Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only one closed-loop step response experiment using a proportional controller (P-controller). To highlight the potential of this method, simulation results have been compared with those obtained with the IMC method as well as other pertinent techniques. Various case studies have also been considered. The comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac- tional controller are directly obtained from the setpoint closed-loop response data without the need of any model of the plant to be controlled.

1. ISA Transaction s 64 (2016) 247–257
Closed-loop step response for tuning
PID-fractional-order-filter controllers
Karima Amoura a
, Rachid Mansouri a,n
, Maâmar Bettayeb b,c
, Ubaid M. Al-Saggaf c
a
L2CSP Laboratory, Mouloud Mammeri University, Tizi Ouzou, Algeria
b
Electrical & Computer Engineering Department, Universit y of Sharjah, United Arab Emirates
c
Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Saudi Arabia
a r t i c l e i n f o
Article history:
Received 7 June 2015
Received in revised form
16 March 2016
Accepted 19 April 2016
Available online 6 May 2016
This paper was recommen ded for publica-
tion by Y. Chen
Keywords:
Fractional order controller
PID controller
Internal model control
Robust control
a b s t r a c t
Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an
empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-
PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has
been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time
delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those
obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only
one closed-loop step response experiment using a proportional controller (P-controller). To highlight the
potential of this method, simulation results have been compared with those obtained with the IMC
method as well as other pertinent techniques. Various case studies have also been considered. The
comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it
might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac-
tional controller are directly obtained from the setpoint closed-loop response data without the need of
any model of the plant to be controlled.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Recently, much interest is devoted to FOPID (Fractional-Order
Proportional-Integral-Derivative) controllers. The FOPID has been
introduced by Podlubny in 1999 [13] and, in the same paper, a
better closed-loop response of this type of controller than the
classical PID was demonstrated when used to control fractional
order systems. Indeed, they provide more flexibility in the con-
troller design. However, the tuning of this kind of controller can be
much more complex because they have five parameters to be
tuned [9,21]. Several research activities are devoted to develop
new effective tuning techniques for non-integer order controllers
by an extension of the classical control theory. In [8], the authors
propose a novel adaptive genetic algorithm (AGA) for the multi-
objective optimization design of a fractional PID controller and
apply it to the control of an active magnetic bearing (AMB) system.
In [11], an auto-tuning method of fractional order controllers is
presented and the experimental platform Basic Process Rig 38-100
Feedback Unit has been used to test the fractional order controllers
n
Correspondin g author.
E-mail addresses: amoura.karima@yah oo.fr (K. Amoura),
rachid_mansouri_ummt o@yahoo.f r (R. Mansouri),
maamar@sharjah.ac.ae (M. Bettayeb), usaggaf@kau.edu.sa (U.M. Al-Saggaf).
designed. Several other tuning techniques have been developed.
Among them, the most well known are the empirical Ziegler–
Nichols tuning rules. Recently, these methods (Ziegler–Nichols-
type rules) are generalized for tuning fractional PID controllers,
see [20] and they have been improved in the course of time and
reported in several papers [1,2,19]. The Ziegler–Nichols method
has some advantages and several disadvantages. Indeed, it can
venture into unstable regions while testing the P-controller caus-
ing the system to be out of control, may not work well on all
processes and it is known that the recommended settings are
quite aggressive for lag-dominant processes [4,17]. Therefore,
Shamsuzzoha and Skogestad presented an alternative empirical
tuning method of an unidentified process [15] for tuning classical
PID parameters. This method, based on closed loop experiment
with proportional only control, is similar to the classical Ziegler–
Nichols experiment, but the process is not forced to its stability
limit. This method works well on a wide range of processes. It
works also for delay-dominant processes because it requires much
more information about the process than the Ziegler–Nichols
method [16]. Note that the majority of methods proposed for
tuning fractional controllers parameters are analytical methods
based on a plant model. In this paper we propose to use an
empirical method for tuning fractional controller parameters to
control an unidentified process which is the setpoint overshoot
http://dx.doi.org/10.1016/j.isatra.2016.04.017
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: w w w .elsevier.com/locate/isatrans

2. 248 K. Amoura et al. / ISA Transactions 64 (2016) 247–257
c
c
c
c
τ
1
method. In this paper the objective is the generalization of the
idea of Shamsuzzoha and Skogestad proposed in [15] to the frac-
tional controllers. Some changes were made to this method in
order to adapt it for tuning the new kind of fractional PID con-
with 0:39oλo 1
3
Ts ð5%Þ � πcosðπ-
-1
Þτλ þ1
ð7Þ
trollers which is the PID-Fractional-Order-Filter Controllers (FOF-
PID) proposed by Bettayeb and Mansouri in [5,6]. The PID-
Fractional-Order-Filter Controller (FOF-PID) has a very interesting
with 0:44oλo 1
ðλþ 1Þ
structure because it can be decomposed in two transfer functions;
an integer PID controller and a fractional filter [6]. The main issue
of the tuning method presented here is to derive correlation
between the setpoint response data and the PID-FOF-controller
settings calculated with the internal model control method (IMC)
[14].
2. Preliminary
2.1. Bode's ideal transfer function
The ideal shape of the open-loop transfer function suggested by
Bode [7] has the form,
1
So, when the performance of the closed-loop is specified in the
time domain, the fractional order λ can be deduced from the
overshoot Mp and the closed-loop constant time τc can be deduced
from the settling time Ts.
Remark 1. The objective of our method is to obtain a system with
the same performances as the performances of the fractional order
model (reference model) and not necessarily all the performances
(in time domain and frequency domain) of the integer order
model used to calculate the fractional order model, because the
integer order model is not exactly equal to the fractional model.
2.2. Internal model control
A more comprehensive model-based design method, internal
LðsÞ ¼
c sλ þ 1
ð1Þ
model control (IMC), was developed by Morari and coworkers [12].
