This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the controller parameters on the achieved control system robustness and performance are investigated. Results show that this kind of controllers is more fragile with respect to the standard proportional-integral-derivative controllers and therefore a significant attention should be paid by the user in their tuning.
On the fragility of fractional-order PID controllers for FOPDT processes
1. Research Article
On the fragility of fractional-order PID controllers for FOPDT processes
Fabrizio Padula a
, Antonio Visioli b,n
a
Dipartimento di Ingegneria dell'Informazione, University of Brescia - Italy, Italy
b
Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia - Italy, Via Branze 38, I-25123 Brescia, Italy
a r t i c l e i n f o
Article history:
Received 2 December 2014
Received in revised form
26 August 2015
Accepted 9 November 2015
Available online 27 November 2015
This paper was recommended for publica-
tion by Dr. Y. Chen
Keywords:
Fractional-order controllers
PID control
Tuning
Fragility
a b s t r a c t
This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers
applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the
controller parameters on the achieved control system robustness and performance are investigated. Results
show that this kind of controllers is more fragile with respect to the standard proportional-integral-
derivative controllers and therefore a significant attention should be paid by the user in their tuning.
& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
It is well known that a properly designed control system must
provide an effective trade-off between performance and robust-
ness. However, it has also been recognized that another important
issue to be addressed is the fragility of the control system to the
variation of the controller parameters, that is, the sensitivity of the
robustness and/or performance of the control system to changes in
the controller parameters.
This issue has been raised in the literature in some papers (see,
for example, [1]) and, in particular, in [2] where it has been
stressed that design techniques based on the minimization of the
H2, H1 and l1 norms can yield to high-order robust, optimal but
also extremely fragile controllers, namely, a very small variation of
the controller coefficients can result in an unstable system. How-
ever, in [3,4] it has been pointed out that this problem can be
solved by using a suitable controller parametrization.
As integer-order proportional-integral-derivative (IOPID)
controllers are the most used controllers in industry, the fragility
of such a kind for controllers has been specifically addressed in
[5,6]. Therein, authors suggest to tune the IOPID controller in
order to maximize the l2 norm of the controller parameter vector
in the stabilizing region for a given plant. However, the typical
industrial performance measures (related to the set-point
following and/or to the load disturbance rejection task) are not
taken into account. Further, it has been shown in [7] that this
kind of approach applied to first-order-plus-dead-time (FOPDT)
and integrator-plus-dead-time (IPDT) processes yields a tuning
similar to that obtained by using the Ziegler–Nichols step
response method [8] which is known to be improvable under
many points of view [9].
Thus, it has been recognized in the literature that one of the
main reasons to investigate the fragility of IOPID controllers is to
give to the user an idea of how a fine tuning of the controller can
be done [10–12]. In other words, as the IOPID parameters have a
clear physical meaning, the operator can modify them in order to
change the control system performance. In this context, it is useful
to evaluate the sensitivity of the robustness/performance behavior
with respect to (small) changes of the parameters. For this pur-
pose, a graphic tool called fragility rings providing a visual aid for
evaluation of the controller robustness/fragility has been proposed
in [13].
In the recent years, there has also been a significant interest
from the academic and industrial communities for fractional-
order-proportional-integral-derivative (FOPID) controllers because
they are capable to provide (as there are five parameters to tune)
more flexibility in the control system design (see, for example,
[14–17]). Many different tuning rules have been proposed in the
literature to facilitate their use (see, for example, [18–23]). In this
context, while the problem of stabilizing a (possibly fractional)
dynamic system using FOPID controllers has been already
addressed in the literature (see, for example, [24–26]), for such a
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2015.11.010
0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.: þ39 030 3715460; fax: þ39 030 380014.
E-mail addresses: fabrizio.padula@unibs.it (F. Padula),
antonio.visioli@unibs.it (A. Visioli).
ISA Transactions 60 (2016) 228–243
2. kind of controllers, a fragility analysis has been only partially
exploited until now. In particular, in [27,28], the tuning of the
FOPID controllers is performed by considering the centroids of the
admissible regions in the parameter space so that a non-fragile
controller results. However, one of the main purposes for evalu-
ating the fragility of the controller is in evaluating the sensitivity of
the robustness/performance indexes to the (possibly fine) tuning
of the parameters.
