Design of Fractional Calculus Free Controllers with Fractional
Behaviors
JOCELYN SABATIER
IMS Laboratory,
Bordeaux University,
351, Cours de la Liberation 33400 Talence,
FRANCE
Abstract: - Faced with the complexity and drawbacks of fractional calculus highlighted in the literature, this
paper proposes simple solutions to avoid its use in the field of feedback control and especially to define
fractional PID- and CRONE-like controllers. It shows that it is possible to generate fractional behaviors, which
are known since the work of Bode to be useful in the field of control, without invoking fractional calculus and
fractional models. Fractional calculus based models and fractional behaviors are indeed two different concepts:
one denotes a particular class of models and the other a class of dynamical behaviors that can be generated and
modelled by a wide variety of mathematical tools other than fractional calculus. Solutions to tune the fractional
PID- and Crone-like controllers defined in this paper are proposed.
Key-Words: - Fractional dynamical behaviors, Fractional PID Controllers, CRONE controllers, Fractional
Calculus, Fractional differentiation, Robust control.
1 Introduction
An intensively studied application area of fractional
calculus is automatic control, achieved primarily
through fractional PID controllers or Crone Control.
Fractional PID controllers are extensions of classical
integer ones, [1], in which integral and derivative
parts are replaced by fractional integral and
derivative operators. In the Laplace domain these
operators are respectively defined by
and ,
and a fractional PID controller is defined by the
transfer function:
(1)
with and .
Different design and tuning methods have been
proposed for this class of controller, [2], [3], [4], [5],
[6], [7], [8], [9] and some industrial process control
applications now exist, [10].
Crone controllers are mainly dedicated to
solving robustness issues in control loops, [11].
Assuming a unity feedback loop in which the
controller (with transfer function ) and the
plant (with transfer function ) are connected in
series in the direct chain, the Crone controller is
deduced (using a fitting algorithm) from the
frequency response of the ratio
, (2)
where is the nominal plant. The transfer
function is defined by:
(3)
where
(4)
being the crossover gain frequency.
In the strict sense, we cannot say that this
transfer function uses fractional operators for its
definition (it is for example not possible to deduce a
fractional differential equation from it), but it admits
a fractional behaviour in the frequency range
, analogous to that of the complex fractional
integrator
, (5)
Received: April 8, 2023. Revised: December 12, 2023. Accepted: December 26, 2023. Published: December 31, 2023.
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and exactly of the real part in relation to of this
complex fractional integrator defined by
(6)
These two classes of controller exhibit
drawbacks and in particular manipulate a complex
mathematical tool which undoubtedly contributes to
limiting their use: fractional calculus. Keeping in
mind that fractional calculus based models and
fractional behaviors are two different concepts:
- the first one denotes a particular class of models,
- the second is a class of dynamical behaviors that
can be generated and modelled by a wide variety
of mathematical tools other than fractional
calculus, [12],
the goal of this paper is to propose a new
formulation of these controllers so that they
maintain fractional behaviors without having to
manipulate fractional calculus.
The paper is organized as follows. The limitations
and drawbacks of fractional PID and Crone
controllers are first described. Then a gain function
that exhibits a fractional behavior but without
involving fractional calculus is introduced. It is
shown that such a function can be used to define a
fractional PID-like or Crone-like controller. An
algorithm is proposed to deduce the minimum phase
corresponding to this gain function and its
efficiency is demonstrated. Finally, solutions to tune
these fractional PID-and Crone-like controllers are
proposed.
2 Limitations and Drawbacks of
Fractional PID and Crone
Controllers
2.1 Fractional PID controllers
If the fractional PID controller (1) is used to solve a
control problem, several drawbacks arise.
