Abstract: This work reveals an Internal Model Control (IMC)-based series
cascade control for the non-minimum phase and time delay process. The
combination of a higher-order fractional IMC filter and inverse response
compensator for designing the outer loop controller illustrates the uniqueness of
this work. For the time delay term, a higher-order approximation is considered.
The standard IMC-PID structure adopts for the inner loop controller design, while
the higher-order fractional filter coupled with inverse response compensator takes
for the design of the outer loop controller. The suggested scheme demonstrates
enhanced exhibition for setpoint tracking and disturbance rejection. Moreover,
the sensitivity analysis is also accomplished to determine the robustness of the
closed-loop system under process parameter variations.
Keywords: Cascade control, fractional filter, Internal Model Control (IMC),
Non-minimum phase and time delay system.
1. Introduction
The Cascade control is used for slow dynamics and actuator nonlinearities like
distillation column, a heat exchanger (Seborg, 2004, Raja and Ali, 2017). In
cascade control, two controllers are prepared like that one can regulate the set
point of the other. There are two loops in a cascade control, i.e., acknowledged as
the inner and outer loop. The cascade control consists of benefits like an
improvement in the dynamic appearance of the structure, capability to earlier
improvement from disturbances and compensation of dead time effect from the
system. Cascade control system consists of two types, i.e., series and parallel
cascade control (Stephanopoulos, 2015).
Moreover, the series cascade control scheme is extensively utilized in process
engineering. An outline survey for series cascade control was addressed (Raja and
Ali, 2017). In series, cascade control minimum no. of parameters are required to
tune. This feature becomes beneficial for users Ranganyakulu et al., (2020). Series
cascade control for integrating the process with time delay was discussed (Padhan
and Manjhi, 2013). IMC-based PID controller tuning for series cascade unstable
systems was discussed (Santosh and Chidambaram, 2013). Vanavil, B. et al.
(2012) was discussed cascade control for an unstable system. Robust IMC –PID
tuning for cascade control was addressed (Azhar and Serrano, 2014).
However, the ubiquity of the PID controller is not just restricted to integer-order
goes further to fractional order (Podlubny,1999). In recent years, fractional PID
(Fergani and Charef, 2016) has been explored potentially for the time delay. In
Bettayeb et al. (2014), the smith predictor embedded fractional filter PID
controller was addressed. Yumuk et al. (2019) was discussed fractional order PID
controller.
Conversely, specific chemical process control systems specify their preliminary
output in the reverse direction in process industries. Notably, when the system
possesses an inverse response, its transfer function has positive zeros, then it is
termed a Non-minimum phase (NMP) system. These NMP zeros demonstrate the
following special effects rather than undershoot. The output of the closed-loop
system display overshoot (ii) predictable deterioration that arises involving a
restricted gain margin for the feedback control system (iii) the restricted gain
margin also involves a constraint on the robustness of the closed-loop system
(Hogg and Bernstein, 2007). However, the occurrence of dead time restricts the
proportional gain of the controller and affects oscillations in output response. On
increasing the controller gain, the closed-loop system is directed towards
instability (Stephanoplous, 2015).
Likewise, the inverse response and dead-time system are mainly challenging to
control, such as boiler drum level. If a step increases the flow rate of the cold
feedwater, the total volume of the boiling water and consequently the liquid level
starts to decrease for a short period and then it starts increasing. However, the
cascade control may not deliver relevant results due to inverse response and time
delay. Thus, the idea for better control has been introduced via inverse response
compensator embed with IMC controller for outer loop of series cascade scheme.
The inverse response compensator predicts the inverse behavior of the process
and provides a corrective signal to eliminate it. The notion of inverse response
and dead-time compensator was woven together with a fractional-order IMC filter
(Nagarsheth and Sharma, 2020). Their work used lower-order fractional IMC
filter, dead-time, inverse response compensator, and setpoint filter for the single
loop feedback system.
Consequently, the above publications offered a lower-order fractional filter with
an inverse response compensator for a single loop feedback system. However, no
unified theory is available that combine higher-order fractional IMC filter and
inverse response compensator for series cascade control. This paper produces a
combined theory of higher-order fractional IMC filter, inverse response
compensator and higher-order time delay approximation with series cascade
control, which prevents the use of setpoint filter and dead-time compensator. In
addition, the proposed method required a two-controller unlike three controllers
for setpoint tracking, dead time and inverse response compensation. Furthermore,
both the loops of series cascade control are tuned with the most familiar tuning
approach, i.e., Internal Model Control (IMC). The Riemann surface does the
stability assessment of the proposed work. The sensitivity analysis is
accomplished to determine the robustness. To validate the efficacy of the
suggested process is applied into two practical problems.
