A novel auto-tuning method for fractional order PID controllers

Research Article
A novel auto-tuning method for fractional order PI/PD controllers
Robin De Keyser a
, Cristina I. Muresan b,n
, Clara M. Ionescu a
a
Ghent University, Department of Electrical Energy, Systems and Automation, Technologiepark 914, B9052 Zwijnaarde, Belgium
b
Technical University of Cluj-Napoca, Department of Automation, Gh. Baritiu, No. 26-28, Cluj-Napoca, Romania
a r t i c l e i n f o
Article history:
Received 5 September 2015
Received in revised form
19 December 2015
Accepted 29 January 2016
Available online 20 February 2016
This paper was recommended for publica-
tion by Prof. Y. Chen
Keywords:
Auto-tuning
Fractional order controller
Robustness
Validation
a b s t r a c t
Fractional order PID controllers benefit from an increasing amount of interest from the research com-
munity due to their proven advantages. The classical tuning approach for these controllers is based on
specifying a certain gain crossover frequency, a phase margin and a robustness to gain variations. To tune
the fractional order controllers, the modulus, phase and phase slope of the process at the imposed gain
crossover frequency are required. Usually these values are obtained from a mathematical model of the
process, e.g. a transfer function. In the absence of such model, an auto-tuning method that is able to
estimate these values is a valuable alternative. Auto-tuning methods are among the least discussed
design methods for fractional order PID controllers. This paper proposes a novel approach for the auto-
tuning of fractional order controllers. The method is based on a simple experiment that is able to
determine the modulus, phase and phase slope of the process required in the computation of the con-
troller parameters. The proposed design technique is simple and efficient in ensuring the robustness of
the closed loop system. Several simulation examples are presented, including the control of processes
exhibiting integer and fractional order dynamics.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The proportional-integrative-derivative (PID) controller is still
the most widely used control strategy [1]. In numerous processes,
the PI(D) controllers account for more than 90% of the control
loops [2]. Sometimes, system modeling may be difficult, especially
in large industrial plants with sub-system interaction. Probably the
most popular method to tune the controllers in this particular
situation, where the process model is unavailable, is that devel-
oped by Ziegler and Nichols [3]. The design proposed by Ziegler
and Nichols is based on the measurement of the critical gain and
critical frequency of the plant and determining the PI/PID con-
troller parameters according to these critical values. The problem
with the design is that it leads to some closed loop systems with
poor robustness [1]. Several methods to avoid this problem and
improvements of the original tuning procedures of Ziegler and
Nichols have been proposed throughout the years. Åström–Häg-
glund used the same general ideas as presented by Ziegler and
Nichols, but combined these with the use of robust loop shaping
for control design, allowing for a clear tradeoff between
robustness and performance [1]. An automatic tuning method
based on a simple relay feedback test which uses the describing
function analysis to give the critical gain and the critical frequency
of the system has also been proposed as an alternative [4]. A
modification of this initial auto-tuning method based on a relay
with hysteresis for noise immunity has also been proposed, with
the new approaches based on an artificial time delay within the
relay closed-loop system to change the oscillation frequency in
relay feedback tests [5]. For the tuning of a PI controller based on
the relay feedback test, two equations for phase and amplitude
assignment are used. For the PID, in the modified Ziegler–Nichols
method [6], a parameter α¼4 is used as the ratio between the
integral and derivative time constants. However, since research
has shown that the control performance is heavily influenced by
the choice of the parameter α [7], improved methods for the auto-
tuning of the PID controller have been proposed, such as for
example the addition of a third condition to make the system
robust to open loop gain variations [2].
The generalization of the classical PID controller to the frac-
tional order one is due to Podlubny [8] and it is based on the use of
a fractional integrator of order μ and a fractional differentiator of
order λ, instead of the classical integer order integrator and dif-
ferentiator. Researchers have shown that this generalization allows
for a better shaping of the closed loop responses, mainly due to the
two supplementary tuning parameters involved, the fractional
Contents lists available at ScienceDirect
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ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.01.021
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: Robain.DeKeyser@ugent.be (R. De Keyser),
Cristina.Pop@aut.utcluj.ro (C.I. Muresan),
Claramihaela.Ionescu@ugent.be (C.M. Ionescu).
ISA Transactions 62 (2016) 268–275

