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Delay-compensated controllers for two-inputtwo-output (TITO) multivariable processes.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
Published online 12 October 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.095
Research Article
Delay-compensated controllers for two-input/two-output
(TITO) multivariable processes
A. Seshagiri Rao, V. S. R. Rao and M. Chidambaram*
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
Received 19 March 2007; Revised 24 July 2007; Accepted 13 August 2007
ABSTRACT: A decentralized controller design based on the direct synthesis method for two-input/two-output (TITO)
processes in the Smith predictor configuration is proposed. The decentralized controllers are designed individually on
the basis of the diagonal individual transfer functions of the process. Using the direct synthesis principle, the two
decentralized controllers are designed using the individual diagonal transfer functions of the process matrix and the
individual desired diagonal closed-loop transfer functions. Further, these direct synthesis controllers are implemented
in the Smith predictor configuration for each output. While converting from the synthesis controller to the Smith
predictor configuration, the decentralized controller parameters are directly obtained in terms of the individual closedloop desired tuning parameters. The method is compared with the recently reported methods. ? 2007 Curtin University
of Technology and John Wiley & Sons, Ltd.
KEYWORDS: Smith predictor; time delay; TITO process; direct synthesis method
INTRODUCTION
In process industries, two-input/two-output (TITO) processes are the most commonly encountered multivariable processes. Moreover, many processes with
inputs/outputs beyond two can be treated as several
two-by-two subsystems in practice.[1] Since different
time delays are present in different control loops, and
owing to the interactions, the difficulty further increases
in multivariable systems. Alevisakis and Seborg[2] have
extended the original Smith delay compensator to multiinput/multi-output (MIMO) processes for a single time
delay. Later, many authors have proposed modified
design methods for the Smith predictor.[3 ? 6]
Decentralized controllers are mostly used for multivariable processes because it is simple, easy to implement and also easy to tune. In view of the fact that
a conventional proportional-integral-derivative (PID)
controller often results in excessive oscillation of the set
point response in contrast with a proportional-integral
(PI) controller, Chien et al .[7] have proposed a modified implementation form of the PID controllers and
a tuning rule for multiloop control systems. Improved
control design methods have also been proposed by
many authors.[8 ? 11] Huang et al .[8] proposed multiloop
*Correspondence to: M. Chidambaram, Department of Chemical
Engineering, Indian Institute of Technology Madras, Chennai 600
036, India.
E-mail: chidam@iitm.ac.in
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
PI/PID controller design formulas by decomposing a
multiloop system into a number of single loop systems
and formulating an effective open loop process. Liu
et al .[11] proposed TITO PI/PID controller design formulas by using the Maclaurin series for approximating
the obtained controller expression for each loop. Lee
et al .[9] proposed analytical PI/PID controller expressions based on IMC principles. However, the methods
proposed involve several rigorous mathematical calculations and the methods are complicated. In the present
work, a simple design method based on the direct synthesis method is proposed. The method of designing
controllers by Lee et al .[9] is modified without approximating the time delay. Direct synthesis controllers are
implemented in the Smith predictor configuration.
THEORETICAL DEVELOPMENTS
The general block diagram of the feedback control for
TITO processes with a decentralized controller is shown
in Fig. 1 where Gp (s) is the transfer function matrix of
the process and Gc is the transfer function matrix of the
decentralized controller.
The closed-loop transfer function matrix for set-point
changes is given by
y(s)
= H(s) = [I + Gp (s)Gc (s)]?1 Gp (s)Gc (s)
yr (s)
(1)
Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES
d
Gc
e
yr
-
Gc11
0
0
u
Gc22
action. A time delay compensator enhances the closedloop performance by compensating for the time delay.
After a time delay compensator is used, the main
controller should be designed on the basis of the
process without considering the time delay. To develop
a compensator and to design a compensated controller,
in the present work, the same direct synthesis controller,
Eqn (5), is used.
Gp
G11
G21
G12
G22
y
Figure 1. Simple decentralized feedback for TITO
processes.
According to the direct synthesis method, the controller
matrix is given by
Gc (s) = Gp ?1 (s)R(s)[1 ? R(s)]?1
(2)
where R(s) [= (y(s)/yr (s))d ] is the desired closedloop transfer function matrix for TITO processes with
time delay. The interaction is a major problem in
multivariable processes. To get decoupled responses,
the off-diagonal elements are dropped to zero i.e. Gp =
diag (G11 , G22 ). With that, the closed loop responses
are decoupled and the controller for i th loop can be
written as
Gci (s) = Gii ?1 (s)Ri (s)[1 ? Ri (s)]?1
DELAY-COMPENSATED CONTROLLER
DESIGN
Without approximating the time delay in Eqn (5) for
controller design, Eqn (5) can be structurally implemented as shown in Fig. 2. Further, Fig. 2 can be written in an alternative way as shown in Fig. 3. Figure 4 is
the Smith predictor structure for process Gii (s), where
Gcii is the predictor controller. Now the controller is
designed for a first order plus time delay (FOPTD) process. For a FOPTD process, Gii ? = kii /(?ii s + 1).
