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Optimal tuning rules for proportional-integer-derivative and fractional-order proportional-integer-derivative controllers for integral and unstable processes

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A set of tuning rules for standard (integer-order) proportional-integral-derivative(PID) and fractional-order PID controllers for integral and unstable processes is presented in this study. Based on a simple model of the process, the tuning rules have been devised in order to minimise the integrated absolute error. Both set-point tracking and load disturbance rejection tasks are considered. The achieved performance indexes can also be used for the assessment of the controller performance. A remarkable feature of the optimisation procedure employed for integral processes is highlighted. The provided results allow the user to quantify, for a given process, the performance improvement that can be obtained by using the fractional controller instead of the integer one.
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Published in IET Control Theory and Applications
Received on 14th July 2011
Revised on 31st October 2011
doi: 10.1049/iet-cta.2011.0419
ISSN 1751-8644
Optimal tuning rules for proportional-integral-
derivative and fractional-order proportional-integral-
derivative controllers for integral and unstable
processes
F. Padula A. Visioli
Dipartimento di Ingegneria dell’Informazione, University of Brescia-Italy, Via Branze 38, I-25123 Brescia, Italy
E-mail: antonio.visioli@ing.unibs.it
Abstract: A set of tuning rules for standard (integer-order) proportional-integral-derivative (PID) and fractional-order PID
controllers for integral and unstable processes is presented in this study. Based on a simple model of the process, the tuning
rules have been devised in order to minimise the integrated absolute error. Both set-point tracking and load disturbance
rejection tasks are considered. The achieved performance indexes can also be used for the assessment of the controller
performance. A remarkable feature of the optimisation procedure employed for integral processes is highlighted. The provided
results allow the user to quantify, for a given process, the performance improvement that can be obtained by using the fractional
controller instead of the integer one.
1 Introduction
Proportional-integral-derivative (PID) controllers are surely
the most adopted controllers in industry because of the
cost/benefit ratio they are capable to provide (note that
they are also used in the context of model predictive
control, which usually provide the set-point to low-level
PID feedback control loops). Indeed, the large numbers of
tuning rules [1] for their three parameters and the presence
of reliable automatic tuning techniques [2] allow the user to
design this kind of controllers with a moderate effort and to
obtain a satisfactory performance for many processes.
In the last years, the design of fractional-order
proportional-integral-derivative (FOPID) controllers has
been the subject of many investigations (see e.g. [37])
because of the additional flexibility they are capable
to provide with respect to standard (integer-order) PID
controllers. Indeed, the presence of five parameters to
select makes the achievement of an increased performance
virtually possible but this also implies that the tuning of the
controller can be much more complex. In order to address
this problem, different methods for the design of a FOPID
controller have been proposed in the literature (where
different objective functions are considered) [813] and
different tuning rules have been proposed [1419]. Among
them, the tuning rules proposed in [20] have very important
features that the provided control action is invariant when
the time scale is changed [19].
However, all the tuning rules proposed in the literature,
at least to the authors’ knowledge, are related only
to self-regulating (i.e. asymptotically stable) processes,
whereas non-self-regulating (i.e. integral) processes and
unstable processes have been overlooked. Actually, unstable
processes have been considered in [21], but, therein, only
the stabilisation issue has been addressed. Indeed, this
kind of processes is frequently encountered in the process
industry. Typical examples of integral processes include
tanks, where the level is controlled by manipulating the
difference between the input and output flow rates, and
batch distillation columns [22], whereas typical examples
of unstable processes are continuous stirred tank reactors,
polymerisation reactors and bioreactors [23].
Thus, by following an approach similar to that proposed
in [20], in this paper we propose a new set of tuning rules,
based on the minimisation of the integrated absolute error
[24], for PID and FOPID controllers for integral and unstable
processes (the case where both an integral and an unstable
mode is present in the process has not been considered).
The peculiar feature of the optimisation procedure employed
for integral processes is highlighted. Both the set-point
tracking and the load disturbance rejection tasks will be
considered explicitly. It is worth noting that these two tasks
are considered separately so that the user can achieve an
optimal performance when only one of two tasks is of
(main) concern. In case, in a given application, both tasks
are important, starting from the results achieved in this
context, a two degree-of-freedom controller can be designed
(namely, a set-point weight can be employed [25,26]) or,
alternatively, a weighted servo/regulation control approach
can be pursued [27]. Further, an analytical expression of the
performance index is also given and this can be exploited in
a performance assessment context. It will be shown that, in
776 IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 776–786
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this context, as for first-order-plus-dead-time (FOPDT) self-
regulating processes, in order to improve the performance
the use of a fractional integral action is not useful whereas
it is worth using a fractional derivative action.