orders μ and λ [9,10]. Due to these benefits, many researchers have
focused on the design problem of fractional order controllers
[9–14]. The majority of the tuning techniques for these fractional
order controllers are based on some frequency domain specifica-
tions that most frequently refer to a certain gain crossover fre-
quency, a certain phase margin of the open loop system, an iso-
damping property, output disturbance rejection and high fre-
quency noise rejection.
The auto-tuning of fractional order controllers has seen less
attention from the research community, with some notable papers
that discuss the design of a classical integer order PID combined
with a fractional order integrator or differentiator s
α
, with αϵ
(À1,1), denoted as the phase shaper [15]. The basic idea in the
design resides on the iso-damping property, also present for
classical PID controllers and for general fractional order PI
μ
D
λ
s
[16]. Here, the relay test is used in the auto-tuning procedure, with
the exact approach being based on an extension of the classical
approach used in the auto-tuning of integer order PID controllers.
The final controller is computed by designing separately a frac-
tional order PI controller (FO-PI) and a fractional order PD (FO-PD)
controller with a filter. The actual tuning of the FO-PI and FO-PD
controller parameters is based on ensuring a flat phase around the
frequency of interest (the iso-damping property), a gain crossover
frequency, and phase margin. The procedure is lengthy and is
based on maximizing the robustness to plant gain variations.
A different approach for the auto-tuning of fractional order PID
controllers has been also proposed [17], which is inspired from
both the classical Zigler–Nichols and Åström–Hägglund tuning
methods. The Ziegler–Nichols tuning procedure is firstly used to
determine the proportional and integrative gains of the controller,
while the initial value of derivative gain is obtained using Åström–
Hägglund method. The same frequency domain specifications are
used here as well. Based on the critical frequency and critical gain
obtained according to the Åström–Hägglund method, two non-
linear equations are obtained and used to achieve a specified
phase margin. A fine tuning of the derivative gain is required to
achieve the best numerical solutions of these two equations. The
fractional orders μ and λ are obtained from these equations using
an optimization technique. In case of a better step response of the
closed loop system, an optimization model, developed using
Simulink MATLAB, is used, which employs the previously com-
puted controller parameters as initial values and produces new
values for the controller parameters.
In the current study, a novel auto-tuning of fractional order PI
and PD controllers is proposed for stable, minimum phase, integer
order or fractional order processes. The tuning equations are based
on the widely used nonlinear system of equations obtained by
specifying a certain phase margin, gain crossover frequency and
iso-damping property. To determine the FO-PI/FO-PD controller
parameters, apart from the performance specifications, the mag-
nitude, phase and phase slope of the process at the gain crossover
frequency are necessary. It is assumed that no process model (e.g.
transfer function) is available. The magnitude and phase can be
obtained directly using frequency domain data, via a sine test
applied to the process. However, the novelty of the approach is
that the phase slope can also be computed, via filtering techniques
applied to the same sine test data. The efficiency of the proposed
auto-tuning method is demonstrated using several simulation
examples. An example that considers a known process transfer
function is also given, where the auto-tuning results are compared
to the analytical method to validate the proposed technique.
Additionally, the method is suitable and demonstrated for pro-
cesses which are described by both integer order model structures,
as well as generalized fractional order models.
The paper is structured as follows. Section 2 describes the clas-
sical tuning technique for FO-PI/FO-PD controllers, for the case of a
known process model. Section 3 details the basis for the auto-
tuning procedure in the case of no process model; it explains the
technique for estimating the process phase slope. Section 4 includes
the pseudo-algorithm for auto-tuning of FO-PI/FO-PD controllers,
while Section 5 presents the simulation results that prove the
efficiency of the proposed auto-tuning technique. The final section
contains the concluding remarks.
2. Classical tuning procedure for fractional order PI/PD
controllers
The transfer functions of the fractional order PI/PD (FO-PI/FO-
PD) controllers are indicated below:
HFOÀ PIðsÞ ¼ kp 1þ
ki
sμ

ð1Þ
HFOÀ PDðsÞ ¼ kp 1þkdsλ

ð2Þ
with the controller parameters defined as follows: μ; λA 0C2ð Þ the
fractional orders and kp, ki and kd the proportional, integrative and
derivative gains, respectively. One of the most widely used tuning
procedures for fractional order PI/PD controllers starts with a set
of three frequency domain performance specifications that are
used to determine the FO-PI/FO-PD controller parameters [9].
These performance specifications refer to:
1. A gain crossover frequency ωgc: the gain crossover frequency is
related to the settling time of the closed loop system and may
be thus used as an important tuning parameter for the con-
troller [18]. A large gain crossover frequency will result in a
smaller closed loop settling time. In order for a system to ensure
the imposed gain crossover frequency, the following condition
must hold:
HopenÀloopðjωgcÞ