(3)
?is+1
e
Where Ri (s) is the desired closed loop transfer function
for i th loop.
Note: The off-diagonal processes are neglected just
for simplicity. It will be shown later that even by
neglecting the off-diagonal elements, the obtained
closed-loop responses are satisfactory.
Let us consider the process as Gii (s) = Gii + (s)
Gii ? (s) where Gii + (s) is non-minimum phase (NMP)
elements of the process i.e. the time delay and right
half plane zero. Gii ? (s) is the process without the NMP
elements. According to the direct synthesis method, the
desired closed-loop transfer function should contain the
process NMP elements. Hence, in the present work, the
desired closed-loop transfer function is considered as
Ri (s) =
(?i s + 1)
(?s + 1)2
Gii + (s)
Figure 2.
Eqn (5).
[(?i s + 1)2 ? (?i s + 1)Gii + (s)]
(5)
As Gii + (s) contains the time delay term, this time delay
is approximated either using a Taylor series expansion
or a Pade approximation. After approximating the
time delay and simplifying, Eqn (4) results in a PID
controller according to the direct synthesis method.
However, because of the time delay approximation,
there will be a limit on the controller gain and therefore
there will be a limit on the possible amount of control
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Structural implementation of Gcii ,
(?is+1)G?1
ii?(s)
e
+
+
u
(?is+1)2
Gii+(s)Gii?(s)
Figure 3.
Fig. 2.
(4)
(?i s + 1)
u
Gii+(s)
Upon substitution in Eqn (3), the controller is
obtained as
Gci (s) = Gii ? ?1 (s)
?1(s)
Gii?
(?is+1)2
+
+
Alternative structure for
Gcii
?1(s)
(?is+1)Gii?
e
+
+
+-
u
[?i(s+1)2?(?is+1)]
Gii?(s)
Gii+(s)Gii?(s)
Figure 4. Delay-compensated controller in the
Smith predictor form.
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
DOI: 10.1002/apj
511
512
A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM
Substituting in Gcii results in
Gcii =
(?i s + 1)(?ii s + 1)
(6)
kii [(?i s + 1)2 ? (?i s + 1)]
Expanding and simplifying yields
Gcii =
(?i s + 1)(?ii s + 1)
kii (2?i ? ?i )s[(?i 2 /(2?i ? ?i ))s + 1)]
(7)
Equation (7) can be further simplified to a PI controller
form by considering ?ii = ?i 2 /(2?i ? ?i ) as:
?i
1
(?i s + 1)
Gcii =
=
1+
(8)
kii (2?i ? ?i )s
kii (2?i ? ?i )
?i s
where ?i = 2?i ? (?i 2 /?ii ). In a conventional PI controller form [kci (1 + (1/?Ii s))], the controller parameters are obtained as:
?i
and ?Ii = ?i
(9)
kci =
kii (2?i ? ?i )
Hence for a given TITO process with time delay,
the two decentralized controllers can be calculated
from Eqn (9) on the basis of the processes G11 and
G22 with the assumption that there is no plant model
mismatch. But, in practice, the effect of the other
manipulated variables affect the process output and
hence the predictor structure should be implemented by
considering the other processes also, as shown in Fig. 5
for TITO processes. Here G ij (i = 1, 2; j = 1, 2) is the
process model with time delay and Giim is the process
model without time delay.
SELECTION OF THE TUNING PARAMETER (?I )
The tuning parameter for the individual closed-loop
responses should be chosen such that the resulting controller should be able to provide both nominal and
[G11m G12m]
+
G11 G12
yr1
+
Gc11
+
-
0
-
+
0
u
yr2
+
y1
G11 G12
G21 G22
Gc22
+
-
G21 G22
[G21m G22m]
y2
+
+
Figure 5. Decentralized Smith predictor configuration.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
robust control performances. Many simulation results
are carried out on different TITO time delay processes
and it is observed that the initial value of the tuning
parameter can be considered as half of the corresponding time constant of the process. So, the selection of
the tuning parameter is purely based on the simulation
results. The proposed method of selection of the tuning
parameter is simple.