The IMC method, as the direct synthesis method usually used in
where ðλþ1Þ is the fractional order, λ is a real, and 0 oλo 1. The
amplitude of L(s) is a straight line of constant slope
- 20ðλþ1Þ dB/dec and its phase is a horizontal line at -ðλþ1Þπ=2
rad. The phase does not depend on the value of the gain but only
on the non-integer order. Thus it exhibits important properties
such as infinite gain margin and constant phase margin. The unit
feedback system with Bode's ideal transfer function inserted in the
forward path is then
1
the conventional feedback control, is based on assumed process
models and leads to analytical expressions for the controller set-
tings. The IMC approach has the advantage that it allows model
uncertainty and tradeoffs between performance and robustness to
be considered in a more systematic way. Furthermore, it guaran-
tees internal stability of the closed-loop system. Indeed, the
closed-loop system is stable if the IMC-controller CIMC(s) and the
system to be controlled G(s) are both stable.
The IMC paradigm is based on the simplified block diagram
f ðsÞ ¼
þ
τ sλ þ 1
ð2Þ shown in Fig. 1.
The block diagram for the conventional feedback control is
This choice of L(s) as an open-loop transfer function gives a closed-
loop system robust to process gain variations and the step
response exhibits iso-damping property (the overshoot depends
given in Fig. 2. The two block diagrams are identical if controllers C
(s) and CIMC(s) satisfy the relation
C ðsÞ
only on λ). It is this feature which is often sought in the fractional CðsÞ ¼ IMC
8Þ
controllers design. In this work, f(s) is used as a reference model to
tune the controller parameters. It has two adjustable parameters,
the closed-loop time constant (τc) and the real λ which intervenes
in the fractional order. When the performance of the closed-loop is
specified in the frequency domain, these parameters are deduced
from the gain crossover frequency ωc and the phase margin φm
[14,15].
π -φm
1-CIMC ðsÞGm ðsÞ
ð
λ ¼
π=2
- 1 ð3Þ
τ
1
c ¼
ωλ þ 1
ð4Þ
because the step response of f(s) (when 0 oλo 1) is similar to that
of an underdamped second-order system for which the damping
ratio is (0 o ζo 1). Some other useful formulae characterizing the
time response of f(s) are given in [3].
1. The overshoot Mp of the step response is
ymax -yð1Þ
Fig. 1. Internal model control structure.
Mp ¼
yð1Þ ¼ 0:8λðλþ 0:25Þ ð5Þ
with 0oλo 1
2. The settling time Ts (for 2% and 5% criteria) is
4
Ts ð2%Þ �
cos π -
π
ðλþ 1Þ
- 1
τλ þ 1
ð6Þ
Fig. 2. Conventional feedback control.

3. K. Amoura et al. / ISA Transactions 64 (2016) 247–257 3
m
m
m ðsÞ
ð þ c Þ
þ
1
i
IMC ð Þ¼
f
-
þ
¼
1
Thus, any IMC-controller is equivalent to a standard feedback con- where τf ¼ τc
, Kc ¼ T
and τi ¼ T, CðsÞ is then a usual PI-controller
θ Kθ
troller, and vice versa. The IMC-controller is designed in two steps [5]:
• Step 1: The process model Gm(s) is factored as
Gm ðsÞ ¼ G þ
ðsÞG -
ðsÞ ð9Þ
cascaded with the fractional filter 1
.
1 þ τf sλ
2.3.2. First-order padé approximation of the e - θs term
When the time delay term is approximated by a first-order Pade
0
approximation
m m
where G þ
ðsÞ contains any time delays and right-half plane
zeros. Gþ
ðsÞ must have a steady-state gain equal to one.
Step 2: The controller is specified as
1
θ
1 s
e- θs 2
θ
1 s
2
ð18Þ
CIM C ðsÞ ¼
G
-
f ðsÞ ð10Þ
the IMC controller becomes
where f(s) is a low pass filter with a steady-state gain of one. In θ
the integer case it typically has the form [14]:
1
ð1þ TsÞ 1þ
2
s
CIMC ðsÞ ¼
K 1 τ sλþ 1
ð19Þ
f ðsÞ ¼ r ð11Þ
ð1 þτcsÞ and the corresponding feedback controller is
τc is the desired closed-loop time constant. Parameter r is a θ
positive integer chosen so that the controller C(s) is realizable. ð1þ TsÞ 1þ
2
s
2.3. PID-fractional-order-filter controller design
CðsÞ ¼
θK τcsλ þ 1 þ
ð20Þ
s
In previous works [5,6], new structure of fractional order con-
This can also be written as
trollers has been proposed. The fractional property of the closed-
loop is not especially imposed by the controller structure but by
CðsÞ ¼ 1
1þτf sλ
Kc 1þ
1
τi s
þτds ð21Þ
the closed-loop reference model of Eq. (2). The resulting controller
is fractional but it has a very interesting structure for its imple- where τf ¼ 2τc
, Kc ¼θ
2 T þ θ
K θ , τi ¼
2 T þ θ Tθ
2 and τd ¼ 2 T θ. Once again, C
mentation. In general, it is an integer PID-controller cascaded with
simple fractional filter. It is called PID-Fractional-Order-Filter
Controller (PID-FOF-controller) and given by Eq. (12). Internal
model control paradigm is used to design the PID-FOF-controller.
(s) is an integer PID-controller cascaded with the fractional filter
1 þ τfsλ. One of the main advantages of the proposed method is that
it has no restriction on the class of process models. Tuning rules by
the IMC method for other more complicated integer and non-
integer process models are listed in tables in [5,6].