¼ 1 ð3Þ
where Hopen-loop(s) is the loop transfer function defined as:
Hopen-loop(s)¼P(s). HFOC(s), where P(s) is the transfer function
of the process to be controlled and HFOC(s) is either the FO-PI or
FO-PD controller defined in (1) or (2), respectively.
2. A phase margin φm: Phase margin is an important measure of
system stability and also an indicator of the closed loop over-
shoot [18,19]. Usually, an interval between 45° and 65° is used
for a proper phase margin selection. In order for a system to
ensure a certain phase margin, the following condition must
hold:
∠Hopen ÀloopðjωgcÞ ¼ Àπþϕm ð4Þ
3. Iso-damping property: This condition ensures that the system is
more robust to gain changes and the overshoot of the response
is almost constant within a gain range. To ensure a constant
overshoot, a constant phase margin needs to be maintained
around the desired gain crossover frequency, which ultimately
implies that the phase of the open-loop system must be kept
constant around the specified ωgc [9]. In order for a system to
ensure the iso-damping property, the following condition
must hold:
d ∠HopenÀloopðjωÞ
À Á
dω

ω ¼ ωgc
¼ 0 ð5Þ
The complex representations in the frequency domain of the
transfer functions describing the FO-PI or the FO-PD in (1) and (2)
R. De Keyser et al. / ISA Transactions 62 (2016) 268–275 269

are given below:
HFOÀPIðjωÞ ¼ kp 1þkiωÀ μ cos
πμ
2
Àj sin
πμ
2
h i
ð6Þ
HFOÀPDðjωÞ ¼ kp 1þkdωλ cos
πλ
2
þj sin
πλ
2
!
ð7Þ
The corresponding modulus and phase for each of these two
controllers are then computed as
HFOÀPIðjωÞ

¼ kp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2kiωÀ μ cos
πμ
2
þki
2
ωÀ 2μ
r
ð8Þ
∠HFOÀPIðjωÞ ¼ Àa tan
kiωÀμ sin πμ
2
1þkiωÀμ cos πμ
2
!
ð9Þ
for the FO-PI controller and
HFOÀPDðjωÞ

¼ kp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2kdωλ cos
πλ
2
þkd
2
ω2λ
r
ð10Þ
∠HFOÀPDðjωÞ ¼ a tan
kdωλ sin πλ
2
1þkdωλ cos πλ
2
!
ð11Þ
for the FO-PD controller.
Then, for a FO-PI controller, the conditions in (3)–(5) may be
further written as
kp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2kiωÀμ
gc cos
πμ
2
þki
2
ωÀ2μ
gc
r
¼
1
PðjωgcÞ

ð12Þ
kiωÀ μ
gc sin πμ
2
1þkiωÀμ
gc cos πμ
2
¼ tan πÀϕm þ∠PðjωgcÞ
À Á
ð13Þ
μkiωÀ μÀ 1
gc sin πμ
2
1þ2kiωÀμ
gc cos πμ
2 þk
2
i ωÀ2μ
gc
þ
d∠PðjωÞ
dω

ω ¼ ωgc
¼ 0 ð14Þ
To tune the FO-PI controller, the system of nonlinear Eqs. (12)–
(14) needs to be solved using either optimization techniques or
graphical methods [9,10,20]. For the optimization techniques, the
Matlab optimization toolbox may be used, namely the fmincon
function, for which the modulus condition in (12) may be used as
the function intended to be minimized, while the phase condition
in (13) and the robustness condition in (14) are specified as the
nonlinear constraints. Initial values for the controller parameters
need also to be specified, with the possibility of setting their lower
and upper bounds, as well. The fmincon function will return the
controller parameters, such that the modulus condition is mini-
mized and the nonlinear constraints are met. An alternative pos-
sibility of determining the controller parameters is based on gra-
phical methods. This approach consists in the evaluation of the ki
parameter as a function of the fractional order μ from (13) and
(14). Then for different values of μϵ(0,2), the corresponding ki
values are computed. The graphical approach then consists in
plotting of the different ki values determined based on (13), as a
function of the fractional order values μϵ(0,2), as well as the cor-
responding ki values computed using (14). The intersection point
of these two graphs gives the final values for the ki and μ para-
meters. Once these have been determined, the modulus condition
in (12) may be used to compute the final value for the kp para-
meter. Similar equations may be obtained using the FO-PD con-
troller, by simply following the substitutions kd ¼ki and λ¼ Àμ, in
(12)–(14). Nevertheless, regardless of the approach taken to
determine the controller parameters, to completely tune the FO-
PI/FO-PD controllers, the modulus, phase and phase slope of the
process at the gain crossover frequency have to be known, as
indicated in the above Eqs. (12)–(14).
3. Proposed methodology
As mentioned in the previous section, in order to tune the FO-
PI/FO-PD controllers, the phase, magnitude and phase slope of the
process at the imposed gain crossover frequency need to be
determined. The phase and magnitude of any stable process at a
specific gain crossover frequency ωgc may be easily determined by
applying a sinusoidal input signal to the process as indicated
below:
uðtÞ ¼ Ai sin ωgct
À Á
ð15Þ
The output signal, denoted as y(t), of the process may be
represented as indicated in Fig. 1, where the input signal in (15)
has also been represented. Assume in what follows that the pro-
cess may be described by a transfer function P(s), considered
unknown. Then, the process magnitude and phase at the test
frequency may be computed according to
M ¼ PðjωgcÞ

