International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
TUNING OF DECENTRALISED PI (PID) CONTROLLERS FOR TITO PROCESS
M.Bharathi,*Dr.C. selvakumar**
*HOD, EIE Dept, Bharath University, Chennai-73
**HOD, ICE Dept, St. Joseph’s College of Engineering , Chennai-119
The PID controller is the most popular
controller used in process control, because of
Abstract
its remarkable effectiveness and simplicity of
This paper presents one of the simplest
implementation.
Although
significant
method to tune decentralised PI (PID)
developments have been made in advanced
controllers for two-input and two-output
control theory, according to the literature,
(TITO) processes. The TITO process was
more than 95% of industrial controllers are
decoupled through a decoupler matrix. To
still PID, mostly PI controllers. PI (PID)
handle loop interactions a model reduction
method with suitable FOPDT and SOPDT
model
for each element of the resulting
diagonal process through fitting the nyquist
plots at particular points.
Simulation
examples of MIMO systems are given to
demonstrate the effectiveness and accuracy
of the proposed algorithm. Compared to
other tuning methods it has less number of
tuning parameters and more over uses only
less number of controllers, therefore it is one
of the cost effective method to control the
process.
control is sufficient for a large number
of
control
processes,
particularly
when
dominant process dynamics are of first
(second) order and there design requirements
are not too rigorous. Although this controller
has only three parameters, it is not easy to
find their optimal values without a systematic
procedure. As a result PI (PID) tuning methods
are extremely desirable due to their wide
spread use.
Generally, most industrial processes are
multi variable systems. When interactions in
different channels of the process are modest,
a diagonal PID controller is often adequate.
Key words: TITO,MIMO system, PID
1.Introduction
Two-input two-output (TITO) systems are one
of the most prevalent categories of multi
variable systems. So an approach for square
systems is to use a decoupler plus a
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decentralized PID controller. One great
Zhuo et.al., (2011) proposed the design of a
advantage of this method is that it allows the
multi-loop PI controller to achieve the desired
use
gain and phase margins for two-input and
of single-input
single-output
(SISO)
controller design methods. One of the
simplest methods to tune decentralised PI
(PID) controllers for TITO process is proposed
in this paper.
An extension of the inverted decoupling
approach that allows for more flexibility in
choosing the transfer functions of the
The TITO process was decoupled through
a decoupler matrix that allows for more
flexibility in choosing the transfer functions
of the decoupled apparent process. A model
reduction method with suitable FOPDT and
SOPDT model for each element of the
resulting diagonal process through fitting the
nyquist
two-output (TITO) processes .
plots
at
particular
points
is
decoupled
apparent
process
(Juan
et.al.,2011).
The idea of an effective open-loop transfer
function
(EOTF)
is
first
introduced
to
decompose a multi-loop control system into a
set of equivalent independent single loops.(
Truong et.al., 2010)
implemented to handle loop interactions.
Simulation examples OF MIMO systems are
incorporated to validate the usefulness of
the presented algorithm. Compared to other
tuning methods it has less number of tuning
parameters and more over uses only less
number of controllers, therefore it is one of
Branislav and Miroslav (2010) designed
multivariable controller based on ideal
decoupler D(s) and PID controller optimization
under constraints on the robustness and
sensitivity to measurement noise
the cost effective method to control the
Nordfeldta and Haddlund(2006) proposed
process
controller consists of a decoupler and a
.Many PI or PIDcontrollers have been
diagonal PID controller.
proposed (Chien and Fruehauf1990; Tyreus
Control Engineering Practice, Volume 14, Issue
and Luyben, 1992).
9, September 2006, Pages 1069-1080
Juan et.al.,(2012)
proposed a generalized
formulation of simplified decoupling to n × n
processes
that
allows
for
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,
different
configurations depending on the decoupler
elements set to unity.
Saeed Tavakoli, Ian Griffin, Peter J. Fleming
