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A comparative study on Tuning of PID controller

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 A comparative study on Tuning of PID controller
Munna Kumar*, Awadhesh K. Tiwari, R.S.Singh
Department of Chemical Engineering and Technology, IIT BHU, Varanasi 221005, UP, India
*munna.kumar.che11@itbhu.ac.in
(Research Program: Process Dynamics and control)
Abstract ID: PC 011

ABSTRACT
A comparative study was done for the selection of better design method of “Proportional-
Integral -Derivative (PID)” controller using various tuning techniques for approximated
FOPDT model. The control parameters of PID controller were calculated for a hypothetical
model using Ziegler- Nichols (Z-N), ITAE, modified Z-N and Tyreus-Luyben techniques and
results were compared for servo and regulator problems. Simulation results shows that the
ITAE method gives non oscillatory response for set point changes but an oscillatory for load
changes. The modified Ziegler- Nichols is satisfactory for load as well as set point changes.

Keywords: PID, pH control, FOPTD, C-H-R, ITAE, Tyreus-Luyben, Ziegler-Nichols.

1. Introduction
Most of the industrial loops use PID controllers till today. These types of controllers are
popular because of their ease in operation, robust behaviour, and easy maintenance. Even
after the invention of advanced process control strategies, predictive controllers, adaptive,
fuzzy controller, neural network etc., use of a PID controller dominates process industries.
A recent literature of Kano and Ogawa, 2010 shows that the ratio of application of
different types of controller, e.g., PI control, conventional advance control and model
predictive control is about 100:10:1. The accuracy and performance of these controllers are
greatly dependent on the method of tuning controller parameters, namely, KC, IJI, and IJD.
In the simple PID controller, the Ziegler-Nichols tuning procedure works quite well,
but this method is laborious and time consuming, particularly for processes with large time
constants or delays (Ziegler et al., 1942). There are many tuning rules to determine the three
tuning parameters systematically. However, poorly tuned PID controllers are often found in
industry. One of the keys to overcome this is that the tuning rules should be simple and
applicable to a wide range of processes (skogestad et al., 2003). The methods of Luyben et
al., 1996, Tyrus and Luyben, 1992 along with the most accepted PID tuning rule given by
Ziegler and Nichols (Z–N), 1942 use the frequency response information in tuning the PID
controller. These simple and attractive methods use only the critical frequency point of the
process for tuning the PID controller.
The Cohen-Coon method (Cohen & Coon, 1953), methods based on gain margin and phase
margin specifications (Ho et al., 1995), methods based on optimization of integral error
criteria (Visioli et al., 2001), an analytical tuning method which is based on finding the
parameters of overall transfer function in some transformed domain to have desired set-point
response (Chidambaram & Sree, 2003), method based on IMC and percentage overshoot
specification (Ali & Majhi, 2009). The internal model control methods (Rivera et al., 1986),
(Shamsuzzoha & Lee, 2007) and direct synthesis method (Chen & Seborg, 2002). The SIMC
method by Skogestad et al., 2003 to tune the PID controller is revisited, and a new method
(K-SIMC) is proposed. The proposed K-SIMC method includes modifications of model
reduction techniques and suggestions of new tuning rules and set point filters (Lee at al.,
2014) are the PID controller design methods which are based on achieving the desired closed-
loop response.
The conventional tuning methods which works based on fixed parameters will result in
lesser performance when system necessitates. Here we have discussed about the three tuning
techniques for open loop system, Ziegler-Nichols (ZN), Modified Ziegler Nichols (MZN) and
Chien Hrones Reswich (C-H-R), for designing the PID controllers. From the above specified
tuning methods, the proportional band, integral time and derivative time can be calculated.
By using those values one can determine the Proportional constant (Kc), Integral constant
(Ki) and Derivative constant (Kd) (Krishnan et al., 2014). In this paper we have studied the
application of these tuning methods to pH control in neutralization processes. The pH
neutralization process shows a strong nonlinear behaviour and time-varying nonlinear
characteristics resulted from the variation of the feed components or total ion concentrations.
The pH neutralization process was approximated to first-order plus time delay model (FOPDT) and
various tuning methods were applied to control the pH of the process and simulated results were
obtained with the help of MATLAB/SIMULINK and compared.
2. Comparison of tuning formulas
Different PID design or tuning techniques found in the literature for a certain process via the
proposed criteria. However, it is possible that a specific method might be good for a specific
plant model, it is difficult to give conclusions on which method is the best (in fact no best at
all). What we can conclude is that some methods show better performance in disturbance
rejection and/or robustness than other methods.
Here we compare the various PID tuning technique under the following conditions:
௞
(1) The process model is first-order with dead time (FOPDT Gp(s) = ఛ௦ାଵ ݁ ିఛ೏ ௦ )
ଵ
In this study a parallel form of PID controller Gc(s) = ‫(ܿܭ‬1+ ൅ ߬஽ ‫ )ݏ‬used to control the
ఛ ಺
FOPDT model and the parameters of controller were obtained by different tuning techniques
in open loop as well as closed loop methods and the results were compared.
The following PID tuning formulas are considered:
(a) The Ziegler-Nichols (ZN) controller settings (Ziegler and Nichols, 1942) are
pseudo-standards in the control field. There are two versions of Z–N method. One depends on
the reaction curve, and the other, the ultimate gain Ku and the ultimate period Pu. One
quarter decay ratio has considered as design criteria for this method.
(b) Tyreus-Luyben method procedure is quite similar to the Ziegler-Nichols method
but gives more conservative settings (higher closed loop damping coefficient) and is more
suitable for chemical process control applications. Having the ultimate gain and frequency
(Ku and Pu) and using table 3.3, the controller parameters can be obtained.
(c) ITAE criteria reduce errors that persist for long period of time. In general the
ITAE is preferred criterion because it usually results in the most conservative controller
settings. Controller tuning relations for ITAE performance index are shown in table 3.4.
These relations were developed for FOPDT process and the parallel form of the PID
controller.
 (d) In Cohen-Coon method the process reaction curve is obtained first, by an open
loop test and then the process dynamics is approximated by a FOPDT model. The PID design
parameters can be calculated as shown in table 2.
(e) The method proposed by Chien, Hrones and Reswich [Ziegler et al., 1942] is a
modification of open loop Ziegler and Nichols method. They proposed to use “quickest
response without overshoot” or “quickest response with 20% overshoot” as design criterion.
They also made the important observation that tuning for set point responses and load
disturbance responses are different. The PID design parameters is shown in table 3 for 0%
overshoot as well as 20% overshoot.
Design relations:
Y=A(Ԧ/߬)B, where Y=KKc for proportional mode, ߬/߬I for integral mode and ߬/߬D for
derivative mode.
b
For set point changes the design relation for integral mode is ߬/߬I=A+B(ԧ/߬).

