Design of Proportional Integral and Derivative (PID) controllers based on IMC principles for various types of integrating systems with time delay is proposed. PID parameters are given in terms of process model parameters and a tuning parameter. The tuning parameter is IMC filter time constant. In the present work, the IMC filter (Q) is chosen in such a manner that the order of the denominator of IMC controller is one less than the order of the numerator. The IMC filter time constant (λ) is tuned in such a way that a good compromise is made between performance and robustness for both servo and regulatory problems. To improve servo response of the controller a set point filter is designed such that the closed loop response is similar to that of first order plus time delay system. The proposed controller design method is applied to various transfer function models and to the non-linear model equations of jacketed CSTR to demonstrate its applicability and effectiveness. The performance of the proposed controller is compared with the recently reported methods in terms of IAE and ITAE. The smooth functioning of the controller is determined in terms of total variation and compared with recently reported methods. Simulation studies are carried out on various integrating systems with time delay to show the effectiveness and superiority of the proposed controllers.

1. Research Article
Tuning of IMC based PID controllers for integrating systems
with time delay
D.B. Santosh Kumar, R. Padma Sree n
Department of Chemical Engineering, AU College of Engineering (A), Visakhapatnam 530003, India
a r t i c l e i n f o
Article history:
Received 30 October 2015
Received in revised form
9 March 2016
Accepted 30 March 2016
Available online 14 April 2016
This paper was recommended for publica-
tion by Panda Rames.
Keywords:
Integrating systems
Unstable system
Time delay
PID controller
IMC method
Ms value
a b s t r a c t
Design of Proportional Integral and Derivative (PID) controllers based on IMC principles for various types
of integrating systems with time delay is proposed. PID parameters are given in terms of process model
parameters and a tuning parameter. The tuning parameter is IMC filter time constant. In the present
work, the IMC filter (Q) is chosen in such a manner that the order of the denominator of IMC controller is
one less than the order of the numerator. The IMC filter time constant (λ) is tuned in such a way that a
good compromise is made between performance and robustness for both servo and regulatory problems.
To improve servo response of the controller a set point filter is designed such that the closed loop
response is similar to that of first order plus time delay system. The proposed controller design method is
applied to various transfer function models and to the non-linear model equations of jacketed CSTR to
demonstrate its applicability and effectiveness. The performance of the proposed controller is compared
with the recently reported methods in terms of IAE and ITAE. The smooth functioning of the controller is
determined in terms of total variation and compared with recently reported methods. Simulation studies
are carried out on various integrating systems with time delay to show the effectiveness and superiority
of the proposed controllers.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Integrating systems are the processes which contain at least
one pole at the origin. They are non-self regulating, which means
that when they are disturbed from the equilibrium operating point
by any environment disturbance/change in input conditions, the
process output varies continuously with time at certain speed. The
phenomenon is very disadvantageous and dangerous in most
occasions. Therefore, efficient control of such kind of processes is
always a challenging task. Different types of integrating systems
exist depending upon the number of poles present at the origin
and location of other poles in transfer function. Accordingly inte-
grating systems are classified as stable First Order plus Time Delay
systems with an Integrator (FOPTDI) Unstable First Order plus
Time Delay systems with an Integrator (UFOPTDI), Pure Integrating
plus Time Delay (PIPTD) systems and Double Integrating plus Time
Delay (DIPTD) systems.
Industrial processes such as composition control loop of a high
purity distillation column [1], bottom level control in a distillation
column [2], storage tank with a pump at the outlet [3], many level
control problems [4], an isothermal continuous copolymerization
reactor [5], the heating of well insulated batch systems [6], totally
heat integrated distillation columns [7] and high pressure steam
flowing to a steam turbine generator in a power plant [8], exhibit
Pure Integrator plus Time Delay transfer function models.
First order systems with an integrator and with/without zero
are frequently encountered in process industries. The occurrence
of such transfer function models is reported for the liquid storage
tanks [9], paper drum dryer cans [4] and a jacketed Continuous
Stirred Tank Reactor (CSTR) carrying out an exothermic reaction
[10]. For First Order plus Time Delay systems with an Integrator
(FOPTDI) and with a positive zero, the system gives inverse
response. The inverse response becomes deeper as the zero moves
towards the origin on the real axis, which is a tough challenge for
process control.
Double integrating systems exist in processes such as aerospace
control systems, vertical take-off of airplanes [11], DC motors and
high speed disk drives [12], oxygen control in feed batch (filament
fungal) fermentation reactors [13].
PID controllers are widely implemented in many of the che-
mical process industries because they are very simple to tune, easy
to understand and robust in control. PID control is the most
common control algorithm used in industry and has been uni-
versally accepted in industrial control. The popularity of PID con-
trollers can be attributed partly to their robust performance in a
wide range of operating conditions and partly to their functional
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.03.020
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail address: padvan@gmail.com (R. Padma Sree).
ISA Transactions 63 (2016) 242–255

2. simplicity, which allows engineers to operate them in a simple,
straight forward manner.
In literature various methods are proposed to tune PID con-
trollers for integrating systems with time delay. They are empirical
method [14,15], Internal Model Control (IMC) method [2,16–22],
direct synthesis method [23–28], equating coefficient method
[29,30], two degree of freedom (2DOF) control scheme [31–33],
stability analysis method [34–37] and optimization method [38–
43]. Various methods to control different types of integrating
processes are given by Visioli and Zhang [44].
Seshagiri Rao et al. [25] proposed PID controller tuning rules for
integrating processes with time delay using direct synthesis
method. For PIPTD system the closed loop is assumed to be second
order system with first order numerator and time delay. For stable/
unstable FOPTDI system and DIPTD systems, the closed loop
transfer function is assumed to be third order with second order
numerator and time delay. From the closed loop transfer function,
using process transfer function controller transfer function is
obtained. First order Pade's approximation is used for time delay
in the denominator of controller transfer function, so that PID
controller parameters are obtained in terms of process model
parameters.
Panda [19] proposed a PID controller for pure integrating plus
time delay system based on IMC method. To avoid singularity
problem, Laurent series is used to derive PID controller settings.
The given tuning rules are logically applied to low order open loop
unstable process plant model with time delay. The tuning factor λ
was selected based on faster/sluggish response. The controller is
robust, stable and can be implemented easily on a real time
processes.
Nageswara Rao and Padma Sree [20] proposed a PID controller
based on IMC principles for integrating systems with time delay.
They have used first order Pade's approximation for process time
delay in the process itself. For that process, IMC controller is
designed and IMC filter is selected such that the numerator order
is equal to or less than the denominator order. From IMC con-
troller, PID controller is designed. PID parameters are given in
terms of model parameters. Anil and Padma Sree [43] proposed
tuning rules for PID controllers using differential evolution algo-
rithm to minimize Integral Time weighted Absolute Error (ITAE).
To reduce overshoot in servo problem, set point weighting is also
suggested.
Ajmeri and Ali [27] have proposed parallel control structure
that decouples servo problem and regulatory problem for PIPTD,
DIPTD and FOPTDI systems. For servo problem, proportional and
derivative (PD) controller is used and for regulatory problem, PID
controller is used. The controller is implemented as parallel form
of PD/PID controllers. Analytical tuning rules are proposed for PD
and PID controllers based on direct synthesis method. The tuning
parameters are tuned in such a way to achieve the desired
robustness. Ajmeri and Ali [27] have reported the PD/PID para-
meters for a maximum magnitude of sensitivity function, Ms¼2
for PIPTD systems. The performance of the method is reported in
terms of Integral Square Error (ISE), Integral Absolute Error (IAE),
TV and settling time.
Shamsuzzoha [45] proposed analytical tuning rules for closed
loop PI/PID controller for stable and integrating process with time
delay. This method requires a closed loop step set point experi-
ment using a proportional (P) only controller with gain Kco
: In P
mode, a step change is given to the system such that the overshoot
is 30% .On the basis of this simulation results for a first order plus
time delay processes, simple correlations are derived to give PI/PID
controller settings. The controller gain (KC=KCo
) is the only func-
tion of the overshoot observed in the set point experiment. The
controller integral (τI) and derivative time (τD) is mainly a function
of the time of reach the first peak (tP).
Cho et al. [28] proposed simple analytical PID controller tuning
rules for unstable process based on direct synthesis method. The
closed loop transfer function model is assumed to be third order
with second order numerator and time delay term. From the
closed loop transfer function, using process transfer function,
controller transfer function is obtained. The process time delay is
approximated using Taylor's series expansion up to two terms. The
tuning rules are derived and PID settings are tuned such that
trade-off between performance and robustness is achieved.