Tavakoli et.al(2006) presented a decentralised
PI (PID) tuning method for two-input two-
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
output processes
based on dimensional
Most of the industrial process have
multivariable
analysis
control
variable
that
are
common properties for the models of
industrial
processes
to
have
significant
Wang et.al.,(2006) considered auto-tuning of
uncertainties, strong interaction and non-
simple lead-lag decoupler plus decentralized
minimum phase behaviour so it is important
PI/PID controllers for effective control of two-
for control engineer, chemical engineer to
input and two-output (TITO) processes.
understand the non-idealities of industrial
processes. The structure of MIMO control
Maghade and Patre (2012) proposed a
system is shown in figure1.
decentralized PI/PID controller design method
based on gain and phase margin specifications
for two-input–two-output (TITO) interactive
processes
Automatica, Volume 31, Issue 7, July
1995, Pages 1001-1010
Z.J. Palmor, Y. Halevi, N. Krasney
Palmor et.al., (1995) present an algorithm for
automatic tuning of decentralized PID control
Fig1. (2×2) Multivariable Model Structure
for two-input two-output (TITO) plants that
fully extends the single-loop relay auto-tuner
Here from fig 1,G11(s) is a symbol used to
to the multiloop case.
represent
the
forward
path
dynamics
between mv1 and cv1, while G22(s) describes
Rajapandiyan and Chidambaram(2012)
proposed the closed-loop identification of
two-input–two-output (TITO) second-order
plus time delay (SOPTD) transfer function
models of multivariable systems is presented
based on optimization method using the
how cv2 responds after a change in mv2. The
interaction effects are modelled using transfer
functions G21(s) and G12(s). G21(s) describes
how cv2 changes with respect to a change in
mv1 while G21(s) describes how cv1 changes
with respect to a change in mv2.
combined step-up and step-down responses.
2.1 Interaction
2. Structure of PI controller design
Interaction is undesirable in a MIMO
system.
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This
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is
true
for
setpoint
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
disturbances.While changing setpoint of one
inputs and outputs. While in full matrix PID
loop the other loop should not be affected
control there are 3n2 parameters. In case of
and if the loop do not interact, each individual
actuator or sensor failure, it is relatively easy
loop can be tuned by itself and the whole
to stabilize manually because only one loop is
system should be stable if each individual loop
directly affected by the failure. Despite its
is
simple structure, decentralized PID control
stable.
Unfortunately,
due
to
this
interactions design of an effective control
has long record of satisfactory performance.
system for multivariable process is difficult.
When interactions are significant the multi
loop PID design most often fails to give
acceptable
responses.
In
other
words,
adjusting control parameters of one loop
affects
the
performance
of
another,
sometimes to the extent of destabilising the
entire system. So one approach for square
Fig 2.Decentralized control of MIMO process
system is to use a decoupler plus a
decentralised PID controller.
2.3 Decoupler design
One great advantage of this method is that it
The design of decoupled control system with
allows
decoupler matrix can be done combining a
the
use
of
single-input
single-
output(SISO) controller design methods.
diagonal
controller
Kd(s)
with
a
block
compensator D(s). With this configuration the
2.2 Decentralised controller
controller see the process as a set of n
Decentralized PID controller is one of
the most common control scheme for
interacting multi-input multi-output (MIMO)
plants in chemical and processing industries.
The main reason for this is its relatively simple
completely independent process or with
interaction minimized.
The objective in decoupling is to compensate
for the effect of interactions brought about by
cross coupling of the process variables.
structure, which is to understand and to
implement. With decentralized techniques,
from figure.2, a multivariable system inputs
and output variables is treated as n
monovariable systems. The number of tuning
parameters is 3n where n is the number of
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 21   22,