In any case, as a very significant feature, the obtained
results allow the user to quantify the improvement of
the performance that can be obtained by using a FOPID
controllers instead of a standard PID controller and therefore
to characterise their cost/benefit ratio.
The paper is organised as follows. In Section 2 the
problem is formulated. The tuning rules for integrator-plus-
dead-time (IPDT) processes are described in Section 3,
whereas those for unstable first-order-plus-dead-time
(UFOPDT) processes are presented in Section 4. Simulation
results are shown in Section 5 and conclusions are drawn in
Section 6.
2 Problem formulation
In this section the problem is formulated for both unstable
and integral processes. We consider the unity-feedback
control scheme of Fig. 1. In case an integral process
is considered, the process is assumed to have an IPDT
dynamics, namely
P(s)=K
seLs (1)
where, evidently, Kis the gain and Lis the dead time.
In case an unstable process is considered, the process is
assumed to have a UFOPDT dynamics, namely
P(s)=K
Ts 1eLs (2)
where Tis the time constant. The process dynamics (2) can
be conveniently characterised by the normalised dead time
defined as L/T. Hereafter we consider unstable processes
with 0.05 L/T1 which is a sensible range within the
stabilisable range by considering a PID controller [28].
Two types of FOPID controllers are considered: that in
series form, defined as
C(s)=Kp
Tisλ+1
Tisλ
Tdsμ+1
Tfs+1(3)
and that in ideal form, defined as
C(s)=Kp1+1
Tisλ+Tdsμ1
Tfs+1(4)
where Kpis the proportional gain, Tiis the integral time
constant, Tdis the derivative time constant, λand μare
the non-integer orders of the integral and derivative terms,
respectively, and Tfis the time constant of an additional
first-order filter whose meaning is explained below. It is
worth noting that, by selecting λ=μ=1, a standard PID
C
ry
eP
d
Fig. 1 Considered control scheme
controller in ideal and series form is obtained, respectively.
In particular, the integer-order series form
C(s)=Kp
Tis+1
Tis
Td+1
Tfs+1(5)
will be considered hereafter, because the resulting optimal
PID controllers have in all the considered cases Ti>4Td.
This means that the PID controller in series form can be
always converted in a PID controller in ideal form by
employing suitable formulae (see e.g. [25]). On the contrary,
by expanding (3), it can be easily noted that, in general,
there are no conversion formulae between the series and
ideal form of a FOPID controller.
The additional first-order filter has been employed in (4)
and (3) in order to make the controller proper [25]. The
value of the time constant Tfhas been selected as
Tf=min T1
i
10 ,T1
d
10 (6)
so that the corresponding pole (which, in the series form,
is ten time faster than the fastest zero of the controller)
does not influence the controller dynamics significantly
and it does filter the high-frequency noise at the same
time (note that this is a typical choice in standard PID
controllers [29]). Also note that the filter is an integer-order
first-order system, because, for the implementation of the
fractional-order controller, the Oustaloup continuous integer-
order approximation [30] has been employed. It consists
in using the following approximation based on a recursive
distribution of zeros and poles
sν
=k
N
n=1
1+(sz,n)
1+(sp,n),ν>0 (7)
which is valid in a frequency range [ωl,ωh]and where the
gain kis adjusted so that both sides of (7) have the same
gain in the mid point of the interval [ωl,ωh]. In this paper
the value N=8 has been chosen, while ωland ωhhave been
selected as 0.001ωcand 1000ωc, respectively, where ωcis
the gain crossover frequency of the loop transfer function.
The specified control requirement is to minimise the (set-
point ror load disturbance d) step response integrated
absolute error [24]
IAE =
0
|e(t)|dt=
0
|r(t)y(t)|dt(8)
Obviously, minimising (8) implies that the process is
stabilised.
For integral processes, aiming at just obtaining the
theoretical minimum integrated absolute error that can be
achieved for the single-loop system might not be sensible in
practical cases because the robustness issue and the control
effort have also to be taken into account. For this reason,
the devised tuning of the parameters aim at minimising the
integrated absolute error by constraining at the same time
the maximum sensitivity (as in the well-known Kappa–Tau
tuning rules for standard PID controllers [31]), which is
defined as
Ms=max
ω∈[0,+∞)
1
1+C(s)P(s)(9)
and which represents also the inverse of the maximum
distance of the Nyquist plot from the critical point (1, 0).
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Obviously, the higher the value of Msis, the less robust is
the system to modelling uncertainties.
This approach has not been pursued for unstable processes
because constraining the maximum sensitivity might prevent
the stabilisation of the control system and, in any case,
minimising the integrated absolute error yields, for an
unstable process, a satisfactory robustness.