Lee et al. [26] revised the tuning rules of SIMC method pro-
posed by Skogestad [47] for PID controller. They suggested mod-
ifications of model reduction techniques proposed by Skogestad
[47].
Jin and Liu [21] have proposed 2DOF scheme using IMC prin-
ciples with an extra set point filter for PIPTD, DIPTD and FOPTDI
systems. They have designed IMC controller with numerator order
one greater than the denominator order. The conventional PID
controller is designed and implemented as parallel form PID
controller. An optimization problem is formulated with an objec-
tive function of IAE for regulatory problem with robustness
(Ms ¼2) as a constraint. The servo performance is improved by
using an extra set point filter. Analytical tuning rules are reported
for PID parameters and for an extra set point filter. The perfor-
mance is reported in terms of IAE and TV for both servo and
regulatory problem. Nageswara Rao and Padma Sree [33] proposed
a design to two degree of freedom PID controller for double
integrator with time delay systems and unstable First Order plus
Time Delay systems with an integrator. DIPTD systems and
UFOPTDI systems are stabilized in the inner loop by using PD
controller. To the stabilized system, PID controllers are designed by
equating the denominator of the stabilized system with the
numerator of the outer loop PID controller. Using the phase angle
criteria for the combined stabilized system and outer loop PID
controller, the cross over frequency (ωC) is obtained. The ultimate
value of controller gain (outer loop) is calculated by using gain
margin (GM) criteria. Usually the GM of 1.5–2.5 is used to obtain
the design value of controller gain of the outer loop controller. Liu
et al. [51] have proposed two degree of freedom control structure
in which set point tracking and load disturbance loops are
decoupled. Set point tracking controller is PD controller and the
parameters are obtained analytically by specifying the Integral
Squared Error (ISE). By proposing the desired closed loop com-
plementary sensitivity function for rejecting load disturbances,
disturbance estimator is designed. Robust stability analysis for the
proposed control structure is provided in the presence of the
process multiplicative uncertainty. Liu and Gao [52] proposed
modified IMC based controller design for step and ramp type load
disturbance. The set point tracking is decoupled with the load
disturbance rejection with separate loops for both. The tuning
parameter is the closed loop time constant for load disturbance
rejection and tuned to meet a good trade-off between perfor-
mance and closed loop stability.
Review of literature reveals that though there are many
methods available to design PID controllers for integrating sys-
tems, still there is a scope to improve the performance and
robustness of the PID controller for integrating systems. Many
authors proposed a complicated structure with more than one
controller for control of integrating processes [12,27,51,52].
Therefore in the present work, design of PID controller with a
compensator for integrating systems with time delay to enhance
the performance for both servo and regulatory problems using
IMC principles is proposed. If the process is represented by a
perfect model with no modeling errors and if the model is inver-
tible, then the IMC controller is the inverse of the process model
and no IMC filter is required. But in the presence of modeling
errors and if the process model contains non-invertible parts like
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 243

3. time delay and non-minimum phase zeros, IMC filters are chosen
to account for the non-invertible part of the process transfer
function and to account for the robustness for parametric uncer-
tainty. Therefore, the selection of IMC filter depends on the process
transfer function model.
IMC controller is the inverse of the invertible part (of the
process transfer function) multiplied by IMC filter. IMC filter con-
sists of both numerator and denominator dynamics. The order of
the numerator is equal to the number of unstable poles (and/or
integrators). Usually the denominator order of the IMC filter is
chosen in such a way that the order denominator of the IMC
controller is the same or more than the order of the numerator, so
that it is realizable.
In the present work, the order of the IMC filter is selected in
such a way that the denominator of IMC controller is one order
less than the numerator order. Actually such IMC controller if
implemented in IMC structure gives an error message as it is not
realizable. But in the present work, from the IMC controller, PID
controller is designed and implemented in the conventional
feedback control loop. The closed loop transfer function model,
which is the non-invertible part of the process transfer function
multiplied by the IMC filter y
yr
¼ Gp
þ
f
, is still realizable. By fol-
lowing this procedure the order of the closed loop transfer func-
tion is reduced and the response is fast. The method is applied to
PIPTD, DIPTD, stable/unstable FOPTDI systems. The method has a
single tuning parameter λ, IMC filter time constant which is to be
selected in such a way that the controller gives good performance
and is robust for parameter uncertainty.
The closed loop response of integrating processes result in large
overshoot and to reduce the overshoot set point filter is suggested.
In literature many authors used set point filter [21,22,28,48] to
improve servo performance. In the present work also set point
filter is designed in such a way that the closed loop response is
similar to the response of a FOPTD system with unity gain [21].
2. Proposed method
2.1. First Order plus Time Delay system with Integrator (FOPTDI)
The process transfer function of FOPTDI system is given by
GP sð Þ ¼
KPeÀ Ls
sðτsþ1Þ
ð1Þ
Using L
τ¼ε and q ¼ τs; the above transfer function model is
written as
GP ¼
K0
PeÀεq
qðqþ1Þ
ð2Þ
where K0
P ¼ KPτ:
The invertible part of the above process transfer function (Eq.
(2)) is given by
GP À
¼
K0
P
qðqþ1Þ
ð3Þ
The non-invertible part of the process is given by
GP þ
¼ eÀεq
ð4Þ
The IMC controller is given by
Q ¼ GP À
À 1
f qð Þ ¼
qðqþ1Þ
K0
P
ðγ0
qþ1Þ
ðλ0
qþ1Þ2
ð5Þ
From IMC controller, the conventional feedback PID controller
is obtained by
GC ¼
Q
1ÀGPQ
ð6Þ
Substituting Eqs. (1) and (5) in Eq. (6), the PID controller for the
process is given by
GC ¼
q qþ1ð Þ γ0
qþ1
À Á
K0
P λ0
qþ1
À Á2
À γ0qþ1
À Á
eÀ ϵq
h i ð7Þ
Using first order Pade's approximation for time delay, the
controller is written as
GC ¼
q qþ1ð Þ γ0
qþ1
À Á
1þ0:5ϵqð Þ
K0
P λ0
qþ1
À Á2
1þ0:5ϵqð ÞÀ γ0qþ1
À Á
1À0:5ϵqð Þ
h i ð8Þ
Rearranging the above equation, the following equation for GC
is obtained.
GC ¼
q γ0
q2
þ 1þγ0
À Á
qþ1
 Ã
½1þ0:5εqŠ
K0
P½q3ð0:5ελ02
Þþq2 λ02
þλ0
εþ0:5γ0ε
þq 0:5εþ2λ0
Àγ0 þ0:5ε
À Á
Š
ð9Þ
In order to represent Gc as PID controller, the constant term of
denominator ϵþ2λ0
Àγ0
À Á
is equated to zero. Thus the value of γ0
is
obtained as
γ0
¼ εþ2λ0
: ð10Þ
GC is written as
GC ¼
1þγ0
À Á γ0
1þ γ0qþ 1
γ0 þ1ð Þq
þ1
!
1þ0:5ϵq½ Š
K0
P λ02
þλ0
ϵþ0:5γ0ϵ
0:5ϵλ02
λ02
þ λ0
ϵþ0:5γ0ϵ
qþ1
! ð11Þ
this is a PID controller with a first order filter as given below
Gc ¼ Kc 1þ
1
τ0
Iq
þτ0
Dq
1þα0
qð Þ
1þβ0
q
À Á ð12Þ
where
K0
pKc ¼
1þγ0
λ02
þλ0
ϵþ0:5γ0ϵ
ð13Þ
τ0
I ¼ γ0
þ1
À Á
ð14Þ
τ0
D ¼
γ0
1þγ0
ð15Þ
β0
¼
0:5ϵλ02
λ02
þλ0
ϵþ0:5γ0ϵ
ð16Þ
α0
¼ 0:5ϵ ð17Þ
and λ0
¼ λ
τ; γ0
¼ γ
ττ0
I ¼ τI
τ ; τ0
D ¼ τD
τ ; α0
¼ α
τ, β0
¼ β
τ:
The closed loop servo response is given by
y
yr
¼ f sð ÞGP þ
¼
γsþ1
À Á
λsþ1
À Á2
eÀ Ls
ð18Þ
A set point filter F sð Þ ¼
λsþ 1ð Þ
γsþ 1ð Þ
is introduced to improve the servo
performance such that the closed loop response resembles the
response of the first order system with time delay.