1,
v1 ( s )   (  ) s
21
22
,  21   22

e
 12   11
1,
v2 ( s )   (  ) s
12
11
 12   11
e
g ( s )  (12 11 ) s
d12 ( s )   12
e
g11 ( s )
Figure 4. General 2×2 system with decouplers
d 21 ( s )  
and single-loop controllers
g 21 ( s )  ( 21  22 ) s
e
g 22 ( s )
Let the TITO process be
 g ( s)e11s
G ( s)   11
 21s
 g 21 ( s)e
g12 ( s)e 12 s 

g 22 ( s)e  22 s 
(6)
(4)
It supposed that off-diagonal elements of G(s)
have no RHP poles. Another assumption is
that diagonal elements of G(s) are no RHP
d11(s)=1 and d22(s)=1
zeros
Q(s)= G(s)D(s) = diag{ q1(s),q2(s) }
From fig 4,
Q(s)
v1 ( s )

D( s )  
d
 21 ( s )v1 ( s )
d12 ( s )v2 ( s ) 

v2 ( s )
 (5)
is
to
be
controlled
(7)
through
a
decentralised PI(PID) controller. In other
words, q1(s) and q2(s), which are two SISO
plants, are controlled through k1(s) and k2(s)
Where:
respectively, where as each leading diagonal
element of decentralised control matrix is a
PI(PID) controller.It is worth noting that
interactions are zero unless additional large
poles are added to the decoupler to make it
proper .
3.0 Model reduction
In this section a first (second) order
plus dead time model is determined for each
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decoupled process. This approximation is
Where the cross over frequency, ωc, of the
done to determine the tuning parameter
original system is determined using
values from the optimum tunung formula
h( jc )  
3.1 FOPDT Approximation Model
As a result, the parameters of FOPDT model
Approximation
of
higher
order
can be calculated.
processes by lower order process plus dead
time model is a common practice. Although a
k p  h(0)
FOPDT model does not capture all the
features of higher order process, it often
By equating h (jωc) and the values of l(jωc)
reasonably describes the process gain, overall
can be given as;
time constant and effective dead time of such
kp
h  jc  
a process.
In order to find a approximate FOPDT model
for h(s), three unknown parameters namely
1   T c 
2
h(0)
h  jc  
1   T c 
kp, τd and T should be determined.
2
The first order system equation is given as
l ( s) 
 T c 
k p e  d s
Ts  1
(8)
k p  h(0)
2
 h(0)

 h  jc 

 T c  
{9)
The steady state gain and gain margin are
same for higher order process and FOPDT
T 
 h(0)

 h  jc 

 h(0)

 h  jc 

c
2

 1


2

 1


2

 1


(10)
model are same.
Hence this equation gives the time constant.
Hence,
l (0)  h(0)
l ( jc )  h( jc )
 l ( jc )   h( jc )
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For finding τd ,
We are equating phase of h(jωc),
 d c  tan 1 Tc   h  jc 
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
Where
In this method, we pick two points s= jωc and
h  jc   
s= jωb,
 d c  tan 1 T c   
where
 d c    tan 1 T c 
h  jb   
 T c 
  tan 1 T c 
d 
 d c    tan
1

2
h  jc   
c
(11)
Where ωb is determined by equating h(jωb)
and
3.2 SOPDT Approximation Model:
Although a large number of industrial
h  jc  
l  jb 
k pn2  cos c d   j sin c d  
c2  2 jnc  n2
processes can be fairly accurately modelled
using FOPDT transfer function, if a process has
an oscillatory step response, FOPDT model
cannot model the process well. In this case a
more accurate model of the process can be
obtained using SOPDT model in
l ( s) 
2  d s
p n
ke
s  2n s  
2
2
n
h  jb  
k pn2  cos b d   j sin b d  
 j h  jb  
h  jb  
,0    1
b2  2 jnb  n2
k pn2  cos b d   j sin b d  

2
n
 b2   2 jnb
k pn2  cos b d   j sin b d  
 j n2  b2   2nb
By equating the real part
(12)
k pn2 cos b d 
Therefore four unknown parameters should
h  jb  
be determined they are kp,τd,ωnandξ. To
k pn cos b d   2b h  jb 
2nb
determine the four unknowns four real
equations are needed and can be constructed
(13)
by fitting the process gain h(s) at to nonzero
frequency points.
By equating the imaginary part
l  jb   h  jb 
l  jc   h  jc 
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h  jb  
h  jc  
k pn2 sin b d 
 n2  b2 
k p sin b d     
2
n
2
n
2
b
 h  j 
k pn2  cos c d  
 n2  c2 
k pn2  cos c d     n2  c2  h  jc 
b
(16)
(14)
By equating the imaginary part
Where ωc is determined using
h  jc   h  jc   1  j  0  
h  jc    h  jc 
h  jc  
k pn2  sin c d  
2 jnc
k pn  sin c d    2 jc h  jc 
(15)
Now by equating
(17)
h  jc  and l  jc 
Thus by equating (13) and (17)
k pn cos b d 
k pn sin c d 

