Classical Proportional Integral Derivative(PID) controller
remains the most popular approach for industrial process
control. Poor tuning of PID controller can lead to mechanical
wear associated with excessive control activity, poor control
performance and even poor quality products. In this paper, we
design procedure for the internal model control(IMC) approach
for tuning of conventional PID controller with proper tuning
rules. Furthermore, with help of analytical rule of step test
obtaining the effective first order time delay model of the process.
A simulation example of continuous stirred tank reactor is used
in which the IMC based PID tuning method implemented and
the step response of the closed loop system is compared with
classical tuning methods like Ziegler-Nichols and Cohen-Coon

Anilkumar, Markana

Parikh, Nishant

Shah, Ankit K.

Nirma University Journal of Engineering and Technology (NUJET)

NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010 51
Performance analysis of IMC based PID controller
tuning on approximated process model
Ankit K. Shah, Markana Anilkumar and Nishant Parikh
Abstract—Classical Proportional Integral Derivative(PID) con-
troller remains the most popular approach for industrial process
control. Poor tuning of PID controller can lead to mechanical
wear associated with excessive control activity, poor control
performance and even poor quality products. In this paper, we
design procedure for the internal model control(IMC) approach
for tuning of conventional PID controller with proper tuning
rules. Furthermore, with help of analytical rule of step test
obtaining the effective first order time delay model of the process.
A simulation example of continuous stirred tank reactor is used
in which the IMC based PID tuning method implemented and
the step response of the closed loop system is compared with
classical tuning methods like Ziegler-Nichols and Cohen-Coon.
Index Terms—Normalized cutset, discrete wavelet transform,
high boost filter
I. INTRODUCTION
NEVERTHELESS , PID controllers are still widely usedin industrial applications including for process control.
The reason is that PID controller has a simple structure which
is easy to be understood by the engineers who design it. This
is not only due to the simple structure which is conceptually
easy to understand and, which makes manual tuning possible,
but also to the fact that the algorithm provides adequate
performance in the vast majority of applications. It is widely
used in process industries because of its simple structure
and robustness to the modeling error. Sophisticated control
algorithms, such as model predictive control, are built on
the basis of the PID algorithm. Even in non-linear control
development, PID control has been used as comparison
reference [1].
According to a survey conducted by Japan electric
Measuring Instruments Manufacturers Association in 1989,
90 percent of the control loops in industries are of PID type
[1] and only small portion of the control loop works well.
Also survey by ender [2] indicates 30 percent of the controller
are operated in manual mode and 20 percent of the loops
use factory tuning. It means that PID controller is widely
used but poorly tuned. Poor tuning can lead to mechanical
wear associated with excessive control activity, poor control
performance and even poor quality products. The present
Ankit K. Shah is with Instrumentation and Control Department
Sardar Vallabhbhai Patel,Inst. of Tech.,Vasad-388306,India.Email: ak-
snishaa@gmail.com Telephone: 9408572742
Markana Anilkumar is with Systems and Control Engineering Indian Insti-
tute of Tech.-Bombay,Mumbai-400076, India Email: anil.markana@iitb.ac.in
Nishant Parikh is with School of Petroleum Technology Pandit Deen-
dayal Petroleum University Gandhinagar, Gujarat-382007, India.Email:
nish23481@gmail.com Telephone: (+91)79-232-75025
work is aimed to provide PID controller tuning guidelines
using Internal Model Control(IMC) approach. Recently much
research effort has been focused on the automatic tuning of
PID controllers, which was first proposed by Astrom and
Hagglund (1984) [2]. They have introduced novel relay tuning
method for finding the critical gain and critical frequency
of closed loop process [2] and proposed several tuning rules
for PID controllers based on this information. The PID
controller tuning is method of computing the three control
parameters Proportional gain, Derivative time and Integral
time, such that the controller meets desired performance
specification. Since the exact dynamics of the plant is
generally unknown, the basic function of autotuners is some
experimental procedure through which plant information
is obtained in order to compute the controller parameters.
Tuning techniques can therefore be classified according to
this experimental procedure. This is particularly true for the
optimal PID controller tuning for time delay processes since
the stability check for a given time-delay closed-loop system
is not a trivial task.
In the second section, the design steps for Internal Model
Control(IMC) will be describe. In the same section tuning
formulas for the conventional PID controller closed loop
system will be given. In the third section simulation results
on continuous flow stirred-tank reactor(CSTR) will be given
with comparison study of closed loop PID controller response