The tuning parameter, IMC filter time constant (λ0
) is tuned in
such a way that the maximum magnitude of sensitivity function
(Ms) is equal to 2. The tuning rules for FOPTDI system that yields
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255244

4. Ms ¼2 for 0:1rεr0:5 are given by
KcKpτ ¼ 81:33ϵ3
À70:66ϵ2
þ6:22ϵþ8:011 ð19Þ
τI
τ
¼ À8:333ϵ3
þ10:21ϵ2
À2:695ϵþ2:056 ð20Þ
τD
τ
¼ À2:65ϵ3
þ2:955ϵ2
À0:748ϵþ0:515 ð21Þ
α
τ
¼ 0:5ϵ ð22Þ
β
τ
¼ 0:483ϵ3
À0:522ϵ2
þ0:214ϵþ0:015 ð23Þ
λ
τ
¼ À4:166ε3
þ5:107ε2
À1:847εþ0:528 ð24Þ
When ε40:5, if PID is tuned to stabilize the system for Ms ¼2,
the performance is sluggish as the tuning parameter λ0
is large.
Therefore for larger time delay to time constant ratio, PID is to be
tuned to larger Ms (i.e. Ms42). The PID controller is tuned for 0:6
rεr1 in such way that good trade-off is made between perfor-
mance and robustness and tuning rules are given for 0:6rεr1 as
a function of ε. The tuning rules for stable FOPTD system with an
integrator for 0:6rεr1 are given below.
KcKpτ ¼ 40:66ε3
À93:98ε2
þ68:58εÀ12:46 ð25Þ
τI
τ
¼ 0:34ϵþ1:956 ð26Þ
τD
τ
¼ 0:068εþ0:496 ð27Þ
α
τ
¼ 0:5ε ð28Þ
β
τ
¼ 0:065εþ0:078 ð29Þ
Remark 1: For First Order plus Time Delay system with an
Integrator and a negative zero, PID controller with a first order
filter is designed for the system without a zero and an additional
PID filter 1
1 þPs
is used. This is implemented on the FOPTDI
system with a zero.
Remark 2: For a system with positive zero KP ð1ÀPsÞeÀ Ls
sðτs þ1Þ , PID con-
troller is designed for a system KP eÀ θs
sðτsþ 1Þ where θ is effective time
delay which is the sum of process time delay and numerator
time constant ‘P’. This is obtained by approximating
1ÀPsð Þ ¼ eÀ Ps
. The designed PID controller is implemented on
the actual system with a positive zero.
2.2. Double Integrating plus Time Delay (DIPTD) system
The process transfer function DIPTD system is given by
GP sð Þ ¼
KPeÀLs
s2
ð30Þ
In Eq. (30), using Ls ¼ p and transfer function model is written
as
GP sð Þ ¼
K0
PeÀp
p2
ð31Þ
where
K0
P ¼ KPL2
ð32Þ
The invertible part of the above process transfer function (Eq.
(31)) is given by
GP À
¼
K0
P
p2
ð33Þ
The non-invertible part of the process is given by
GP þ
¼ eÀp
ð34Þ
The IMC controller is given by
Q ¼ GP À
À 1
f qð Þ ¼
p2
K0
P
γ0
1p2
þγ0
2pþ1
À Á
λ0
pþ1
À Á3
ð35Þ
where λ0
¼ λ
L; γ0
1 ¼
γ1
L2 and γ0
2 ¼
γ2
L
The conventional feedback PID controller is obtained by using
Eq. (6) and is given by
GC ¼
p2
γ0
1p2
þγ0
2pþ1
À Á
K0
P pλ0
þ1
À Á3
À γ0
1p2 þγ0
2pþ1
À Á
eÀ p
h i ð36Þ
Using first order Pade's approximation for time delay, the PID
controller is written as
GC ¼
p2
γ0
1p2
þγ0
2pþ1
À Á
1þ0:5pð Þ
K0
P pλ0
þ1
À Á3
1þ0:5pð ÞÀ γ0
1p2 þγ0
2pþ1
À Á
1À0:5pð Þ
h i ð37Þ
To derive PID controller from Eq. (38), the coefficients of p and
p2
in denominator of Eq. (38) are equated to zero. Thus the value of
γ0
1 and γ0
2 is obtained as
γ0
2 ¼ 3λ0
þ1 ð39Þ
γ0
1 ¼ 3λ02
þ1:5λ0
þ0:5γ0
2 ð40Þ
Rearranging Eq. (38), the following equation for PID controller
is obtained.
GC ¼
γ0
2p3 γ0
1
γ0
2
pþ1þ 1
γ0
2
p
1þ0:5pð Þ
K0
P λ03
þ1:5λ02
þ0:5γ0
1
p3 0:5λ03
λ03
þ 1:5λ02
þ0:5γ0
1
pþ1
! ð41Þ
Rearranging the above equation,
GC ¼
γ0
2
K0
P λ03
þ1:5λ02
þ0:5γ0
1
γ0
1
γ0
2
pþ1þ
1
γ0
2
p
0:5pþ1ð Þ
0:5λ03
λ03
þ1:5λ02
þ 0:5γ0
1
pþ1
!
ð42Þ
This is a PID controller with a first order filter (i.e. Eq. (12)),
where
Kc ¼
γ0
2
K0
P λ03
þ1:5λ02
þ0:5γ0
1
ð43Þ
GC ¼
p2
ðγ0
1p2
þγ0
2pþ1Þð1þ0:5pÞ
K0
P 0:5λ03
p4 þ 3γ02 þ1:5λ02
þ0:5γ1'
p3 þ 3λ02
þ1:5λ0
þ0:5γ0
2 Àγ0
1
p2 þð3λ0
þ0:5Àγ0
2 þ0:5Þp
h i ð38Þ
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 245

5. τ0
I ¼ γ0
2 ð44Þ
τ0
D ¼
γ0
1
γ0
2
ð45Þ
β0
¼
0:5λ03
λ03
þ1:5λ02
þ0:5γ0
1
ð46Þ
α0
¼ 0:5 ð47Þ
where τ0
I ¼ τI
L ; τ0
D ¼ τD
L ; α0
¼ α
L, and β0
¼ β
L
The tuning parameter, IMC filter time constant (λ) is tuned in
such a way that the maximum magnitude of sensitivity function
(Ms) is equal to 2. The tuning rules for DIPTD system that yields
Ms ¼2 (for a value of λ0
¼ 2:82Þ are given by
KCKPL2
¼ 0:1864 ð48Þ
τI
L
¼ 9:46 ð49Þ
τD
L
¼ 3:469 ð50Þ
α
L
¼ 0:5 ð51Þ
β
L
¼ 0:2209 ð52Þ
The closed loop servo response is given by
y
yr
¼ f sð ÞGP þ
¼
γ1s2
þγ2sþ1
À Á
λsþ1
À Á3
eÀLs
ð53Þ
A set point filter F sð Þ ¼
λ2
s2
þ2λsþ 1
À Á
γ1s2 þγ2sþ 1ð Þ
is introduced to improve the
servo performance such that the closed loop response resembles
the response of the first order system with time delay.
2.3. Pure Integrating plus Time Delay (PIPTD) system
The process transfer function for PIPTD system is given by
GP sð Þ ¼
KP
s
eÀLs
ð54Þ
In the Eq. (54), using Ls ¼ p and the transfer function is written
as
GP sð Þ ¼
K0
PeÀp
p
ð55Þ
where
K0
P ¼ KPL ð56Þ
The invertible part of the process transfer function (Eq. (55)) is
given by
GP À
¼
K0
P
p
ð57Þ
The non-invertible part of the process is given by
GP þ
¼ eÀp
ð58Þ
The IMC controller for the above process is given by
Q ¼ GÀ1
P À
f pð Þ ¼
p
K0
P
γ0
pþ1
À Á
λ0
pþ1
À Á2
ð59Þ
where γ0
¼ γ
L and λ0
¼ λ
L
From IMC controller, the conventional feedback PID controller
is obtained by Eq. (6) and is given by
GC ¼
p γ0
pþ1
À Á
K0
P λ0
pþ1
À Á2
À γ0pþ1
À Á
eÀ p
h i ð60Þ
Using first order Pade's approximation for time delay, the PID
controller is written as
GC ¼
p γ0
pþ1
À Á
1þ0:5pð Þ
K0
P λ0
pþ1
À Á2
1þ0:5pð ÞÀ 1À0:5pð Þ γ0pþ1
À Áh i ð61Þ
Rearranging the above equation,
GC ¼
ðγ0
pþ1Þ½1þ0:5pŠ
K0
P½p2ð0:5λ02
Þþp λ02
þλ0
þ0:5γ0
þ 2λ0
þ1Àγ0
À Á
Š
ð62Þ
To represent Gc as PID controller, the constant of denominator
(2λ0
þ1Àγ0
) of Eq. (62) is equated to zero. Thus the value of γ0
is
obtained as
γ0
¼ 2λ0
þ1: ð63Þ
Eq. (63) is rearranged to give PID controller with a first order
filter.