2b h  jb 
2c h  jc 
cos  jb d  b h  jb 

sin  jc d  c h  jc 
h  jb  
k pn2  cos b d   j sin b d  
 n2  b2   2 jnb
(18)
By merging Eq. (14) and (16)
    h  j 

      h  j 
k pn2 sin b d 
k p cos c d
2
n
2
n
2
n
2
b
2
c
sin b d  h  jc 
n2  b2

2
2
n  c
cos c d  h  jb 
b
c
19
By equating the real part
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
Eq. (19) gives the value of ωn ,then kp and ξ are
determinedfromtheEq(13)and(14)
respectively.
 h  jc  
k pn2  cos c d   j sin c d  

2
n
 c2   2 jnc
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formulae resulting from the NDT method
capable of coping with FOPDT lag dominant
4.0 OPTIMAL TUNING FORMULA
and integrating processes.
4.1 Optimal PI controller for FOPDT process
The tuning formulae is given by
In order to propose a set of PI tuning formulae
for FOPDT model, the PI parameters should
be defined based on model parameters,
kc k p 
T
1

2 d 14
kc  f1  k p , d , T  ,
(20)
Ti  f 2  k p , d , T  .
Ti

 1
 min 1  d ,9 d
T
T
 7T

.

Functions f1 and f2 should be determined so
(21)
that a set of performance criteria is optimised.
The optimisation problem may also be include
some constrains. Clearly it is very difficult to
determine these functions because each
parameter of the control is a function of three
parameter of the model. To simplify the
procedure of determining these functions,
non-dimensional tuning(NDT) method for
tuning PI controllers for FOPDT process is
4.2 Optimal PID Controller for SOPDT Process
Considering a step change in the setpoint, the
NDT formulae for a SOPDT process minimising
the IAE and satisfying the GM and PM
constraints are shown,
kc 

,
k p d n
used. The objective function was to minimize
the integral of absolute error(IAE) for a step
change in the setpoint. Since the load
Ti 
2
n
, Td 
1
2n
.
disturbanceresponses may be poor if the ratio
of the time delay to time constant is too
(22)
small, say less than one-ninth, the integral
time should be modified for such processes
which are referred to as lag dominant
processes. In addition, robustness is a key
issue in control systems. Gain and phase
margins are often used aas measures of
It is well known that despite good robustness
and
good
setpoint
responses,
load
disturbance responses may be poor if the
cancelled poles are shown in comparison with
dominant poles.
robustness. Considering GM≥3 and PM≥60 as
the robustness constrains. Optimal PI tuning
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6.6e 7 s
d 21  s    10.9s 31s
19.4e
14.4s  1
0.34 14.4s  1 e 4 s
d 21  s  
10.9s  1
5.0 SIMULATION RESULTS
Three
simulation
examples
By substituting all values in Eq.(5)
are
now
The decoupler matrix is given as
considered to demonstrate the closed-loop
performances of decentralized PID designed
with the proposed method.
5.1 EXAMPLE 1

1


D s 
 0.34 14.4s  1 e4 s

10.9s  1

1.477 16.7 s  1 e2 s 

21s  1


1


The wood-berry binary distillation column
plant is a multivariable system with strong
interaction and significant time delays, it is
described by the following process transfer
matrix:
v2  s   1
Q  s   diag q1  s  , q2  s  ,
q  s   G1  s  D  s 
 12.8e s 18.9e3s 


16.7
s

1
21.0
s

1

G1  s   
 6.6e7 s 19.4e3s 


 10.9s  1 14.4s  1 
v1  s   1
The resulting diagonal system is
6.43 14.4s  1 e7 s
12.8e s
q1  s  