The work arrangement is prepared as: Section 2 expresses preliminaries useful in
practical problems, and Section 3 indicates case studies. The conclusion is
available in Section 4.
2. Preliminaries
This section comprises mathematical preliminaries that are useful for carrying out
Theoretical development & application of the proposed controller to the practical
problems
2.1 Bode's ideal transfer function:
In 1945 H. Bode introduced an idea on the fractional filter, which is termed
Bode's ideal transfer function. In an open-loop, Bode's ideal transfer function
can be exhibit as, Yumuk et al., (2019).
1
1
)(
s
sM
,
R
Where and are the fractional filter parameters. The Bode's ideal transfer
function indicates basic properties variations in gain and constant phase margin.
2.2 A generalized cascade scheme with IMC control
The universal form of the series cascade scheme in the IMC framework displays
in figure1. In this scheme, two controllers
)(
1
sG
c
)(
2
sG
c
are existed, i.e., master and
slave controllers. This control scheme contains two plant transfer function
)(
1
sG
M
,
).(
2
sG
M
In the IMC structure, plants have their process model transfer
function
)(
~
1
sG
M
and
).(
~
2
sG
M
In this scheme, the effects of the inner loop input for
the outer loop. In the generalized scheme
1
r
,
1
y
,
1
d
and
2
r
,
2
y
,
2
d
the input,
output disturbance for outer and inner loop respectively (Stephanopoulos,
2015).
)(
1
sG
c
)(
2
sG
c
)(
2
sG
M
)(
1
sG
M
)(
~
2
sG
M
)(
~
1
sG
M
1
r
2
r
2
y
1
y
+
-
+
-
-
-
2
d
1
d
Inner loop
Outer loop
+
+
+
+
+
Figure 1: Conventional series cascade scheme with IMC control.
From figure 1, the transfer function of both loops can be determined easily which
can be written as
.))()(1()()()(
222
2222
dsGsGrsGsGsy
cMcM
(1)
.))()()()(1((
)()()()(1(
))()(1(
)()()()()(
1
2
11
1122
1122
22
1122
dsGsGsGsG
dsGsGsGsG
sGsG
rsGsGsGsGsy
cMcM
cMcM
cM
cMcM
(2)
2.3 Proposed series cascade control scheme
The traditional series cascade control structure is reviewed in section 2.2, but it
fails to produce an excellent closed-loop presentation if extensive dead time and
NMP zeros occur in the outer loop. Figure 2 illustrates the proposed control
scheme, which combines inverse response compensator
)(sI
with higher-order
fractional filter for the outer loop in series cascade control. The inverse response
compensator predicts the inverse response behaviour and deals with a
counteractive signal to discard the inverse response, while a higher-order
approximation is adopted for the time delay term. The transfer function of the
proposed series cascade control is obtained as,
.))()()()()(1((
)()(){()()(1(
))()(1()()}()(){()()()(
1
2
11
1122
1122
221122
dsIsGsGsGsG
dsIsGsGsGsG
sGsGsrsIsGsGsGsGsy
cMcM
cMcM
cMcMcM
(3)
)(
1
sG
C
)(
2
sG
M
)(
~
2
sG
M
1
y
+
-
+
-
)(sI
1
d
2
d
+
+
+
+
+
+
+
-
+
-
2
y
2
r
1
r
Inner loop
Outer loop
)(
~
1
sG
M
)(
1
sG
M
)(
2
sG
C
Figure 2: Proposed series cascade control scheme.
In IMC framework, inner loop controller (
))(
2
sG
c
can be written as (Seborg, 2004),
,
)(
~
)(
)(
2
2
sG
sf
sG
M
c
(4)
Where the term
)(
~
2
sG
M
has an interpretation of the minimum phase part &
)(sf
shows
the low pass filter. Consequently, the overall transfer function of the system becomes,
).(
~
)()(
~
12
sGsHsG
MrM
(5)
Now from figure 2, the output of the inner loop becomes,
.
))(
~
)()((1
)(
~
)(
222
22
2
sGsGsG
sGsG
H
MMc
Mc
r
Now the arrangement of the proposed scheme in the IMC framework with inverse
response compensator for outer loop in series cascade control is demonstrated in
figure 3.
)(
1
sG
c
)(sG
M
)(sR
-
)(sY
)(sI
+
Inverse response
compensator
+
-
-
+
+
-
)(
~
sG
M
)(sG
c
Figure 3. Simplified proposed scheme in IMC framework
Here equation (6) demonstrates the controller transfer function after accountability
of inverse response compensator (Ogunnaike, 1994),
The generalized IMC control scheme is presented in figure 4 (Seborg, 2004).
