756 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 2, MARCH 2016
Theoretical Analysis and Experimental Validation of a Simplified Fractional
Order Controller for a Magnetic Levitation System
Silviu Folea, Member, IEEE, Cristina I. Muresan, Robin De Keyser, and Clara M. Ionescu, Member, IEEE
Abstract—Fractional order (FO) controllers are among the
emerging solutions for increasing closed-loop performance and
robustness. However, they have been applied mostly to stable
processes. When applied to unstable systems, the tuning
technique uses the well-known frequency-domain procedures
or complex genetic algorithms. This brief proposes a special
type of an FO controller, as well as a novel tuning procedure,
which is simple and does not involve any optimization routines.
The controller parameters may be determined directly using
overshoot requirements and the study of the stability of
FO systems. The tuning procedure is given for the general
case of a class of unstable systems with pole multiplicity. The
advantage of the proposed FO controller consists in the simplicity
of the tuning approach. The case study considered in this
brief consists in a magnetic levitation system. The experimental
results provided show that the designed controller can indeed
stabilize the magnetic levitation system, as well as provide
robustness to modeling uncertainties and supplementary loading
conditions. For comparison purposes, a simple PID controller is
also designed to point out the advantages of using the proposed
FO controller.
Index Terms— Control design, fractional calculus, robustness,
stability analysis, unstable systems.
I. INTRODUCTION
T
HE control of magnetic levitation systems has received
considerable scientific interest not only because of the
highly nonlinear and unstable behavior of such systems [1]
but also because of their ability to eliminate friction,
decrease maintenance cost, and achieve high-precision
positioning [1], [2]. These advantages make magnetic levita-
tion systems a viable choice for high-speed trains, magnetic
bearings, vibration isolation systems, and wind tunnel
levitation [1], [3], [4].
Numerous control solutions have been proposed for such
systems, such as feedback linearization techniques [5], which
Manuscript received October 27, 2014; revised March 2, 2015; accepted
May 3, 2015. Date of publication July 8, 2015; date of current version
February 17, 2016. Manuscript received in final form May 21, 2015. This work
was supported by the Romanian National Authority for Scientific Research
National Council for Development and Innovation–Executive Unit for
Financing Higher Education, Research, Development and Innovation under
Project PN-II-RU-TE-2012-3-0307 and Project TE 59/2013. Recommended
by Associate Editor Y. Chen.
S. Folea, C. I. Muresan, and C. M. Ionescu are with the Department of
Automation, Technical University of Cluj-Napoca, Cluj-Napoca 400604,
Romania (e-mail: silviu.folea@aut.utcluj.ro; cristina.pop@aut.utcluj.ro;
claramihaela.ionescu@ugent.be).
R. De Keyser is with the Department of Electrical Energy,
Systems and Automation, Ghent University, Ghent 9052, Belgium (e-mail:
robain.dekeyser@ugent.be).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCST.2015.2446496
require a very accurate model for the magnetic levitation
system. This may represent a major problem, since a precise
dynamic model may be difficult to obtain [1], [5] because
of the prevailing nonlinearities that characterize the system,
including the variation of the gain of the magnetic
levitation system as a function of the distance to the
magnet [6]. Other control approaches are based on designing
the controllers for the linearized dynamic model at nominal
operating points. Nevertheless, the tracking performances of
such control strategies deteriorate drastically with increasing
deviation from nominal operating points [1]. To account for
the changing parameters and dynamics of magnetic levitation
systems, sliding mode, nonlinear, μ-synthesis, PIDs combined
with notch filters, gain scheduling, backstepping, and
fuzzy neural network-based controllers have been proposed,
providing robustness against unmodeled nonlinearities present
in the system [1], [3], [5]–[12]. A networked control system
application on an unstable triple-magnetic-levitation setup has
also been reported, with the controllers tuned according to the
H
∞
theory [13].
Fractional calculus has been listed among the current
trends in control engineering, with a wide application in
both modeling [14], [15] and control design [16]–[19]. The
main advantage of fractional order (FO) controllers is their
ability to enhance the performance of closed-loop systems and
increase the robustness [6], [19], [20]. The tuning approach
for FO PI
λ
D
μ
controllers consists in specifying a set of
performance criteria and transposing these specifications into
several equations. Using optimization routines or graphical
methods the controller parameters are determined [21]–[23].