16.7 s  1 10.9s  1 21.0s  1
9 s
19.4e3s 9.745 16.7 s  1 e
q2  s  

14.4s  1 10.s  1 21.0s  1
In order to determine a PI controller using the
From Eq.(6)
18.9e 3 s
1
d12  s    21.0s 
12.8e  s
16.7 s  1
1.477 16.7 s  1 e 2 s
d12  s  
21s  1
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NDT formulae, q1(s) andq2(s) should be
expressed
as
FOPDT
processes.
It
is
determined by finding ωc from the nyquist
plot of the original system.
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k p  9.655,
T  4.684,  d  2.157
Therefore by substituting the parameters in
Eq.(8)
9.655e2.157 s
l2  s  
4.684s  1
Fig 6. Nyquist plot of the original system(example 1)
From the fig 6,
From fig 7 & 8 shows the nyquist plots of
c  1.58
original system and FOPDT models process.
h  jc   0.740
h  0   6.37
By substituting all values in Eq.(9),(10) and(11)
k p  6.37
Fig 7. Nyquist plots of q1(s) and l1(s)
2
 6.37 

 1
74.099  1
 0.740 
T

 5.411
1.58
1.58
d 
  tan 1  5.4111.58
1.58

  tan 1 1.454 
1.58
 d  1.065
By substituting all the parameters in the Eq.(8)
l1  s  
1.065 s
Fig 8.Nyquist plots of q2(s) and l2(s)
6.37e
5.411s  1
Similarly from the Nyquist plot of q2(s),the
Optimal PI controller for FOPDT process:
values are given by,
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By substituting values in Eq.(20),
The output response is given as
1  5.411
1
kc1 

 
6.37  2 1.065 14 
kc1 
1
  2.54037  0.0714 
6.37
kc1 
1
  2.6117 
6.37
kc1  0.41
by substituting all values in Eq.(21)
 1 1.065 1.065 
Ti1  5.411 min 1  
,9

 7 5.411 5.411 
Fig 9.Response of example 1(WOOD-BERRY
Ti1  5.411 min 1.028,1.771
distillation column)
Ti1  5.4111.028
Ti1  5.56
The BLT and MV method does not use any
decoupling strategy. NDT and wang method
ki1  0.41
1
5.56
use the same decouplers but, the advantage is
that NDT use only four tuning parameters
ki1  0.074
whereas wang method uses six tuning
parameters.
Simillarly,
kc2  0.12
Ti2  0.024
The NDT controller is given as
5.2 EXAMPLE 2
K NDT
0.074


0
 0.41  s



0.024 

0
0.12 


s 

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Process control is an essential part of
desalination
operation
industry
at
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the
that
requires
optimum
for
operating
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
conditions, an increase in life time of the plant
and reduction of unit product cost.
The Alatiqi subsystem transfer matrix is given
as

0.51e7.5 s

2
 32s  1  2s  1
G2  s   
1.25e2.8 s

  43.6s  1 9s  1



 28s  1  2s  1 

4.78e1.15 s

 48s  1 5s  1 
1.68e2 s
2
1.25e 2.8 s
 43.6s  1 9s  1
d 21  s   
4.78e 1.15 s
 48s  1 5s  1
0.262  48s  1 5s  1 e 1.65 s
d 21  s  
 43.6s  1 9s  1
The diagonal system is given as
q  s   G2  s  D  s 
From Eq.(5)
The diagonal elements of Q(s) are as follows
The decoupler is given as
2

3.294  32s  1 
1


2
28s  1 


D s  

1.65 s
 0.262  48s  1 5s  1 e

5.5 s
e


43.6
s

1
9
s

1





q1  s  
0.51e7.5s

0.439  48s  1 5s  1 e3.65s
 32s  1  2s  1  28s  1  43.6s  19s  1 2s  1
2
4.118  32s  1 e2.8 s
4.78e6.65 s
q2  s  

 48s  1 5s  1  28s  12  43.6s  1 9s  1
2
2
In order to determine a PI controller using the
Where,
NDT formulae, q1(s) andq2(s) should be
v1  s   1
expressed
as
FOPDT
processes.
It
is
determined by finding ωc from the nyquist
v2  s   e5.5 s
plot of the original system.
1.68e2 s
d12  s 
 28s  1  2s  1