Indeed, in order to foster a widespread use of FOPID controllers
in industrial plants, in addition to well-established tuning rules,
clear guidelines on how to modify the controller parameters
should be given to the operator in order for him/her to be con-
fident with them. Thus, the aim of this paper is to provide a fra-
gility analysis for FOPID controllers and to make a comparison
with IOPID controllers in order to understand the differences that
should be taken into account in the adjustment of the parameters
starting from a given tuning. For this purpose, the tuning rules
proposed in [23,29], which aim at minimizing the integrated
absolute error subject to constraints on the maximum sensitivity,
are used, both for FOPID and IOPID controllers. Both the tuning
rules for the set-point following and the load disturbance rejection
tasks are considered. They also have the significant feature of
providing a control action that is invariant when the time unit is
changed. These tuning rules are therefore suitable to perform a
fragility analysis with respect to both robustness and performance.
It is worth stressing that the calculated fragility depends on the
nominal parameters of the control system and for this reason, in
order to obtain a fair comparison, we select tuning rules that solve
the same optimization problem, so that the possible additional
complexity of adjusting the parameters of a FOPID controller, with
respect to a IOPID one, starting from a given tuning is clearly
addressed.
The fragility is evaluated by changing all the parameters at the
same time or just one of them by keeping the other ones fixed. The
latter case is performed in order to investigate which parameter
has more influence on the controller fragility.
The paper is organized as follows. The basic definitions
employed for the fragility evaluation are reviewed in Section 2, in
addition to the description of the tuning rules used for both integer-
order and fractional-order PID controllers. The fragility analysis
related to the robustness is presented in Section 3 while that related
to the performance is presented in Section 4. A discussion is made
in Section 5, while conclusions are drawn in Section 6.
2. Fragility indices
The fragility indices proposed in [10–12] are briefly reviewed in
this section for the sake of clarity and in order to introduce the
notation used in presenting the results.
Consider a unity feedback control system (see Fig. 1) where the
process (which is assumed to be self-regulating) is denoted as P
and the controller as C. In this paper, the controller is a FOPID
controller, which can be expressed either in series form, i.e.,
CðsÞ ¼ Kp
Tisλ þ1
Tisλ
Tdsμ þ1
Tf sþ1
ð1Þ
or in parallel (ideal) form, i.e.,
CðsÞ ¼ Kp 1þ
1
Tisλ
þTdsμ
1
Tf sþ1
: ð2Þ
In both expression, Kp is the proportional gain, Ti is the integral
time constant, Td is the derivative time constant and λ and μ are
the noninteger orders of the integral and derivative terms
respectively.
Note that it is important to consider both forms (1) and (2)
because it is not possible to transform (2) into an equivalent form
(1) and vice versa unless Ti Z4Td and λ ¼ μ [29]. In order to
implement the fractional-order controller, the well-known Ous-
taloup continuous integer-order approximation [30] has been
employed to approximate the fractional differintegrator. In this
paper 16 poles and zeros have been used in order to approximate
the fractional differintegrator in a frequency range ½ωl; ωhŠ, where
ωl and ωh have been selected as 0:0001ωc and 10000ωc
respectively, with ωc being the gain crossover frequency. It is
worth noting that the used number of poles and zeros leads to a
computationally demanding controller and, actually, the frac-
tional controller could be approximated with a lower order
integer one. Nevertheless, considering that the purpose of this
paper is the fragility analysis of the fractional controller, a higher
computational cost is accepted in order to achieve an improved
approximation. The approximated and the ideal open loop
transfer function in this way are virtually indistinguishable at
those frequencies that have an appreciable impact on the closed-
C
r ye
P
d
Fig. 1. The considered control scheme.
Table 1
The controller parameters for the considered example with L=T ¼ 0:5 and for the different control tasks, set-point (SP) following and load disturbance (LD) rejection with a
maximum sensitivity of 1.4 and 2.0 respectively.