)(sG
M
)(sR
+
-
)(sY
)(
~
sG
M
+
-
-
+
)(sG
c
Figure 4: Traditional internal model control scheme
From Figure 4, the expression for traditional feedback controller
)(sC
can be
described as
)(
~
)(1
)(
)(
)(
)(
sGsG
sG
sR
sY
sC
Mc
c
.
(7)
The IMC controller
)(
~
)(
)(
1
sG
sf
sG
M
c
consists of a filter
r
s
sf
)1(
1
)(
and the
reciprocal of
)(
~
sG
M
. The term exhibit tuning parameter and
r
make the
controller transfer function realizable. The term
)(
~
sG
M
interprets the minimum
phase term of the system. The controller design methodology of this paper
utilizes the equivalent system with the IMC notion to develop a generalized IMC
controller for a non-minimum phase system. As a result, combining equations
(6) and (7). The overall controller transfer function is obtained as,
.
)()()(
~
)(1
)(
)(
11
1
sIsGsGsG
sG
sC
cMc
c
(8)
Notably, the embedded transfer function, i.e. internal model control transfer
function, can be viewed as consisting of the minimum phase property, while
)(
~
sG
M
.
)()(1
)(
)(
1
1
sIsG
sG
sG
c
c
c
(6)
comprises zeros at right–hand part. The inverse response compensator is designed
as
)(
~
2)( sGbssI
M
(Ogunnaike, 1994).
b
indicates right half plane zero. After
embedding the internal model control and inverse compensator transfer function,
equation (8) can be re-formed as,
.
)]()(
~
))()[((
~
)(
~
)(
1
sIsGsfsG
sG
sC
MM
M
(9)
Design of controllers for cascade loop: FOPTD system in the inner loop and
Integer order plus time delay (IOPDT) system in the outer loop
Inner loop controller design with FOPDT model: Consider a FOPDT model,
,
1
~
2
2
2
2
s
M
e
sT
K
G
(10)
Where
2
K
is the gain,
2
is the time delay and
2
T
is the time constant.
Inner loop IMC filter.
1
1
)(
1
s
sf
,
where is the tuning parameter According to IMC notion, the process model is
depreciated into two parts, i.e., minimum phase part
1
)(
~
2
2
2
sT
K
sG
M
and non-
minimum phase part
2
2
)(
~
esG
M
After the utilization of equation (4), the inner loop
controller becomes,
r
c
sK
sT
sG
)1(
)1(
)(
2
2
2
.
(11)
Here
r
is chosen in such a way to make
)(
2
sG
c
proper. Moreover, it can be tuned
based on maximum sensitivity (Patel B., 2019),
.
006.1
55.17289.0
s
s
M
M
Here and
s
M
illustrates the time delay and maximum sensitivity.
Outer loop controller with IOPDT system:
,
)1(
)(
~
1
1
1
s
M
e
s
bsK
sG
Where
1
K
is the gain,
1
is the time delay and
b
is the NMP zero. After considering
the inner loop using equation (5) to obtain outer loop process,
s
M
e
ss
bsK
sG
)1(
)1(
)(
~
1
;
.
21
The higher-order fractional IMC filter is (Seborg, 2004),
31
2
)1(
1
)(
s
s
sf
.
Where and are the fractional filter parameters which can be tuned via phase
margin and gain cross over frequency (Bode, 1945),
1
2
..MP
,
.
)(
1
1
gc
(12)
is the additional degree of freedom which the following expression can tune
(Patel et al.,2019),
.)1(1
3
e
To nullify the effect of NMP zero (
)b
the inverse response compensator
)(
~
2)( sGbsI
M
has been utilized. A higher-order approximation is considered for the
time delay (Seborg, 2004). After the utilization of equation (9), the outer loop
controller transfer function with higher-order fractional IMC filter and higher-order
time delay approximation becomes,
.
)(1
)
2
(3)
2122
(
)
21212
(3
12
1
212
)(
111
2
22
222223
32
2
2
1
KsKK
s
bsb
b
s
bb
sss
sb
ss
sC
proposed
.
(13)
Now first-order fractional IMC filter
1
1
)(
1
2
s
sf
& higher-order Pade approximation
1
212
1
212
2
2
ss
ss
are considered for time delay considered. After utilization equation (9),
the outer loop controller transfer function becomes,
.
1
)
2
()
122
(
12
1
212
)(
11
222
2
2
2
KK
s
bs
b
s
sb
ss
sC
proposed
(14)
Design of controller for cascade loop: FOPDT system in the inner loop, FOPDT
system in the outer loop
The design of the inner loop controller is the same as the previous case,
r
c
sK
sT
sG
)1(
)1(
)(
2
2
2
.