using Ziegler-Nichols, Cohen-coon and IMC tuning methods.
The fourth section concludes the our approach of IMC based
PID controller parameter tuning.
II. BASIC DESIGN OF INTERNAL MODEL
CONTROL(IMC)
In this section, we will develop the IMC approach for
PID controller tuning. The name comes from the fact that
the controller has explicit model of the plant as its part [3].
The premise of IMC is that in reality, we only have an
approximation of the actual process. Even if we have the
correct model, we may not have accurate measurements of
the process parameters. Thus the imperfect model should be
factored as part of the controller design.
In the block diagram shown in Fig. 1 [4] implementation
of IMC on the process transfer function Gp is given. In that
G˜p is the approximate transfer function of the process Gp
and G¯c is the model controller. In Fig. 2 block diagram of
the conventional feedback controller shown. By comparison of
Fig. 1 and Fig. 2, conventional feedback controller Gc consists
of G¯c and G˜p. We first need to derive the closed-loop functions
52 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010
Fig. 1. Block diagram of internal model control structure
Fig. 2. Block diagram of conventional PID structure
for the IMC system. Based on the block diagram of the IMC,
the error is E = R-(C -C¯) where C¯ = G˜p. We can rearranged
the equations and get the controller output P as
P =
Gc
1−GcG˜p
(R− C) (1)
This step is to shows the relationship between the conven-
tional feedback controller transfer function Gc shown in Fig.
2 and IMC structure in Fig. 1:
Gc =
Gc
1−GcG˜p
(2)
The poles of Gc are inherited from the zeros of Gp. If Gp
has positive zeros, it will lead to a Gc function with positive
poles. To avoid that, we split the approximate function as a
product of two parts:
G˜p = G˜p+ + G˜p− (3)
with G˜p+ containing all the positive zeros, if present. The
controller will be designed on the basis of G˜p−.. only. We
now define the model controller transfer function G¯c as
Gc =
Gf
G˜p−
(4)
where Gf is first order low pass filter [5] used to avoid
model mismatch. The filter transfer function Gf defined as
Gf =
1
τfs+ 1
(5)
where τf is the filter time constant.
III. PID TUNING USING IMC
For conventional PID controller transfer function is given
by
Gc(s) = Kp[1 +
1
τis
+ τds] (6)
where Kp is the proportional gain, τi is the integral time
and τd is the derivative time or lead time.
To model our process by fitting the open-loop step test data
as a first order function with time delay, our measured or
approximate model transfer function G˜p given by
G˜p =
Ke−tds
τps+ 1
(7)
where K represents the dc gain, τp and td the process time
constant and time delay, respectively. Replacing the time delay
of by a first order Pade rational approximation expressed by
[5]
e−tds ≈ 1−
td
2 s
1 + td2 s
(8)
The approximate model transfer function represent in (7)
can be factorized in to invertible and noninvertible factors as
G˜p− =
K
(τps+ 1)(1 +
td
2 s)
(9)
G˜p+ = (1− td
2
s) (10)
From (9) and (4), IMC controller transfer function can be
derive as
Gc =
(τps+ 1)(1 +
td
2 s)
K(τfs+ 1)
(11)
Comparing equation (11) with the ideal PID controller
equation represented by (6), which will lead to the tuning
parameters of an ideal PID controller based on IMC approach.
Controller parameters are given as
Kp =
2
τp
td
+ 1
K(2
τf
td
+ 1)
(12)
τı =
td
2
+ τp (13)
τd =
τp
2
τp
td
+ 1
(14)
IV. SIMULATION RESULTS ON CONTINUOUS
STIRRED TANK REACTOR(CSTR)
Fig. 3 shows the continuous stirred tank reactor for reactant
A. In this example, we have a stirred tank with a volume
V of 4m3 being operated with an inlet continuous flow rate
Q of 0.4m3/sec and which contains an inlet reactant A at a
concentration Cin.
The model equation for continuous flow stirred-tank reactor
with chemical reaction of the reactant A given as
V
d
dt
CA = Q(Cin − CA)− V λCA (15)
where CA is the molar concentration of reactant A and λ is
the first order reaction rate constant of value 0.1sec−1. Above
equation can be written in simplified manner by
NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1,NO.2, JUL-DEC 2010 53
Fig. 3. Continuous stirred tank reactor
V
Q
d
dt
CA + (1 +
KV
Q
)CA = Cin (16)
By taking laplace transformation of above equation the
transfer function of CSTR can be written as
Gps =
CA
Cin
=
1
1+KVQ
V
Q+KV s+ 1
(17)
The analog output of the concentration detector is trans-
mitted to a controller, which in turn sends a signal to the
injection regulating valve at input stream. Photodetector is
used to monitor the concentration of reactant A. The magic
photodetector is extremely fast and the response is linear
over a large concentration range. Unite of concentration CA
in terms of gmol/m3 now it converted into mV using the
measurement gain 2.6mVm3/gmol and transport lag td =
0.7sec. The regulating valve is especially designed so that inlet
concentration of the reactant A in gmol/m3 varies linearly
with the valve position. The regulating valve is thus first order
with a time constant of 0.2 sec and a steady state gain of
0.6 gmol/m3mV . Now complete system open loop block
diagram show in Fig. 4, in that Gm is the measurement transfer
function, Ga is the valve transfer function and Gps is the actual
transfer function. Transfer function of Gm , Ga and Gps are
given by
Fig. 4. Complete open loop block diagram of CSTR