¼
Ao
Ai
ð16Þ
ϕ ¼ ∠PðjωgcÞ ¼ ωgcτ ¼ ωgc ti Àtoð Þ ð17Þ
where Ao is the output amplitude and τ¼ti Àto is the time shift
between the input u(t) and output y(t) signals, M is the magnitude
and φ is the phase of the process at the specific gain crossover
frequency ωgc.
Thus, if the imposed gain crossover frequency ωgc is used, then
by applying the signal u(t) in (15) to any stable unknown process,
the experimental results obtained may be further used to deter-
mine the process modulus and phase at the gain crossover fre-
quency, using (16) and (17). However, to further tune either the
FO-PI controller according to (12)–(14) or a FO-PD controller
according to a similar set of equations, the phase slope
d∠PðjωÞ
dω

ω ¼ ωgc
has to be determined as well.
Assuming a signal x(t) as the output to an input signal v(t)¼t.u
(t) applied to the process P(s), it follows that
L tUuðtÞð ÞUPðsÞ ¼ XðsÞ ð18Þ
which leads to
À
dUðsÞ
ds
UPðsÞ ¼ XðsÞ ð19Þ
where the property of the Laplace transform has been used,
L ÀtUuðtÞð Þ ¼ dUðsÞ
ds
.
Consider also that if a signal u(t) is applied at the input of the
process derivative dPðsÞ
ds
, then the corresponding output would be
Fig. 1. Input and output signals of a stable process (blue line—input signal, green
line—output signal); arbitrary time units on X-axis and arbitrary amplitude units on
the Y-axis. (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
R. De Keyser et al. / ISA Transactions 62 (2016) 268–275270

yðtÞ, and in the Laplace domain this leads to
UðsÞU
dPðsÞ
ds
¼ YðsÞ ð20Þ
as well as
UðsÞUPðsÞ ¼ YðsÞ ð21Þ
where Y(s) is the Laplace transform of the output y(t), when the
signal u(t) is now applied to the process P(s).
Then, consider also the derivative of the process output signal Y
(s):
dYðsÞ
ds
¼ L ÀtUyðtÞð Þ ) ÀtUyðtÞ ¼ LÀ1 dYðsÞ
ds

¼ LÀ1 d PðsÞUUðsÞ½ Š
ds

ð22Þ
where relation (21) has been used. However
d PðsÞUUðsÞð Þ
ds
¼
dPðsÞ
ds
UUðsÞþ
dUðsÞ
ds
UPðsÞ ð23Þ
If in this last equation, we use (20) and (19), then (23) may be
rewritten as
d PðsÞUUðsÞð Þ
ds
¼ YðsÞÀXðsÞ ð24Þ
Using (24) in (22) and assuming zero initial conditions, leads to
Àt UyðtÞ ¼ LÀ1
YðsÞÀXðsÞ
À Á
ð25Þ
which may be written in the time domain as
xðtÞÀtUyðtÞ ¼ yðtÞ ð26Þ
where yðtÞ is the output of the process derivative, as denoted in
(20).
If we now consider the sinusoidal input signal in (15), its
Laplace transform is given by
UðsÞ ¼
Aiωgc
s2 þω2
gc
ð27Þ
The derivative of this input signal with respect to the Laplace
variable s is computed according to (27):
dUðsÞ
ds
¼ À
2Aiωgcs
s2 þω2
gc
2
ð28Þ
which means that in this case, the signal X(s) in (19) may be fur-
ther computed as
XðsÞ ¼ À
dUðsÞ
ds
UPðsÞ ¼
2Aiωgcs
s2 þω2
gc
2
UPðsÞ ¼
Aiωgc
s2 þω2
gc
2s
s2 þω2
gc
UPðsÞ
ð29Þ
where if we now use (27) and (21), we have the final Laplace
transform for the x(t) signal:
XðsÞ ¼
2s
s2 þω2
gc
UYðsÞ ð30Þ
The experimental scheme in Fig. 2 may be used in the auto-
tuning of FO-PI/FO-PD controllers. If a sinusoidal signal u(t) as
indicated in Fig. 2 is applied at the input of the process with
unknown transfer function, then the resulting experimental data
for the process output y(t), as well as for the process derivative
yðtÞ, are later used to determine the process magnitude, phase and
phase slope at the imposed gain crossover frequency ωgc. As
indicated in (26), the signal yðtÞ is computed as the difference
between the signal x(t) and the term tUyðtÞ, also shown in the right
part of Fig. 2. To determine the signal x(t), (30) is used and thus, as
indicated in Fig. 2, the process output signal y(t) is passed through
the filter 2s
s2 þω2
gc
. In Fig. 2, M and φ denote the modulus and phase
of P(s) at a specific frequency ω. Moreover, based on (20), M and ϕ
denote the modulus and phase of the process derivative dPðsÞ
ds
.
In the frequency domain, the derivative dPðsÞ
ds
at the gain cross-
over frequency may be computed by replacing the Laplace variable
s¼jωgc:
dPðjωÞ
dðjωÞ