Controller Kp Ti Td λ μ
FOPID series
SP 1.4 1.1060 0.9839 0.1554 1 1.2
SP 2.0 1.6698 1.0281 0.1975 1 1.1
LD 1.4 0.7818 0.4683 0.2617 1 1.1
LD 2.0 1.1182 0.4236 0.3105 1 1.1
FOPID parallel
SP 1.4 1.3307 1.1765 0.1384 1 1.1578
SP 2.0 2.0850 1.2507 0.1757 1 1.1351
LD 1.4 1.2786 0.8824 0.1686 1 1.1351
LD 2.0 2.3611 0.9079 0.1440 1 1.1525
IOPID series
SP 1.4 0.8676 0.8127 0.2074 – –
SP 2.0 1.4708 0.9568 0.2347 – –
LD 1.4 0.6369 1.0081 0.3031 – –
LD 2.0 1.007 0.4106 0.3304 – –
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 229
3. loop system dynamics. It can be also noted that an additional
first-order filter has been employed in both (1) and (2) in order to
make the controller proper. The selection of the time constant Tf
is done in such a way that the high-frequency noise is filtered
without influencing the dynamics of the controller significantly
[29,31,32]. Eventually, considering the Oustaloup approximation
and that only the fractional part μ (μÀ1 if μ41) of the derivative
action is approximated, an integer filter is enough to guarantee
the properness of the controller.
Then, it has also to be noted that by selecting λ ¼ μ ¼ 1, an
IOPID controller is obtained. In this paper, just for the sake of
comparison (as the analysis will be focused on FOPID controllers),
we consider the IOPID controller in series form, i.e.,
CðsÞ ¼ Kp
Tisþ1
Tis
Tdsþ1
Tf sþ1
ð3Þ
(note that with the employed tuning rules described below it
results in Ti 44Td and therefore an equivalent IOPID controller in
ideal form can always be considered, i.e., the optimal IOPID
controller is unique). The typical control specification requires
that a predefined performance is obtained in the set-point fol-
lowing and load disturbance rejection task. In both cases, a
typical performance index related to the step responses is the
integrated absolute error [33], which yields, in general, a small
overshoot and a small settling time at the same time and is
defined as
Je ¼
Z 1
0
jeðtÞj dt ¼
Z 1
0
jrðtÞÀyðtÞj dt; ð4Þ
where r is the set-point signal and y is the process variable.
From another point of view, it is often essential that the
control system is also robust to changes in the process dynam-
ics. A commonly employed measure of the robustness of the
system is the maximum sensitivity, which represents the
inverse of the minimum distance of the loop transfer function
from the critical point (À1, 0) in the Nyquist plot and it is
defined as
Ms ¼ max
ωA½0;þ1Þ
1
1þCðsÞPðsÞ
: ð5Þ
For this reason, specific tuning rules have been devised for
each controller (1)–(3) in order to minimize the Je value subject
to constraints on the maximum sensitivity [23,29]. In parti-
cular, both the set-point following and the load disturbance
Fig. 2. Resulting values of RFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243230
4. rejection tasks have been considered separately and, for each
task, the values of Ms ¼1.4 and Ms ¼2.0 have been selected
(note that tuning rules related to integral and unstable pro-
cesses have been proposed in [34]). In general, it has been
shown that the FOPID controller provides a better performance
than the IOPID controller and the improvement is achieved by
using an integer-order integrator and a fractional derivative
order μ41 [23].
The fragility of the controller can be evaluated with respect to
either the robustness or the performance. By denoting as θ
0
c the
vector of the controller parameters (that is, θ
0
c ¼ ½Kp; Ti; Td; λ; μŠ for
the FOPID controller and θ
0
c ¼ ½Kp; Ti; TdŠ for the IOPID controller),
the loss of robustness of the control system when the controller
parameters are perturbed can be expressed by the so-called Delta-
Epsilon-Robustness-Fragility Index which is defined as
RFIΔε ¼
Mm
sΔε
M0
s
À1 ¼
maxfMsðð17δεÞθ
0
c Þg
Msðθ
0
c Þ
À1; ð6Þ
where Mm
sΔε is the extreme maximum sensitivity, that is, the
highest loss of robustness of the control system that occurs
when all the parameters of the controller can vary of the
same δε quantity with respect to their nominal values θ
0
c ,
considering all the possible combinations of the perturbed
parameters. On the contrary, Msðθ
0
c Þ is the nominal sensitivity,
that is, the sensitivity obtained with the nominal controller. It
appears that, an index RFIΔε ¼ 0 implies that the controller is
absolutely robustness-non-fragile. It is however recognized
that a reasonable variation of the parameters is up to 20%. For
this reason, a controller is considered to be robustness resilient
if its delta 20 robustness fragility index is less than 0.10 (that is,
RFIΔ20 o0:10), robustness non-fragile if RFIΔ20 r0:50 and
robustness fragile if RFIΔ20 40:50.