1
CðsÞ ¼ HðsÞ Kc 1þ
τ s
þτds ð12Þ
fract
|
io
ffl{
n
z
al
ffl}
fil t er |fflffl fflfflffl fflffl fflfflffl fflffl fflfflffl{zfflfflffl fflffl fflfflffl fflffl fflfflfflfflffl}
Kc is the gain, τi the integral time constant and τd the derivative
time constant of the PID-controller. H(s) is the fractional part of C
(s). Details of the PID-FOF-controller design are given in [5] for
different kinds of processes. The method proposed in this paper is
based only on first order with time delay model presented by
K e - θ s
3. Tuning based on closed-loop setpoint experiment
The tuning method used here is the setpoint overshoot method
which is a simple method for tuning the controller of an uni-
dentified process, developed by Shamsuzzoha et al. in [15,16] for
tuning classical PID parameters. Here the setpoint overshoot
method has been adapted to the PID-FOF-Controller, by making
some changes to the basic method. Indeed, in this work, we use
Gm ðsÞ ¼
þT s
ð13Þ IMC method instead of SIMC method, and contrary to what is
proposed in [15,16], we let two synthesis parameters available to
Two approximation methods are then employed to approximate
the dead time term e- θ s
. The first one is first order Taylor
expansion which leads to a PI-controller and the second one is the
first order Padé approximation which leads to a PID-controller.
2.3.1. First order Taylor expansion of the e- θs
term
In this case, the time delay term is approximated by
e - θs
¼ 1-θs ð14Þ
Substituting the time delay approximation in Gm(s) and using Eq. (2)
as reference model, the IMC controller given by Eq. (10), is equal to
1þTsÞ
adjust the overshoot and the settling time. The setpoint experi-
ment is based on simulations for a range of first-order with delay
processes in order to derive correlations to give PID-FOF-
Controller settings similar to those of the IMC tuning rule.
3.1. Principle of the proposed method
The setpoint overshoot method is based on the closed-loop
setpoint experiment. This experiment is based on simulations and
observations for a wide range of first order with time delay pro-
cesses. From the setpoint experiment, one can observe many
values, like time and magnitude of the first peak (overshoot),C s
ð
Kð1þτc sλ þ 1Þ
Thus, the corresponding feedback controller is
ð15Þ
steady state value, rise time, period of oscillations and settling
time. This method requires only steady state value, time and
magnitude of the overshoot and controller gain ðKc0 Þ. Thus the
CðsÞ ¼
ð1þ T sÞ ¼ 1þ T s ð16Þ goal is to derive correlation between the setpoint response data
Kðτc sλ þ 1 þθ sÞ
which can be written as
Kθ s 1
τc
þ
θ
sλ
and the controller settings given by Eqs. (17) and (21) Steps of this
method are summarized bellow:
1 1
• Switch the controller to P-only mode as shown in Fig. 3.
CðsÞ ¼
þτ sλ Kc 1þ
τis
ð17Þ • Adjust the value of the controller gain Kc0 to give an overshoot
between 0.10 and 0.60. Note that the controller gain to get
Integer PID Controller
•
2
1

4. 250 K. Amoura et al. / ISA Transactions 64 (2016) 247–257
1 1
1
1
p
Fig. 3. Feedback control structure with P-only mode.
Table 1
Values of Kc0 in terms of overshoot.
Fig. 4. Feedback control structure with PID-FOF-controller.
Fig. 5. Closed-loop step setpoint response with P-only control.
from delay-dominant process (T=θ ¼ 0:1) to lag-dominant (inte-
grating) T=θ ¼ 100, to cover a wide range of processes.
T=θ ¼ 0:1; 0:2; 0:4; 0:8; 1; 1:5; 2; 2:5; 3; 5; 7:5; 10; 20; 50; 100
ð22Þ
The experiment consists of switching the controller to P-only
mode for each of the 15 process models and setting the value of
the controller gain Kc0 that gives different overshoot (Mp), with
Mp ¼ 0:10; 0:20; 0:30; 0:40; 0:50 and 0.60. Therefore we have 6 set-
point responses for each process and four data for each setpoint
response which are: controller gain Kc0 used in experiment,
overshoot Mp ¼ ðymax -y ÞÞ=y , time (tp) of the first peak (time of
0.3 overshoot is about half of the ultimate controller gain nee-
ded in the Ziegler–Nichols closed-loop experiment and to get a
large overshoot, increase the controller gain Kc0 [16].
• Calculate the controller parameters (τf, Kc, τi) of the PI-FOF-
controller or (τf, Kc, τi and τd) of the PID-FOF-controller by
correlation between the setpoint response data and the con-
troller settings calculated with IMC tuning rules. Note that λ is a
parameter fixed as a function of the desired overshoot by using
Eq. (5).
• Insert the controller C(s) in the forward path as shown in Fig. 4
and adjust the response by varying the η value (η is a parameter
described in the next section).
As mentioned in Section 3, from the closed loop setpoint
response experiment, we obtain the following values (Fig. 5).
• Controller gain, Kc0.
overshoot) and relative steady state output change, b ¼ y1 =r.
Furthermore, for each process, we calculate analytically the para-
meters (τf, Kc and τi) of the PI-FOF-controller (Eq. (17)) or those (τf,
Kc, τi and τd) of the PID-FOF-controller (Eq. (21)). The values of Kc0
obtained in the experiment for different values of overshoot are
given in Table 1.
3.3. Correlation between results of experiment and IMC settings
Our objective is to seek a relationship between the controller
settings calculated using IMC method and the four data obtained
from the experiment. Thus, we plot KKc as a function of KKc0 as
shown in Fig. 6, where the values of Kc are calculated by Eqs. (17)
and (21) obtained using IMC method.
As shown in Fig. 6, the curve of KKc as a function of KKc0 can be
approximated by straight lines with different slopes A.
K Kc ¼ A K Kc0 ð23Þ
• Overshoot, Mp ¼ ðymax - y1ÞÞ=y .