3 Optimal tuning rules for integral processes
In order to find the tuning rules for the minimisation of
the integrated absolute error by constraining the maximum
sensitivity value, the following approach has been used.
First, the set-point tracking and the load disturbance
rejection tasks have been considered separately. Second, the
values of the parameters of the FOPID and PID controllers
have been found by means of a genetic algorithm [32],
which is known to provide a global optimum of a problem in
a stochastic frame. The objective function to be minimised is
the integrated absolute error in a step response, whereas two
typical values of the maximum sensitivity, namely, Ms=1.4
and =2 are used as constraints [31]. From a practical point
of view, selecting Ms=1.4 implies that a more robust and
less aggressive controller result (from another point of view,
a slower response with a lower control effort is obtained),
whereas selecting Ms=2.0 implies that the controller is less
robust and more aggressive (that is, a faster response with
a more significant control effort is obtained).
Formally, the optimisation to be solved can be stated as
follows
min
Kp,Ti,Td,λ,μ
0
|e(t)|dt(10)
s.t.
MsM(11)
where the two cases M=1.4 or 2.0 have been considered
and where, obviously, λand μare set equal to one and they
are not selected as optimisation parameters if an integer-
order PID controller is considered.
It is worth stressing that a solution of the optimisation
problem (10)–(11) always exists. In fact, a stabilising
(unconstrained) PID controller always exists and the
stabilising Kpcan be reduced up to zero, whereas the
stabilising Tican be freely increased [28,33]. Taken this
into account, it is clear that, by reducing Kpand increasing Ti
(and by noting that Td,λand μare optimisation parameters)
it is always possible to reduce the maximum sensitivity level
up to 1, preserving the closed-loop stability.
In this context, the genetic algorithm can be applied to a
normalised process transfer function where ¯s=Ls, that is
¯
P(s)=KL
¯se−¯s(12)
so that the optimisation can be performed just once on
the process (12) and then the resulting parameters have
simply to be scaled by Lin order to obtain a set of general
tuning rules (note that the gain Kcan be neglected in
the optimisation procedure provided that the value of the
proportional gain Kpis eventually divided by K).
The genetic algorithm applied for this purpose is the one
implemented in the Matlab Global Optimization Toolbox
[34] with an initial population of 600 individuals, 250
generations and with default values for the other options.
For each individual of the genetic algorithm, the stability
of the closed-loop system is checked before determining the
step response. If the selected parameters do not stabilise the
system, the fitness function assumes a very high value so
that unstabilising parameters are automatically discarded by
the algorithm. The constraint on the maximum sensitivity is
guaranteed to be satisfied by using an augmented Lagrangian
approach [34]. Further, the overall genetic algorithm has
been applied a few times in order to be sure that the global
optimum is indeed achieved (as the results are the same in
the different repetitions).
The tuning rules and the performance indexes obtained in
the different cases are reported in the next subsections. It is
worth noting that the analytical expressions of the optimal
performance indexes can also be used for the assessment
of the controller performance. In fact, the performance
assessment of a control loop is generally performed by
first calculating a performance index based on the available
data and then by evaluating the current control performance
against a selected benchmark, which represents the desired
performance [25,35].
3.1 Set-point tracking task
If only the set-point tracking task is of concern, the results
obtained by applying the optimisation procedure show that
there is no need of using the (possibly fractional) integral
action. Indeed, the pole at the origin of the complex plane
in the loop transfer function that ensures a null steady-state
error with a constant set-point value is already present in
the process and therefore there is no need to add it in the
controller. The optimal tuning rules obtained by means of
the genetic algorithm for the fractional-order PD controller
are (note that, if there is no integral action the ideal and
series forms of the fractional controller are equivalent)
Kp=a
KL (13)
Td=bLμ(14)
and the optimal value of the performance index IAE is
IAEopt =AskL (15)
where the values of the parameters a,b,kand of μare
shown in Table 1and Asis the amplitude of the set-point
step.
Regarding the integer-order PD tuning rules, the following
expressions have been obtained
Kp=a
KL (16)
Td=bL (17)
and the optimal value of the performance index IAE is again
IAEopt =AskL (18)
where the values of the parameters are shown in Table 2.
Table 1 Tuning rules and performance index parameters for
FOPD controllers for set-point tracking task for integral
processes
Msabμk
1.4 0.5962 0.3354 1.20 1.804
2.0 0.8699 0.4494 1.15 1.344
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Table 2 Tuning rules and performance index parameters for
PD controllers for set-point tracking task for integral processes
Msabk
1.4 0.4745 0.3300 2.110
2.0 0.7399 0.5061 1.435
By comparing (15) and (18) (with the related values of k),
it appears that the presence of the fractional derivative action
allows a performance improvement of 17.2% for Ms=1.4
and of 6.34% for Ms=2.0.