Outer loop controller with FOPDT system,
s
M
e
sT
bsK
sG
1
1
1
)1(
)(
~
1
1
.
Now, by considering equation (5), the outer loop process becomes,
s
M
e
ssT
bsK
sG
)1)(1(
)1(
)(
~
1
1
.
The higher-order fractional IMC filter is (Seborg, 2004),
.
)1(
1
)(
21
3
s
s
sf
The detailed process for tuning of fractional filter parameters (
),
and are
available in the previous case. The expression for inverse response compensator is
also presented in the previous case. After the use of equation (9), the outer loop
controller transfer function becomes,
.
1
)(
)
2
(2
)1222
(
)
21212
(
12
)1)(1
212
(
)(
111
2
22
2222
32
1
2
2
3
K
s
sKK
bs
b
bs
bb
ss
sb
sTss
sC
proposed
(15)
Table 1: Filters and approximations for the series cascade control scheme.
Table 1 illustrates different approximations with higher and lower order filters for
the inner and outer loop of series cascade control. Now first-order fractional IMC
filter
1
1
)(
4
s
sf
and higher-order approximation for a time delay are considered.
After utilization equation (9), the outer loop controller transfer function becomes,
.
1
)
(
)
2
()
1212
(
12
1
212
)(
1
1
11
1
2
2
2
2
2
4
K
sT
sKK
T
bs
b
ss
b
ss
sC
proposd
(16)
3. Case studies
To evaluation of the suggested scheme, numerical simulations are carried out for
four systems. In case studies nominal and +15%, process parameters deviations are
discussed. The sensitivity analysis confirms the attenuation of disturbance occurring
at the sensitivity function
1|)(| js
and the amplification of disturbances occurring
1|)(| js
. The fractional filter parameter and are chosen via the notion of
Bode's ideal transfer function. Here equation (12) is accomplished in the preferred
situation. Equation (12) Phase Margin can be found based on maximum sensitivity
and gain cross-over frequency, respectively. The stability inequality
)21arcsin(2
.max
SPM
is utilized to evaluate the phase margin. For
4.1
.max
S
we
Method
Fractional IMC filters and Pade
approximation
used
Proposed 1
1
1
)(
1
s
sf
(inner loop) &
31
2
)1(
1
)(
s
s
sf
(outer loop)+2
nd
order Pade approximation.
Proposed 2
1
1
)(
1
s
sf
(inner loop) &
)1(
1
)(
1
2
s
sf
(outer loop)+2
nd
order Pade approximation.
Proposed 3
1
1
)(
1
s
sf
(inner loop) &
21
3
)1(
1
)(
s
s
sf
(outer loop)+2
nd
order Pade approximation.
Proposed 4
1
1
)(
1
s
sf
(inner loop) &
)1(
1
)(
1
4
s
sf
(outer loop)+2
nd
order Pade approximation.
obtain the
38PM
Further,
1
)3(
2
has utilized for tuning of fractional
filter parameters (Astrom and Murray, 2008).
3.1 Case study I
Consider a boiler drum level system that has an inverse response due to reprocessing
loop and composition loops. The influence of boiler feed water is utilized to control
the level of the drum. (Nagarsheth and Sharma, 2020) recommended the transfer
function of the boiler steam drum level. It can be followed as,
s
es
sG
M
07.0
)418.01(
)(
~
1
,
.
106.1
547.0
)(
~
03.0
2
s
e
sG
M
For the inner loop controller setting utilizing equation (11),
)13.0(547.0
106.1
)(
2
s
s
sG
c
.
After the utilization of equation (5), the overall plant transfer function can be
reformed as,
.
)13.0(
)418.01(
)(
~
1.0
ss
es
sG
s
M
(17)
The suggested controller settings for the outer loop in the fractional IMC framework
can be obtained via equation (13),
,0336.0
1
38.0
568.0
33267.00369.0
105.008.0
)(
45.090.0290.2
2
1
s
ssssss
ss
sC
proposed
(18)
Where the term
568.033267.00369.
105.0008.0
45.090.0290.2
2
sssss
ss
represents the
fractional IMC filter and the term
s
s
0336.0
1
38.0
interprets the PID controller.
While the suggested other controller settings for the outer loop in fractional IMC
framework with lower-order fractional IMC filter can be obtained via equation (14)
,
1
38.0
368.0021.000068.0
105.0008.0
)(
45.02
2
2
s
sss
ss
sC
proposed
(19)
Where the term
368.0021.000068.0
105.0008.0
45.02
2
sss
ss
represents the fractional IMC filter
and the term
s
1
38.0
interprets the PI controller. Here
3.0
,
14.0
,
1
and
45.0
with
45PM
.