It is also important to evaluate the relative influence of a single
parameter on the robustness fragility. In order to do that, the
Parametric-Delta-Epsilon-Robustness-Fragility Index has been defined
as
RFI
pi
δε ¼
M
pi
sδε
M0
s
À1 ¼
maxfMsðð17δεÞpi; θ
0
c Þg
Msðθ
0
c Þ
À1: ð7Þ
Similar to the previous case, the loss of performance of the
control system when the controller parameters are per-
turbed (note again that the tuning rules minimize the int-
Fig. 3. Resulting values of RFI
Kp
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 231
5. egrated absolute error) can be expressed by the so-called
Delta-Epsilon-Performance-Fragility Index which is defined as
PFIΔε ¼
Jm
eΔε
J0
e
À1 ¼
maxfJeðð17δεÞθ
0
c Þg
Jeðθ
0
c Þ
À1; ð8Þ
where Jm
eΔε is the extreme performance and Je
0
is the nominal
performance. The relative influence of a δε variation of a single
controller parameter pi on the performance fragility of the control
system can be expressed by the following Parametric-Delta-Epsi-
lon-Performance-Fragility Index:
PFI
pi
δε ¼
J
pi
eδε
J0
e
À1 ¼
maxfJeðð17δεÞpi; θ
0
c Þg
Jeðθ
0
c Þ
À1: ð9Þ
Similarly again to the robustness case, by assuming a reasonable
threshold of 20%, a controller is considered to be performance
resilient if its delta 20 performance fragility index is less than 0.10
(that is, PFIΔ20 o0:10), performance non-fragile if RFIΔ20 r0:50
and performance fragile if RFIΔ20 40:50.
Remark 1. It is worth stressing that the fragility indices
obviously depend on the tuning of the IOPID or FOPID
parameters. For this reason, in order to provide meaningful
results, it is important to compare the FOPID and IOPID con-
trollers with parameters selected in order to optimize the same
performance index.
3. Robustness fragility
The robustness fragility have been evaluated for the FOPID
controllers in both series and parallel form and the results are
compared with the IOPID controller. In particular, FOPDT pro-
cesses have been considered. They are described by the
transfer function
PðsÞ ¼
K
Tsþ1
eÀ Ls
: ð10Þ
Then, for the normalized gain K¼1 and for different values
of the normalized dead time L=T in the interval ½0:1; 1Š, the
tuning rules for the minimization of integrated absolute error
of the set-point step response and of the load disturbance step
response have been applied. In both cases, the values of the
target maximum sensitivity are set to either Ms ¼1.4 or Ms ¼2.0.
Fig. 4. Resulting values of RFITi
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243232
6. Thus, for each process and for each of the three controllers,
four cases have been considered and the RFIΔε index has been
calculated for different values of δε, by iteratively considering
all the possible variations of the parameters.
For the sake of brevity, only the results related to the process
with normalized dead time L=T ¼ 0:5 are shown. In this case,
assuming K¼1, T¼1 and L¼0.5, we obtain, for the different con-
trol specifications, the controller parameters shown in Table 1.
Actually, the results related to the other processes are very similar
to them. Results are shown in Fig. 2 where it has to be stressed
that when the data is missing it means that the overall control
system is unstable. Thus, it can be easily noted that the FOPID
controller (both in series and parallel form) is much more
robustness fragile (thus, the fine tuning is more critical) than the
IOPID controller.
In order to evaluate better the influence of the single con-
troller parameters, the Parametric-Delta-Epsilon-Robustness-
Fragility Index RFI
pi
δε has also been computed for the different
controller parameters. Results are shown in Figs. 3–7 (note that
the IOPID controller does not include the λ and μ parameters).