ω ¼ ωgc
¼ Me j φ ) Àj
d MðωÞUejφðωÞ
À Á
dω

ω ¼ ωgc
¼ Me j φ ð31Þ
Extending the left hand side of (31) leads to
Àj
dM
dω

ω ¼ ωgc
ejϕ þM
dejϕ
dϕ
dϕ
dω
j
ω ¼ ωgc
0
@
1
A ¼ Mejϕ
or
Àj
dM
dω

ω ¼ ωgc
ejϕ þMUejϕdϕ
dω

ω ¼ ωgc
¼ Mejϕ ð32Þ
Dividing (32) by ejϕ, the next equation is obtained as
Àj
dM
dω

ω ¼ ωgc
þM
dϕ
dω

ω ¼ ωgc
¼ Mej ϕÀ ϕ
À Á
¼ M cos ϕÀϕ

þjM sin ϕÀϕ

ð33Þ
Equating the real and imaginary parts of the left and right hand
sides of (33), leads to the final equalities that represent the slopes
of the process magnitude and phase at the specified gain crossover
frequency ωgc:
dM
dω

ω ¼ ωgc
¼ ÀM sin ϕÀϕ

ð34Þ
dϕ
dω

ω ¼ ωgc
¼
M
M
cos ϕÀϕ

ð35Þ
Then, if the test frequency used for the sinusoidal input signal u
(t) in Fig. 2 is the gain crossover frequency, (35) may be used to
determine the phase slope of the process P(s) at the gain crossover
frequency:
d∠PðjωÞ
dω

ω ¼ ωgc
¼
dϕ
dω

ω ¼ ωgc
¼
M
M
cos ϕÀϕ

ð36Þ
where
M ¼
Ay
Ai
ð37Þ
ϕ ¼ ωgcτy ¼ ωgc ti Àty
À Á
ð38Þ
with Ay the amplitude of the sinusoidal signal yðtÞ, τy the time shift
between the two signals u(t) and yðtÞ.
Note: the theory presented in this section and the related
experimental scheme in Fig. 2 describe the underlying concept of
obtaining the process phase slope (as well as the magnitude slope)
via a simple sine test. However, the scheme in Fig. 2 is prone to
stochastic disturbances that are acting on the measured process
output y(t), if these disturbances have a frequency spectrum close
Fig. 2. Experimental scheme to obtain the (sine) signals y(t) and yðtÞ and compute
the phase slope of the process at the gain crossover frequency.

to the test frequency ωgc. The scheme can however be converted to
an equivalent scheme which is very robust against such stochastic
disturbances. This conversion - although important for practical
implementation - is of a pure technical matter which has no
impact on the fundamental idea presented in this paper. It is
beyond the scope of this paper. These practical implementation
issues as well as their demonstration on real-life test processes -
corrupted with stochastic noise - are the subject of a subsequent
paper. For that reason, the fundamental idea of this paper is illu-
strated in the following section on noise-free simulated processes.
4. Illustrative simulation examples
To auto-tune the FO-PI or alternatively the FO-PD controller,
the following steps are necessary:
1. Given the imposed gain crossover frequency, ωgc, apply a
sinusoidal input signal of the form (15) to the process to be
controlled.
2. Perform the experimental test described in Fig. 2 and determine
Ao and Ay, τ and τy. Compute M and ϕ using (16) and (17) as
well as M and ϕ using (37) and (38).
3. The tuning of the fractional order controller is then based on a
specified phase margin φm and on an optimization routine for
the system of equations given in (12)–(14), where
PðjωgcÞ