• Time of the first peak (time of overshoot), (tp). The ratio Kc =Kc0 is the value of the slopes of lines which is inde-
pendent of the value of T
and only depends on the overshoot. Thus
• Relative steady state output change, b ¼ y =r1 . θ
if we plot the value of A as a function of the overshoot we obtain
where ymax is the peak output at time tp, Mp is the overshoot, y is
1
steady state output and r is the setpoint (step).
3.2. Closed-loop setpoint experiment
The objective of this paper is to provide a one step procedure in
closed-loop for fractional controller tuning similar to the Sham-
suzzoha and Skogestad and Ziegler–Nichols methods. For this
purpose, we considered 15 first-order with time delay models of
(Eq. (13)) where K ¼ θ ¼ 1 with 15 different values of the ratio T=θ,
the curve shown in Fig. 7. Table 2 lists the values of the slopes A as
function of the values of the overshoot Mp.
From Fig. 7, we can establish the relationship between A and
the overshoot, which can be approximated by a quadratic poly-
nomial of Eq. (24) obtained by using cftool function of MATLAB
curve fitting toolbox.
A¼2:267M2
-3:173Mpþ 2:008 ð24Þ
The setpoint overshoot method is an empirical technique to con-
trol unknown systems. Thus θ must be determined according to
the information of the step response. Fig. 8 shows the variation of
T=θ Mp ðλÞ
0.1 0.2 0.3 0.4 0.5 0.6
(0.25) (0.38) (0.50) (0.60) (0.68) (0.75)
0.1 0.10 0.20 0.30 0.40 0.50 0.60
0.2 0.107 0.201 0.309 0.410 0.511 0.611
0.4 0.177 0.293 0.405 0.515 0.626 0.735
0.8 0.376 0.544 0.700 0.850 0.998 1.142
1.0 0.482 0.675 0.855 1.026 1.196 1.363
1.5 0.752 1.008 1.245 1.472 1.692 1.911
2.0 1.031 1.345 1.637 1.915 2.193 2.460
2.5 1.314 1.684 2.030 2.363 2.690 3.014
3.0 1.600 2.026 2.423 2.810 3.190 3.560
5.0 2.750 3.405 4.010 4.601 5.183 5.770
7.5 4.192 5.130 6.000 6.845 7.683 8.520
10 5.640 6.860 8.000 9.090 10.180 11.260
20 11.450 13.810 15.980 18.100 20.200 22.250
50 28.870 34.630 39.930 45.050 50.200 55.210
100 57.900 69.300 79.900 90.100 100.200 110.200

5. K. Amoura et al. / ISA Transactions 64 (2016) 247–257 5
Fig. 6. KKc as a function with KKc0. Fig. 8. Variation of θ=tp as a function of the overshoot.
time delay, the value of θ=tp is greater than 0.3 and for processes
with a smaller time delay, the value of θ=tp is less than 0.3.
3.3.1. PI-FOF-controller parameters tuning
The IMC based controller design yields a PI-FOF-controller (Eq.
(17)) composed by a fractional filter cascaded with an integer PI-
controller where the parameters are
T
Kc ¼
Kθ
ð25Þ
τi ¼ T ð26Þ
The objective is to derive expressions of Kc, τi and fractional filter
of the PI-FOF-Controller from the data read from the closed-loop
step response.
Table 2
Fig. 7. Variation of A with overshoot.
• The controller gain Kc: From Eq. (23), we can calculate the gain Kc
as follows
Kc ¼ A Kc0 ð27Þ
where A is the value of the slopes of lines shown in Fig. 7.
Knowing the value of the overshoot, the value of A can be
Values of slopes A with the overshoot Mp.
Mp 0.1 0.2 0.3 0.4 0.5 0.6
A 1.726 1.4 43 1.254 1.113 1.0 0 0.91
θ=tp as a function of the overshoot for 15 different processes (with
15 different values of the ratio T=θ that cover a wide range of
processes).
In Fig. 8, we observe that for process with T=θ ¼ 0:1, θ=tp is
equal to 0.5 with all overshoots. For process with T=θ ¼ 100, θ=tp
varies between 0.25 for an overshoot equal to 0.1 and 0.32 for an
calculated using Eq. (24).
• The integral time constant τi: From Eqs. (25) and (26) we get τi
τi ¼ KKC θ ð28Þ
When the value of K Kc Eq. (23) is substituted in Eq. (28), we
obtain
τi ¼ A K Kc0 θ ð29Þ
Considering the closed-loop transfer function for Fig. 2, the
steady state value is
K Kc0
overshoot equal to 0.6. Thus for all processes (from delay-
dominant process (T=θ¼ 0:1) to lag-dominant (T=θ¼ 100)), θ=tp
b
1þ K Kc0
ð30Þ
varies between 0.25 and 0.5 (0:25rθ=tp r0:5), so we canwrite:
ηmintp rθrηmaxtp , where ηmin ¼ 0:25 and ηmax ¼ 0:5.
Then
b
Note that in Shamsuzzoha [15], the parameter η was set to two K Kc0 ¼ ð31Þ
intermediate values of the variation range ½ηmin ηmax], which are
0.43 for processes with smaller time delay and 0.305 for lag-
dominant process. In this paper, we propose to give the designer
the choice to set the parameter η according to the process to be
controlled, with η included in the range ½0:25 0:5]. Indeed from
Fig. 8, we can observe that for processes with a relatively large
1 -b
in Fig. 8, θ=tp is plotted as a function of the overshoot and we
observe that the θ=tp value varies between 0.25 and 0.5. So we
can write θ as follows:
θ ¼ η tp ð32Þ
¼

6. 252 K. Amoura et al. / ISA Transactions 64 (2016) 247–257
2
1 b
2
f
b
p
τd ¼
2
ηtp 1-
A
C
τ T 39
1
b
where η has a value in the range 0:25rη r0:5. From Eqs. (28),
(31) and (32) we can get the integral time constant τi as
From Eqs. (43) and (39), the integral time constant τi is given by
τi ¼ 1
KKc θ ð44Þ
b
τi ¼ A
-
ηtp ð33Þ
Substituting Eqs. (23) and (50) leads to
Thus, we managed to estimate the parameters Kc ; τi and θ based
only on the information of the step response (Mp,tp and b).