3.2 Load disturbance rejection task
When the (constant) load disturbance rejection task is
considered, the (possibly fractional) integral action has to be
employed to ensure a null steady-state error. The resulting
optimal tuning rules obtained by means of the genetic
algorithm for the FOPID controller are
Kp=a
KL (19)
Ti=cLλ(20)
Td=bLμ(21)
and the optimal value of the performance index IAE is
IAEopt =AdkKL2(22)
where Adis the amplitude of the load disturbance step and
the values of the parameters λand μare shown in Table 3
for the FOPID controller in ideal form and in Table 4for
the FOPID controller is series form. Note that in both cases
λ=1 and μ= 1, that is, it is worth using a fractional-order
derivative action and an integer-order integral action.
Regarding the integer-order PID tuning rules, the
following expressions have been obtained
Kp=a
KL (23)
Ti=cL (24)
Td=bL (25)
and the optimal value of the performance index IAE is again
IAEopt =AdkKL2(26)
Table 3 Tuning rules and performance index parameters for
FOPID controllers in ideal form for load disturbance rejection
task for integral processes
Msabcλμ k
1.4 0.5636 0.4170 4.6206 1 1.15 8.75
2.0 1.0357 0.3723 3.1698 1 1.18 3.19
Table 4 Tuning rules and performance index parameters for
FOPID controllers in series form for load disturbance
rejection task for integral processes
Msabcλμ k
1.4 0.5106 0.4489 3.9856 1 1.15 8.79
2.0 0.8015 0.4863 1.9206 1 1.15 3.53
Table 5 Tuning rules and performance index parameters for
PID controllers in series form for load disturbance rejection
task for integral processes
Msabck
1.4 0.4058 0.5267 3.5035 10.817
2.0 0.6718 0.5099 2.2727 4.129
where the values of the parameters are shown in Table 5.
In this case the presence of the fractional derivative action
allows a performance improvement, with respect to the
standard integer-order PID controller, of 19.1% for Ms=1.4
and of 22.7% for Ms=2.0 for the FOPID controller in ideal
form and of 18.8% for Ms=1.4 and of 14.6% for Ms=2.0
for the FOPID controller in series form [see (22) and (26)].
4 Optimal tuning rules for unstable
processes
The optimisation procedure employed to solve the
optimisation problem (10) for unstable processes is more
complex than that employed for integral processes. Indeed,
in order to find the tuning rules that give the values of
the controller parameters which minimise the integrated
absolute error an approach similar to that employed in [20]
has been used. In particular, the set-point tracking and the
load disturbance rejection tasks have been considered again
separately and different processes with different values of
the normalised dead time have been considered. For each of
them, the values of the parameters of the FOPID and PID
controllers have been found by means of a genetic algorithm.
The same considerations done in Section 3 can be applied
also in this case: a large population has been employed and
the application of the genetic algorithm has been repeated in
order to ensure that a global optimum is achieved. Further,
the stability of the control system has been checked for each
trial of the genetic algorithm and, in case an unstable system
results, the value of the fitness function has been increased
significantly. It is worth stressing in any case that a solution
of the optimisation problem exists because a stabilising PID
controller exists for the processes considered [28] and λand
μare tuning parameters.
Eventually, for each considered controller, the optimal
coefficients found for the different values of L/Thave
been interpolated in order to derive suitable tuning rules.
The obtained performance indexes have been interpolated as
well. In this context, different interpolating functions have
been considered [36], by taking into account the aim of
providing tuning rules where, as for the integral case, the
resulting performance is scaled by the time constant T.In
other words, with the same normalised dead time, the value
of the integrated absolute error that is obtained for a given
value of Tis equal to Ttimes the value of the integrated
absolute error that is obtained for T=1. The tuning rules
and the performance indexes obtained in the different cases
are reported in the next subsections.
4.1 Set-point tracking task
If only the set-point tracking task is of concern, the optimal
tuning rules obtained by means of the genetic algorithm for
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the FOPID and PID controller are
Kp=1
Kaexp bL
T+cL
Td(27)
Ti=aexp bL
T+cexp dL
TTλ(28)
Td=aL
Tb
+cTμ(29)
where the values of the tuning rule parameters are shown
in Tables 6and 7for FOPID controllers in series and
ideal form, respectively, and in Table 8for PID controller.