If both the nominal and robust control performances
are achieved with these values, these values can be
taken as the final tuning parameters; else, the tuning
parameters should be tuned around this value till robust
control performances are achieved. In practice, the
selection of the tuning parameter is based on the
complete knowledge of the plant and its dynamics. In
general, the tuning parameters selected on the basis
of the diagonal processes are able to provide robust
control performances. However, when there is large
process model mismatches in all the processes, the
tuning parameter selected on the basis of the diagonal
processes may not give robust control performances.
In that case, only the tuning parameters need to be
detuned to get robust performances. In the present work,
no detuning is done for uncertainties. The final values
for the tuning parameters are calculated for the nominal
processes only after doing many simulation studies to
get improved performances.
SIMULATION STUDY
For the purpose of simulation studies, different TITO
time delay processes are considered and the closed-loop
performance is compared with the recently reported
methods in the literature.
Example 1: Consider the widely studied Wood and
Berry distillation process[12]
?
?
12.8e?s
?18.9e?3s
+1
21s + 1 ?
G(s) = ? 16.7s ?7s
6.6e
?19.4e?3s
10.9s + 1 14.4s + 1
Using Eqn (9), the decentralized controllers parameters
are calculated on the basis of the principal diagonal
process transfer functions as kc1 = 0.248, ?I 1 = 12.157,
kc2 = ?0.134, ?I 2 = 11.5517. The tuning parameters
are selected as ?1 = ?2 = 8. With these controller
parameters, the proposed method is simulated after giving a unit step change in set points for the two controlled
variables and a negative step input of magnitude 0.1 in
the load at t = 200 s for two manipulated variables,
respectively. For comparison, the methods proposed by
Lee et al .,[9] Liu et al .[11] and Lee and Edgar[10] are
considered. The corresponding responses are shown in
Fig. 6. From the responses, it can be seen that the proposed method gives better control performances. The
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES
corresponding control action responses are shown in
Fig. 7. It can be seen that the proposed method gives
a smoother control action compared to the other methods. It should be noted that Refs [9?11] have already
shown that the performance of their methods are better
than those of many previous methods.
In practice, there always exists a model plant
mismatch, and therefore the closed-loop performance
should be evaluated by considering the process uncertainties. The uncertainty may exist as an input uncertainty, an output uncertainty or a process uncertainty.
Process uncertainty is considered in the present work.
Usually the delay compensators are sensitive to the time
delay mismatches in a process if the ratio between time
delay and time constant is large, but if the controller
design is robust, the delay-compensated controller is
able to give robust control performances.[13] In the
present work, +30% perturbation in each process time
delay is considered, and the corresponding closed-loop
responses are shown in Fig. 8. It is clear from the figure
that the proposed method performs better. Unlike in
other methods, the responses for the proposed method
are smooth, even though there exist delay perturbations. Figure 9 shows the corresponding control action
responses. From the figure, it can be seen that the control action responses of the proposed method are smooth
compared to the other methods.
1.5
1.5
1
y1
y1
1
0.5
0.5
0
0
0
50
100
150
200
250
300
100
150
200
250
300
0
50
100
150
Time
200
250
300
1.5
y2
1.5
y2
50
2
2
1
1
0.5
0.5
0
0
?0.5
?0.5
0
0
50
100
150
Time
200
250
300
Figure 8. Responses for the Wood and Berry distillation
Responses for the Wood and Berry distillation column for perfect parameters: solid line ? proposed;
dash ? Ref. [11]; dot ? Ref. [9]; dash dot ? Ref. [10]. This
figure is available in colour online at www.apjChemEng.com.
Figure 6.
column for a perturbation of +30% in each process time
delay: solid ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash
dot ? Ref. [10]. This figure is available in colour online at
www.apjChemEng.com.
1
0.5
0.5
U1
U1
1
0
0
?0.5
0
50
100
150
200
250
?0.5
300
100
150
200
250
300
0
50
100
150
Time
200
250
300
0.1
0.05
0.05
0
U2
U2
50
0.15
0.1
?0.05
?0.1
0
0
?0.05
0
50
100
150
Time
200
250
300
Figure 7.
Control action responses for the Wood
and Berry distillation column for perfect parameters: solid ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash
dot ? Ref. [10]. This figure is available in colour online at
www.apjChemEng.com.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
?0.1
Figure 9. Control action responses for the Wood and
Berry distillation column for a perturbation of +30% in
each process time delay: solid ? proposed; dash ? Ref. [11];
dot ? Ref. [9]; dash dot ? Ref. [10]. This figure is available in
colour online at www.apjChemEng.com.