1 – The definition of a fractional PID controller
does not take into account the fact that the fractional
differentiation or integration operators are doubly
infinite operators, [12] and that it is necessary to
approximate them, or more precisely to truncate
their frequency behavior at low and high
frequencies. Many methodologies have been
developed for the implementation of fractional
operators but all lead to the above-mentioned
truncation. Often, the high and low frequency
asymptotic behaviors which result from these
approximations are poorly controlled. This is for
instance the case using the Grünwald-Letnikov
definition to approximate a fractional integrator
with . If and are
respectively the input and the output of this
fractional integrator, the approximation of the
sampled output is defined by:
. (7)
A discrete transfer function approximation of a
fractional integrator is thus
. (8)
As tends towards 1 (to evaluate the steady
state behavior), this transfer function no longer
tends to infinity. The effect of the integration is lost
in the approximation/discretisation process.
2 - Although it is interesting to use a fractional
differentiation transfer function for a lead effect, a
fractional integral transfer function is of low
interest. A fractional integrator of order ,
, , is no more efficient for steady-state
error cancellation than an integer integrator of order
(as can be proved using the final value theorem).
3 - The use of a non-band-limited differentiator
leads to an infinite control effort value and to a great
sensitivity to measurement noise. We consider,
therefore, that the fractional differentiation part of
the controller needs to be band-limited before being
tuned, thus leading to an additional tuning
parameter. Only then will the controller be proper.
4 – The above-mentioned approximations lead to
discrete time or continuous-time approximations
that require computer resources much greater than
those necessary for a classic PID.
5 – Several papers in the field claim a greater
efficiency of fractional PID controllers in
comparison to classic PID, but they forget to
mention that a fractional PID controller has 5 tuning
parameters while a classic PID has only 3. Wouldn't
a classical controller with 5 parameters defined by
( being the gain crossover frequency)
(9)
be just as effective?
As mentioned in [13], to take into account the
previous two drawbacks, it would be better to define
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a fractional PID controller, with the same number of
tuning parameters, by the transfer function:
, (10)
where is the desired open loop crossover gain
frequency (which is part of the specifications as it
controls the loop rapidity). This form allows a
fractional behaviour where it is necessary, i.e.
around the corner frequency . The tuning
parameters are then the gain , the corner
frequencies and , the fractional order and
the parameter .
2.2 CRONE Controllers
Despite its 30 years of existence, CRONE control
has had difficulty establishing itself in the industry.
Applications of the resulting controllers very often
remain at the level of research and development
departments. This control methodology uses a
mathematical tool, complex fractional
differentiation, which is seldom taught in higher
education, which may explain this situation. It is
precisely the use of fractional differentiation that
allows the parameterization of the open-loop
transfer function with a small number of parameters
(4 independent parameters), which is necessary
because the search for the optimal value of these
parameters in a control robustness problem is
carried out using a non-linear optimization
algorithm. Even if in practice this control
methodology gives good results, it remains
suboptimal concerning a given robustness problem
as the structure of the open-loop transfer function
for the nominal behavior of the process to be
controlled is imposed (not the case with H∞ control
for instance). We can ask the following questions
for the same problem: would a different structure of
this open loop among the existing infinity not have
led to a better result? Or again: would not the choice
of another nominal behaviour for the calculation of
the controller have led to better results? Aware of
this sub-optimality, the authors of [14], [15], [16]
proposed to introduce in the definitions of the open
loop transfer function a curvilinear template, that is
to say roughly, to add central terms such as the one
which appears in relation (3) in this same relation.
However the interest of using fractional
differentiation is then lost, since as shown in [14],
the search for 10 independent parameters is then
required for the resolution of a robustness problem.
3 Introduction of a New Controller
and Open Loop with Fractional
Behaviors but without Involving
Fractional Calculus
This section gives the definition of a new controller
and/or new open loop in a unity feedback loop
context
- that exhibit fractional behaviors
- without requiring fractional calculus.