Obviously it is λ=μ=1 for the standard integer-order PID
controllers. For the FOPID controller, it is
λ=1 (30)
and
μ=−0.008233 L
T2
0.05605 L
T+1.205 (31)
for the series form, and
μ=0.1301 L
T2
0.1996 L
T+1.216 (32)
for the ideal form. As an illustrative example, the optimal
values of the Kpparameter found by the genetic algorithm
for the FOPID case and the corresponding interpolating
function are shown in Fig. 2. Similar results have been
obtained for the other parameters and they are therefore not
shown for the sake of brevity.
Table 6 Tuning rules parameters for FOPID controllers in
series form for set-point tracking task for unstable
processes
Parameter ab c d
Kp2.363 0.6377 2.693 0.6977
Ti0.4528 3.127 7.359 ×1016 25.32
Td0.5011 1.303 0.004218
Table 7 Tuning rules parameters for FOPID controllers in
ideal form for set-point tracking task for unstable processes
Parameter ab c d
Kp1.065 0.9063 1.051 1.088
Ti0.5659 2.942 0.6172 4.655
Td0.4884 1.35 0.001938
Table 8 Tuning rules parameters for PID controllers in
series form for set-point tracking task for unstable processes
Parameter abcd
Kp0.07809 0.9958 1.035 0.9305
Ti6.107×10712.49 0.4247 3.031
Td0.5522 1.026 0.006063
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5
10
15
20
L/T
Kp
Fig. 2 Optimisation results (‘+’) and interpolation curve for Kp
with unstable processes and FOPID controller
Table 9 Parameters of the IAE performance index
expression for the set-point tracking task for unstable
processes
Controller abcd
FOPID series 0.0429 5.9277 2.8148 1.1667
L/T0.4
FOPID series 0.3255 3.8420 5.3195 4.5204
L/T>0.4
FOPID ideal 0.1137 4.3033 0.2588 0.2205
L/T0.4
FOPID ideal 4.9722 ×10613.2717 4.9926 2.7626
L/T>0.4
PID 0.0288 7.1112 3.0542 1.1122
L/T0.4
PID 0.1153 4.6010 2.7518 1.2484
L/T>0.4
The optimal value of the performance index IAE can be
expressed as
IAEopt =Asaexp bL
T+cL
TdT(33)
where the values of the parameters a,b,cand dfor FOPID
and PID controllers are shown in Table 9and Asis the
amplitude of the set-point step. The unit step response
IAE values obtained for different normalised dead times
are shown in Fig. 3. It appears that the FOPID controller
(both in ideal or series form) is capable of providing
a better performance than the standard integer-order PID
controller. The improvement of the performance [which
can be quantified by considering expression (33) for the
different cases] for series FOPID is more significant when
the normalised dead time of the process increases, as it is
shown in Figs. 3and 4.
The ideal FOPID controller shows a behaviour similar to
the series one when the value of the normalised dead time
is in the middle of the admissible range. On the contrary,
for small values of the normalised dead time it is convenient
to use the ideal form, whereas the opposite is true for big
values of the normalised dead time.
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
L/T
IAE
Fig. 3 Values of the normalised IAE obtained for FOPID in series
form (solid line), FOPID in ideal form (dash-dot line) and PID
(dashed line) controllers for the set-point tracking task for unstable
processes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5
10
15
20
25
30
35
40
45
L/T
Δ IAE [%]
Fig. 4 Percentage improvement obtained by the FOPID in series
form (continuous line) and the FOPID in ideal form (dashed line)
controllers with respect to the PID controller for the set-point
tracking task for unstable processes
It is worth noting that, as for stable FOPDT systems
and for IPDT systems the value λ=1 results for FOPID
controllers, that is, it is just the fractional derivative action
that is useful in improving the performance.
4.2 Load disturbance rejection task
If the load disturbance rejection task is considered, the same
expressions (27)–(29) for the tuning rules can be employed.