GC ¼
γ0
þ0:5
 à 0:5γ0
0:5þ γ0
ð Þ
pþ1þ 1
0:5þ γ0
ð Þp
!
K0
P λ02
þλ0
þ0:5γ0
h i
0:5λ02
λ02
þ λ0
þ0:5γ0
pþ1
! ð64Þ
Eq. (64) is PID controller with a first order lag filter as given
below.
Gc ¼ Kc 1þ
1
τ0
Ip
þτ0
Dp
1
1þβ0
p
À Á ð65Þ
where
Kc ¼
0:5þγ0
À Á
KP
0
λ02
þλ0
þ0:5γ0
ð66Þ
τI
0
¼ 0:5þγ0
À Á
ð67Þ
τD
0
¼
0:5γ0
0:5þγ0
À Á ð68Þ
The lag filter time constant is
β0
¼
0:5λ02
λ02
þλ0
þ0:5γ0
ð69Þ
where τ0
I ¼ τI
L ; τ0
D ¼ τD
L , β0
¼ β
Land λ0
¼ λ
L
The tuning parameter, IMC filter time constant (λ) is tuned in
such a way that the maximum magnitude of sensitivity function
(Ms) is equal to 2. The tuning rules for PIPTD system that yields
Ms ¼2 (for a value of λ0
¼ 1:62Þ are given by
KCKPL ¼ 0:7448 ð70Þ
τI
L
¼ 4:74 ð71Þ
τD
L
¼ 0:4473 ð72Þ
β
L
¼ 0:2062 ð73Þ
The set point filter is selected as λs þ1
γs þ1
, so that the closed loop
response is resembles the response of First Order plus Time Delay
system.
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255246

6. 2.4. Unstable First Order plus Time Delay systems with Integrator
(UFOPTDI) system
The process transfer function of Unstable First Order plus Time
Delay systems with Integrator is given by
GP sð Þ ¼
KPeÀ Ls
s τsÀ1ð Þ
ð74Þ
Using L
τ¼ε and q ¼ τs in Eq. (74), the transfer function model is
written as
GP ¼
K0
PeÀεq
qðqÀ1Þ
ð75Þ
where
K0
P ¼ KPτ ð76Þ
The invertible part of the process transfer function is given by
GP À
¼
K0
P
qðqÀ1Þ
ð77Þ
The non-invertible part of the process is given by
GP þ
¼ eÀ εq
ð78Þ
The IMC controller for the above process transfer function is
given by
Q ¼ GP À
À 1
f qð Þ ¼
q qÀ1ð Þ
K0
P
γ0
1q2
þγ0
2qþ1
À Á
λ0
qþ1
À Á3
ð79Þ
The PID controller for the process is given by Eq. (6). Sub-
stituting Eqs. (79) and (75) in Eq. (6), the following equation for Gc
is obtained
GC ¼
q qÀ1ð Þ γ0
1q2
þγ0
2qþ1
À Á
K0
P λ0
qþ1
À Á3
À γ0
1q2 þγ0
2qþ1
À Áh
eÀ ϵS
i ð80Þ
Using first order Pade's approximation for time delay, the Eq.
(81) is written as
GC ¼
q qÀ1ð Þ γ0
1q2
þγ0
2qþ1
À Á
1þ0:5ϵqð Þ
K0
P λ0
qþ1
À Á3
1þ0:5ϵqð ÞÀ γ0
1q2 þγ0
2qþ1
À Á
1À0:5ϵqð Þ
h i ð81Þ
Rearranging Eq. (82), the following equation is obtained
To derive PID settings, the coefficient of q term 3λ0
þmÀγ0
2
À Á
in
denominator of Eq. (82) is equated to zero. Thus the value of γ0
2 is
obtained as
γ0
2 ¼ 3λ0
þϵ ð83Þ
Rearranging Eq. (77), the following equation is obtained for Gc.
The part of denominator, À 0:5ϵλ03
3λ02
þ1:5λ0
ϵþ 0:5ϵγ0
2
Àγ0
1
q2
À
λ03
þ1:5λ02
ϵþ0:5ϵγ0
1
3λ02
þ1:5λ0
ϵþ 0:5ϵγ0
2
Àγ0
1
qÀ1Š of Eq. (84) is equated to qÀ1ð Þ 1þβ0
q
À Á
and comparing the corresponding coefficients of q2
and q, the
following equations are obtained for β.
β0
¼
À0:5ϵλ03
3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
ð85Þ
1Àβ0
¼
À λ03
þ1:5λ02
ϵþ0:5ϵγ0
1
3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
ð86Þ
From the Eq. (87), β0
is given by
β0
¼ 1þ
ðλ03
þ1:5λ02
þ0:5εγ0
1Þ
3λ02
þ1:5λ0
εþ0:5εγ0
2 Àγ0
1
ð87Þ
By using Eqs. (85) and (87), γ0
1 is obtained as
γ0
1 ¼
3λ02
þ1:5λ0
εþ0:5εγ0
2 þλ03
þ1:5λ02
εþ0:5ελ03
Þ
1À0:5ε
ð88Þ
Then Eq. (85) is written as
Gc ¼
À γ0
1q2
þγ0
2qþ1
À Á
qÀ1ð Þ 1þ0:5ϵqð Þ
K0
Pq 3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
qÀ1ð Þ β0
qþ1
À ÁÂ Ã ð89Þ
Upon pole zero cancellation, Gc is given by
Gc ¼
Àγ0
2
γ0
1
γ0
2
qþ 1
γ0
2
qþ1
1þ0:5ϵqð Þ
K0
P 3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
β0
qþ1
À Á ð90Þ
This is a PID controller with a first order filter as given in Eq.
(12).where
K0
c ¼
Àγ0
2
K0
P 3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
ð91Þ
τ0
I ¼ γ0
2 ð92Þ
τ0
D ¼
γ0
1
γ0
2
ð93Þ
β0
¼
À0:5ϵλ03
3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
ð94Þ
α0
¼ 0:5ε ð95Þ
and τ0
I ¼ τI
τ ; τ0
D ¼ τD
τ ; α0
¼ α
τ, β0
¼ β
τ
GC ¼
q qÀ1ð Þ γ0
1q2
þγ0
2qþ1
À Á
1þ0:5ϵqð Þ
K0
P 0:5ϵλ03
q4 þ λ03
þ1:5λ02
ϵþ0:5ϵγ0
1
q3 þ 3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
q2 þ 3λ0
þϵÀγ0
2
À Á
q
h i ð82Þ
GC ¼
À γ0
1q2
þγ0
2qþ1
À Á
q qÀ1ð Þ 1þ0:5ϵqð Þ
K0
Pq 3λ02
þ1:5λ0
ϵþ0:5ϵγ0
2 Àγ0
1
À 0:5ϵλ03
3λ02
þ1:5λ0
ϵþ 0:5ϵγ0
2
Àγ0
1
q2 À
λ03
þ 1:5λ02
ϵþ 0:5ϵγ0
1
3λ02
þ 1:5λ0
ϵþ0:5mϵÀγ0
1
qÀ1
! ð84Þ
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 247

7. The tuning parameter, IMC filter time constant (λ) is tuned in
such a way that the maximum magnitude of sensitivity function
(Ms) is equal to 2. The tuning rules for unstable FOPTDI system that
yields Ms ¼2 for 0:1rεr0:4 are given by
KcKpτ ¼ À1381ε3
þ1317ε2
À413:5εþ43:25 ð96Þ
τI
τ
¼ À495ε3
þ360ε2
À57:35εþ3:72 ð97Þ
τD
τ
¼ À396:3ε3
þ314:4ε2
À57:13εþ3:433 ð98Þ
α
τ
¼ 0:5ε ð99Þ
β
τ
¼ À0:287ε2
þ0:366εÀ0:012 ð100Þ
λ
τ
¼ À165ε3
þ120ε2
À19:45εþ1:24 ð101Þ
A set point filter F sð Þ ¼
λsþ 1ð Þ
2
γ1s2 þ γ2s þ1ð Þ
is introduced to improve the
servo performance such that the closed loop response resembles
the response of the first order system with time delay.