For unstable systems, few papers deal with the design
of FO controllers [24], [25]. For magnetic levitation systems,
FO controllers have been previously designed [26], [27].
However, all of these papers propose the general form of the
FO PI or PD controllers and the tuning technique uses the
well-known frequency-domain procedures [27] or complex
genetic algorithms [26].
In this brief, a special type of an FO controller, different
than the usual FO PID, is proposed for the magnetic levitation
system. The tuning procedure is simple and does not involve
any optimization routines. The controller parameters may be
determined directly using overshoot requirements and the
study of the stability of FO systems. The advantage of the
proposed FO controller consists in the simplicity of the tuning
approach. The novelty of this brief resides in the new tuning
technique for FO controllers, as well as in the different form
of the FO controller proposed for a class of unstable systems
1063-6536 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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FOLEA et al.: THEORETICAL ANALYSIS AND EXPERIMENTAL VALIDATION OF A SIMPLIFIED FO CONTROLLER 757
with pole multiplicity. In addition, this brief presents the
implementation and experimental validation of the proposed
control algorithm on a pilot scale magnetic levitation stand.
An analysis regarding the robustness of the proposed controller
is also included.
This brief is structured into five main parts. Section II
presents the algorithm for tuning stabilizing FO controllers for
a set of unstable systems with pole multiplicity. Section III
details the experimental setup, as well as the controller
tuning procedure and the experimental results. The robustness
analysis is carried out in Section IV, including several
operating points, as well as two case studies in which
supplementary loads are added to the permanent disk magnet.
The experimental results show that the designed controller
is robust against modeling uncertainties and intrinsic
nonlinearities. Section V presents the comparison of the
proposed control strategy with a simple PID controller,
while the final section summarizes the main outcome of
this brief.
II. S
TABILITY ANALYSISOFAFRACTIONAL ORDER
CONTROLLER FOR A CLASS OF UNSTABLE PROCESSES
The transfer function of the class of unstable systems with
pole multiplicity considered in this brief is
G(s) =
1
(s − z)(s + z)
(1)
with z > 0. A simple PD controller may be designed to
stabilize the system
C
PD
(s) = k(s + z) (2)
with z chosen to compensate the stable pole of the system
in (1). The resulting closed-loop system will exhibit an over-
damped response, for a value of the gain k
stab
that will make
the system stable. However, such a PD controller will exhibit
steady-state offsets, which could be eliminated with a simple
PID controller. An alternative solution to the classical integer
order PID controller is proposed in this brief, consisting in
a PD controller and a fractional integrator that eliminates the
steady-state errors
C(s) = k
(s + z)
s
λ
(3)
with λ ∈[0, 1]. For certain values of k and λ, the closed-loop
system will become stable. The choice of the FO controller
in (3), apart from its simplicity, is also justified by the lower
number of tuning parameters. The controller in (3) may be
rewritten as an FO I
λ
D
1−λ
controller
C(s) = k
s
1−λ
+
z
s
λ
. (4)
The open-loop transfer function would then be given as
H
open−loop
(s) = C(s)G(s) =
k
s
λ
(s − z)
. (5)
The characteristic equation used to determine the stability
of the closed-loop system is then
k + s
λ
(s − z) = 0. (6)
The equation in (6) is a function of the complex variable s
whose domain can be seen as a Riemann surface, with the
principal sheet defined as −π<arg(s)<π[22]. This
definition of the principal sheet assumes a cut along R
−
and
is associated to the Cauchy principal value of the integral
corresponding to the inverse transformation of Laplace, or to
that obtained by direct application of the residue
theorem [22]. In the case of λ = (1/n),wheren is a
positive integer, the n sheets of the Riemann surface will be
given by
s =|s|e
jφ
,(2i + 1)π<φ<(2i +3)π (7)
with i =−1, 0, 1,...,n − 2. For example, for i =−1, the
principal Riemann sheet is obtained, whereas for i = 0, the
secondary Riemann sheet is obtained. If a mapping is used,
defined as w = s
λ
, then the Riemann sheets become the
regions in the w plane defined by
w =|w|e
jθ
,λ(2i + 1)π<θ<λ(2i +3)π. (8)
The characteristic equation in (6) will have an infinite
number of roots, among which only a finite number of roots
will be on the principal sheet of the Riemann surface [28].
