1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
115
TUNING OF AN I-PD CONTROLLER USED WITH A HIGHLY
OSCILLATING SECOND-ORDER PROCESS
Galal A. Hassaan
Emeritus Professor, Department of Mechanical Design & Production,
Faculty of Engineering, Cairo University, Giza, EGYPT
ABSTRACT
High oscillation in industrial processes is something undesired and controller tuning has to
solve this problems. I-PD is a controller type of the PID-family which is suggested to overcome this
problem with improved performance regarding the spike characteristics associated with certain types
of controllers. This research work has proven that using the I-PD is capable of solving the dynamic
problems of highly oscillating processes.
A second order process of 85.45 % maximum overshoot and 8 seconds settling time is
controlled using an I-PD controller (through simulation). The controller is tuned by minimizing the
sum of square of error (ISE) of the control system using MATLAB. The MATLAB optimization
toolbox is used assuming that the tuning problem is an unconstrained one. The result was cancelling
completely the 85.45 % overshoot and decreasing the settling time from 8 seconds to only 1.46
seconds without any undershoot. The performance of the control system using an I-PD controller
using the present tuning technique is compared with that using the ITAE standard forms tuning
technique.
Keywords: Highly oscillating processes – I-PD controller - Optimal controller tuning – Improving
control system performance.
1. INTRODUCTION
Highly oscillating response is present in a number of industrial processes incorporating low
damping levels. Conventionally, the PID controller is used and tuned for better performance of the
control system. The I-PD controller is one of the next generation of PID controllers where research
and application is required to investigate its effectiveness compared with PID controllers.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 5, May (2014), pp. 115-121
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
116
Katsumata (1994) described a method for the auto-tuning of an I-PD controller for processes
with long dead time using neural networks [1]. Seraji (1998) introduced a class of simple nonlinear
PID-type controllers comprising a sector-bounded nonlinear gain in cascade with a linear fixed-gain
P, PD, PI or PID controller [2]. Lelic (1999) extracted the essence of the most recent development of
PID control during the 1990’s based on a survey of 333 papers published in various journals [3].
Hamdan and Gao (2000) developed a modified PID (MPID) controller to control and minimize the
hysteresis effect in pneumatic proportional valves. The modified controller showed better command
following and disturbance rejection qualities than other types [4]. El-Sousi (2001) presented the
analysis, design and simulation of 1DOF and 2 DOF controllers for the indirect field orientated
control of an induction motor drive system [5]. Tagami and Ikeda (2002) proposed a design method
of robust I-PD controllers based on a genetic algorithm [6]. Araki and Taguchi (2003) surveyed the
important results about two degree of freedom PID controllers including equivalent transformations,
the effect of 2DOF structure and relation to the preceded derivative PID and the I-PD controllers [7].
Astrom and Hagglund (2004) presented a design method used to maximize the integral gain subject
to a robust constraint giving the best reduction of load disturbance. They revised tuning of PID
controllers in the spirit of Ziegler and Nichols technique [8]. Su, Sun and Duan (2005) proposed an
enhanced nonlinear PID controller with improved performance than the conventional linear fixed-
gain PID controller. They incorporated a sector-bounded nonlinear gain in cascade with the
conventional PID controller [9]. Li, Ang and Chong (2006) presented remedies for problems
involving the integral and derivative terms of the PID controller. They studied the PI-D and I-PD
control structures [10]. Arvanitis, Pasgianos and Kalogeropoulos (2007) investigated the control of
unstable second order plus dead-time process using PID-type controllers. They proposed tuning rules
based on the satisfaction of gain and phase margin specifications [11]. Madady (2008) proposed a
PID type with iterative learning control update law to control discrete-time SISO linear time-
invariant systems performing repetitive tasks. He proposed an optimal design method to determine
the PID parameters [12]. Coelho (2009) proposed a tuning method to determine the parameters of
PID control for an automatic regulator voltage system using chaotic optimization approach based on
Lozi map [13].
Khare (2010) developed an internal model mode based PID controller to control the
temperature of outlet fluid of the heat exchanger system. His controller demonstrated 84 %
improvement in overshoot and 44 % improvement in settling time compared to the classical
controller [14]. Ntogramatzidis and Ferrante (2011) introduced a range of techniques for the exact
design of PID controllers for feedback control problems involving requirements on the steady-state
performance and standard frequency domain specifications. The control parameters had to be
calculated on-line meaning that their techniques appear convenient with adaptive and self-tuning
control strategies [15]. Shiota and Ohmari (2012) proposed an adaptive I-PD controller using
augmented error method for SISO systems [16]. Prasad, Varghese and Malakrishnan (2012) designed
I-PD controller and optimized its parameters using particle swarm intelligence for a first order lag
integrating plus time delayed model [17]. Rajinikanth and Latha (2012) proposed a method to tune
the I-PD controller for the time-delayed unstable process using bacterial foraging optimization
algorithm [18]. Agalya and Nagaraj (2013) studied using a nonlinear feedback controller to control
continuous stirred tank reactors. They used PID and I-PD controller structures with integral square
error (ISE) criterion to tune the controllers [19]. Shiota and Ohmori (2013) proposed an adaptive I-
PD control scheme with variable reference model as one of the control schemes which overcome
model uncertainty. They showed the validity of their scheme through numerical simulation [20].
Hassaan (2014) presented a simple tuning approach to tune PID controllers used with both
underdamped and overdamped equivalent processes based on the integral square error (ISE) criterion
[21,22].