Figure 5 reveals a comparative investigation of closed-loop response fashioned from
the suggested technique. Figure 5(a) demonstrates the step response, where
controllers are embedded with inverse response compensator.
Figure 5(b) specifies that after +15% mismatch suggested structure is represented
convincible consequences in terms of overshoot and undershoot.
Figure 5: Comparative analysis for boiler drum level
Figure 5(a) and (b) display that, available controller suffers from overshoot & under
-shoot. The proposed method is revealed the smallest peak after disturbance & quite
less time to settle. The transient analysis and performance indices are listed in Table
2. This investigation shows that proposed controller 1 and proposed controller 2
have less overshoot and less settling time in both (Nominal and +15% mismatch)
cases
Table 2: Transient and performance indices for boiler drum level
The IAE and ISE values are also demonstrated in Table 2. The proposed controller1
and proposed controller 2 shows minimum IAE and ISE value in both (Nominal and
+15% mismatch) cases. The proposed controllers are demonstrated convincible
outcome.
Nominal
Case
+15%
mismatch
Method
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time,
s
IAE
ISE
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time
s
IAE
ISE
Nagarsheth
– Sharma
(2020)
1.47
8.29
4.26
1.3
1.7
2.16
3.69
1.87
3.4
11.72
Proposed
controller 1
1.061
13.37
9.87
0.16
0.027
1.364
4.17
2.82
0.76
0.58
Proposed
controller 2
1.094
13.66
9.73
0.651
0.42
2.08
1.67
1.07
2.5
6.25
Furthermore, stability analysis of proposed method joints on the using Riemann
surface. Here we have used the FOMCON toolbox of MATLAB for graphical
assessment of fractional characteristic polynomial The generalized expression for
stability analysis is illustrated in Table 3.
Table 3 Generalized expression for fractional characteristic polynomial of
IOPDT system
The specific structure of the fractional characteristic polynomial associated with
the closed-loop of boiler drum level is illustrated in Table 4.
Table 4: Fractional quasi characteristic polynomial for boiler drum level
The roots of fractional order characteristic polynomial lie in mapping
planes
a
plane linked with an equivalent characteristic polynomial. Choose
s
, then linked
characteristic polynomial is available in Table 5.
Table 5: Mapping of
planes
with characteristic polynomial
Proposed 1
.568.033267.00369.000048.01)(
1000145019002000290030004000
sf
Proposed 2
.368.0021.000068.01)(
1000145020003000
sf
Where
001.0
s
. For the stability of the proposed fractional closed-loop system, the
absolute values of the angles of roots of the associated characteristic polynomials
should be more significant than
2
for the stable system. Table 6 illustrates the
stability analysis of the boiler drum level.
Proposed 1
.
2
3
2122
3
2121212
1)(
1
2
2
222
22
3333
4
bssb
b
s
s
bb
ss
sb
sf
Proposed 2
.
221212
1)(
1
2
2
32
bss
b
s
bs
sf
Proposed 1
.568.033267.00369.000048.1)(
45.190.1290.234
ssssssssf
Proposed 2
.368.0021.000068.01)(
45.123
sssssf
The roots of natural degree quasi characteristic polynomial inside the Riemann sheet
then reveal instability, while roots lie outside the Riemann sheet then reveal the
system's stability. In Table 6 zoomed plot shows that all the roots are lying outside of
the Riemann sheet(Monje et al., 2010).
Table 6: Stability analysis of boiler drum level
(a) Proposed 1 (b) Proposed 2
Proposed 1
Proposed 2
Robustness Analysis: The partial characterization of the feedback system can
change with any variation in process dynamics and disturbances. The
generalized sensitivity function can be written as (Goodwin et al., 2001).
)()(
~
1
1
)(
sCsG
sS
proposedM
.
(20)
For that purpose, substitute the value of
)(
~
sG
M
and
)(
1
sC
proposed
,
)(
2
sC
proposed
from equation (17), (18) and (19) into (20). The graphical interpretation of the
robustness analysis of the proposed controller can be found in figure 6. The
proposed controllers reveal less rise in maximum sensitivity in nominal and 15%
model mismatch cases
Figure 6: Robustness analysis for boiler drum level
Remarks: In (Nagarsheth-Sharma, 2020), lower-order fractional IMC filter,
setpoint filter, dead time and inverse response compensator have been utilized
with single loop feedback. However, in their work, a dead time compensator and
setpoint filter has been utilized. In this work, the higher-order fractional IMC
filter, inverse response compensator has been used with series cascade control to
prevent setpoint filter and dead-time compensator. The proposed controllers have
demonstrated improved performance as compared to the existing controller.
