained by solving a Riccati
equation as below:
2
110
TT
PAA PPPC C

 


where [A,C] is observable.
4. Simulation
After The binary distillation model used in the simula-
tion is the same as the one developed by [2]. The nomi-
nal values of outputs are 10 0.98(mol%)
Y
(mol%)
and
20 . The nominal values of reflux flow
and boil-up flow are L = 2.28625 (kmol/min) and
V = 2.78625 (kmol/min). The feed flow and feed compo-
sition are F = 1 (kmol/min) and Zf = 0.5 (mol%).
0.02Y
90
09
A



10
,1
.1
01
C




0.0555 0
00
.0555
P


The distillation column is controlled using the control
strategy (13) developed in this paper. The tacking prop-
erty and the disturbance attenuation property of the con-
trol system are demonstrated through simulation. To
make the transient response be more elegant, the dy-
namic neural network based identifier is trained for some
time offline.
In the simulation, we choose
,
and the positive definite P we obtained is
.
1) Servo properties
Let the setpoint of the top light component change
from nominal value 0.98 to 0.995 and the bottom com-
position remain nominal value 0.02. The augmentations
of system output are demonstrated in Figure 1. The
curves of the control inputs are in Figure 2.
2) Robustness
The responses of the closed-loop system are illus-
Figure 1. The augmentations of outputs.
Figure 2. The curves of control inputs.
Figure 3. The augmentations of outputs (Feed increases
10%).
Figure 4. The augmentations of outputs (Feed composition
increases 10%).
trated in Figures 3 and 4 when feed flow F and feed
composition Zf increase 10% respectively.
C
opyright © 2011 SciRes. ICA
F. LI387
5. Conclusions
A dynamic neural network based online nonlinear identi-
fier for a binary distillation column is designed. The
learning algorithm of the network weights is established
in detail, which can guarantee the boundedness of the
identification error. To deal with the modeling error, a
nonlinear H∞ controller based on the identifier is given
by choosing the penalty variables for the column. The
effectiveness of the proposed strategy is demonstrated in
simulation results. The algorithm developed in this paper
can be applied to other chemical processes as well.
6. Acknowledgements
The author thanks the reviewers for very helpful com-
ments. Views expressed in this paper are the author’s
professional opinions and do not necessarily represent
the official position of the US Food and Drug Admini-
stration.
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Copyright © 2011 SciRes. ICA