The definition is first given for the controller and
the diversity of frequency response shapes that can
be obtained is illustrated. Some of these shapes are
similar to those obtained with complex fractional
calculus in CRONE control. The idea used to define
the controller is then applied to the definition of a
new open loop which allows shapes similar to those
provided by the third generation CRONE control
with and without an extra template (central term in
relation (3)).
3.1 Definition of the New Controller
To propose a new controller with fractional
behavior and without involving fractional calculus,
let us start from a classical filtered PID controller of
the form:
. (11)
This form is classically taught to students as it
permits an easy computation of its parameters using
two specifications that model the closed loop
response:
- the gain crossover frequency of the open loop
, that impacts the closed loop rapidity,
- the phase margin that impacts the closed
loop damping.
To satisfy these two specifications it is necessary to
impose the following equalities
where and denote
respectively the magnitude and the phase of the
plant to be controlled (defined by the transfer
function ) at the frequency .
The magnitude in decibels of the controller is
defined by:
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(12)
To generalise such a controller, the idea is to
make a kind of series expansion of the lead-lag part
gain. It is proposed to replace relation (12) by an
expression of the form:
(13)
with
(14)
and
.
3.2 Phase Computation
For implementation purposes, the phase associated
to relation (13) is required. To compute the phase,
Bode relationships will be used as described in [17].
Under the assumption that is analytic and has
no zeros for (minimum-phase systems),
then:
. (15)
In relation (15), phase is uniquely
determined from the gain (in nepers) from the
following relation, [17]:
. (16)
For an easier numerical computation using
Gaussian quadrature, the previous integral is split
into two parts:
. (17)
Using the change of variable and
thus integral
becomes:
. (18)
Using the changes of variables and then
, integral becomes:
. (19)
Numerically, considering
and using the trapezoidal rule, the phase can
be approximated by the following sums:
, (20)
, (21)
with
,
.
As an example, this method is applied to a low pass
filter whose gain is defined by (in nepers):
. (22)
To compute the phase , the following
parameters were chosen:, ,
. The estimated and the exact phases are
compared in Figure 1, which shows that the
estimation is accurate.
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Fig. 1: Comparison of the exact and the estimated
phase for the low pass filter
As another example, the method is applied to
the fractional transfer function:
. (23)
The gain (in nepers) of this transfer function is
defined by
.
(24)
The phase (in degrees) defined analytically by
(25)
is compared in Figure 2 with the phase estimated by
the algorithm described at the beginning of this
section with , ,
and . This comparison again
reveals a very good accuracy of the estimated phase.
Fig. 2: Comparison of the exact and the estimated
phase for the transfer function
3.3 Frequency Behaviors Generated
To illustrate the impact of the parameters and the
terms of relations (13) and (14), the Nichols
diagrams of the functions whose gain (in decibels)
are
(26)
with
and
. (27)
with
, are represented respectively by
Figure 3 and Figure 4. The phase is computed using
the algorithm described in section 3.2.
Fig. 3: Nichols diagram of the function whose gain
is showing the impact of parameter
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Fig. 4: Nichols diagram of the function whose gain
is showing the impact of parameter
Figure 3 and Figure 4 illustrate the diversity of
shapes that can be obtained with only two
parameters to model the frequency response of the
controller around the crossover gain frequency,
without involving fractional calculus. They also
highlight the low number of parameters required to
obtain this diversity of shapes:
- with only two or three free additional parameters
(, and/or ) in relation to the controller
(11) which has two free parameters (, ),
- with only one or two additional free parameters
(, ) in relation to the fractional controller
(10) which has three free parameters (, , ),
- with the same number of parameters as a
fractional PID controller (relation (1)).
This diversity offers a large number of degrees of
freedom to solve several regulation problems
simultaneously. Beyond phase margin and
bandwidth specifications, many other constraints
can be taken into account as shown in section 3.6.