The related parameters are shown in Tables 1012 for the
FOPID and PID controllers, respectively. Again, for FOPID
controllers it is λ=1 while it is useful to employ a fractional
derivative action where the derivative order is
μ=−0.04783 L
T2
0.01017 L
T+1.218 (34)
Table 10 Tuning rules parameters for FOPID controllers in
series form for load disturbance rejection task for unstable
processes
Parameter ab c d
Kp0.4346 0 1.947 0.7747
Ti0.2736 3.491 0.5366 13.33
Td0.5038 1.298 0.008506
Table 11 Tuning rules parameters for FOPID controllers in
ideal form for load disturbance rejection task for unstable
processes
Parameter ab c d
Kp1.602 1.255 1.075 1.126
Ti0.5207 2.875 0.5913 3.752
Td0.4939 1.383 0.002607
Table 12 Tuning rules parameters for PID controllers in
series form for load disturbance rejection task for unstable
processes
Parameter ab c d
Kp5.418 8.539 1.268 0.6704
Ti0.2125 3.758 0.4673 16.62
Td0.5786 0.9149 0.0006551
Table 13 Parameters of the IAE performance index
expression for the load disturbance rejection task for
unstable processes
Controller abcd
FOPID series 0.0017 11.2262 2.0663 2.0247
L/T0.4
FOPID series 0.0365 6.3840 15.0042 7.4056
L/T>0.4
FOPID ideal 7.6922×10410.4503 2.8310 2.2116
L/T0.4
FOPID ideal 8.5461×10612.7409 4.9206 2.7406
L/T>0.4
PID 0.0024 11.0222 2.5593 2.0682
L/T0.4
PID 0.0503 6.1840 14.2533 7.3901
L/T>0.4
for FOPID controllers in series form, and
μ=0.06326 L
T2
0.1508 L
T+1.245 (35)
for FOPID controllers in ideal form. The optimal value of
the performance index IAE can be expressed as
IAEopt =Adaexp bL
T+cL
TdTK (36)
where Adis again the amplitude of the load disturbance step
and the values of the parameters a,b,cand dfor FOPID
and PID controllers are shown in Table 13. The unit step
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
0
2
4
6
8
10
12
L/T
IAE
Fig. 5 Values of the normalised IAE obtained for FOPID in series
form (solid line), FOPID in ideal form (dash-dot line) and PID
(dashed line) controllers for the load disturbance rejection task
for unstable processes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
20
30
40
50
60
70
L/T
Δ IAE [%]
Fig. 6 Percentage improvement obtained by the FOPID in series
form (continuous line) and the FOPID in ideal form (dashed
line) controllers with respect to the PID controller for the load
disturbance rejection task for unstable processes
response IAE values obtained for different normalised dead
times and the resulting improvement of the performance
obtained by using the fractional derivative action are plotted
in Figs. 5and 6, respectively. It appears that in this case the
use of the ideal form is more convenient for large range of
the values of the normalised dead time.
5 Simulation results
5.1 Example 1
As a first illustrative example, we consider the following
integral process [37]
P1(s)=0.0506
se6s(37)
Table 14 Results related to the process P1(s)
Controller KpTiTdλμ IAE Ms
FOPD SP 1.4 1.96 2.88 1.2 10.74 1.43
PD SP 1.4 1.57 1.98 12.65 1.42
FOPD SP 2.0 2.87 3.53 1.15 8.09 1.99
PD SP 2.0 2.44 3.04 8.60 1.98
series FOPID LD 1.4 1.68 23.91 3.52 1 1.15 15.92 1.43
ideal FOPID LD 1.4 1.86 27.72 3.27 1 1.15 15.88 1.42
PID LD 1.4 1.34 21.02 3.16 19.68 1.42
series FOPID LD 2.0 2.64 11.52 3.82 1 1.15 6.29 2.07
ideal FOPID LD 2.0 3.41 19.02 3.08 1 1.18 5.80 2.05
PID LD 2.0 2.21 13.64 3.06 7.52 2.02
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
time
process variable
0 5 10 15 20 25 30 35 40
−10
0
10
20
30
time
control variable
Fig. 7 Set-point unit step response for process P1(s)
Solid line: FOPD with Ms=1.4; dashed line: PD with Ms=1.4
The different controllers considered in Section 3 have been
tuned by the corresponding tuning formulae. Then, both
the set-point step response and the load disturbance step
response have been evaluated separately in all the cases
(note that a unit step signal has been always applied). The
results for the different cases are summarised in Table 14,
where the obtained values of the controller parameters as
well as the obtained value of the integrated absolute error
and of Msin both the set-point and load disturbance step
response are shown. Note that the tuning rule employed
is described as SP or LD (which means that the set-point
tracking or the load disturbance rejection task is addressed,
respectively) followed by the target maximum sensitivity.
The set-point and load disturbance step responses are shown
in Figs. 710 for the different cases.
It appears that, as expected, the FOPID controller provides
a better performance than the integer-order one, even if
with a more nervous control action (with the same level
of robustness) which has to be possibly taken into account
in a given application.
The obtained performance can be compared with the
one obtained in [37] (an ideal form is employed in this
case), where, when the tuning for the integrated square
error minimisation is employed, we have IAE =11.3 with
Ms=3.09 for the set-point step response and IAE =4.85
with Ms=7.58 for the load disturbance step response. It
appears that if the maximum sensitivity is not constrained,
then the robustness of the control system can be very poor.