¼ M ð39Þ
∠PðjωgcÞ ¼ ϕ ð40Þ
d∠PðjωÞ
dω

ω ¼ ωgc
¼
M
M
cos ϕÀϕ

ð41Þ
The autotuning method presented above will be illustrated via
some simulation examples, considering both processes that may
be described by integer order or fractional order models. For the
integer order case, a first order time delay process and a higher
order process are considered, as illustrative examples from the 131
benchmark processes proposed by Åström and Hägglund [21]. In
the first example, it is shown that the classical fractional order
controller tuning procedure and the proposed auto-tuning method
produce similar results. In the second example, the proposed auto-
tuning method is detailed for a higher order process, including the
closed loop step response results, demonstrating the robustness of
the design procedure against gain variations. Finally, a third
example is included which presents the auto-tuning of the frac-
tional order PI controller for a process described by a fractional
order model. This last example extends the auto-tuning procedure
to general processes.
The effectiveness of the proposed auto-tuning procedure in
terms of closed loop results greatly depends upon the perfor-
mance specification set. This is similar to the classical tuning
procedure for fractional order controllers. Several rules regarding
the feasibility range of performance specifications may be found in
[22] and [23]. In this paper, the purpose is to show that the clas-
sical tuning procedure for fractional order controllers can be suc-
cessfully extended to situations where a mathematical process of
the model is not available. Therefore, in the numerical examples,
rather than an optimal set of performance specifications to obtain
excellent closed loop results, a set of random, but feasible, fre-
quency domain performance specifications are selected, without
any major concern regarding the final closed loop results.
4.1. First order time delay process
Consider the plant given by the following transfer function:
P1ðsÞ ¼
1
6sþ1
eÀs
ð42Þ
The performance specifications are: a gain crossover frequency
ωgc ¼0.1 rad/s, a phase margin φm ¼55° and a flat open loop phase
around the gain crossover frequency to ensure the iso-damping
property. To tune the controller, the modulus, phase and phase
Table 1
Process modulus, phase and phase slope at the gain crossover frequency.
P1ðjωgcÞ

∠P1ðjωgcÞ (deg) ∂∠P1 ðjωÞ
∂ω

ω ¼ ωgc
Analytical computation 0.8575 À36.69 À5.4118
Experimental identification 0.8574 À37.81 À5.4551
Fig. 3. Graphical solution for the classical tuning approach.
Table 2
Fractional order controller parameter values using the classical method and the
auto-tuning method.
Integral gain
ki
Proportional gain kp Fractional order
μ
Classical approach 0.1412 0.5065 1.245
Proposed auto-tuning
method
0.1407 0.5142 1.238
0 50 100 150 200 250 300 350 400
-6
-4
-2
0
2
4
6
Time (s)
Amplitude
Input u(t)
Output y(t)
Output y
bar
(t)
Fig. 4. Experimental results for first order time delay process ybar tð Þ ¼ yðtÞ
À Á
.

slope of the process P1(s) at the gain crossover frequency may be
determined analytically based on the process transfer function in
(42). The results are given in Table 1.
To tune the fractional order PI controller, using the classical
approach described in Section 2, the modulus, phase and robust-
ness conditions in (12)–(14) may now be expressed as
kp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2ki0:1À μ
cos
πμ
2
þki
2
0:1À 2μ
r
¼
1
0:8575
ð43Þ
ki0:1À μ
sin πμ
2
1þki0:1À μ
cos πμ
2
¼ tan ð1:5413Þ ð44Þ
μki0:1À μÀ 1
sin πμ
2
1þ2ki0:1Àμ
cos πμ
2 þk
2
i 0:1À2μ
À5:4118 ¼ 0 ð45Þ
Fig. 3 shows the values obtained for the ki parameter, as a
function of the fractional order μ, as resulting from (44) and (45).
The intersection point gives the final controller parameter values:
ki ¼0.1412 and μ¼1.245. Using now (43), the proportional gain is
determined to be kp¼0.5065. A summary of the controller para-
meter values is given in Table 2.
The experimental auto-tuning procedure described in Section 3
is now used to obtain the modulus, phase and phase slope of the
process, assuming that the process transfer function (42) is
unknown. A sinusoidal signal is used with unit amplitude and test
frequency taken as ωgc ¼0.1 rad/s. The experimental results are
given in Fig. 4. The amplitude of the input signal at ti ¼329.8 s is
Ai ¼1, as indicated in Fig. 4. The amplitude of the output signal
at to¼336.4 s is Ao ¼0.8574. Using (16) and (17), the parameters
M and φ are computed, with M¼0.8574 and φ ¼ 0:1U
329:8À336:4ð Þ ¼ À0:66 rad ¼ À 37:81o
. Then, according to (39)
and (40), P1ðjωgcÞ

¼ 0:8574 and ∠P1ðjωgcÞ ¼ À37:81o
. The
amplitude of yðtÞ is also determined from Fig. 3, Ay ¼ 5:1659 at
ty ¼ 372:2 s. Then, using (37) and (38), M ¼
Ay
Ai
¼ 5:1659 and
ϕ ¼ 0:1U 329:8À372:2ð Þ ¼ À4:24 rad or ϕ ¼ À 242:93o
. Finally,
using (41), the process phase slope is computed as d∠P1ðjωÞ
dω