• The fractional filter parameters. In the first case, using IMC
method we obtain a PI-FOF-Controller where the fractional
τi ¼ 1
AKKc0 θ ð45Þ
Also, from Eq. (30), from the steady state of the closed-loop, we
have
filter is
bKKc0 ¼ ð46Þ
1 1-b
HðsÞ ¼ τc
ð34Þ
1þ
θ
sλ
τ c and λ are reference model parameters fixed by the designer
(Eq. (3)). θ is a parameter deducted from the experiment and it
is given by Eq. (32). Substituting the expression for θ in Eq. (32),
we obtain
1
As explained before, the time delay θ can be expressed as
follows
θ¼ η tp 0:25oηo 0:5 ð47Þ
From Eqs. (45) to (47), we can get the integral time constant τi
which is given by
HðsÞ ¼
þ
with
τ sλ ð35Þ
1
τi ¼
2
A
1
b ηtp ð48Þ
-
τ
τc
f ¼
ηtp
ð36Þ • The derivative time constant τd: The expression of the derivative
time arising from the use of the IMC tuning rule is
3.3.2. PID-FOF-controller parameters tuning
In order to find the expressions of the PID-FOF-Controller
τ
T θ
d ¼
2T þθ
¼
T
T
2
θ
þ 1
ð49Þ
parameters from the data read from the closed-loop step
response, we perform the same experiment done for the PI con-
By substituting the expression for T (Eq. (42)), τd becomes
troller case. The results are similar to those obtained with the first
experiment, but with different numerical values. Indeed the value
τd ¼
1
θ 1
2
1
KKc
ð50Þ
of the gain Kc calculated analytically by using IMC method chan-
ges. Thus, the value of slope A of lines changes, and the coefficients
From Eqs. (23) and (50), we obtain
of the equation which give relationship between the slope and
overshoot changes also, as follows:
τd ¼
1
θ 1
2
1
AKKc0
ð51Þ
A¼4:709M2
-6:531Mp þ 4:064 ð37Þ From Eqs. (46), (47) and (51), we can get the derivative time τd
Recall that the PID parameters of PID-FOF-controller (Eq. (21))
0
1
1
1 C
obtained with IMC method are
2Tþ θ
BB
@
1
A ð
b
52Þ
Kc ¼
Kθ
ð38Þ -
θ
i ¼ þ
2
ð Þ
• The fractional filter. The fractional filter PID-FOF-Controller is
1
Tθ HðsÞ ¼ 2τc
λ
ð53Þ
τd ¼
2T
ð40Þ
þ 1þ
θ
s
By substituting the expression given by Eq. (47), we obtain:
• The controller gain Kc0: The controller gain Kc can be obtained HðsÞ ¼
1
1
τ sλ ð54Þ
from Eq. (23) as follows:
Kc ¼ AKc0 ð41Þ
þ f
where
where A is given by Eq. (37).
• The integral time constant τi: From the analytical expression of
τ
1 2τc
f ¼
η tp
ð55Þ
the gain Kc obtained using IMC rule (Eq. (38)), we can get T as
function of KKc and θ.
τ c and λ are reference model parameters fixed by the designer.
Kc ¼
2T þθ
Kθ
) T¼
KKc θ-θ
2
ð42Þ
3.3.3. Final controller
As mentioned above, the last step of the proposed method is
the insertion of the fractional controller in the forward path, as
Adding θ=2 to both terms of Eq. (42), we obtain shown in Fig. 4, Section 3. Thus the final tuning formulas for the
θ KK c θ -θ θ proposed method “setpoint overshoot method” are summarized as
T þ
2
¼
2
þ
2
ð43Þ follows:
-
-
θ

7. K. Amoura et al. / ISA Transactions 64 (2016) 247–257 253
f
f
j j
p
¼ - ref ¼
• PI-FOF parameter tuning: The final controller with setpoint
overshoot method is
1 1
4.1.1. Effect of η and ρ
It is interesting to study the influence of parameters η and ρ on
the behavior of the controlled system. For this purpose, we plot
CðsÞ ¼
1 τ sλKc 1þ τis
ð56Þ the step response for different values of η and ρ. In the first case
we set the parameter ρ at 0.32 and we plot the step response for
where τf ¼ τc
, Kc ¼ A Kc0 , τi ¼ A b
ηtp with A ¼ 2:267M2
- different values of η. Controller parameters, obtained by using the
ηtp 1 - b p
3:173Mp þ 2:008 and 0:25rη r0:5:
Mp ; tp ; b: Data obtained from the setpoint experiment using a
P-controller with gain Kc0.
τc ; λ: reference model parameters chosen by the designer.
• PID-FOF parameter tuning: The final controller with setpoint
overshoot method is
1 1
IMC tuning rule and the setpoint overshoot method, are listed in
Table 3, and the step responses are illustrated in Fig. 9.
It can be seen that the overshoot depends on η. Indeed, for η
¼ 0:32 the step response obtained with the proposed method and
the step response obtained using the IMC rule are almost the
same. By increasing the parameter η ðη ¼ 0:5Þ the overshoot
decreases, and inversely, by decreasing the parameter η ðη¼ 0:25Þ
the oscillations becomes larger, showing that the overshoot is
inversely proportional to the parameter η.