3. Simulation results
In this section, simulation results on various types of integrat-
ing transfer function models are reported. The performance of the
controller is measured in terms of Integral Absolute Error (IAE)
and smooth functioning of the controller is given in terms of total
variation (TV) is sum of the differences between the current
controlled output and previous controlled output, over a time
equal to settling time of the process. In the present work, PID
controllers are tuned in such a way that they will give the same Ms
value as given by the literature reported methods, so that perfor-
mance comparison can be made for the same robustness level. For
stable systems, if Ms value is between 1.4 and 2, the controller is
said to be robust. But integrating processes can be considered as a
sub class of unstable systems. Hence the Ms value can exceed 2 for
robust control. The present methods are also implemented for
higher order systems. The higher order systems are reduced to
integrating systems with time delay using model reduction tech-
niques. To the reduced model, PID controller is designed based on
the present method and implemented on the original system. Jin
and Liu [21] implemented PID controller in parallel mode, there-
fore in the present work also it is implemented in parallel mode.
3.1. Stable First Order plus Time Delay system with Integrator
(FOPTDI)
A stable First Order plus Time Delay with Integrator (FOPTDI)
system GP ¼ 0:2
sð4sþ 1ÞeÀ s
[21] is considered. The proposed method is
compared with Jin and Liu [21] method. The IMC filter time con-
stant λ is tuned to the value of 1.4 such that maximum magnitude
of sensitivity function (Ms) is 1.98 which is reported by Jin and Liu
[21]. The PID controller parameters of the proposed method and of
Table 1
PID parameters for different methods Kc 1þ 1
τI sþτDs
h i
αsþ 1
βsþ 1
.
Transfer function Method KC τI τD α β Set point filter
Case study 1 0:2eÀ s
sð4sþ 1Þ
Proposed 7.415 7.8 1.9487 0.5 0.1863 0:7s þ1
3:8s þ1
Jin and Liu [21] 3.686 10.39 2.473 – – 9:7915s2 þ 6:2664s þ 1
25:6994s2 þ10:392sþ1
Case study 2 Jacketed CSTR [10] Proposed 2.7 16.72 3.195 0.5 0.2087 0:17sþ 1
4:4sþ1
Jin and Liu [21] 1.108 15.27 4.5147 – – 22:63s2
þ 9:51s þ1
68:95s2 þ 15:27sþ 1
Case study 3 eÀ 0:5s
sðsþ 1Þðsþ 2Þðsþ 3Þ
Proposed 4.095 7.64 1.4087 0.5385 0.2473 2:35sþ 1
5:777s þ1
Lee et al. [26] 3.6823 7.363 1.3916 – – 3:44s þ1
5:5sþ 1
Case study 4 0:2eÀ 7:4s
s
Proposed 0.5169 34.1 3.2985 – 1.4836 17:25sþ 1
30:4sþ1
Lee et al. [26] 0.4226 26.2 1.145 – – 15:63sþ 1
25sþ 1
Jin and Liu [21] 0.384 35.788 – – – 14:2sþ 1
35:788sþ 1
Case study 5 eÀ s
s2
Proposed 0.1883 9.4 3.4489 0.5 0.2199 7:84s2
þ5:6sþ1
32:42s2 þ 9:4sþ 1
Lee et al. [26] 0.1275 7.5649 3.7878 – – 7:16s2 þ5:35sþ1
28:62s2 þ 10:7sþ 1
Case study 6 Jacketed CSTR a
[46] Proposed 30,6517.5 9.4 3.448 0.5 0.22 7:84s2
þ5:6sþ1
32:42s2 þ 9:4sþ 1
Lee et al. [26] 20,7620 7.565 3.7878 – – 7:16s2
þ 5:35sþ 1
28:6s2 þ 16:7sþ 1
Case study 7 eÀ 0:2s
sðsÀ 1Þ
Proposed 1.9421 2.9 1.5459 0.1 0.0488 0:81s2 þ 1:8sþ 1
4:4832s2 þ 2:9sþ 1
Cho et al. [28] 0.8594 4.4 2.7 – – 1:93s2
þ2:8sþ1
11:88s2 þ 4:4sþ 1
Liu et al.b
[51] 1.4738 2.1752 1.683 – – s2 þ sþ1
0:36s2 þ 1:2sþ 1
a
Along with the PID controller with filter an additional filter 1
ð766:0752sþ1Þ is considered.
b
Two loop control scheme. Set point tracking controller is PD controller with Kc ¼1 and τD ¼2. The controller parameters given in the above table are for disturbance
rejection.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
0 10 20 30 40 50 60 70 80 90 100
-10
0
10
20
30
40
t
u
Proposed
Jin and Liu
Fig. 1. The response of transfer function model 0:2eÀ s
sð4sþ 1Þ (case study 1).
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255248

8. Jin and Liu [21] are listed in Table 1. The response of the system for
unit step change in set point at t¼0 s and for the unit step change
in load at t¼50 s with the proposed controller is shown in Fig. 1.
The servo and regulatory response of the system with the con-
troller designed by Jin and Liu [21] is also shown in Fig. 1.
The performance of both the controllers is reported in terms of
IAE, ITAE and the smoothness of the controller which is given in
terms of total variation (TV) for both servo and regulatory pro-
blems (refer to Table 2).When compared with the Jin and Liu [21]
method, the performance of the proposed method is superior with
less IAE and ITAE values for servo and regulatory problems
respectively. The proposed method gives less overshoot and peak
error for servo and regulatory problems respectively. The con-
troller action for the present method is smooth (measured in
terms of TV) for servo problem compared with the method of Jin
and Liu [21]. The proposed controller gives fast response compared
to the controller by Jin and Liu [21].
The performance of the system for parameter uncertainty is
checked by Kharitonov's theorem (refer to Appendix A) and the
range of process parameters for which the controller (designed for
nominal model parameters) can stabilize the system are given in
Table 3. The performance of the controller designed by proposed
method and the controller designed by the method of Jin and Liu
[21] for þ20% uncertainty simultaneously in Kp, τ and L is shown
in Fig. 2 and for À20% uncertainty simultaneously in Kp, τ and L is
shown in Fig. 3. The performance of both the controller in terms of
IAE, ITAE and TV for 720% uncertainty in Kp, τ and L individually
is also reported in Table 4. The proposed controller performs
better than the controller designed by the method of Jin and Liu
[21] even in the presence of uncertainty in process parameters.
3.2. Non-linear Jacketed CSTR
Consider a non-linear Jacketed CSTR carrying first order irre-
versible exothermic reaction proposed by Hovd and Skogestad [10]
and the model equations are
dCA
dt
¼ cAF ÀcAð Þ
F
V
Àr ð102Þ
Table 2
Performance comparison of different methods.
Case study Method Ms Phase marginZ Gain marginZ Servo problem Regulatory problem
IAE ITAE TV IAE ITAE TV
1 Proposed 1.98 29.15° 2.01 3.51 8.667 23.13 1.05 61.15 1.917
Jin and Liu [21] 1.99 28.56° 2.03 4.136 12.9 37.528 2.81 170.3 1.69
2 Proposed 1.95 28.68° 1.98 31.59 346.1 45.05 27.13 455.3 8.00
Jin and Liu [21] 1.96 28.32° 1.98 47.96 536.9 9.42 60.37 918.32 7.478
3 Proposed 1.97 29.37° 2.02 3.984 12.89 6.75 1.87 15.0 1.634
Lee et al. [26] 1.97 29.41° 2.03 4.385 18.68 4.46 2.03 15.35 1.71
4 Proposed 2 28.27° 1.95 18.45 261.49 0.8417 65.96 15,346 1.796
Lee et al. [26] 2 29.39° 2.0 21.45 404.1 0.6317 68.39 15,851 2.05
Jin and Liu [21] 2 28.97° 2.0 21.58 306.73 0.322 93.26 21,973 1.915
5 Proposed 2.01 28.78° 1.98 4.8 18.438 0.4861 49.92 3479 2.121
Lee et al. [26] 2.01 28.85° 1.99 6.1 36.76 0.2458 68.962 4796 2.785
6 Proposed 2.0 28.8° 1.99 12.61 86.66 3165.8 0.0332 0.2882 15.61
Lee et al. [26] 2.0 28.8° 1.99 16.6 124.76 3529 0.0654 0.7844 37.8
7 Proposed 1.94 30.1° 2.06 1.423 1.74 5.63 1.4932 41.75 2.8
Cho et al. [28] 1.94 29.8° 2.06 2.097 3.92 2.138 5.1199 150.58 3.2882
Liu et al. [51] – – – 1.4056 1.3645 4.5409 1.7446 48.118 3.7599
Table 3
Stability regions for model parameters (Kharitonovs Theorem).
