Backstepping technique based nonlinear controller
for a textile dyeing process
B. Bandyopadhyay
Systems & Control Engineering
Indian Institute of Technology Bombay
Mumbai, India
Abstract
This paper describes a control scheme for a
MIMO system using backstepping method. Backstepping method is preferred in case of uncertainties in the process and constraints acting on the
control input. In the present work, backstepping
method for a PMTD process is investigated in order to maintain the concentration and volume of
the dye bath thereby reduce the tailing effect. The
proposed control strategy is designed and simulated
using Matlab/ simulink and results are presented to
validate the proposed control scheme.
Keywords:
Backstepping ,
PMTD
(Padding mangle for textile dyeing) process,
Tailing effect.
1
Introduction
In many real world systems, there are nonlinearities, unmodeled dynamics, immeasurable noise,
multiloop etc. which pose problems to engineers in
trying to implement control strategies. During the
past two decades a large number of control strategies such as adaptive, optimal and robust control
have been evolved. Most of these techniques need
to use linear model of the plant. In the application
of such techniques, development of mathematical
models is a prior necessary. However, such mathematical modeling which is largely based on the
assumptions of linearization of systems might not
reflect the true physical properties the system. Improving or understanding a chemical process operation is a major overall objective for developing a
dynamic process model.. These models are often
J. V. Desai & C. D. Kane
Textile & Engineering Institute
Ichalkaranji, India
used for (i) Operator training (ii) Process design
(iii) Safety system analysis or design or (iv) Control system design.
Continuous methods of dyeing and processing of
fabric have the merits of saving time and producing a uniform result, though they have the disadvantage especially in continuous dyeing, that it is
necessary to process considerably large amounts of
fabric to be dyed in any one shade to justify the
initial high cost of the plant. In such continuous operations the basic operation common to all
is the padding operation which consists of passing
carefully scoured and bleached fabric in open-width
through a small trough containing the processing
solution and then removing the excess solution by
passing between positively driven loaded rollers,
which constitute the padding mangle.
A simple padding mangle consists of two squeezing bowls (rollers) the upper one of iron and covered
with rubber and the lower one of brass or vulcanite,
arranged over a shallow trough, provided with two
or more freely rotating guide rollers. Fig.1 shows
the line diagram of padding mangle. If the dyes or
chemicals used in padding on the fabric have affinity, their concentration in the liquor contained in
the trough goes on decreasing, resulting in the tailing effect.. The tailing effect will finally result in
reduction in the volume and concentration.
2
Mathematical model
In the present work, while developing the model it
is found that the whole dyeing phenomenon cannot be fully modeled. The process is governed
by many non-measurable parameters.. Parameters
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Plant dynamics
10
Concentration
9
8
7
Conc
6
5
4
1
2
3
4
5
6
7
8
9
Samples
Figure 1: Line diagram of PMTD process
Figure 2: Plant dynamics
like dye exhaustion, electrolyte concentration, dye
affinity towards cellulose, fabric construction, pre
CB = Concentration of the solution in the bath.
treatment of fabric and liquor ratio are not measurable. Measurable parameters like temperature,
MB = Volume of the solution in the dye bath.
pH , and mangle pressure have greater significance
on the behaviour of the process. But the relation
V= Velocity of the fabric = Feed rate
of these parameters with the change in the concenRv = Liquor retention by the fabric
tration is not known. Again, ‘Rv’ known as ‘liquor
retention in fabric’ is a key parameter to be conFM = Flow rate of concentrated feed.
sidered in the model. ‘Rv’ depends on many of the
FW = Flow rate of diluted feed.
earlier stated process parameters and its relation
with the latter cannot be established. The model
CM =Concentration of the solution in make up
of the PMTD process can be obtained using mass
tank.
balance equations. However the above limitation
makes the mass balance equations based mode approximate.. Experimentation is carried out on a
laboratory model and using the measured data and Table 1: Variations in process parameters during
mass balance formulations, a nonlinear model is de- dyeing in a PMTD Process
veloped. The developed mode lis shown below.