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
DOI: 10.1002/apj
513
A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM
y1
The decentralized controller parameters are calculated on the basis of the principal diagonal transfer
functions of the process as kc1 = 34.4, ?I 1 = 36.56,
kc2 = ?24.07, ?I 2 = 26.74. The tuning parameters are
selected as ?1 = 22.5 and ?2 = 18 respectively.
With these controller parameters, the performance
of the closed-loop system is evaluated after giving
unit step input to the set points of the two controlled
variables simultaneously and a negative step input of
magnitude 10 at t = 1000 s in the load for the two
manipulated variables. The responses are shown in
Fig. 10. For comparison, the methods of Huang et al .[8]
and Lee and Edgar[10] are considered. From the figure,
it can be seen that the proposed method gives improved
performance compared to the other two methods. A
perturbation of +30% in each process time delay is
considered, and the performance is evaluated for all
the methods, and corresponding responses are shown
in Fig. 11. It is clear from the figure that the proposed
method performs better.
The methods of Huang et al .[8] and Lee and Edgar[10]
result in more overshoot and settling time under nominal and mismatch conditions, whereas the proposed
method results in significantly less overshoot and settling time under both nominal and mismatch conditions.
A similar improvement in closed-loop performance is
observed when the mismatch is considered in all the
process gains and time constants. Also, it is observed
that the control action responses (not shown) for the
1.2
1
0.8
0.6
0.4
0.2
0
0
500
0
500
1000
1500
1000
1500
1.5
1
0.5
0
?0.5
Time
Figure 10. Responses for the Wardle and Wood column
for perfect parameters: solid ? proposed; dash ? Ref. [10] ;
dot ? Ref. [8]. This figure is available in colour online at
www.apjChemEng.com.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
1.5
y1
1
0.5
0
0
500
0
500
1000
1500
1000
1500
1.5
1
y2
Example 2: Consider the Wardle and Wood distillation
column[14]
?
?
?0.101e?12s
0.126e?6s
1
(48s + 1)(45s + 1) ?
G(s) = ? 60s +?8s
?0.12e?8s
0.094e
38s + 1
35s + 1
y2
514
0.5
0
?0.5
Time
Figure 11. Responses for the Wardle and Wood column
for a perturbation of +30% in each process time delay:
solid ? proposed; dash ? Ref. [10]; dot ? Ref. [8]. This figure
is available in colour online at www.apjChemEng.com.
proposed method are smooth compared to that of the
other two methods.
Example 3: To show the advantage of the proposed
method, the Wood and Berry[12] distillation process is
modified in which the time delay of each process is
considered as 3 times the corresponding time constant.
The modified process matrix is given by:
?
12.8e?50.1s
+1
G(s) = ? 16.7s?32.7s
6.6e
10.9s + 1
?
?18.9e?63s
21s + 1 ?
?19.4e?43.2s
14.4s + 1
For the proposed method, using Eqn (9), the decentralized controller parameters are calculated on the basis
of the principal diagonal process transfer functions as
kc1 = 0.14, ?I 1 = 15.37, kc2 = ?0.072, ?I 2 = 14. The
tuning parameters are selected as ?1 = ?2 = 12. It is
to be noted that the tuning parameters are increased in
this case compared to those in Example 1. The reason
is that with the same controller settings as in Example 1, the proposed method is able to provide good
control responses at the cost of more control action.
To minimize the control action, in this case, the tuning
parameters are slightly increased. One can see that there
is an almost 50% increase in the tuning parameters. But
this increase in tuning parameters is to reduce the control action only. In the Lee et al .[9] method, the tuning
parameters are selected as 80 and 80, which provided
the best performance for their method.
With these controller parameters, the proposed
method is simulated after giving a unit step change at
t = 0 s in set point r1 and at t = 300 s in set point
r2 for the two controlled variables and a negative step
input of magnitude 0.1 in the load at t = 700 s for
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES
Y2
Y1
the two manipulated variables respectively. For comparison, the method of Lee et al .[9] is considered. The
corresponding responses are shown in Fig. 12. From
the responses, it can be seen that the proposed method
gives significantly better control performances at higher
time delays. The corresponding control action responses
are shown in Fig. 13. A similar improvement in control action responses is obtained compared to the other
method. When the closed-loop performance is evaluated
by considering perturbations in the process parameters,
the proposed method is able to track the set points
and reject the disturbances whereas the method of Lee
et al .[9] is not able to track the set point; in other words,
their responses are completely oscillatory but will be
stable if enough time is given.