3.2 Case study II:
In process industries, a distillation column is utilized to separate low boiling
material from the final product. Owing to the limpid demand for products, the
process of the distillation column is rather complicated. Since distillation columns
have slow dynamics with long-dead time due to composition and temperature,
inverse response arises from reboiler swell and froth density effects. These
detrimental effects bound the system's stability and control. The proposed method
is utilized to compensate for these effects. As a result. The transfer function of the
distillation column is given as (Nagarsheth and Sharma, 2020),
1
)2.01(
)(
~
1.0
1
s
es
sG
s
M
,
.
1
)(
~
1.0
2
s
e
sG
s
M
After utilizing equation (11), the IMC controller for the inner loop structure is
given as,
)135.0(
1
)(
2
s
s
sG
c
.
After the utilization of equation (5), the overall plant transfer function can
be reformed as
.
)135.0)(1(
)2.01(
)(
~
2.0
ss
es
sG
s
M
(21)
While from equation (9), the suggested controller settings for the outer loop
is obtained as,
,
1
14.014.1
24.02
054.000386.0000093.0
)1)(11.0003.0(
)(
28.0
256.23
2
3
s
s
s
ssss
sss
sC
proposed
(22)
Where the term
24.02
054.000386.0000093.0
)1)(11.0003.0(
28.0
256.23
2
s
ssss
sss
indicates
fractional IMC filter, while the term
s
s
1
14.014.1
demonstrates PID
controller. Moreover, from equation (9), the suggested controller (lower-
order fractional IMC filter with an inverse response compensator)
settings for the outer the loop is obtained as,
.35.0
1
35.1
1.0023.00006.0
11.0003.0
)(
28.02
2
4
s
ssss
ss
sC
proposed
(23)
Figure 7 Comparative analysis for distillation column
Where the term
1.0023.00006.0
11.0003.0
28.02
2
sss
ss
indicates lower order fractional
filter. While the term
s
s
35.0
1
35.1
demonstrates PID controller. Here
35.0
1.0
1
and
28.0
with
65PM
.
Figure 7 reveals a comparative investigation of closed-loop response fashioned
from the suggested technique. Figure 7(a) demonstrates the step response, where
controllers are embedded with inverse response compensator. Figure 7(b) specifies
that after +15% mismatch suggested structure is represented convincible
consequences in terms of overshoot and undershoot.
Figures 7(a) and (b) display that the available controller suffers from overshoot and
undershoot. The proposed method is revealed the smallest peak after disturbance
and relatively less time to settle.
The transient response and performance indices are depicted in Table 7. After
accountability for the +15% mismatch case in process parameters, the proposed
controller is demonstrated convincible outcome.
Table 7: Transient and Performance indices for distillation column
The generalized expression for stability analysis is illustrated in Table 8.
Table 8: Generalized expression for stability analysis of
FOPDT system
The specific structure of the fractional characteristic polynomial associated with
the closed-loop of the distillation column is presented in Table 9.
Nominal
Case
+15%
mismatch
Method
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time,
s
IAE
ISE
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time
s
IAE
ISE
Nagarsheth
– Sharma
(2020)
1.5
5.1
2.7
1.8
3.3
1.74
4.6
2.4
1.5
2.3
Proposed
controller 1
1.40
9.9
6.18
0.43
0.18
1.46
9.1
5.3
0.34
0.11
Proposed
controller 2
1.41
5.8
3.7
1.1
1.3
1.54
6.5
3.7
1.03
1.064
Proposed 3
.
2
2
21222121212
1)(
1
2
2
22
3322
42
bss
b
b
s
bb
ss
sb
sf
Proposed 4
.
221212
1)(
1
2
2
32
bss
b
s
bs
sf
Table 9: Fractional characteristic polynomial for distillation column
Table 10: Mapping of
planes
with characteristic polynomial
The roots of fractional order characteristic polynomial lie in the
js
.
The mapping of a
planes
linked with an equivalent characteristic
polynomial. Choose
s
, then linked characteristic polynomial is available
in Table 10.
Table 11: Stability analysis for distillation column
The roots of natural degree quasi characteristic polynomial inside the Riemann
sheet then it reveals instability, while roots lie outside the Riemann sheet, then it
reveals the stability of the system. In Table 11 all the roots are lying on outside of
Riemann sheet.
Proposed 3
.24.02
054.000386.000093.01)(
100012800
2000300035604000
sf
Proposed4
.1.0023.0006.01)(
1000128020003000
sf
Proposed 3
.24.0054.0
00386.000093.021)(
2
3428.156.3
ss
sssssf
Proposed 4
.1.0023.0006.01)(
28.123
sssssf
Proposed 3
Proposed 4
Robustness Analysis: To examine the robustness of proposed controllers,
substitute the value of
)(
~
sG
M
and
)(
3
sC
proposed
,
)(
4
sC
proposed
from equations (21),
(22) and (23) into (20). The graphical interpretation in figure 8 reveals that the
proposed controllers have less peak of maximum absolute sensitivity. The lower
peak confirms that proposed controllers are more robust as compare to other
controllers.