782 IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 776–786
© The Institution of Engineering and Technology 2012 doi: 10.1049/iet-cta.2011.0419
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0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
time
process variable
0 5 10 15 20 25 30 35 40
−10
0
10
20
30
40
time
control variable
Fig. 8 Set-point unit step response for process P1(s)
Solid line: FOPD with Ms=2.0; dashed line: PD with Ms=2.0
0 50 100 150
−0.2
0
0.2
0.4
0.6
time
process variable
0 50 100 150
−1.5
−1
−0.5
0
0.5
time
control variable
Fig. 9 Load disturbance unit step response for process P1(s)
Solid line: series FOPID with Ms=1.4; dash-dot line: ideal FOPID
with Ms=1.4; dashed line: PID with Ms=1.4
0 50 100 150
−0.1
0
0.1
0.2
0.3
0.4
time
process variable
0 50 100 150
−2
−1.5
−1
−0.5
0
0.5
time
control variable
Fig. 10 Load disturbance unit step response for process P1(s)
Solid line: series FOPID with Ms=2.0; dash-dot line: ideal FOPID
with Ms=2.0; dashed line: PID with Ms=2.0
Table 15 Results related to the process P2(s)
Controller KpTiTdλμ IAE Ms
FOPD SP 1.4 0.07 4.07 1.2 15.33 1.45
PD SP 1.4 0.06 2.64 16.87 1.42
FOPD SP 2.0 0.11 4.91 1.15 12.06 1.73
PD SP 2.0 0.09 4.05 12.26 1.78
series FOPID LD 1.4 0.06 31.88 4.91 1 1.15 566.2 1.55
ideal FOPID LD 1.4 0.07 36.96 4.56 1 1.15 560.2 1.54
PID LD 1.4 0.05 28.03 4.21 706.4 1.51
series FOPID LD 2.0 0.10 15.36 5.31 1 1.15 261.6 2.49
ideal FOPID LD 2.0 0.13 25.36 4.33 1 1.18 229.8 2.32
PID LD 2.0 0.08 18.18 4.08 286.0 2.19
0 5 10 15 20 25 30 35 40 45 50
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
time
process variable
0 5 10 15 20 25 30 35 40 45 50
0
0.5
1
1.5
time
control variable
Fig. 11 Set-point unit step response for process P2(s)
Solid line: FOPD with Ms=1.4; dashed line: PD with Ms=1.4
5.2 Example 2
As a second illustrative example, the following high-order
integral process is considered
P2(s)=1
s(s+1)8es(38)
In order to apply the tuning rules proposed in Section 3, an
IPDT model (1) has been estimated with K=1 and L=
8. Then, as for Example 1, the different controllers have
been tuned by the corresponding tuning formulae and the
set-point unit step response and the load disturbance unit
step response have been evaluated. The results obtained for
the different cases are summarised in Table 15 (note that
the obtained value of the maximum sensitivity is obviously
different from the target one because of the low-order model
approximation), whereas the set-point and load disturbance
step responses are shown in Figs. 1114 for the different
cases.
It appears that the provided tuning rules can address the
robustness issue satisfactorily.
5.3 Example 3
As a third illustrative example, we consider the following
unstable process [37]
P3(s)=1
s1e0.2s(39)
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doi: 10.1049/iet-cta.2011.0419 © The Institution of Engineering and Technology 2012
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0 5 10 15 20 25 30 35 40
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
time
process variable
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
time
control variable
Fig. 12 Set-point unit step response for process P2(s)
Solid line: FOPD with Ms=2.0; dashed line: PD with Ms=2.0
0 50 100 150
0
5
10
15
time
process variable
0 50 100 150
−1.5
−1
−0.5
0
time
control variable
Fig. 13 Load disturbance unit step response for process P2(s)
Solid line: series FOPID with Ms=1.4; dash-dot line: ideal FOPID
with Ms=1.4; dashed line: PID with Ms=1.4
where it is evident that K=1, T=1 and L=0.2 [see
(2)]. The tuning rules presented in Section 4have then been
applied for the FOPID and PID controllers and the results for
the different cases are summarised in Table 16, together with
the integrated absolute errors obtained in the unit set-point
and load disturbance step responses (considered separately).
Figs. 15 and 16 show the process and control variables for
the different cases.
It appears that, as for integral processes, the FOPID
controller provides a better performance than the integer-
order one (with a more nervous control action).
Also in this case we can compare the performance
obtained with the one obtained in [37] (with the tuning
selected in order to minimise the integrated square error).
For the set-point step response we have IAE =0.8, whereas
for the load disturbance step response we have IAE =0.148.
Obviously, the integrated absolute error values are higher
in this case because a different objective function has been
employed in the optimisation procedure.