CðsÞ ¼
1 τ sλKc 1þ τis þτds ð57Þ
Fig. 10 shows the step response of the closed-loop for different
values of ρ, with parameter η set at 0.32. This figure shows that the
parameter ρ has influence on the settling time. The values of
where τf ¼ 2τc
, Kc ¼ A Kc0 , τI ¼ 1
A b
ηtp , τd ¼ 1
ηtp 1- 1
ηtp 2 1 - b 2 A b
1 - b settling time are listed in Table 4. By increasing the value of ρ
with A ¼ 4:709M2
- 6:531Mp þ 4:064 and 0:25rη r0:5. (ρ ¼ 0:5) the settling time decreases, and inversely, by decreasing
the parameter ρ ðρ¼ 0:25Þ the response becomes slow, as shown
Remark 2. After applying this method to several numerical in Table 4. So we have two adjustable parameters. First set
examples, we noted that the parameter τf of the fractional filter
can improve the settling time of the closed-loop. However, this η¼ρ ¼ 0:3, then we adjust the parameter ρ to fix settling time and
parameter depends on the constant η, and PID parameters (τi and
τd) also depend on η. Therefore we propose to separate the con-
stant η in two. It will be kept as it is in the PID parameters, and we
call them ρ in the fractional filter. Obviously the choice of η and ρ
is always between 0.25 and 0.5. This will give more degrees of
freedom to the designer. More details about the influence of
parameters are given in Example 4.1 in the next section.
4. Tests and results
In order to evaluate the final controllers, three examples will be
next we adjust the value of η to obtain the desired overshoot.
This example demonstrates that, by adjusting the parameter η
and ρ, the proposed method can give better results than those
obtained with the IMC method. Several tests were made on several
models of first order systems with time delay, and the results of
these tests were compared with those obtained by IMC method
with good agreement. Therefore, we can say that, for such systems,
the proposed method offers accurate results as the IMC method.
Moreover, it might provide better results if the parameters η and ρ
are well adjusted.
Table 3
Tuning for process (Eq. (58)) (with IMC method: Kc ¼ 10, τi ¼ 5 and Mp ¼ 0:21).
used here to test the performance of the proposed closed-loop
tuning method. The first is a first-order process with time delay,
the second example is a high order process assumed unknown and
in the last example we use simulation results of an experimental
example presented in [11]. But before this, we study the influence
of parameters η and ρ on the behavior of the controlled system.
For this reason we plot the step response for different values of η
and ρ.
4.1. Example 1
Assume that the plant to be controlled is a first-order with time
delay model G(s) where
e- 0:5s
0.5 10.08 7.87 0.11
GðsÞ ¼
þ
5s
ð58Þ
The time delay is approximated by a first order Taylor expansion
(e - θs
1 θs). The reference model is: G 1
1 þ τc sλ þ 1 with λ ¼ 0:4
corresponding to the value of the overshoot Mp ¼ 0:2, τc is the
parameter that determines the dynamics of the desired closed-
loop system. If we want to speed up the response, τc must be
chosen less than θ and if we want a less aggressive response, τc
must be chosen greater than θ [15]. In other words, we can impose
the closed loop (specific dynamic) by choosing τc, and this is one of
the advantages of the proposed method [15,16]. In this example,
we choose τc greater than θ, ðτc ¼ 2Þ. For the experiment we can
choose different overshoots (about 0.1–0.6) for tuning the PI-
parameters. The objective is to obtain a closed-loop step response
with overshoot Mp ¼ 0:2 which corresponds to λ ¼ 0:4.
Fig. 9. Effect of the parameter η on the step response. (1: Proposed method
(η ¼ 0:25), 2: Proposed method (η ¼ 0:32), 3: IMC method, 4: Proposed method
(η ¼ 0:37), 5: Proposed method (η ¼ 0:50)).
þ
þ
1
η Proposed method
Kc τi MP
0.25 10.08 3.93 0.29
0.32 10.08 5.03 0.20
0.37 10.08 5.82 0.17

8. 254 K. Amoura et al. / ISA Transactions 64 (2016) 247–257
ð Þ¼
c
1
Fig. 10. Effect of the parameter ρ on the step response. (1: Proposed method
(ρ ¼ 0:5), 2: Proposed method (ρ ¼ 0:37), 3: IMC method, 4: Proposed method
(ρ ¼ 0:32), 5: Proposed method (ρ ¼ 0:25)).
Fig. 12. Step responses of the closed-loop system. The specification s are τc ¼ 10,
λ ¼ 0:4 and gain variation. (1: K ¼ 1:25, 2:K ¼ 1, 3: K ¼ 0:75).
Table 5
Overshoot values for different values of τc.
Table 4
Settling time for different values of ρ (with IMC method: tsð3%Þ ¼ 9:4 s).
ρ 0.25 0.32 0.37 0.5
t sð3%Þ 12.2 10.3 9.4 7.73
4.2. Example 2
The setpoint overshoot method works well on a wide range of
processes, and it does not need a model of the plant to be con-
trolled. In this example, the empirical setpoint overshoot method
will be compared with the IMC method, which is an efficient
analytical method. So let us now apply the proposed method to
the high-order plant transfer function assumed unknown [18].
1
G s
ðs þ1Þð0:2sþ1Þð0:04s þ1Þð0:008sþ1Þ
The desired closed-loop (reference model) is
1
ð59Þ
Gref ðsÞ ¼
1 τ sλþ 1
ð60Þ
Fig. 11. Step responses of the closed-loop system. The specification s are τc ¼ 5, λ ¼
0:4 and gain variation. (1: K ¼ 1:25, 2: K ¼ 1, 3: K ¼ 0:75).
4.1.2. Effect of gain variation
In order to study the robustness of the system given by Eq. (58),
we introduce a variation of 25% in the nominal plant gain K such
that K ¼ 0:75; 1; 1:25. The step responses are illustrated in Fig. 11,
The objective is to find a controller C(s) which allows for the closed
loop system an overshoot equal to 0.1 (which corresponds to
λ ¼ 0:25) with τc chosen equal to 1. The closed-loop experiment is
used with an overshoot of about 0.1, which gives the following
information: kc0 ¼ 3; Mp ¼ 0:10; tp ¼ 0:922 and b ¼0.75. Controllers
obtained with the IMC method are C1 ðsÞ and C2 ðsÞ and with the
proposed method are C3 ðsÞ and C4 ðsÞ where
1 1
with reference specification τc ¼ T ¼ 5 and λ ¼ 0:4 which corre- C1 ðsÞ ¼ τc 0:25 4:74 1þ
1:07s
ð61Þ
sponds to the overshoot Mp ¼ 0:20. The results highlighted in
Fig. 12 correspond to the step responses with reference specifica-
tion τc ¼ 2T ¼ 10 and λ ¼ 0:4.