Transfer function Method KP (%) L (%) τ (%)
Case study 1 0:2eÀ s
sð4sþ 1Þ
Proposed 753 725 733
Jin and Liu [21] 755 762 755
Case study 2 Jacketed CSTR [10] Proposed 763 726 732
Jin and Liu [21] 753 770 755
Case study 3 eÀ 0:5s
sðsþ 1Þðsþ 2Þðsþ 3Þ
Proposed 756 724 734
Lee et al. [26] 757 758 755
Case study 4 0:2eÀ 7:4s
s
Proposed 721 744 –
Lee et al. [26] 758 752 –
Case study 5 eÀ s
s2
Proposed 746 740 –
Lee et al. [26] 730 766 –
Case study 6 Jacketed CSTR [46] Proposed 746 740 –
Lee et al. [26] 732 765 –
Case study 7 eÀ 0:2s
sðsÀ 1Þ
Proposed 742 760 753%
Cho et al. [28] 735 768 755%
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Response
Proposed
Jin and Liu
Fig. 2. Response of the system in case study 1 for þ20% uncertainty in Kp, L and τ.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time
Response
-Proposed
Jin and Liu
Fig. 3. Response of the system in case study 1 for À20% uncertainty in Kp, L and τ.
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 249

9. dcB
dt
¼ ÀcB
F
V
þr ð103Þ
dT
dt
¼ TF ÀTð Þ
F
V
À
ΔHr
ρCP
À
hA
ρVCP
T ÀTCð Þ ð104Þ
r ¼ KcAeÀ E
RT ð105Þ
The process parameters are given in Table 5. The above non-
linear equations are linearized around the operating point
CA ¼10 K mol/m3
, CBS
¼5.62 K mol/m3
and TS(steady state reactor
temperature)¼590 K and the transfer function model is given by
TðsÞ
CAF ðsÞ
¼
0:07803sþ0:007803
s3 þ0:1766s2 þ0:00735sÀ0:00031
ð106Þ
On simplifying Eq. (106), the following equation is obtained
TðsÞ
CAF ðsÞ
¼
0:007803ðsþ0:1Þ
ðsþ0:1Þðsþ0:0805ÞðsÀ0:0039Þ
ð107Þ
Upon pole zero cancellation, Eq. (107) reduces to,
TðsÞ
CAF ðsÞ
¼
0:07803
0:0805 Â 0:0039ð12:4224sþ1Þð256:41033sÀ1Þ
ð108Þ
In the above equation, the unstable time constant (256.4103) is
very large and can be considered into process gain. On simplifying
the equation, stable FOPTDI transfer function is obtained.
TðsÞ
CAF ðsÞ
¼
0:9693
sð12:4224sþ1Þ
ð109Þ
A measurement delay of one minute is considered and the
transfer function is written as
TðsÞ
CAF ðsÞ
¼
0:9693eÀ1s
sð12:4224sþ1Þ
ð110Þ
A PID controller is designed for the above transfer function
model of jacketed CSTR using Eqs. (102)–(105) by tuning λ in such
a way that it gives same Ms value as Jin and Liu [21]. The PID
Table 4
Performance comparison under parameter uncertainty for case study 1.
Problem Error criteria Method KP ; τ and L KP τ L
þ20% À20% þ20% À20% þ20% À20% þ20% À20%
Servo IAE Proposed 3.26 3.82 3.34 3.94 3.67 3.5 3.42 3.57
JL 4.13 4.52 4.13 4.41 3.93 4.21 4.0 4.13
ITAE Proposed 6.55 12.28 8.11 11.78 9.26 9.86 8.1 9.77
JL 15.24 17.92 15.25 14.05 12.9 15.23 12.9 13.03
TV Proposed 28.62 21.3 27.2 22.36 23.1 27.76 27.4 21.71
JL 40.07 37.62 40 37.62 38.2 39.83 39.4 37.8
Regulatory IAE Proposed 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05
JL 2.82 2.86 2.82 2.82 2.82 2.83 2.81 2.82
ITAE Proposed 61.14 61.5 61.1 61.4 61.2 61.14 61.1 61.4
JL 170.2 173.5 170.3 170.8 170.2 170.9 170.3 170.3
TV Proposed 3.8 2.38 2.4 2.71 1.87 3.43 2.55 2.48
JL 3.14 2.4 2.8 2.64 2.8 2.67 2.9 2.54
Table 5
Parameter values for jacketed CSTR [10].
Parameter Value
Volume, V 50 m3
Feed flow rate, F 5 m3
/min
Feed Temperate,TF 500 K
Feed Concentration, CAF 15.61 K mol/m3
Product of heat Transfer coefficient and heat transfer area
(hA)
2600 kJ/min K
Specific Heat, Cp 1.8 kJ/kg K
Heat of reaction, ÀΔH À20,000 kJ/K mol
Universal gas constant, R 8.314 kJ/K mol
Frequency factor, Ko 680,000 minÀ1
Density, ρ 800 kg/m3
Temperature, Tc 401.6 K
0 10 20 30 40 50 60 70 80
590
591
592
593
594
595
596
597
T
0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
Time in min
CAFTCAF
Proposed
Jin and Liu
0 10 20 30 40 50 60 70 80
590
590.5
591
591.5
592
592.5
593
593.5
594
0 10 20 30 40 50 60 70 80
12
14
16
18
20
Time in min
Proposed
Jin and Liu
Fig. 4. (a) The servo response of Jacketed CSTR (non-linear model simulation) (case
study 2). (b) The regulatory response of response of Jacketed CSTR [non-linear
model simulation] (case study2).
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255250

10. controller and filter parameters of the proposed method and of Jin
and Liu [21] are listed in Table 1.The servo response of the pro-
posed controller with the non-linear process model (Eqs. (102)–
(105)) for a step change in operating temperature from 590 to
595 K are shown in Fig. 4(a).The servo response of the system with
the controller designed by Jin and Liu [21] is also shown in Fig. 4
(a). The performance of both the controllers is reported in terms of
IAE, ITAE and the smoothness of the controller is given in terms of
total variation (TV) for servo problem (refer to Table 2). When
compared with the Jin and Liu [21] method, the performance of
the proposed method is superior with less IAE, ITAE values (refer
to Table 2) and less overshoot. The proposed controller gives fast
response compared to the controller by Jin and Liu [21].
The regulatory performance of the proposed controller and
controller designed by Jin and Liu [21] for a change of feed con-
centration from 15.61 to 20 K mol/m3
is reported in Table 2 in
terms of IAE, ITAE and the response is shown in Fig. 4(b). The
response of the proposed controller gives less peak error. The
proposed controller gives fast response compared to the controller
by Jin and Liu [21].
The stability regions for the model parameters (considering the
transfer function model in Eq. (110)) for which the controller can
stabilize the system are obtained by using Kharitonov's theorem
and reported in Table 3.
3.3. Higher order system
The transfer function model eÀ 0:5s
s sþ 1ð Þ sþ2ð Þ s þ3ð Þ [41] is considered.
The model is reduced to FOPTDI model 0:167eÀ 1:077s
sð1:863sþ 1Þ [32]. The PID
parameters for the proposed method and Lee et al. [26] method
are given in Table 1. The response of the proposed controller for a
unit step change in set point is shown in Fig. 5(a). The servo
response of the system with the controller designed by Lee et al.
[26] is also shown in the Fig. 5(a). The performance of both the
controllers is given in terms of IAE, ITAE and the smoothness of the
controller which is given in terms of total variation (TV) for servo
problem (refer to Table 2). The IMC filter time constant λ is tuned
to the value of 2.35 to obtain the same robustness level of
0 5 10 15 20 25 30 35 40 45 50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
0 5 10 15 20 25 30 35 40 45 50
-2
0
2
4
6
t
u
Proposed
Lee et al.
0 5 10 15 20 25 30
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
y
0 5 10 15 20 25 30
-0.5
0
0.5
1
t
u
Proposed
Lee et al.