3.4 New Open Loop Definition
Figure 5 shows the Nichols diagram of the transfer
function
(28)
with
,
which contains the part that defines the “generalised
template” in the CRONE open loop transfer
function given by relation (3). This figure shows
that the shapes obtained are similar to those in
Figure 4. Relation (13) dedicated to the definition of
a new controller can thus be used to define a new
open loop without fractional calculus in a control
design strategy similar to CRONE control. The new
open loop gain definition (in nepers) is:
(29)
with
(30)
.
The corresponding phase can be computed
using the algorithm defined in section 3.2. In
relation (29), the parameters and can be
defined as in CRONE Control, [11], [15]:
- if denotes the order of the asymptotic
bbehaviorof the plant at low frequency (
), is defined by if and
if , as cancels the position
error, and cancels the hauling error;
- if denotes the order of the asymptotic
bbehaviorof the plant at high frequency ,
order is given by .
The definition of this new open loop means that the
following free parameters: , , with
have to be defined. According to the
comparison of Figure 3 and Figure 5, open loop
shapes similar to those obtained with the third
generation CRONE control can be obtained with
and thus four independent parameters. More
complex shapes, similar to those obtained with the
third generation CRONE control with a curvilinear
template, can be obtained with .
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Fig. 5: Nichols diagram of the function whose gain
is to show the impact of parameter
3.5 Implementable Controller
The controller and the open loop behaviors
proposed in the previous sections are
defined only by their frequency responses: the gains
are respectively defined by relations (13) and (29)
and the corresponding phases are computed using
the algorithm described in section 3.2.
For the fractional PID-like controller (13), the
implementable controller can be obtained by
using a frequency identification method such as the
ones described in [18]. These frequency
identification methods permits to fit the frequency
response in the form of a rational transfer
function to get .
From the Crone-like open loop of relation (29),
the frequency response of the controller is defined
from the relation (as in the CRONE control
strategy):
, (31)
where denotes the nominal frequency
response of the plant. Then, again, the
implementable controller in the form of a
rational transfer function can also be obtained by
using a frequency identification method such as the
ones described in [18].
For a minimum phase and stable plant ,
the resulting controller is also stable and
minimum phase as it is possible to constrain le pole
and zeros location of in the complex plane
with the identification method used in [19]. For non
minimum phase or unstable plant , terms can
be added in relation (29) to unsure the stability of
the loop and of the controller , as it is done in
CRONE control, [18].
3.6 Parameters Tuning
To tune the parameters of the fractional-like
controller or the Crone-like open loop
behaviour , the following specifications can
be used, where denotes the frequency
response of the plant to be controlled.
- Steady-state errors in relation to reference unit
step or ramp signal. To ensure steady state error
cancellation relation to reference unit step or ramp
signal, conditions on parameter must be chosen
as described just after relation (30).
- Gain crossover frequency. To ensure a specified
gain crossover frequency the following equality
must be met:
. (32)
- Phase margin. To ensure a specified phase margin
, the following equality must be met:
. (33)
- Phase crossover frequency. To ensure a specified
phase crossover frequency the following
equality must be met:
. (34)
- Gain margin. To ensure a specified gain margin
, the following equality must be met:
. (35)
- Rejection of high frequency noise. Measurement
noise rejection can be adjusted using the following
condition:
. (36)
- Rejection of output disturbance. Output
disturbance rejection can be adjusted using the
following condition:
. (37)
- Robustness to the plant phase variation. For the
control loop to ensure robustness to the plant phase
variation, the following constraint can be imposed:
(38)
in which denotes the phase of the
open loop function. Through this condition, the
phase is forced to be almost constant around the
frequency and thus the closed loop system is
more robust to gain changes.
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- Robustness to any plant variation. To ensure
robustness to plant variation, a solution consists in
using the stochastic robustness concept introduced
in [19]. The stochastic robustness of a closed loop
system can be evaluated using Monte Carlo
simulations to draw a statistical portrait of
parameter variations and their effect on the closed
loop. In this work, the uncertain parameters are
assumed to have a bounded, continuous,
uncorrelated and uniform probability distribution.