Table 1 Tuning relationships for different tuning techniques (Seborg 2003)

Techniques Kc ߬I ߬D
ZN Ku/1.7 Pu/2 Pu/8
TL Ku/2.2 2.2Pu Pu/6.3
Modified ZN 0.2Ku Pu/2 Pu/3

Table 2 Tuning relationships for ITAE (Seborg 2003)

Type of input Mode A B
Disturbance P 1.357 -0.947
I 0.842 -0.738
D 0.381 -0.995
Set point P 0.965 -0.850
I 0.796b -0.1465b
D 0.308 0.929

Table 3. Tuning relationships for different tuning techniques (Seborg et al., 2003)
Methods kc IJI IJD
ଵǤଶ த
Ziegler-Nichols ௄೘
.ௗ 2d 0.5d

Cohen-Coon ଵ த ସ த೏ ͵ʹ ൅ ͸ɒௗ Ȁɒ Ͷ
( + ) ɒௗ ɒௗ
௄ த೏ ଷ ସத ͳ͵ ൅ ͺɒௗ Ȁɒ ͳͳ ൅ ʹɒௗ Ȁɒ
0% overshoot 20% overshoot

C-H-R kc IJI IJD kc IJI IJD
଴Ǥ଺ த೘ ଴Ǥଽହ த೘
Ǥ  ɒ௠ 0.5d Ǥ ͳǤͶɒ௠ 0.4d
௄೘ ௗ ௄೘ ௗ

3. Results and Discussion

To illustrate performance of these tuning techniques we applied these methods to control pH
system, approximated as FOPTD model (Skogested et al., 1996) and simulated results were
compared to both servo problem as well as regulatory problem.
Example (first order plus time delay process).
Consider a first order plus time delay process (FOPDT) of eq. 1 shown below and PID
parameters were obtained by various tuning methods given in table 4.
௞

G(s) = e-ԧsGd(s) =  e-ԧs 1
ఛ௦     
Here k = 1500, kd = 1000,  = 5 min. & ԧ = 5 sec. (0.083 min.)

Table 4. PID controller parameters for servo and regulatory problem using various techniques
Methods Servo problem Regulatory problem
kc I (min.) d (min) kc I (min.) d (min)
Ziegler- Nichols 0.0381 0.1649 0.04122 0.05715 0.1688 0.0422
Modified 0.0127 0.1649 0.1099 0.039 0.1649 0.1099
Ziegler- Nichols
ITAE 0.0209 6.3006 0.0346 0.06578 0.2884 0.0323
Tyreus-Luyben 0.0286 0.72556 0.0523 0.0428 0.7255 0.0523

Fig. 1 Responses of servo problem Fig . 2 responses of regulatory problem

From the above response graph we can notice that for hypothetical steady state values of the
process model the control of pH process using PID controller for set point changes were more
satisfactory than load changes. From the result we can predict that the ITAE method shows a
less oscillatory and give quick response compared to another tuned methods. The Tyreus-
Luyben method shows less overshoot and response quickly than the modified ZN and ZN
methods. For load changes ZN and modified ZN shows less overshoot and settles down
quickly.

4. Conclusions
In the present study the tuning of PID controller were done using various techniques and the
result were compared for set point change and load change. For hypothetical steady state
values of the process model the control of pH process using PID controller for set point
changes were more satisfactory than load changes. From the result we can predict that the
 ITAE method shows a less oscillatory and give quick response compared to another tuned
methods. The Tyreus-Luyben method shows less overshoot and response quickly than the
modified ZN and ZN methods. For load changes ZN and modified ZN shows less overshoot
and settles down quickly. An experimental studies of pH neutralisation processes are required
to validate these tuning techniques.
References
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specification to controller set point change. ISA Transactions, 48, pp.10-15, 2009.
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