Gps =
0.67
6.67s+ 1
(18)
Gm = 2.6e
−0.7s (19)
Ga =
0.6
0.2s+ 1
(20)
Complete open loop transfer function of the CSTR given as
Gp =
(0.67)(0.6)(2.6)e0.7s
(6.67s+ 1)(0.2s+ 1)
(21)
However, to use the empirical tuning relations, we need
to fit the data to a first order transfer function with dead
time. Thus at this stage, we probably would have obtained
the approximation of the CSTR by giving step signal to the
input injection regulating valve. From the Fig. 5, first order
approximate transfer function of complete CSTR system is
given by
Gp =
1.04e−0.85
7.1s+ 1
(22)
From equation (12), (13), (14) and (22) values of PID
controller parameters are
Kp = 6.87, τı = 7.52sec, τd = 0.4sec (23)
For the Ziegler-Nichols [6] and Cohen-Coon [7] methods
conventional PID parameters are evaluated using tuning rules
given in Table 1. The values of PID controller parameters
for IMC, Ziegler-Nichols and Cohen-Coon tuning rules are
given in Table 2. From Table 2, IMC tune PID controller have
proportional gain less and integral time more as compare to
other two methods. The unit step response for the close loop
control of CSTR shown in Fig. 6 using above setting
54 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010
Fig. 5. Open loop step response of CSTR Structure
Fig. 6. Closed loop response using different PID tuning methods
of PID controller parameters for given three methods. The
comparison between the Ziegler-Nichols, Cohen-Coon and
IMC PID tuning rules closed loop step response given in Table
3.
V. CONCLUSION
Internal model control methodology elementary notions
were reviewed, with particular relevance being given to the
conversion from the IMC structure to a conventional PID
controller configuration. From these we can able tune the
gains PID controller by using IMC parameters. From Fig. 6
and Table 3, IMC based PID controller unit step response
required less settling time to reach desired concentration and
give less overshoot in the response compared to other two
methods. We can conclude that the IMC based tuning of PID
controller outperformed the Ziegler-Nichols and Cohen-Coon
tuning methods.
REFERENCES
[1] K. Astrom and B. Wittenmark, Computer Controlled Systems: Theory
and Design, NJ: Englewood Ciffs, Prentice-Hall, 1984.
[2] K. J. Astrom and T. Hagglund, Automatic tuning of simple regulators
with specifications on phase and amplitude margins, Automatica, Vol.
20, 645-693, 1984.
[3] K. Moudgalya, Digital Control, John Wiley and Sons Ltd., England, 2007.
[4] P. B. Oliveira and J. B. Cunha, Teaching PID controller tuning through
internal model control, Control Eng. Practica, Vol.6, 155-165, 1998.
[5] M. Sharusuzzoha, M. Lee and J. Lee, IMC-PID controller tuning for
improved disturbance rejection of unstable time delay processes, Theories
and application of Chemical Engineering, Vol. 11, 2005.
[6] C.C. Yu, Autotuning of PID controllers, Springer, 1999.
[7] M. A. Garcia-Alvarado, I. I. R. Lopez and T. T. Ramos Tuning of
muultivariate PID controller based on characteristic matrix eigenvalues,
Lyapunov functions and robustness creteria, Chemical Engineering Sci-
ence, Vol. 60, 897-905, 2005.
Anil Markana received his B.E. degree in In-
strumentation and Control engineering from GEC
Gandhinagar, Gujarat University, India, in 2000,
M.Tech degree in Systems and Control engineering
from Indian Institute of Technology, Bombay, India,
in 2008, and currently pursuing Ph.D. degree in
Systems and Control engineering from the Indian
Institute of Technology, Bombay, Mumbai. He is
currently a lecturer in the School of Petroleum
Technology, Pandit Deendayal Petroleum University,
Gandhinagar, Gujarat, India. His current research
interests include advance Process Control, Optimal Control strategies like
Model Predictive Control, Linear Quadratic Gaussian Control, GPC, Minimum
Variance Control, Industrial Automation, Distributed Control Systems, and
Modelling etc.
Ankit Shah received his B.E. degree in Instru-
mentation and Control engineering from L. D. Col-
lege of Engineering, Ahmedabad, Gujarat, India, in
2002 and M.Tech degree in Systems and Control
engineering from Indian Institute of Technology,
Bombay, India, in 2010. He is currently a lecturer
in the Instrumentation and control dept. at Sardar
Vallabhbhai Inst. Of Tech., Vasad, Gujarat, India.
His current research interests include areas of the
hybrid system identification, advance process control
and digital control.
Nishant Parikh received his B.E. degree in Instru-
mentation and Control engineering from Shantilal
Shah College of Engineering and Technology, Bhav-
nagar, India, in 2002, M.Tech degree in Systems
and Control engineering from Indian Institute of
Technology, Bombay, India, in 2008, and currently
pursuing Ph.D. degree in Chemical engineering from
the Indian Institute of Technology, Bombay, Mum-
bai. He is currently a lecturer in the School of
Petroleum Technology, Pandit Deendayal Petroleum
University, Gandhinagar, Gujarat, India. His current
research interests include areas of system identification, advance process
control and digital control.


Performance analysis of IMC based PID controller
tuning on approximated process model

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Performance analysis of IMC based PID controller tuning on approximated process model

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