To ensure closed loop stability degree robustness
while guaranteeing a given loop damping level, it is
proposed to minimize the probability of the first
overshoot value of the sensitivity function to be
out of a given interval due to plant
variation. The cost function to be minimized can
thus be defined by
. (39)
denotes the number of times the sensitivity
function is evaluated in the Monte Carlo simulation
to sweep uncertainty intervals. Minimization of such
a criterion reduces the variations of the first
overshoot value of the sensitivity function and thus
minimizes the impact of the plant variation on the
loop stability degree. Many other criteria can be
defined on the same principle.
Using the previous specifications, algorithms
that enable the fractional behaviour controller
or the Crone-like open loop behaviour
to be tuned can be defined as follows.
Algorithm for parameters tuning
1 – Impose parameters and as described just
after relation (30).
2 – Impose the gain crossover frequency
3 – Impose the phase margin
4 – Minimise the criterion
under the
constraints (36) and (37), and thus, at each
optimisation algorithm step with a new set of
parameters :
4.1 - Compute to ensure
4.2 - Compute using the algorithm
described in section 3.2
4.3 – Compute
and
4.4 – Compute the cost function and check the
constraints (36) and (37).
5 – For the obtained optimal values of parameters
, fit the rational controller
according to the comments in section 3.5
using the frequency response of .
Algorithm for parameters optimisation and
computation
1 – Impose parameters and as described just
after relation (30).
2 – Impose the gain crossover frequency .
3 – Minimise the cost function (40)
under the constraints (37)
and (38), and thus, at each optimisation algorithm
step with a new set of parameters
:
3.1 - Compute using relation (30)
3.2 - Compute
3.3 - Compute using the algorithm
described in section 3.2
3.4 – Compute and
using relation (31)
3.5 – Compute
and
3.6 – Compute the cost function (39) and check
the constraints (36) and (37).
4 – For the obtained optimal values of parameters
:
4.1 - Compute using relation (30)
4.2 - Compute
4.3 - Compute using the algorithm
described in section 3.2
4.4 – Compute and
using relation (31)
4.5 - Fit the rational controller according
to the comments in section 3.5 using the
frequency response of .
4 Conclusion
This paper proposes alternative solutions to
fractional PID controllers and Crone controllers.
Without resorting to fractional differentiation or
integration notions, it introduces a fractional PID-
like controller and a Crone-like open loop function
(that is then used to define a Crone-like controller)
with the same restricted number of parameters and
similar generated frequency responses. First, the
gains of the fractional PID-like controller and a
Crone-like open loop function are defined in the
form of a kind of series expansion of the lead-lag
part of the classical fractional PID controller and
Crone open loop function. The corresponding phase
is computed with an algorithm based on Bode phase
relationships and specially developed in this work.
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Some solutions are also presented for parameter
tuning of the proposed fractional PID-like controller
and a Crone-like open loop function. This work
makes it possible to overcome the limits and
drawbacks inherent to fractional PID and Crone
controllers, in particular the use of fractional
calculus, which is a limiting factor in the diffusion
of these control strategies.
This work reinforces the idea already mentioned by
the author that fractional calculus based models and
fractional behaviors are two different concepts:
- the first one denotes a particular class of
models
- the second is a class of dynamical behaviors
that can be generated and modeled by a wide
variety of mathematical tools other than
fractional calculus, [12].
Due to space constraints, applications of these new
control strategies are not presented here and will be
described in coming papers. However, beyond the
topic of controller synthesis, considering fractional
behaviors without being limited to fractional models
opens up countless avenues of research in the field
of model analysis and identification, and more
generally in the understanding of the physical
phenomena that induce fractional behaviors.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Jocelyn Sabatier has contributed to the
conceptualization, le methodology development,
simulations, writing of the original draft, review and
editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare that
are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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