Figure 8: Robustness analysis for distillation column
Remarks: The proposed controllers are compared with the same set of parameters
(Nagarsheth and Sharma, 2020). The proposed controllers have demonstrated
improved performance as compared to the existing controller.
3.3 Case study III:
A higher-order integrating process is considered. The transfer function model of
the higher-order integrating process is given as (Begum et al., 2017).
)11.0)(14.0)(15.0(
)5.01(5.0
)(
7.0
ssss
es
sG
s
p
.
This process is modelled as (Pai et al., 2010).
s
p
e
s
s
sG
51.0
)4699.01(
)(
1
,
s
p
e
s
sG
30.0
11609.1
5183.0
)(
2
.
For the inner loop controller setting utilizing equation (11),
)195.0(5183.0
11609.1
)(
2
s
s
sG
c
.
After the utilization of equation (5), the overall plant transfer function can be
restructured as,
.
)195.0(
)4699.01(
)(
~
7.0
ss
es
sG
s
M
(24)
The suggested controller settings for the outer loop in the fractional IMC
framework can be obtained via equation (13),
,82.1
1
874.0
.15.213.0
16.26.0168.0017.0
135.040.0
)(
65.0
30.165.123
2
1
s
s
ss
ssss
ss
sC
proposed
(25)
Where the term
.15.213.0
16.26.0168.0017.0
1035.0040.0
65.0
30.165.123
2
ss
ssss
ss
represents the fractional
IMC filter and the term
s
s
82.1
1
874.0
interprets the PID controller. While the
suggested other controller settings for the outer loop in fractional IMC
framework with lower-order fractional IMC filter can be obtained via equation
(14)
,
1
95.0
11.06.021.0019.0
1035.0040.0
)(
65.02
2
2
ssss
ss
sC
proposed
(26)
The term represents the fractional IMC filter and
s
1
95.0
interprets the PI
controller Here
95.0
92.
6.0
and
65.0
with
39PM
.
Figure 9 reveals a comparative investigation of closed-loop response fashioned
from the suggested technique. Figure 9(a) demonstrates the step response, where
controllers are embedded with inverse response compensator.
Figure 9: Comparative analysis for higher-order integrating process
Figure 9(b) specifies that after +15% mismatch suggested structure is represented
convincible consequences in terms of overshoot and undershoot. Figures 9(a) & (b)
display that the available controller suffers from overshoot & under-shoot. The
Proposed method is revealed the smallest peak after disturbance & relatively less
time to settle. The transient analysis and performance indices are offered in Table
12. This analysis displayed that suggested controllers 1 and 2 have less overshoot
and less settling time in both (Nominal and +15% mismatch) cases. The IAE and
ISE values are also revealed in Table 12.
Table 12: Transient and performance indices for higher-order integrating process
Likewise, stability investigation of suggested scheme joints on the using Riemann
surface. A detailed description of stability analysis is available in the case study I.
Table 13 shows stability analysis for a higher-order integrating process (Monje et
al., 2010).
Table 13: Stability analysis for higher-order integrating process
(a) Proposed 1 (b) Proposed 2
Nominal
Case
+15%
mismatch
Method
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time,
s
IAE
ISE
Peak
overshoot
ratio, %
Peak
time, s
Rise
time
s
IAE
ISE
Nagarsheth
– Sharma
(2020)
1.7
7.9
5.2
8.7
76.7
1.84
8.15
5.46
6.6
44.54
Proposed
controller 1
1.33
14.6
8.5
0.017
0.002
1.53
12.56
7.88
0.016
0.002
Proposed
controller 2
1.54
7.7
5.6
0.79
0.62
1.72
7.9
5.5
0.53
0.28
Proposed 1
Proposed 2
Moreover, for the robustness analysis, substitute the value of
)(
~
sG
M
and
)(
1
sC
proposed
,
)(
2
sC
proposed
from equation (24), (25) and (26) into (20). The graphical interpretation
of the robustness analysis of the suggested controller can be found in figure 10. The
proposed controllers reveal less rise in maximum sensitivity in nominal and 15%
model mismatch cases.
Figure 10: Robustness analysis for higher-order integrating process
Remarks: To examine the efficacy of the proposed controller is compared with
Begum et al. (2017). However, a single loop feedback control scheme was utilized in
their work and an inverse response compensator was not present. The proposed
controllers have demonstrated improved performance as compared to the existing
controller.