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
117
1. ANALYSIS
Process:
The process is a second order process having the parameters:
Natural frequency: ωn = 10 rad/s
Damping ratio: ζ = 0.05
The process has the transfer function:
Mp(s) = ωn
2
/ (s2
+ 2ζωn s + ωn
2
) (1)
The time response of this process to a unit step input is shown in Fig.1 as generated by MATLAB:
Fig.1 Step response of the uncontrolled process.
The severity of the process oscillations is measured by its maximum percentage overshoot. It
has a maximum overshoot of 85.4 % and an 6 seconds settling time.
Controller:
The controller used in this study is an integral (I) - proportional derivative (PD) controller. In
this controller, the integral part acts only on the error e of the closed loop control system
incorporating the controller and the controlled process. The proportional and derivative parts act on
the system output y. By this it is possible to get red of the kick following a reference input change
(set-point kick) as quoted by Prasad, Varghese and Balakrishnan [17]. The block diagram of the
closed-loop control system incorporating the I-PD controller is shown in Fig.2 [17].
controller
Fig.2 I-PD controller.

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
118
The controller model is function of the control system three variables:
- Control system error, E(s).
- Controller output, U(s).
- Control system output, Y(s).
Using the block diagram of Fig.2, the 3 variables are related through the model:
U(s) = (Ki/s)R(s) – (Kpc + Kds + Ki/s)Y(s) (2)
Where:Kpc = Proportional gain
Ki = Integral gain
Kd = Derivative gain
i.e. the controller has 3 parameters to be identified to control the process and produce a satisfactory
performance.
Control System Transfer Function:
U(s) is related to Y(s) through the process transfer function, Gp(s). That is:
Gp(s) = Y(s) / U(s) (3)
Combining Eqs.2 and 3 gives the transfer function of the closed-loop system as:
M(s) = b0 / {a0s3
+ a1s2
+ a2s + a3} (4)
where:
b0 = ωn
2
Ki , a0 = 1
a1= 2ζωn + ωn
2
Kd , a2 = ωn
2
(1 + Kpc)
a3 = ωn
2
Ki
System Step Response:
A unit step response is generated by MATLAB using the numerator and deniminator of Eq. 3
providing the system response c(t) as function of time [23].
2. CONTROLLER TUNING
The sum of square of error (ISE) is used an objective function, F of the optimization process.
Thus:
F = ∫ [c(t) – css]2
dt (5)
where css = steady state response of the system.
The performance of the control system is judged using two time-based specifications:
(a) Maximum percentage overshoot, OSmax
(b) Settling time, Ts
Tuning Results:
The MATLAB command "fminunc" is used to minimize the optimization objective function
given by Eq.5 without any parameters ot functional constraints [24]. The results are as follows:
Controller parameters:

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
119
Kpc = 1.7523
Ki = 5.3314
Kd = 0.1113
The time response of the closed-loop control system to a unit step input is shown in Fig.3.
Fig.3 Step response of the I-PD controlled second order process.
Characteristics of the control system using the tuned I-PD controller:
- Maximum percentage overshoot: 0 %
- Maximum percentage undershoot: 0 %
- Settling time: 1.46 s
3. COMPARISON WITH STANDARD FORMS TUNING
The control system in terms of its transfer function is a fourth order one. The optimal
characteristic equation of such a system with a second-order numerator is given using an ITAE
critertion by [25]:
s3
+ 1.75ωo s2
+ 2.15ωo
2
s + ωo
3
(6)
Comparing Eq.6 with the corresponding one in Eq.4 we get 3 equations in ωo, Kpc, Ki and Kd
i.e. 4 unknowns and 3 equations. To be able to get the controller parameters using this tuning
technique, one of the parameters has to be assumed. It was reasonable from the equations to assign
Ki (it was taken as 5.3314 as obtained in the present tuning technique using the ISE criterion). The
rest of the controller parameters were calculated as:
Kpc = 0.4136
Ki = 5.3314
Kd = 0.1319
The time response of the control system using this standard forms tuning technique is shown in
Fig.4:

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
120
Fig.4 Step response of the I-PD controlled second order process using the ITAE standard forms.
Characteristics of the control system using the standard forms tuning technique:
- Maximum percentage overshoot: 1.98 %
- Maximum percentage undershoot: 2.35 %
- Settling time: 0.44 s
4. CONCLUSIONS
- It is possible to suppress completely the higher oscillations in processes through using the I-PD
controller.
- It was possible to overcome the set-point kick problem associated with the standard PID.
- Through using an I-PD controller it was possible reduce the settling time from about 8 seconds
to about 1.46 seconds indicating the fast settlement of the controlled process.
- The maximum and minimum overshoot were cancelled completely .
- Tuning the controller using standard forms produced a time response of the closed loop system
having more overshoot and undershoot (1.98 % and 2.35 % respectively).
- However, the settling time was better in the standard forms technique than the present
technique based on ISE error criterion.
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Electronics, Vol.2, pp.1112-1118.

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME
121
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International Journal of Mechanical Engineering & Technology (IJMET), Volume 5, Issue 3,
2014, pp. 91 - 103, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.