2.5
2
1.5
1
0.5
0
?0.5
3
2.5
2
1.5
1
0.5
0
?0.5
0
200
400
600
800
1000
1200
0
200
400
600
Time
800
1000
1200
Figure 12. Responses for Example 3 for the perfect model:
solid ? proposed method; dash ? Lee et al.[9] This figure is
available in colour online at www.apjChemEng.com
DISCUSSION
The design of controllers in the proposed method
is straightforward and hence the proposed method
can be implemented practically easily. With the proposed method of design of controllers, both nominal and robust control performances are achieved.
Also, the compensated structure and the controller?s
parameters are obtained simultaneously unlike in any
other compensated controller, in which the controllers
are designed after designing the compensator.[15] The
method can be extended to MIMO systems, which
needs further studies. Also, the advantage of the present
method depends on the diagonal dominance of the process transfer function matrix.
The performance of the controllers is clearly given
in the graphs. The same difference can be given by
calculating the integral of absolute error (IAE) or
integral of squared error (ISE) values. In the present
work, the set point changes and load disturbances are
given simultaneously and therefore the error values
are calculated as a sum of both servo and regulatory
problems. ISE values are calculated for all the examples
for perfect parameters and are shown in Table 1. From
the ISE values one can see that the proposed method
shows improved control action responses for all the
examples.
It can be observed that for Example 2 the time
constants of the diagonal processes are 60 and 35,
respectively. Thus the tuning parameters are considered
differently. However, in Examples 1 and 3, the time
constants are very close. Therefore, same value for the
tuning parameters (?1 and ?2 ) is chosen.
In the literature, the closed-loop response is usually chosen as a FOPTD model. However, it is shown
here that by selecting the closed-loop specification as
a second-order model, the closed-loop performance
0.2
U1
0.15
Table 1. ISE values for the control action of all methods
for perfect parameters.
0.1
0.05
Example 1
0
?0.05
0
200
400
600
800
1000
1200
0.1
U2
0.05
Method
U1
U2
Total
Proposed
Liu et al .[11]
Lee et al .[9]
Lee and Edgar[10]
9.24
9.90
9.7
9.95
1.01
2.10
1.8
3
10.26
12.00
11.5
12.95
2.4
4.5
6.0
7.3
11.2
13.0
2.61
4.01
27.2
42.98
0
Example 2
?0.05
?0.1
0
200
400
600
Time
800
1000
1200
Control action responses for Example 3 for
the perfect model: solid ? proposed method; dash ? Lee
et al.[9] This figure is available in colour online at
www.apjChemEng.com
Proposed �?5
Lee and Edgar[10] �?5
Huang et al .[8] �?5
Example 3
Figure 13.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
4.9
6.7
7.0
Proposed
Lee et al .[9]
24.58
38.97
Asia-Pac. J. Chem. Eng. 2007; 2: 510?516
DOI: 10.1002/apj
515
Y1
A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
?0.2
Asia-Pacific Journal of Chemical Engineering
CONCLUSIONS
0
50
100
150
200
250
300
200
250
300
Time
Y2
516
0
50
100
150
Time
Figure 14. Responses for Example 1 for the perfect model:
solid ? second order; dash ? first order desired closed-loop
transfer functions. This figure is available in colour online at
www.apjChemEng.com.
can be improved both for perfect model conditions
and for perturbations. Hence in the present work,
the closed-loop trajectory is chosen as a second-order
model instead of the traditional FOPTD. The controller parameters obtained by considering FOPTD
model (exp(?? s)/?s + 1) as the desired closed transfer function is kci = ?ii /kii ?i , ?Ii = ?ii . The controller
parameters calculated for Example 1 are kc1 = 0.1631,
?i 1 = 16.7, kc2 = ?0.0928 and ?i1 = ?14.4. The performance comparisons between the two for Example 1
are shown in Fig. 14 after giving a unit step change in
set points for the two controlled variables and a negative
step input of magnitude 0.1 in the load at t = 200 s for
the two manipulated variables, respectively. It can be
seen from the responses that the second-order desired
closed-loop model gives better performances. A similar
improvement is obtained when there are perturbations
in the process parameters.
? 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
A simple method of designing the controllers for TITO
time delay processes in the Smith predictor configuration is proposed with direct design of the predictor
controllers. The method has two tuning parameters and
these two can be selected to get robust control performances. Significant improvement in control performances is obtained when compared to recently reported
methods, particularly when the individual time delays of
the processes are large compared to the corresponding
time constants.
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DOI: 10.1002/apj
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two, inputtwo, output, controller, multivariable, titov, processes, dela, compensate
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