Volume of the bath Conc. in the Bath
2000
10.00
1960
9.87
dCB
= k1 CB + k3
(1)
1920
9.43
dt
1880
8.20
dMB
= FM + FW − Rv V
1840
7.33
dt
1801
6.87
Rv V
k1 = −
(2)
1762
6.08
MB
1723
5.26
FM CM
k3 =
(3)
1686
4.95
M
B
where
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Table 2: K/S Values of a dyed fabric
Sr.no K/S Values
1
96.85
2
94.68
3
94.10
4
93.04
5
91.96
6
85.57
7
62.83
.
x2 = f2 (x, u1 , u2 ).
It is desired that
.
x1 = f3 (x1 ),
f3 (x1 ) − f1 (x, u1 ) = 0
The following two-step procedure illustrates the
design procedure for a backstepping controller for
the PMTD process.
3 Nonlinear control of PMTD Step1: Define z1 = x1 − x1d as the first tracking
error variable.
Process
where x1 =Concentration of the dye bath
x1d = Desired concentration of the dye bath.
As seen from the plant dynamics in Fig 2, concenThe derivative of z1 is given by
tration and volume decreases with time and therefore both these states need to be controlled and
Rv V (z1 + x1d ) + u1 CM
.
z1 =
.
maintained at desired values. Here the design of
z2 + x2d
control law for the PMTD process using backstepStep 2: Define z2 = x2 − x2d , as the second variping technique is discussed. In traditional backable.
stepping, the output is selected as x1 (t), which
where x2 = Volume of the dye bath
might be required to follow a prescribed trajectory
x2d = Desired Volume of the dye bath.
x1 d(t) : In this method one starts at the desired
The
derivative of the variable z2 is given by
output, and backsteps through the system selecting
desired values of the state components until the ac.
z2 = u1 + u2 − Rv V.
tual control input u(t)is reached.
Consider the nonlinear model of the PMTD
Considering the desired dynamics, the derivaplant.
tives will be made equal to stable states like -z1,
or -z2, and the control inputs are derived.
dCB
= k1 CB + k3
(4)
dt
z1 = x1 − x1d ,
dMB
= FM + FW − Rv V
dt
z2 = x2 − x2d .
Rv V
CB = x1, MB = x2 , FM = u1, FW = u2
k1 = −
MB
dx1
Rv V
u1
.
x1 =
=−
x1 + CM
FM CM
dt
MB
x2
k3 =
.
MB
x2 = u1 + u2 − Rv V
where CB = x1 = Concentration of the dye bath
x1 = z1 + x1d
MB = x2 =Volume of the dye bath
x2 = z2 + x2d
FM = u1 =Control input 1
Rv V (z1 + x1d ) + u1 CM
.
FM = u2 =Control input 2
z1 =
z2 + x2d
.
z2 = u1 + u2 − Rv V
dx1
Rv V
u1
.
x1 =
=−
x1 + CM
dt
x2
x2
Let
.
.
x2 = u1 + u2 − Rv V
z1 = −z1
.
x1 = f1 (x, u1 ),
.
z2 = −z2 .
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Figure 3: Set point tracking of concentration
Then upon simplifying , the two control inputs
obtained are as shown below.
Figure 4: Set point tracking of Volume.
coeffecient for disperse dyes”, Textile Research
Journal, 71(4), pp 357 -361, 2002.
1
[(x1d − x1 ) x2 + Rv V x1 ] ,
CM
(5) [4] E.Cleve, E.Bach, U.Denter, H.Duffner, and
E.Schollmeyer “ New mathematical model for
determining time dependent adsorption and diffusion of dyes into fibers through dye sorption
1
u2 = x2d −x2 −
[(x1d − x1 ) x2 + Rv V x1 ]+Rv V.
curves in combination shades” Textile Research
CM
Journal, 67(10), pp 701 -706, 1997.
(6)
u1 =
3.1
Simulation results
Nonlinear controller using backstepping technique
is designed. Fig 3 and Fig.4 indicate the simulated
results for set point tracking of concentration and
volume respectively.
References
[1] Denn. M . M ., “ Process Modelling”, Longman,
Newyork, 1986.
[2] Cheng-chun Huang and W en-Hong Yu.,“ Control of dye concentration, pH , temperature in
dyeing processes ”, Textile Research Journal, 69
(12), 9 14-9 18, 1999.
[3] MathildeCasetta, Vladan Koncharand Claude
Caze., “ Mathematical modeling of the diffusion
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