These roots are generally referred to as the structural roots
of the system and are responsible for the closed-loop system
exhibiting either damped oscillation, oscillation of constant
amplitude, oscillation of increasing amplitude with monotonic
growth. The roots that are located in the secondary sheets of
the Riemann surface are related to solutions that are always
monotonically decreasing functions [22], thus a system that
has all poles in the secondary Riemann sheets is closed-loop
stable.
For the case of commensurate-order systems, where all the
orders of derivation are integer multiples of a base order,
the characteristic equation is a polynomial of the complex
variable w = s
λ
. The stability condition for these types of
systems is expressed as [22], [28], [29]
|θ| >λ
π
2
. (9)
Using(8)andmakingi =−1, the condition for the
principal Riemann sheet is obtained as
−λπ<θ <λπ. (10)
Intersecting (9) and (10), yields the final stability condition
for all roots lying in the first Riemann sheet
θ
λ
π
2
,λπ
∪
− λπ, −λ
π
2
. (11)
In the case of λ = (1/n),wheren is a positive integer, the
stability condition in (11) is modified to be
θ
π
2n
,
π
n
∪
−
π
n
, −
π
2n
. (12)
Considering the above mapping, w = s
λ
,andλ = (1/n),
the characteristic polynomial in (6) becomes
w
n+1
− z ∗w
1
+ k = 0. (13)
The tuning of the controller parameters k and λ is performed
to meet the overshoot requirements (%OS) and it is based on
758 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 2, MARCH 2016
Fig. 1. Equivalence between (a) s plane and (b) w plane.
the root locus analysis of (13) in the w plane. The equivalence
of the s and w planes is given in Fig. 1. For k = 0, the roots
of (13) are
w = 0andw = z
1
n
e
2mπ
n
(14)
with m ∈ Z. It can be easily seen from (14) that for m = 0,
the roots that lie in the principal Riemann sheet correspond
to w = 0andw = z
(1/n)
, with all other roots, corresponding
to m = 0, located in the secondary sheets. The root
locus corresponding to the two roots located in the principal
Riemann sheet is given in Fig. 1(b). According to Fig. 1(a),
in order for the closed-loop system to have a certain overshoot,
denoted %OS, the poles must lie on the %OS line. The
following relations hold:
sin α = ζ and %OS = e
−πξ/
√
1−ζ
2
(15)
where ξ is the damping factor. The %OS line and its corre-
sponding angle α in the s plane are translated into the w plane
as indicated in Fig. 1(b), with the angle β = ((α + (π/2))/n).
In order for the closed-loop poles to have an exact
overshoot %OS, the angle of the asymptotes and the angle β
must meet the following requirement:
β<
π
n + 1
or
α +
π
2
n
<
π
n + 1
. (16)
Using (15) and (16), a simple tuning procedure may be
derived to meet an imposed overshoot %OS
∗
.
1) Compute damping factor ξ as
ζ =
|ln(%OS
∗
)|
√
π
2
+ ln
2
(%OS
∗
)
. (17)
Fig. 2. (a) Experimental setup of the magnetic levitation system. (b) Block
diagram of the magnetic levitation system.
2) Evaluate the angle α, corresponding to the imposed
%OS
∗
α = sin
−1
ξ. (18)
3) Using (16), determine the range for n and the
corresponding FO λ
n >
π
2
+ α
π
2
− α
. (19)
Based on (17)–(19), for a small overshoot, α → (π/2),
whichinturnleadston →∞⇒λ → 0. Select n as
the minimum n
min
that obeys (19).
4) Based on stability analysis in the w plane, deter-
mine k such that the roots of (13) located in the
principal Riemann sheet lie on the imposed %OS
∗
line.