Fig. 5. (a) The servo response of transfer function model eÀ 0:5s
s sþ1ð Þ sþ 2ð Þ sþ 3ð Þ (case study
3). (b) Regulatory response of transfer function model eÀ 0:5s
s sþ 1ð Þ sþ2ð Þ sþ3ð Þ (case study 3).
0 50 100 150 200 250 300 350 400
0
0.5
1
1.5
2
2.5
3
3.5
4
y
0 50 100 150 200 250 300 350 400
-0.5
0
0.5
1
1.5
u
Proposed
Jin and Liu
Lee et al.
Fig. 6. The response of transfer function model GP ¼ 0:2
s eÀ 7:4s
(case study 4).
0 50 100 150
0
2
4
6
8
10
y
0 50 100 150
-1
-0.5
0
0.5
1
1.5
u
Proposed
Lee et al.
Fig. 7. The response of transfer function model eÀ s
s2 (case study 5).
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 251

11. maximum magnitude of sensitivity function (Ms) 1.97 of Lee et al.
[26]. When compared with Lee et al. [26] method, the perfor-
mance of the proposed method is superior with less IAE and ITAE
values and less overshoot. The proposed controller gives fast
response compared to the controller by Lee et al. [26]. The reg-
ulatory response of the proposed controller for a 0.1 change in load
is shown in Fig. 5(b). The regulatory response of the system with
the controller designed by Lee et al. [26] is also shown in the Fig. 5
(b). When compared with the Lee et al. [26] method, the perfor-
mance of the proposed method is superior with less IAE and ITAE
values for both servo and regulatory problems. For regulatory
problem the proposed method gives less peak error. The controller
action is smooth for regulatory problem for the present method
compared with the method of Lee et al. [26]. The proposed con-
troller gives fast response compared to the controller by Lee et al.
[26].
The stability regions for the model parameters (considering the
FOPTDI transfer function model) for which the controller can
stabilize the system are obtained by using Kharitonov's theorem
and reported in Table 3.
3.4. Pure Integrator plus Time Delay (PIPTD) system
The transfer function model 0:2
s eÀ7:4s
[21,26,42] is considered.
For this system PID controller with a lag filter is designed by using
the present method ((Eqs. (70)–73)) for Ms ¼2. The PID controller
and filter parameters for various methods are reported in Table 1.
The response of the system for unit step change in set point at
t¼0 s for the unit step change in load at t¼200 s with the pro-
posed controller is shown in Fig. 6. The servo and regulatory
performances of the system with the controller designed by Lee
et al. [26] and Jin and Liu [21] is also shown in Fig. 6.
The performance of the three controllers is given in terms of
IAE, ITAE and the smoothness of the controller is given in terms of
total variation (TV) for both servo and regulatory problems (refer
to Table 2). The controller action is smooth for regulatory problem
for the present method compared with the method of Lee et al.
[26] and Jin and Liu [21]. When compared with the reported
methods, the performance of the proposed method is superior
with less IAE, ITAE values for both servo and regulatory problems.
The present method gives less overshoot and peak error for servo
and regulatory problems respectively. The proposed controller
gives fast response compared to the controller by the Lee et al. [26]
and Jin and Liu [21].
The stability regions for the model parameters (Kp, L) for which
the controller can stabilize the system are obtained by using
Kharitonov's theorem and reported in Table 3. 3.5. Double Integrator plus Time Delay (DIPTD) system
The transfer function model e À s
s2 [26] is considered. For this
system PID controller with a lead lag filter is designed by using the
present method ((Eqs. (48)–52)) for Ms ¼2. The PID controller and
filter parameters for various methods are reported in Table 1. The
response of the system for unit step change in set point at t¼0 s
and for the unit step change in load at t¼60 s with the proposed
controller is shown in Fig. 7. The servo and regulatory perfor-
mance of the system with the controller designed by Lee et al. [26]
is also shown in Fig. 7. The performance of both the controllers is
given in terms of IAE, ITAE and the smoothness of the controller is
given in terms of total variation (TV) for both servo and regulatory
problems (refer to Table 2). When compared with the Lee et al.
[26] method, the performance of the proposed method is superior
with less IAE, ITAE values for servo and regulatory problems. The
Table 6
Parameter values for jacketed CSTR [46].
Parameter Value
Volume, V 1 m3
Feed flow rate, F 0.00065 m3
/s
Feed Temperate, To 300 K
Feed Concentration, CAo 7.5 K mol/m3
Product of overall heat Transfer coefficient and heat transfer
area (UA)
1.4 kJ/s K
Specific Heat, Cp 3.5 kJ/kg K
Heat of reaction, ÀΔH 50,000 kJ/K mol
Universal gas constant, R 8.345 kJ/K mol K
Frequency factor, Ko 1.8 Â 107
sÀ1
Density, ρ 850 kg/m3
Activation energy E 69,000 kJ/K mol
0 10 20 30 40 50 60
344
344.5
345
345.5
346
346.5
347
347.5
0 10 20 30 40 50 60
-500
0
500
1000
1500
2000
Proposed
Lee et al.
Time, s
0 10 20 30 40 50 60
343.992
343.994
343.996
343.998
344
344.002
344.004
344.006
344.008T,K
0 10 20 30 40 50 60
310
315
320
325
TJ,K
T,KTJ,K Proposed
Lee et al.
Time, s
Fig. 8. (a) The servo response of Jacketed CSTR (non-linear model simulation) (case
study 6). (b) The regulatory response of Jacketed CSTR (non-linear model equations
of CSTR) (case study 6).
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255252

12. proposed method gives less peak error for regulatory problem. The
proposed controller gives fast response compared to the controller
by the Lee et al. [26]. Comparison of TV value shows that the
proposed controller action is smoother than Lee et al. [26]
controller.
The stability regions for the model parameters (Kp, L) for which
the controller can stabilize the system are obtained by using
Kharitonov's theorem and reported in Table 3.
3.6. Non-linear Jacketed CSTR
Consider a jacketed CSTR carrying out first order irreversible
exothermic reaction [46] and the model equations are
dCA
dt
¼
F
V
CAo
ÀCA
À Á
ÀKOCAexp À
E
RT
ð111Þ
dT
dt
¼
F
V
TO ÀTð Þþ
ðÀΔHÞ
ρCP
KOCAexp À
E
RT
þ
UA
ρVCP
ðTJ ÀTÞ ð112Þ
Process parameters are given in Table 6. The above non-linear
equations are linearized around unstable operating point
CAs ¼3.734 K mol/m3
, TS ¼344 K and TJS ¼317.4 K and transfer
function model is given by
TðsÞ
TJðsÞ
¼
0:0004706sþ6:143 Â 10À 7
s2 À0:0004483sÀ4:055 Â 10À 7
ð113Þ
Rearranging Eq. (113)
TðsÞ
TJðsÞ
¼
6:143 Â 10À 7
ð1þ766:0752sÞ
ðsÀ0:8992 Â 10À3
Þðsþ0:4509 Â 10À3
Þ
ð114Þ
TðsÞ
TJðsÞ
¼
6:143 Â 10À 7
ð1þ766:0752sÞ
ð0:8992 Â 10À 3
 0:4509  10À 3
Þð1112:099sÀ1Þð2217:79sþ1Þ
ð115Þ
In the above transfer function, stable and unstable time con-
stants (1112.099, 2217.79 respectively) are very large and can be
considered in to process gain. On simplifying the transfer function
model, DIPTD system with a zero is obtained.
TðsÞ
TJðsÞ
¼
6:143 Â 10À 7
ð1þ766:0752sÞ
s2
ð116Þ
The above transfer function model, a model time delay of 1 s is
considered and the transfer function is written as
TðsÞ
TJðsÞ
¼
6:143 Â 10À 7
ð1þ766:0752sÞ
s2
eÀ1s
ð117Þ
For this process PID controller is designed using Eqs. (48)–(52).
The PID parameters for the proposed method and literature
reported method is given in Table 1. The response of the proposed
controller with the non-linear process model ((Eqs. (111) and 112))
for a step change in reactor temperature from 344 to 346 K is
shown in Fig. 8(a).The servo response of the Jacketed CSTR with
the controller designed by Lee et al. [26] is also shown in Fig. 8(a).