1þ
0:224
s
C s
10:492 ð Þ¼
2τc 11
1:18s
þ 0:10s ð62Þ
As can be observed in Fig. 11, with τc ¼ 5, the overshoot of the
closed-loop output is not invariant to the gain variation, but in
Fig. 12, with τc ¼ 2T, the overshoot remains almost constant under
1
0:224
1
s0:25
1
gain variations, i.e, the iso-damping property is exhibited. The C3 ðsÞ ¼ τc 0:25 5:11 1þ
1:17s
ð63Þ
overshoot values for different values of τc are summarized in
Table 5 and it can be seen that, the more we increase the value of
1þ
0:230
s
τc, the more the overshoots becomes closer and closer, till they are
equal from τc 42T . To obtain a robust closed-loop system to gain
variation, we must choose τc sufficiently greater than T.
C4 ðsÞ ¼
1þ
10:32
2τc
s0:25
0:230
1
þ
1:33s
þ
0:12s ð64Þ
þ
þ
þ
τc K
0.75 1 1.25
T=2 0.19 0.22 0.25
T 0.20 0.22 0.23
2 T 0.20 0.20 0.21
3 T 0.21 0.21 0.21

9. K. Amoura et al. / ISA Transactions 64 (2016) 247–257 255
ð Þ¼
Table 7
Load disturbance performanc e (disturbance amplitude 0.5 at t ¼ 20 s).
Fig. 13. Closed-loop step responses with noise and disturbanc e responses with
C1 ðsÞ; C2 ðsÞ, C3 ðsÞ and C4 ðsÞ.
Table 6
Setpoint performance (desired closed-loop with Mp ¼ 0:10 and tsð3%Þ ¼ 6:6 s).
PID-FOF 0.13 6.4 0.10 7.4 Fig. 14. Step response for the system controlled by C3 ðsÞ, with different values of
the gain K ðK ¼ 1:5; 1:25; 1; 0:75; 0:5Þ, and τc ¼ 1.
It is to be noted that the controllers C1 ðsÞ and C2 ðsÞ are calculated
with the IMC method by using the approximated model0:2248
Gapp ðsÞ ¼ e -
. Eqs. (61)–(64) show that although the controllers
1 þ 1:067s
C3 ðsÞ and C4 ðsÞ are obtained by using the step response of the
model of large dimension assumed unknown and C1 ðsÞ and C2 ðsÞ
are obtained by using the approximated model (first order with
delay), the values of the parameters ðτf ; Kc ; τi Þ of C3 ðsÞ and C4 ðsÞ
obtained by the proposed method are close to those of C1 ðsÞ and
C2 ðsÞ obtained by the IMC method.
Fig. 13 illustrates these results and shows the step responses of
the closed-loop with the controllers C1ðsÞ;C2ðsÞ, C3ðsÞ and C4ðsÞ. To
evaluate the performance of the controllers, a disturbance of
amplitude 0.5 at t ¼ 20 s and a Gaussian noise with variance 1 are
added. The disturbance transfer function is arbitrarily chosen as
g s 1
1:07s þ 1
. Data on setpoint responses are summed up in Table 6
and data on load disturbance are summarized in Table 7. In this
and in the following tables, the settling time tsð3%Þ is reckoned
according to the 73% rule and the overshoot is Mp.
These results demonstrate that the proposed method gives
good results with acceptable disturbance rejection and is not
sensitive to noise. We can say that the results obtained with the
proposed method are similar to or better than those obtained with
the IMC method.
Fig. 14 shows the effect of the gain variation on the perfor-
mance of the closed-loop with τc ¼ 1. We observe from this figure,
that the overshoot changes if we vary the value of the gain.
However, if we choose τc sufficiently large (τc 42T ), for example
for τc ¼ 4, we find that the system is robust to gain variation as
shown in Fig. 15. Fig. 16 shows the effect of the T variation, where T
is a dominant (large) time constant of the plant (Eq. (59)).
As can be observed in Fig. 15, the overshoot of step responses is
almost constant for the variation considered from 0.5 to 1.5 ð-50
% to þ 50%Þ in the gain of the plant. Thus we can say that with the
proposed method, we get closed-loop systems robust to gain
variation and step responses exhibiting an iso-damping property
for system without time delay. This example has demonstrated
Fig. 15. Step response for the system controlled by C3 ðsÞ, with different values of
the gain K ðK ¼ 1:5; 1:25; 1; 0:75; 0:5Þ, and τc ¼ 4.
another advantage of the proposed method. Indeed, this method is
easily applicable to high order systems (real system), unlike other
methods which require approximation of the high order system
with a reduced order system, such as the IMC method. Further-
more, with this method, we have the possibility of tuning the
controller for a desired response just by adjusting the parameters
η and ρ in the range ½0:25 0:5].