III. T
UNING AND EXPERIMENTAL VALIDATION OF A
SIMPLIFIED FRACTIONAL ORDER CONTROLLER
FOR A
MAGNETIC LEVITATION SYSTEM
An experimental laboratory scale magnetic levitation
unit has been developed and built, as given in Fig. 2(a),
consisting of a small permanent disk magnet suspended in
a voltage-controlled magnetic field. Several modules are
used to implement the control algorithm, as indicated in
Fig. 2(b), such as the NI cRIO-9014 embedded real-time
controller, the NI 9103 reconfigurable chassis, and the
NI 9215/NI 9263 input/output modules. The vertical position
of the levitating permanent magnet is measured using an
SS495A ratiometric linear Hall sensor, which generates an
output voltage that is proportional with the magnetic field.
The sensor is placed at the bottom of the experimental unit,
away from the coil to eliminate the influence of the magnetic
field generated by the coil on the magnetic field generated
by the permanent magnet. A low-pass filter has also been
FOLEA et al.: THEORETICAL ANALYSIS AND EXPERIMENTAL VALIDATION OF A SIMPLIFIED FO CONTROLLER 759
Fig. 3. Free body diagram of the magnetic levitation system.
implemented to filter the noisy signals at high frequencies.
A current driver was used to control the current running
through the coil, yielding a linear relation between voltage and
current
i =
u
67.83
(20)
where u is the voltage [V] and i is the current applied to the
coil [A].
The electromagnet coil is a 24 V dc solenoid having the iron
core replaced with an aluminum one. The suspended object is a
permanent magnet and under this circumstances, it is preferred
a material having a smaller magnetic permeability such that
the magnet will not get stuck at the bottom of the coil. The
levitating permanent disk magnet, made out of neodymium–
ferric–boric, weighs 1.79 g, with a diameter of 10 mm and
a height of 3 mm. The coil has a diameter of 22 mm,
with a height of 13 mm, and resistance of 95 ,being
made out of 2600 turns, with each wire having a diameter
of 0.18 mm.
To model the magnetic levitation system, the free body
diagram given in Fig. 3 is used, as well as electromag-
netic and mechanical equations. The resulting net force
acting on the permanent disk magnet, denoted F
net
,is
computed using Newton’s law of motion while neglecting
friction
F
net
= F
g
− F
l
= mg − c
i
2
x
2
(21)
where F
l
is the electromagnetic force (levitation force) [6]
and F
g
is the gravitational force, m is the mass of the
permanent magnet [kg], g = 9.81 is the gravitational speed
constant [m/s
2
], x is the distance [m], ¨x is the acceleration
of the permanent magnet [m/s
2
], and c is the magnetic force
constant. The resulting net force is equal to
F
net
= m ¨x. (22)
Replacing (22) into (21) leads to
m ¨x − mg + c
i
2
x
2
= 0. (23)
Denoting f (x, ¨x, i) = m ¨x − mg + c(i
2
/x
2
), the equation
in (23) is then linearized with respect to an equilibrium
point (x
0
, i
0
) by applying the Taylor series expansion
f (x, ¨x, i) = f (x
0
, ¨x
0
, i
0
) +
∂ f
∂x
(x
0
, ¨x
0
,i
0
)
(x − x
0
)
+
∂ f
∂ ¨x
(x
0
, ¨x
0
,i
0
)
( ¨x −¨x
0
) +
∂ f
∂i
(x
0
, ¨x
0
,i
0
)
(i −i
0
).