ω ¼ ωgc
¼
À 5:4551 s. These results are also included in Table 1, clearly
showing that the experimental test in Fig. 2 produces similar
results as those obtained analytically.
A FO-PI controller is tuned next according to the design pro-
cedure presented previously using the experimentally computed
values in Table 1 and based on (12)–(14). The graphical approach
yields in this case a proportional gain kp¼0.5142, an integrative
gain ki ¼0.1407 and a fractional order μ¼1.238. These controller
parameter values - obtained without knowledge of the process
transfer function - are included in Table 2, along with the con-
troller parameter values as computed using the classical tuning
procedure - i.e. with knowledge of the process transfer function.
The results in Table 2 indicate that the auto-tuning procedure
described in Section 3 produces similar results as the classical
tuning procedure described in Section 2.
The transfer function of this FO-PI controller, based on the
auto-tuning approach, is
HFOÀ PIðsÞ ¼ 0:5142 1þ
0:1407
s1:238

ð46Þ
The Bode diagram of the open loop system considering the
designed FO-PI controller, as well as the process transfer function
in (42) is given in Fig. 5, showing that in nominal conditions the
phase margin and the gain crossover frequency meet the imposed
performance specifications. Also, the phase around the gain
crossover frequency is flat, which implies that for 730% gain
variations the phase margin remains unchanged, as indicated in
the Bode diagram. The same conclusions may be drawn from the
closed loop responses in Fig. 6, where the overshoot remains the
same as in the nominal case, despite the 730% gain variations.
Fig. 5. Bode diagram of the open loop system for example 1.
0 20 40 60 80 100
0
0.5
1
1.5
Time (s)
Output
0 20 40 60 80 100
0.5
1
1.5
2
Time (s)
Input
nominal
+30% uncertainty
-30% uncertainty
Fig. 6. Closed loop step responses for example 1: output and input signals.
0 50 100 150 200 250 300
-6
-4
-2
0
2
4
6
Time (s)
Amplitude
Input u(t)
Output y(t)
Output y
bar
(t)
Fig. 7. Experimental results for higher order process ybar tð Þ ¼ yðtÞ
À Á
.

4.2. Higher order process
As it has been shown in the previous example, the analytical
and auto-tuning methods produce similar results. Consider now
the plant given by the following transfer function:
P2ðsÞ ¼
1
sþ1ð Þ6
ð47Þ
ωgc ¼0.1 rad/s, a phase margin φm¼50° and a ﬂat open loop phase
property. The auto-tuning of a FO-PI controller, as described in
Section 3, starts with an experimental test similar to Fig. 2, where
the input signal is applied of the form (15), with ωgc ¼0.1 rad/s and
Ai ¼1. The experimental results are given in Fig. 7, where the
amplitude Ai of the input signal is measured at ti ¼141.4 s. Simi-
larly, the amplitude of the output signal is Ao¼ 0.9706 at
to¼147.4 s. Using (16) and (17), the parameters M and φ are
computed, with M¼0.9706 and φ ¼ 0:1U 141:4À147:4ð Þ ¼
À0:6 rad ¼ À 34:37o
. Then, according to (39) and (40), P1ðjωgcÞ

¼ 0:9706 and ∠P1ðjωgcÞ ¼ À34:371. The amplitude of yðtÞ is also
determined from Fig. 6, Ay ¼ 5:7932 at ty ¼ 180 s. Then, using (37)
and (38), M ¼
Ay
Ai
¼ 5:7932 and ϕ ¼ 0:1U 141:4À180ð Þ ¼ À3:86 rad
or ϕ ¼ À 221:16o
. Finally, using (41), the process phase slope is
computed as d∠P1ðjωÞ
dω

ω ¼ ωgc
¼ À 5:927 s.
With the process modulus, phase and phase slope determined
experimentally, the system of equations in (12)–(14) is solved to
determine the controller parameters: kp ¼0.4627, ki ¼0.1210 and
μ¼1.32. The transfer function of the designed FO-PI controller is
HFOÀ PIðsÞ ¼ 0:4627 1þ
0:121
s1:32