2
0.51e 7.5 s
 32s  1  2s  1
2
d12  s  
3.294  32s  1 e5.5
2
 28s  1
2
Fig 10.Nyquist plot of the original
system.(example 2)
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From the fig 10,
c  0.26
h  jc   0.0045
h  0   0.071
By substituting values in Eq.(9), (10)
Fig 11.Nyquist plots of q1(s) and l1(s)
and (11),
k p  0.071
T  58.761
 d  77.24
By substituting these values in Eq.(9)
0.071e77.24 s
l1  s  
58.761s  1
Fig 12.Nyquist plots of q2(s) and l2(s)
Similarly
0.662e61.981s
l2  s  
32.133s  1
Optimal PI controller for FOPDT process:
By substituting values in Eq.(20),
Fig 11 & 12 shows the Nyquist plots of original
system and FOPDT model of the process
kc1 
1
1
 58.76

 
0.071  2  77.24 14 
kc1 
1
  0.3803  0.0714 
0.071
kc1 
1
  0.45180 
0.071
kc1  6.393
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
By substituting values in Eq.(21),
 1 77.24 77.24 
Ti1  58.76  min 1  
,9

 7 58.76 58.76 
Ti1  5.411 min 1.1877,11.83
fig 13
Ti1  5.4111.1877
Fig 13.Response of example 2(Alatiqi subsystem)
Ti1  6.427
ki1  6.393 
1
6.427
ki1  0.092
5.3 EXAMPLE 3
Similarly,
Let the process be:
kc2  0.499
0.5s

2
2

 0.1s  1  0.2s  1

G3  s  

1

  0.1s  1 0.2s  12

Ti2  0.012
1


 0.1s  1 0.2s  1


2.4

2
 0.1s  1 0.2s  1  0.5s  1 
2
The NDT controller is given as
Due to large interactions in this process, the
K NDT
0.092


0
 6.393  s



0.012 

0
0.499 


s 

performance
of
the
decentralised
PID
controller based on the critical point is not
satisfactory and so is not considered for
comparison
The response is given as
Using Eq.(5), the decoupler is given by:

1
2  0.1s  1 


D  s   5
  0.5s  1

1
 12

Where:
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1
d12  s   
v1  s   1
 0.1s  1 0.2s  1
2
0.5s
2
2
 0.1s  1  0.2s  1
 2  0.1s  1
determined by finding ωc from the nyquist
plot of the original system.
v2  s   1
1
 0.1s  1 0.2 s  1
d 21  s   
2
2.4
 0.1s  1 0.2 s  1  0.5s  1

5
12
 0.5s  1
2
Using additional poles with small time
Fig 14.Nyquist plot of the original system.(example 3)
constants, a practical decoupler is given by:
From the Fig 14,

1


D s  
 5  0.5s  1

 12 0.05s  1
2
0.1s  1 
0.01s  1 


1


c  19.89
h  jc   0.0777
h  0   0.917
The diagonal elements of
By substituting values in Eq(9) (10)

Q s  G s D s
and (11)
k p  0.0777
Are given as
T  0.591
5

 0.5s  1 
 0.5
12
q1  s  


2 
 0.1s  1 0.2s  1  0.1s  1 0.05s  1 




1
2.4
2
q2  s  


2 
 0.2s  1   0.1s  1 0.5s  1 0.01s  1 
1
 d  0.083
By substituting all values in Eq.(8)
The FOPDT model for q1(s) is given by:
In order to determine a PI controller using the
NDT formulae, q1(s) andq2(s) should be
expressed
as
FOPDT
ISSN: 2231-5381
processes.
It
is
0.917e0.083s
l1  s  
0.591s  1
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
Similarly
sin 19.89  d   0.139
The FOPDT model for q2(s) is given by:
 d  0.029
l2  s  
From Eq.(19), substituting the values
4.4e0.252 s
3.003s  1
sin  5.572  0.029   0.0777
n2  5.5722

2
2
n  19.89
cos 19.89  0.029   0.510155
SOPDT model reduction:
Since FOPDT model cannot model the process
well. In this case, a more accurate model of
the process can be obtained using the SOPDT
n2  31.047
0.002820  0.0777