0 20 40 60 80 100 120
−5
0
5
10
15
time
process variable
0 20 40 60 80 100 120
−2
−1.5
−1
−0.5
0
0.5
time
control variable
Fig. 14 Load disturbance unit step response for process P2(s)
Solid line: series FOPID with Ms=2.0; dash-dot line: ideal FOPID
with Ms=2.0; dashed line: PID with Ms=2.0
Table 16 Results related to the process P3(s)
Controller KpTiTdλμ IAE
series FOPID SP 6.1965 0.8462 0.0658 1 1.1930 0.5659
ideal FOPID SP 6.9415 0.7759 0.0576 1 1.1808 0.5462
PID SP 4.7209 0.7786 0.1119 0.6253
series FOPID LD 6.3406 0.5126 0.0709 1 1.2136 0.0947
ideal FOPID LD 7.8350 0.6462 0.0560 1 1.2169 0.0868
PID LD 4.7125 0.4338 0.1334 1 1 0.1105
00.5 1 1.5 2
0
0.5
1
1.5
2
2.5
time
process variable
00.5 1 1.5 2
−20
0
20
40
60
80
time
control variable
Fig. 15 Set-point unit step response for process P3(s)
Solid line: series FOPID; dash-dot line: ideal FOPID; dashed line:
PID
784 IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 776–786
© The Institution of Engineering and Technology 2012 doi: 10.1049/iet-cta.2011.0419
www.ietdl.org
0 0.5 1 1.5 2 2.5
−0.1
0
0.1
0.2
0.3
time
process variable
0 0.5 1 1.5 2 2.5
−3
−2
−1
0
time
control variable
Fig. 16 Load disturbance unit step response for process P3(s)
Solid line: series FOPID; dash-dot line: ideal FOPID; dashed line:
PID
Table 17 Results related to the process P4(s)
Controller KpTiTdλμ IAE
series FOPID SP 1.4440 10.3229 0.5053 1 1.1403 9.8645
ideal FOPID SP 1.4815 10.7177 0.4903 1 1.1460 11.2495
PID SP 1.2460 8.9608 0.5582 14.2484
series FOPID LD 1.5128 8.9809 0.5123 1 1.1595 6.7274
ideal FOPID LD 1.5319 9.2172 0.4965 1 1.1570 7.8309
PID LD 1.2692 9.1123 0.5792 10.2791
0 5 10 15 20
0
1
2
3
4
5
6
time
process variable
0 5 10 15 20
−10
−5
0
5
10
time
control variable
Fig. 17 Set-point unit step response for process P4(s)
Solid line: series FOPID; dash-dot line: ideal FOPID; dashed line:
PID
0 5 10 15 20
−1
0
1
2
3
4
time
process variable
0 5 10 15 20
−6
−5
−4
−3
−2
−1
0
time
control variable
Fig. 18 Load disturbance unit step response for process P4(s)
Solid line: series FOPID; dash-dot line: ideal FOPID; dashed line:
PID
5.4 Example 4
As a last illustrative example, we consider an unstable
process with a greater normalised dead time
P4(s)=1
s1es(40)
where it is evident that K=1, T=1 and L=1. The
application of the proposed tuning formulae give the results
shown in Table 17. The set-point and load disturbance step
responses are shown in Figs. 17 and 18 for the different
cases.
Obviously, the peak errors are significant in the different
cases because of the large normalised dead time, but in any
case the performance provided by the FOPID controller is
much better than that provided by the PID controller.
6 Conclusions
Tuning rules for FOPID controllers for integral and unstable
processes have been proposed in this paper. At least to
the authors’ knowledge, this case has not been considered
in the literature until now. The aim of the rules is to
minimise the integrated absolute error for either the set-point
tracking or the load disturbance rejection task. A constraint
on the maximum sensitivity has been considered for integral
processes. It has to be remarked that, as for FOPDT
processes, the use of a fractional-order integral action does
not provide any improvement in the performance, whereas
the use a fractional-order derivative action is convenient.
By comparing the results with those obtained for standard
PID controllers (for which tuning rules have been also
determined) the improvement of the performance that can
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doi: 10.1049/iet-cta.2011.0419 © The Institution of Engineering and Technology 2012
www.ietdl.org
be achieved by employing a FOPID controller has been
specified quantitatively so that the user can characterise
cost/benefit ratio of such controllers for a given application.
Indeed, analytical expressions of the performance index have
been provided so that they can be employed effectively for
the purpose of performance assessment.
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... The method is based on the minimization of the integrated absolute error. Two dedicated FO-PIDs are computed in [46] to control the process in (25) ...
... An unstable process is considered next [46]: ...
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