ð48Þ
The closed loop simulation results considering a unit step
reference are given in Fig. 8, considering both the nominal gain
equal to 1, as indicated in (47), as well as 730% gain variations,
with the process gain now considered 0.7 and 1.3 respectively. The
results show that the designed FO-PI auto-tuner is robust to gain
variations as it manages to maintain a constant overshoot, despite
these modeling uncertainties.
4.3. Fractional order process
Consider now the plant given by the following transfer function:
P3ðsÞ ¼
1
5s0:5 þ1
ð49Þ
ωgc ¼1 rad/s, a phase margin φm ¼70° and a ﬂat open loop phase
property. The auto-tuning of a FO-PI controller, as described in
Section 3, starts with an experimental test similar to Fig. 2, where
the input signal is applied of the form (15), with ωgc ¼1 rad/s and
Ai ¼1. The experimental results are given in Fig. 9, where the
amplitude Ai of the input signal is measured at ti ¼39.27 s. Fig. 9
also shows that the yðtÞ signal at the output of the process deri-
vative will exhibit a very long transient regime, due to the half
order integrator involved [9]:
d P3ðsÞ½ Š
ds
¼ À
2:5
5s0:5 þ1
À Á2
s0:5
ð50Þ
The amplitude of the y(t) signal is computed in the transient
regime as Ao ¼ AoðtoÞ þAoðt1Þ
2 ¼ 0:1771 þ 0:171
2 ¼ 0:1741, where
to¼39.932 s and t1 ¼43.1 s. Then, using (16) and (17), the para-
meters M and φ are computed, with M¼0.1741 and
φ ¼ 1U 39:27À39:932ð Þ ¼ À0:662 rad ¼ À 37:93o
. Then, according
to (39) and (40), P1ðjωgcÞ

¼ 0:1741 and ∠P1ðjωgcÞ ¼ À37:931. The
amplitude of yðtÞ is also determined from Fig. 9, similarly to how
0 10 20 30 40 50
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
Amplitude
Input u(t)
Output y(t)
Output ybar
(t)
Fig. 9. Experimental results for fractional order process ybar tð Þ ¼ yðtÞ
À Á
.
0 5 10 15
0
0.5
1
1.5
Time (s)
Output
0 5 10 15
0
2
4
6
Time (s)
Input
nominal
+30% uncertainty
-30% uncertainty

Ao has been determined. An analysis of the yðtÞ signal in Fig. 9,
gives the following Ay ¼
AyðtyÞþ Ayðt2Þ
2 ¼ À 0:0465 þ 0:1999
2 ¼ 0:0767, with
ty ¼ 44:523 s and t2 ¼47.67 s. Then, using (37) and (38), M ¼
Ay
Ai
¼
0:0767 and ϕ ¼ 1U 39:27À44:523ð Þ ¼ À5:235 rad or ϕ ¼ À 301o
.
Finally, using (41), the process phase slope is computed as
d∠P1ðjωÞ
dω

ω ¼ ωgc
¼ À 0:0533 s.
With the process modulus, phase and phase slope determined
experimentally, the system of equations in (12)–(14) is solved to
determine the controller parameters: kp¼0.38, ki ¼14.77 and
μ¼0.8418. The transfer function of the designed FO-PI controller
is:
HFOÀPIðsÞ ¼ 0:38 1þ
14:77
s0:8418

ð51Þ
The closed loop simulation results considering a unit step refer-
ence are given in Fig. 10, considering both the nominal, as well as
730% gain variations. The results show that the designed FO-PI auto-
tuner is robust to gain variations as it manages to maintain a con-
stant overshoot, despite these modeling uncertainties.
5. Conclusions
One of the most important aspects in control engineering
consists in an accurate modeling of the process to be controlled. In
a wide range of applications, an accurate model of the process is
difficult to be obtained if not impossible. In such case, the tuning of
controllers becomes a tedious task. In this paper, a novel auto-
tuning method for fractional order PI or PD controllers has been
presented, that yields a robust controller despite the lack of an
actual process model. A simple experiment is sufficient in order to
determine the process parameters that are required for the tuning
of the fractional order controller: the process phase, the process
magnitude and the slope of the process phase at a specified gain
crossover frequency. The auto-tuning technique is exemplified
considering some benchmark processes, a first order time delay
process, a higher order process, as well as a process exhibiting
some fractional order dynamics. The closed loop simulation results
show that the proposed technique meets the performance criteria
and ensures the robustness of the closed loop system despite the
lack of an actual process model.
A comparison between the tuning of a fractional order PI
controller based on the classical approach (with knowledge of the
process model) versus the tuning based on the proposed auto-
tuning method (with lack of a process model), indicates that the
two tuning procedures lead to similar results.
Results are available for a future paper that will demonstrate
the application of the proposed method on real-life processes,
which are subject to stochastic disturbances.
Acknowledgements
Clara Ionescu is a post-doctoral fellow of the Flanders Research
Centre (FWO), grant no. 12B345N. Cristina I. Muresan is financed
by a grant of the Romanian National Authority for Scientific
Research and Innovation, CNCS – UEFISCDI, project number PN-II-
RU-TE-2014-4-0598, TE 86/2015.
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