2
n  395.6121
0.999  0.510155
n2  31.047  n2  395.6121  0.0004299
model.
n2  41.229
It is given as
l ( s) 
Then kp and ξare determined from Eqs.(14)
k p n2 e d s
s  2 n s  
2
2
n
, 0  1
and (13), respectively
k p  41.29  cos 19.89  0.029    41.229  31.047   0.077
b  5.572 and c  19.89
k p  0.7925
h  jb   0.510155
h  jc   19.89
0.7925  41.229  cos  5.572  0.029  2    0.510155
By substituting values in Eq.(18)
cos  5.572  d  5.572  0.510155

sin 19.89  d 
19.89  0.0777
  0.88374
Thus by substituting all four unknown values
in Eq.(12)
0.510155
cos  5.572  d  
19.89
cos  5.572  d   0.0256
and sin 19.89  d  
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0.0777
5.572
l1  s  
32.674e0.029 s
s 2  11.349s  41.229
Similarly SOPDT model for q2 is
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l 2 s 
46.601e0.008 s
s 2  7.838s  9.443
and
Ti1  0.591  min 1.02006,1.2695   0.5911.02006
Ti1  0.6028
ki1  3.946 
1
0.6028
ki1  6.551
Fig 15.The Nyquist plots of q1(s) and its appxoximation
models(example 3)
Similarly,
kc2  1.368
ki2  0.451
The NDT-PI controller is given by
K NDT  PI
6.551


3.946

0


s


0.451 

0
1.368 


s 

Fig 16.The Nyquist plots of q2(s) and its approximation
models (example 3)
The response of it is given as
OPTIMAL TUNING:
Optimal PI controller for FOPDT process
kc1 
0.917  0.591 1

2  0.083
14
kc1  3.946
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Td1  0.088113
kd1  6.088  0.088113
kd1  0.536
The NDT-PID controller is given by
Fig 17. Response of example 3(with PI controller)
Optimal PID controller for SOPDT process:
K NDT  PID
22.113


0
 6.088  s  0.536s



12.686


0
10.529


1.344
s


s


The response is given as
By substituting all values in Eq.(22)
kc1 
0.88374
0.7925  0.029  6.42098
kc1 
0.88374
0.14757
kc1  6.088
Fig 18. Response of example 3(with PID controller)
Ti1 
2  0.88374
6.42098
Due to adding poles with small time
constants to off-diagonal elements of the
Ti1  0.275266
1
ki1  6.088 
0.275266
decoupler, the NDT controllers have non-zero
interactions. Clearly the NDT-PID controllers
gives the best response interms of set point
regulation and load disturbance rejection.
ki2  22.11
Td1 
1
2  0.88374  6.42098
1
Td1 
11.34895
ISSN: 2231-5381
6. CONCLUTION
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 5 Number3–Sep 2012
This method is one of the simplest
method to tune decentralised
variable PID controllers from
PI(PID)
controllers for TITO process. The TITO process
decentralized relay feedback”.
Wang, Q.G.,
5.
was decoupled through a decoupler matrix. A
Lee, T.H., Fung, H. W., Bi, Q.,&Zhang,
model reduction method was done to find the
Y.(1999). “PID tuning for improved
suitable FOPDT(SOPDT)
performance.”
model
for each
element of the resulting diagonal process
Wood, R. K., &
6.
through fitting the nyquist plots at particular
Berry, M.W. (1973). “Terminal
points.
composition control of a binary
The
performance
of
the
NDT
technique was investigated through examples.
distillation column. Chemical
Compared to other methods it has less
Engineering Science”.
number of tuning parameters and more over
uses
only
less
number
of
controllers,
therefore it is one of the cost effective
www.Waset.com
www.archivesofcontrolscience.com
method to control the process
REFERENCE
1. Luyben, W.L. (1986). “Simple method
for tuning for SISO controllers in
multivariable systems”.
SaeedTavakoli,
2.
Ian Griffin, Peter . Fleming (2006).
“Tuning of decentralised PI(PID)
controllers for TITO process”.
Tavakoli, S.,
3.
Griffin, I., & Fleming, P.J.(2003).
“Optimal tuning of PI controlles for
first order plus dead time/long dead
time models using dimension
analysis”.
Wang,
4.
Q.G.,Zou,B.,Lee,T.H., &Bi,
Q.(1997).“Auto-tuning of multi-
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