It appears that the robustness fragility of the FOPID controllers
is less critical if one parameter at a time is fine tuned and, in
any case, the fractional order of the derivative term is the most
dangerous parameter, especially when the employed tuning
rule aims at achieving a more aggressive controller (namely,
the target Ms ¼2.0 has been selected) and when a FOPID con-
troller in parallel form is used. A discussion about this issue
will be done in Section 5.
4. Performance fragility
The same analysis done for the robustness fragility has been
performed also for the performance fragility, that is, for each
FOPDT process and for each of the three controllers with the four
different considered tuning rules, the PFIΔε index has been cal-
culated for different values of δε. Results related again to the
process with a normalized dead time L=T ¼ 0:5 are shown in Fig. 8.
The same conclusions as for the robustness fragility can be made
for the performance fragility. The FOPID controllers (especially
that in parallel form) are more sensitive than the IOPID controller
with respect to changes in the parameters so that their fine tuning
can be more critical. By evaluating the Parametric-Delta-Epsilon-
Performance-Fragility Index PFI
pi
δε for each single parameter (see
Fig. 5. Resulting values of RFITd
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 233
7. Figs. 9–13), it can be deduced that the FOPID controllers are more
sensitive for changes in the fractional order of the integral and,
most of all, of derivative terms than in the other parameters.
5. Discussion
In the previous sections it has been pointed out that the fra-
gility of the FOPID controllers is mainly motivated by the presence
of the fractional-order derivative term. The reasons for this are
analyzed in this section by means of an illustrative example.
Consider the process
PðsÞ ¼
1
sþ1
eÀ0:5s
ð11Þ
and the tuning rules applied devised for the load disturbance
rejection with a target maximum sensitivity of Ms ¼2.0. For the
parallel FOPID controller (the series form is omitted for the
sake of brevity, but results are very similar to the parallel case)
these yield Kp ¼2.361, Ti ¼0.908, Td ¼0.144, λ ¼ 1, μ ¼ 1:153,
while for the IOPID controller we obtain Kp ¼1.008, Ti ¼0.411,
and Td ¼0.330. From the analysis of the Bode plots obtained in
the nominal case, shown in Fig. 14, it is evident that the FOPID
controller allows an increment of the bandwidth with respect
to the IOPID controller (the gain crossover frequency is ωgc ¼
2:01 for the FOPID controller and ωgc ¼ 1:73 for the IOPID
controller), with the same level of robustness. This is achieved
by exploiting the phase advance introduced by the fractional
derivative of order greater than one. While this implies a better
performance in the step response, it is also evident that the
frequency response function monotonicity is no longer guar-
anteed (see Fig. 14) because of the increased high frequency
roll-up that the fractional differentiator may exhibit and this
implies an incremented fragility of the controller. This can be
better analyzed by considering the frequency derivative of the
magnitude of the loop transfer function, which results in
d CðjωÞPðjωÞ
35. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N2
r ðωÞþN2
i ðωÞ
D2
r ðωÞþD2
i ðωÞ
s
ð13Þ
Fig. 6. Resulting values of RFIλ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243234
44. dω
¼
À Kj jT2
ω
1þT2
ω2
3
2
ð15Þ
and
where the controller CðjωÞ has been expressed as
CðjωÞ ¼
NrðωÞþjNiðωÞ
DrðωÞþjDiðωÞ
ð17Þ
and
NrðωÞ ¼ 1þTiωλ cos
π
2
λ
þTiTdωλþ μ cos
π
2
ðλþμÞ
NiðωÞ ¼ Tiωλ sin
π
2
λ
þTiTdωλþ μ sin
π
2
ðλþμÞ
DrðωÞ ¼ Tiωλ cos
π
2
λ
þTiTf ωλþ 1
cos
π
2
ðλþ1Þ
DiðωÞ ¼ Tiωλ sin
π
2
λ
þTiTf ωλþ1
sin
π
2
ðλþ1Þ
ð18Þ
and
dNrðωÞ
dω
¼ TiλωλÀ1
cos
π
2
λ
þTiTdðλþμÞωλþ μÀ 1
cos
π
2
ðλþμÞ
dNiðωÞ
dω
¼ TiλωλÀ1
sin
π
2
λ
þTiTdðλþμÞωλþμÀ 1
sin
π
2
ðλþμÞ
Fig. 7. Resulting values of RFIμ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
d CðjωÞ
53. dDrðωÞ
dω
¼ TiλωλÀ1
cos
π
2
λ
þTiTf ðλþ1Þωλ cos
π
2
ðλþ1Þ
dDiðωÞ
dω
¼ TiλωλÀ1
sin
π
2
λ
þTiTf ðλþ1Þωλ sin
π
2
ðλþ1Þ
: ð19Þ
Actually, a deeper analysis can be performed by evaluating the
frequency response function when one parameter at a time changes in
the FOPID controller. Results are shown in Figs. 15–19. As expected
from the results shown in Figs. 7 and 13, it appears that the (fractional)
derivative action is the most critical one. In particular, the increment of
the fractional order μ leads to high-frequency peaks in the sensitivity