The performance of both the controllers is given in terms of IAE,
ITAE and the smoothness of the controller is given in terms of Total
Variation (TV) for servo problem (refer to Table 2). When com-
pared with the Lee et al. [26] method, the performance of the
proposed method is superior with less IAE and ITAE values and
less overshoot. The proposed controller gives fast response com-
pared to the controller by Lee et al. [26]
The regulatory response of the proposed controller for a load
change in reactor jacket temperature from 317.4 to 310 K is shown
in Fig. 8(b).The regulatory response of the system with the con-
troller designed by Lee et al. [26] is also shown in Fig. 8(b). The
performance of both the controllers is given in terms of IAE, ITAE
and the smoothness of the controller is given in terms of total
variation (TV) for servo problem (refer to Table 2). The perfor-
mance of the proposed method is superior with less IAE, ITAE
values for both servo and regulatory problems. The proposed
method gives less overshoot and less peak error when compared
with the method reported by Lee et al. [26]. The TV values for
servo and regulatory problems are less than the reported method.
The proposed controller gives smoother control action. The pro-
posed controller gives fast response compared to the controller by
Lee et al. [26].
The stability regions for the model parameters [considering the
transfer function model in Eq. (117)] for which the controller can
stabilize the system are obtained by using Kharitonov's theorem
and reported in Table 3.
3.7. Unstable First Order plus Time Delay system with Integrator
(UFOPTDI)
The transfer function model e À 0:2s
sðsÀ 1Þ [28,51] is considered. The PID
controller parameters are reported in Table 1. The response of the
system for unit step change in set point at t¼0 s for the unit step
change in load at t¼40 s with the proposed controller is shown in
Fig. 9. The servo and regulatory performances of the system with
the controller designed by Cho et al. [28] and Liu et al. [51] is also
shown in Fig. 9. Liu et al. [51] decoupled set point tracking pro-
blem with regulatory problem using two degree of freedom con-
trol scheme. For servo response PD controller with a set point filter
is used and for regulatory problem PID controller is used. The PID
setting for both the controllers are listed in Table 1.
The performance of the three controllers is given in terms of
IAE and the smoothness of the controller is given in terms of total
variation (TV) for both servo and regulatory problems (refer to
Table 2). The IMC filter time constant λ is tuned to the value of
0.9 to obtain the same robustness level of maximum magnitude of
sensitivity function (Ms) 1.94 of Cho et al. [28]. The performance of
the proposed method is superior to Cho et al. [28] with less IAE,
ITAE values for both servo and regulatory problems. The control
scheme proposed by Liu et al. [51] gives better performance for
servo problem compared to the proposed controller and the reg-
ulatory response of the proposed controller is better than Liu et al.
[51]. The proposed method gives less overshoot and peak error for
0 5 10 15 20 25 30 35 40 45 50
-2
-1
0
1
2
3
t
u
0 5 10 15 20 25 30 35 40 45 50
0
0.5
1
1.5
2
2.5
y
t
Proposed
Cho et al.
Liu et al.
Fig. 9. The response of transfer function model Gp ¼ eÀ 0:2s
sðsÀ1Þ (case study 7).
D.B.S. Kumar, R. Padma Sree / ISA Transactions 63 (2016) 242–255 253

13. servo and regulatory problems respectively. The controller action
is smooth for regulatory problem for the present method com-
pared with the method of Cho et al. [28].
4. Conclusions
PID controller design using IMC principles for PIPTD, DIPTD,
stable FOPTDI with/without a zero and unstable FOPTDI systems is
proposed. Tuning rules are given for stable/unstable FOPTDI sys-
tem, PIPTD system and DIPTD system for Ms ¼2 in terms of model
parameters. First order Pade's approximation is used for time
delay to design PID controller. The controller is tuned by a single
tuning parameter λ which governs the trade-off between the
performance and robustness of the control system. For larger
values of time delay to time constant, λ is tuned in such a way that
a trade-off is made between performance and robustness and the
tuning rules for stable FOPTDI system are given. In all the case
studies, PID controllers are tuned in such a way that they will give
the same maximum magnitude of sensitivity function Ms value as
given by the literature reported methods, so that performance
comparison is made for the same robustness level. The designed
controller performed well for both disturbance rejection and servo
problem and reduced overshoot and peak error. The proposed
controllers are implemented on various transfer function models,
non-linear model equations of jacketed CSTR and the process with
higher order dynamics. The higher order system is reduced to
integrating processes with time delay system using model reduc-
tion techniques. To the reduced model, PID controller is designed
based on the present method and implemented on the original
higher order system. The controller designed by the present
method gives fast response. The performance comparison in terms
of IAE, ITAE and TV show that the proposed controller performs
better than the recently reported methods.
Appendix A
Kharitonov's theorem
The closed loop characteristic equation is given by
G sð Þ ¼ 1þGcGP ¼ 0 ðA1Þ
where,
Gc ¼ kc 1þ
1
τIs
þτDs
!
1þαs
1þβs
ðA2Þ
GP ¼
kPeÀ Ls
sðτsþaÞ
ðA3Þ
Case (i): τ¼0 and a¼1, the process Gp is a pure integrating
system.
Case (ii): τ¼1 and a¼0, the process is Gp a double integrating
system.
Case (iii): a¼1, the process Gp is an Stable First Order plus Time
Delay system with an Integrator.
Case (iv): a¼ À1, the process Gp is an Unstable First Order plus
Time Delay system with an Integrator.
Using second order Pade's approximation for time delay, the
characteristic equation is given by
G sð Þ ¼ a1s6
þa2s5
þa3s4
þa4s3
þa5s2
þa6sþa7 ðA4Þ
where,
a1 ¼ cL2
τIβτ ðA5Þ
a2 ¼ cL2
ττI þaτβcL2
þ0:5LττIβþcL2
τIτDαKCKP ðA6Þ
a3 ¼ KcKP cL2
τIαþcL2
τIτD À0:5LτIτDα
þcaL2
τI
þ0:5LτIτþ0:5LaτIβþτIβτ ðA7Þ
a4 ¼ KcKP cL2
αþcL2
τI À0:5LτIαÀ0:5LτIτD þτIτDα
þ0:5LaτI þτIτþaτIβ ðA8Þ
a5 ¼ KcKP cL2
À0:5LαÀ0:5LτI þτIαþτIτD
þaτI ðA9Þ
a6 ¼ KcKPðαþτI À0:5LÞ ðA10Þ
a7 ¼ KcKP ðA11Þ
c ¼ 1=12 ðA12Þ
The Kharitonov's polynomials are given below. Here aÀ
i and aþ
i
are the lower bound and upper bound of ai respectively.
aþ
1 s6
þaþ
2 s5
þaÀ
3 s4
þaÀ
4 s3
þaþ
5 s2
þaþ
6 sþaÀ
7 ¼ 0 ðA13Þ
aÀ
1 s6
þaÀ
2 s5
þaþ
3 s4
þaþ
4 s3
þaÀ
5 s2
þaÀ
6 sþaþ
7 ¼ 0 ðA14Þ
aþ
1 s6
þaÀ
2 s5
þaÀ
3 s4
þaþ
4 s3
þaþ
5 s2
þaÀ
6 sþaÀ
7 ¼ 0 ðA15Þ
aÀ
1 s6
þaþ
2 s5
þaþ
3 s4
þaÀ
4 s3
þaÀ
5 s2
þaþ
6 sþaþ
7 ¼ 0 ðA16Þ
For fixed value of kP and τ, a perturbation in time delay L i.e.
(LÀΔL)rLr(LþΔL) is substituted in the above coefficients and
Kharitonov's polynomials are checked for stability by Routh–Hur-
witz method [49,50]. Similar procedure is repeated to find stability
regions for kP and τ.
For PIPTD system, since a1 ¼0 as α¼0 in Eq. (A2), the Khar-
itonov's polynomials are given below.
aþ
2 s6
þaþ
3 s5
þaÀ
4 s4
þaÀ
5 s3
þaþ
6 s2
þaþ
7 s ¼ 0 ðA17Þ
aÀ
2 s6
þaÀ
3 s5
þaþ
4 s4
þaþ
5 s3
þaÀ
6 s2
þaÀ
7 s ¼ 0 ðA18Þ
aþ
2 s6
þaÀ
3 s5
þaÀ
4 s4
þaþ
5 s3
þaþ
6 s2
þaÀ
7 s ¼ 0 ðA19Þ
aÀ
2 s6
þaþ
3 s5
þaþ
4 s4
þaÀ
5 s3
þaÀ
6 s2
þaþ
7 s ¼ 0 ðA20Þ
For fixed value of kP; a perturbation in time delay L i.e. (LÀΔL)
rLr(LþΔL) is substituted in the above coefficients and Khar-
itonov's polynomials are checked for stability by Routh–Hurwitz
method [49,50]. Similar procedure is repeated to find stability
region for kP.
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