3.4 Case study IV
Consider the following third-order process with the inverse response and time delay (Jeng
and Lin, 2012).
)15.0)(1)(12(
)21(
)(
5.0
sss
es
sG
s
p
.
Adopt the actual process is unidentified and the subsequent model was achieved through
process reaction curve for controller design.
s
M
e
s
s
sG
4.0
171.2
)62.11(
)(
1
,
s
M
e
s
sG
2.0
150.1
94.0
)(
2
.
For the inner loop controller setting utilizing equation (11),
)16.0(94.0
150.1
)(
2
s
s
sG
c
.
After the utilization of equation (5), the overall plant transfer function can be
restructured as,
.
)171.2)(16.0(
)62.11(
)(
~
6.0
ss
es
sG
s
M
(27)
The suggested controller settings for the outer loop in the fractional IMC
framework can be obtained via equation (13),
,35.0
1
19.1
.01.1385.07216.05.00012.0
)171.2)(103.03.0(
)(
42.0284.23
2
3
s
s
sssss
sss
sC
proposed
(28)
Where the term
.01.1385.07216.05.00012.0
)171.2)(103.030.0(
42.0284.23
2
sssss
sss
represents the
fractional IMC filter and the term
s
s
35.0
1
19.1
interprets the PID controller.
While the suggested other controller settings for the outer loop in fractional IMC
framework with lower-order fractional IMC filter can be obtained via equation (14)
,32.0
1
21.3
32.05.011.0048.0
103.030.0
)(
42.02
2
4
ssss
ss
sC
proposed
(29)
The term represents the fractional IMC filter and
32.0
1
21.3
s
interprets the PI
controller Here
5.0
59.0
5.0
and
42.0
with
52PM
.
Figure 11: Comparative analysis for third-order process
Figure 11 reveals a comparative investigation of closed-loop response fashioned from
the suggested technique. Figure 11(a) demonstrates the step response, where controllers
are embedded with inverse response compensator. Figure 11(b) specifies that after
+15% mismatch suggested structure is represented convincible outcomes in terms of
overshoot and undershoot.
Figures 11(a) and (b) display that the available controller suffers from overshoot and
undershoot. The proposed method is revealed the smallest peak after disturbance and
quite less time to settle.
Table 14: Transient and performance indices for the third-order process
The transient response and IAE and IAE values depicted in Table 14 indicate the
proposed controller's adequacy. After accountability for the +15% mismatch case in
process parameters, the proposed controller is demonstrated convincible outcomes. The
roots of natural degree quasi characteristic polynomial inside the Riemann sheet then it
reveals instability, while roots lie outside the Riemann sheet then it reveals the stability
of the system
Table 15: Stability analysis for the third-order process
To examine the robustness of proposed controllers, substitute the value of
)(
~
sG
M
and
)(
3
sC
proposed
,
)(
4
sC
proposed
from equations (27), (28) and (29) into (20). The graphical
interpretation in figure 12 reveals that the proposed controllers have less peak of
Nominal
Case
+15%
mismatch
Method
Peak
overshoot
ratio, %
Peak
time,
s
Rise
time,
s
IAE
ISE
Peak
overshoot
ratio, %
Peak
time, s
Rise
time
s
IAE
ISE
Nagarsheth
– Sharma
(2020)
1.22
9.8
7.5
3.16
9.6
1.35
10.03
7.5
3.3
11.28
Proposed
controller 1
1.13
31.3
19.43
0.16
0.028
1.16
21.89
18.41
0.20
0.043
Proposed
controller 2
1.14
14.28
9.8
2.1
4.7
1.20
17.49
11.75
3.0
9.0
Proposed 3
Proposed 4
maximum absolute sensitivity. The less peak confirms that proposed controllers are
more robust as compare to other controllers.
Figure 12 Robustness analysis for third-order process
Remarks: In (Jeng and Lin. 2012), has been utilized single loop feedback. However,
in this work, the higher-order fractional IMC filter, inverse response compensator has
been utilized with series cascade control to prevent setpoint filter and dead-time
compensator.
4. Conclusions:
This paper establishes a combined theory for fractional IMC-based series cascade control
with inverse response compensator. The integer order IMC filter has been used for the
controller design of the inner loop. The higher-order fractional IMC filter with inverse
compensator has been utilized to design the controller of the outer loop. The enhanced
performance has been found in the outer loop controller, particularly in inverse response
and dead time behaviour compensation. The simulation results have revealed the
supremacy of the proposed controller in terms of performance indices. This work has
provided prevention of setpoint filter and dead-time compensator. The sensitivity
analysis has been accomplished to examine the robustness of the suggested controller.
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