(24)
Using (23) into (24), leads to
f (x, ¨x, i) =
−
2ci
2
x
3
(x
0
, ¨x
0
,i
0
)
(x − x
0
) + m|
(x
0
, ¨x
0
,i
0
)
( ¨x −¨x
0
)
+
2ci
x
2
(x
0
, ¨x
0
,i
0
)
(i −i
0
). (25)
Considering now the linearization point (x
0
, i
0
),and
denoting i = i − i
0
and x = x − x
0
, (25) may be
rewritten as
f (x, ¨x, i) =−
2ci
2
0
x
3
0
x +m ¨x +
2ci
0
x
2
0
i = 0. (26)
Applying the Laplace transform to the linearized (26),
leads to
−
2ci
2
0
x
3
0
x(s) + ms
2
x(s) +
2ci
0
x
2
0
i(s) = 0 (27)
which leads to the final transfer function of the mechanical
subsystem
x(s)
i(s)
=−
2ci
0
x
2
0
ms
2
−
2ci
2
0
x
3
0
. (28)
For the electromagnetic subsystem, the relation between the
voltage and the current in (20) is used, leading to the final
transfer function for the magnetic levitation system
x(s)
u(s)
=−
2ci
0
x
2
0
67.83
ms
2
−
2ci
2
0
x
3
0
. (29)
To determine the unknown parameter c in (23), several
experiments have been performed by placing the magnet at
different distances from the coil and supplying the coil with
a voltage large enough to raise the magnet. The experiments
and resulting values for c, computed using (23), with ¨x = 0,
are listed in Table I, in the Appendix. The final value for c is
computed as the mean value, averaging over the interval for x
0
between 3 and 6.5 mm. In this interval, the value for c does not
vary much from the average computed value of 8.05 × 10
−5
,
and may be thus assumed as constant. For an equilibrium point
chosen as (x
0
, i
0
) = (4.2 × 10
−3
m, 49.8 × 10
−3
A), the
transfer function in (29) becomes
x(s)
u(s)
=−
6.1
(s − 70)(s + 70)
. (30)
The transfer function in (30) is rewritten so as to be similar
to (1)
x(s)
u(s)
=−
1
(s − 70)(s + 70)
(31)
760 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 2, MARCH 2016
Fig. 4. Root locus for n = 5andλ = 0.2.
where the gain 6.1 in (30) will be considered as part of
the controller gain k, in (3). To tune a simplified stabilizing
controller for the magnetic levitation system in (31), the
stability analysis in Section II, as well as an overshoot
requirement of %OS
∗
= 1% are employed. First, based
on (17), ζ = 0.8261, while according to (18), α = 0.97 rad.
Finally, using (19), the following result is obtained:
n > 4.2. (32)
Since n is a positive integer, the minimum value that
satisfies (32) is n = 5, or λ = 0.2, which is further used
in the tuning of the parameter k by plotting the root locus
of (13). This is given in Fig. 4, for several values of k. It can
be easily noted that for k = 600, the root locus crosses the
imposed %OS
∗
line.
For k < 600, the closed-loop system will exhibit an
oscillatory behavior, while for larger values of the gain k,
the control effort will be increased, as also demonstrated
in Figs. 5 and 6.
Fig. 5(a) and (b) shows the position of the disk, according
to different reference step signals, and considering the open-
loop gain k = 600 in comparison with k = 300 and k = 1000.
As expected from Fig. 4, the lower the value of k, the closer
the system is to the unstable region. The experimental results
demonstrate this accurately, with high oscillatory behavior of
the closed-loop system in the case of k = 300, as indicated
in Fig. 5(a). A higher value of the gain, k = 1000, should
result in roots of the characteristic equation closer to the
secondary sheet of the Riemann surface.
This implies that the closed-loop system should exhibit
an overdamped response. The experimental results given
in Fig. 5(b) show that for k = 1000, the magnetic disk tracks
its reference position with less overshoot and less oscillations
in comparison with the case when k = 600. The required
control signals are also given in Fig. 5(c) and (d). The results
show that the larger the open-loop gain k, the larger the control
effort. Thus, a value k = 600 has been selected with the
resulting FO controller indicated in (33). Considering that the
process exhibits a 6.1 gain, the final FO controller gain is 98.5,
with the transfer function
C(s) =−98.5
s
0.8
+
70
s
0.2
. (33)
Fig. 5. Experimental results with the FO controller considering different
open-loop gains. (a) Disk position for k = 600 in comparison with k = 300.
(b) Disk position for k = 600 in comparison with k = 1000. (c) Control
effort disk for k = 600 (black line) in comparison with k = 300 (gray line).
(d) Control effort for k = 600 (black line) in comparison with k = 1000
(gray line).
To implement the FO controller, a digital approximation was
derived using the recursive Tustin method, of ninth order, with
a sampling time T
s
= 2 ms [30].