function, to a loop gain with multiple gain crossover frequencies and,
eventually, to instability. Indeed, μ is the only parameter that is able to
destabilize the loop in spite of variations smaller than 30%. This hap-
pens because increasing μ also means an increased non-monotonic
behavior of the frequency response function (see Fig. 19) as a con-
sequence of the increased high frequency roll-up. Another critical
parameter is the derivative time constant Td. Indeed the optimal FOPID
controller has a derivative action with a derivative order μ greater than
1. This means that the optimal frequency response function is already
non-monotonic and an increased derivative time constant pushes up
the frequency response close to the 0 dB axes (see Fig. 17) creating
again high frequency peaks in the sensitivity function with a con-
sequent loss of robustness and performance. Again, this behavior is
expected from the results shown in Fig. 5 where robustness fragility is
considered. It can be appreciated that the IOPID controller always
results in less fragile compared to the FOPID one. This happens
because of its monotonic behavior.
On the contrary, variations in the integrator order do not generate
dramatic changes in the frequency response function (see Fig. 16).
Indeed, the monotonicity of the frequency response function is inde-
pendent from the selected value of λ, unless Ti⪢Td and a series FOPID
controller is considered, but, evidently, this is not a meaningful tuning
of the controller. This is inde66pc4.68 pendent from the fact that the
optimal FOPID controller is obtained with λ ¼ 1.
Summarizing, the relevant difference between FOPID and IOPID
controllers is that the former ones are capable (indeed because of
the fractional-order derivative action) of providing a reduction of
the integrated absolute error but their fragility should be carefully
considered when the tuning is performed and, in particular, when
the fractional-order derivative term is changed, as a small varia-
tion can modify the performance significantly.
Fig. 8. Resulting values of PFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243236
54. Fig. 9. Resulting values of PFI
Kp
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 237
55. Fig. 10. Resulting values of PFITi
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243238
56. Fig. 11. Resulting values of PFITd
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 239
57. Fig. 12. Resulting values of PFIλ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243240
58. Fig. 13. Resulting values of PFIμ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 241
59. Fig. 14. Magnitude Bode plots in the nominal case for the illustrative example.
Solid line: FOPID controller. Dashed line: IOPID controller.
Fig. 15. Magnitude Bode plots for the illustrative example (FOPID controller) when
the proportional gain Kp changes in the range 730%.
Fig. 16. Magnitude Bode plots for the illustrative example (FOPID controller) when
the integral time constant Ti changes in the range 730%.
Fig. 17. Magnitude Bode plots for the illustrative example (FOPID controller) when
the derivative time constant Td changes in the range 730%.
Fig. 18. Magnitude Bode plots for the illustrative example (FOPID controller) when
the fractional integral order λ changes in the range 730%.
Fig. 19. Magnitude Bode plots for the illustrative example (FOPID controller) when
the fractional derivative order μ changes in the range 730%.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243242
60. 6. Conclusions
As the parameters of IOPID controllers have a clear physical
meaning and the relative easiness of their manual tuning has
determined (among other factors) their success, it is believed that
the same feature should be provided for FOPID controllers in order
to allow a more widespread use of them in industry. A key role in
this context is played by the robustness and performance fragility
of this kind of controllers which have been analyzed in this paper
in order to evaluate the criticalness of the fine tuning of the
parameters. It has been highlighted that FOPID controllers are
more fragile than IOPID controllers and a special attention should
be paid especially in changing the fractional order of the derivative
term. Thus, research effort should be provided in the future in
order to devise tuning rules for FOPID controllers that are able to
guarantee a resilient robustness and performance.
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