4.3. Example 3
In [11], an experimental platform is used to test fractional order
controller. It consists a low pressure flowing water circuit which is
bench mounted and completely self-contained. According to
Monje et al. the liquid level system is modeled by a first order
IMC method Proposed method
Mp tsð3%Þ Mp tsð3%Þ
PI-FOF 0.25 23.16 0.24 23.16
PID-FOF 0.22 23 0.23 23.20
IMC method Proposed method
Mp tsð3%Þ Mp tsð3%Þ
PI-FOF 0.12 6.4 0.10 6.4

10. 256 K. Amoura et al. / ISA Transactions 64 (2016) 247–257
ð Þ¼
Fig. 16. Step response for the system controlled by C3 ðsÞ, with different values of T
ðT ¼ 1; 1:25; 1:5; 0:75; 0:25Þ, and τc ¼ 4. Fig. 17. Step response of closed-loop system with C2 ðsÞ (solid line) and the refer-
ence model (Gref(s)) (dashed line).
transfer function with time delay given by:
3:13e-50s
Gapp ðsÞ ¼
433:33s
1
ð65Þ
The design specifications required for the system are
• Gain crossover frequency, ωcg ¼ 0:008 rad=s.
• Phase margin, φm ¼ 601.
In [11], the author's objectives are frequency performance (phase
margin and crossover frequency), contrary to our method which
use temporal objectives (settling time and overshoot). For this
reason, we set as a goal to obtain a closed-loop step response
having the same temporal performance as that of the response
obtained in [11], i.e. Mp ¼ 0:12 and ts ð3%Þ ¼ 680 s. The reference
model obtained by using Eq. (5) and satisfying almost these
requirements is
1 Fig. 18. Closed-loop step response with disturbanc e: solid line shows proposed
Gref ðsÞ ¼
þ500s1:29 ð66Þ
method and dashed line shows the Monje et al. method.
Thus with our method the objective is the time specifications of
the model equation (66). Using the FMINCON function of Matlab
optimization toolbox [10], the PIλ
Dμ
controller obtained by Monje
et al. is
Table 8
Setpoint and load disturbance performance.
Setpoint Load disturbance
0:01 0:48 Mp tsð3%Þ MP tsð3%Þ
C1 ðsÞ ¼0:62þ
s0:90 þ 4:39s
ð67Þ
Monge et al. method 0.12 680 0.15 2650
To calculate the PI-FOF controller parameters ðτf ; Kc ; τi Þ using the
proposed method, we seek to achieve the same time specifica-
tions. Thus the PI-FOF-controller obtained using the proposed
method is
Proposed method 0.12 684 0.18 2360
established at ts ð3%Þ ¼ 680 s with Monje et al. method and at
ts ð3%Þ ¼ 684 s with the proposed method. Thus the overshoot and
1 1
settling time obtained are almost the same with both methods.
C2 ðsÞ ¼
1
2:74 1
þ 8:88s0:29 505:13 s ð68Þ
Fig. 19 shows the control laws obtained with each controller. We
The step responses of the closed-loop system with C2 ðsÞ and
reference model are illustrated in Fig. 17. As can be observed in
the step responses it is seen that temporal specifications of
overshoot and settling time are met. Thus we can say that the
controller has met the objectives of the control.
Fig. 18 shows the step response of the closed-loop system with
an PID-FOF-controller tuned with the closed-loop overshoot
method and the step response of the system controlled with
Monje et al. controller, with an added disturbance of magnitude
0.3 at t ¼ 2000s: The disturbance transfer function is arbitrarily
can observe the advantage of our method according to the peak
values of these two signals. Indeed, with the method proposed, the
peak value of control law is 0:95 V. However, with Monje et al.
method the peak value is very high and is 110:87 V. This is due to
the derivative action of the Monje et al. controller. This problem is
recurrent especially when the reference input is a step one. Two
methods can be used to resolve this problem. One method is to
cascade the differentiator with a low pass filter with a chosen time
constant to eliminate high frequencies. Second method is to cas-
cade the differentiator with a low pass filter with tunable time
chosen as G s 1
1 þ 100s. Data on setpoint responses and load dis- constant.
turbance are summed up in Table 8. As can be seen, the overshoots
are the same with both methods (Mp ¼ 0:12). The step response is
In Fig. 20, a Gaussian noise with unit variance and zero mean is
added to the output signal. With PIλ
Dμ
controller given in [11], we
þ
þ
1

11. K. Amoura et al. / ISA Transactions 64 (2016) 247–257 257
of the proposed method and results showed their potential in
tuning a wide range of processes.
Indeed, the simulation results have revealed that the PID-FOF-
controller offers better control performance when coupled with
the proposed method. The combination of PID-FOF with the pro-
posed tuning technique can ensure a robust performance of the
controlled system even if the gain and/or noise fluctuate. Fur-
thermore, it also guarantees the iso-damping property of the step
response.
The comparison with other tuning method such as the IMC and
Monje et al. has shown the effectiveness of our method. Good
agreement has been found between the previous technique and
the one proposed herein. Moreover, slight advantages have been
observed such as higher speed, easier to use and adaptability to
control wide range of processes. Further studies are planned to
apply this method to control a real system.
Fig. 19. Magnitude of the control law: solid line shows proposed method and
dashed line shows the Monje et al. method.
Fig. 20. Closed-loop step response with noise: solid line shows proposed method
and dashed line shows the Monje et al. method.
obtain a noisy signal, due to the presence of derivative function,
the method being sensitive to noise. However, with our method,
we obtain a robust control performance immune to signal noise,
because of the absence of derivative control function and the
presence of a low pass filter. This is another advantage of our
method combined with ease and simplicity compared to the
complexity of the nonlinear optimization problem of the method
of Monje et al.
5. Conclusion
In this study novel tuning method for a new class of fractional
controllers has been presented and their performance successively
tested against pertinent techniques. Unlike the other methods, the
proposed one in this paper does not require any model of the
system to be controlled. It is based only on a single closed loop
setpoint experiment using a P-controller with gain Kc0 and the
controller can be directly obtained from the closed-loop data. The
reference model consists of an ideal closed loop system whose
open-loop is given by the Bode's ideal transfer function. Three
examples are presented to test and demonstrate the effectiveness
Acknowledgment
This project was funded by the Deanship of Scientific Research
(DSR), King Abdulaziz University, Jeddah, under grant no. Gr/34/5.
The authors, acknowledge with thanks DSR technical and financial
support.
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