Design of retuning fractional PID controllers for a closed-loop magnetic levitation control system

Abstract

In this paper, we study the problem of fractional-order PID controller design for an unstable plant - a laboratory model of a magnetic levitation system. To this end, we apply model based control design. A model of the magnetic lévitation system is obtained by means of a closed-loop experiment. Several stable fractional-order controllers are identified and optimized by considering isolated stability regions. Finally, a nonintrusive controller retuning method is used to incorporate fractional-order dynamics into the existing control loop, thereby enhancing its performance. Experimental results confirm the effectiveness of the proposed approach. Control design methods offered in this paper are general enough to be applicable to a variety of control problems.

Full text

Design of Retuning Fractional PID Controllers for a
Closed-loop Magnetic Levitation Control System
Aleksei Tepljakov and Eduard Petlenkov
Department of Computer Control
Tallinn University of Technology
Ehitajate tee 5, 19086 Tallinn, Estonia
E-mail: aleksei.tepljakov at ttu.ee
and eduard.petlenkov at ttu.ee
Juri Belikov
Institute of Cybernetics
Tallinn University of Technology
Akadeemia tee 21, 12618 Tallinn, Estonia
E-mail: jbelikov at cc.ioc.ee
Emmanuel A. Gonzalez
Jardine Schindler Elevator Corporation
8/F Pacific Star Bldg.,
Sen. Gil Puyat Ave. cor. Makati Ave.,
Makati City 1209 Philippines
E-mail: emm.gonzalez at delasalle.ph
Abstract—In this paper, we study the problem of fractional-
order PID controller design for an unstable plant—a laboratory
model of a magnetic levitation system. To this end, we apply
model based control design. A model of the magnetic levitation
system is obtained by means of a closed-loop experiment. Several
stable fractional-order controllers are identified and optimized
by considering isolated stability regions. Finally, a nonintrusive
controller retuning method is used to incorporate fractional-order
dynamics into the existing control loop, thereby enhancing its
performance. Experimental results confirm the effectiveness of
the proposed approach. Control design methods offered in this
paper are general enough to be applicable to a variety of control
problems.
Index Terms—fractional-order calculus, PID control, unstable
plant, stability analysis
I. INTRODUCTION
Fractional-order calculus offers novel mathematical tools
applicable to dynamical system modeling and control. This
allows to achieve more accurate process models and more
flexible controllers, thereby enhancing the quality of control
loops. Since the majority of industrial control loops are of
PI/PID type [1], it is of significant interest to study the problem
of enhancing conventional PID controllers by introducing
additional dynamical properties arising from making use of
fractional-order integrators and differentiators. A controller of
this type, called the fractional-order PIλDµcontroller (FOPID),
was proposed by Podlubny in [2] and has since been a topic
of active discussion in the control community [3]. Indeed, the
additional freedom in tuning the controller allows to consider
multiple robustness criteria. Therefore, a set of controller
parameters can be obtained such, that fulfills several design
specifications, which cannot be achieved by using a conven-
tional PID controller [4]. More importantly, however, using
fractional controllers grants the ability to obtain a wider set of
stabilizing controller parameters, which is critical in case of
unstable plants.
The Magnetic Levitation System (MLS) is a nonlinear,
open-loop unstable, and time-varying system [5]. Therefore,
designing a stabilizing controller for it is a challenging prob-
lem. Yet it is also of significant importance, since MLS has a
considerable range of real-life applications—it is used in, e.g.,
high-speed magnetic levitation passenger trains and vibration
isolation of sensitive machinery [6]. Corresponding nonlinear
control design methods were proposed in, e.g., [7], [8], [9].
However, few research papers deal with control design for
unstable systems [10], and, in particular, for the MLS, which
forms the motivation for our present research effort.
We now summarize the contribution of this paper. First, a
nonlinear model of the MLS is proposed, which is constructed
based on several modeling approaches offered in literature [5],
[10]. It is used in a model based control design method, which
includes linear analysis around a working point, selecting
random stabilizing FOPID controllers, heuristically detecting
rectangular-shaped stability regions for pairs of controller
gains, and obtaining suboptimal FOPID controller settings. The
FOPID controller is then integrated into the control loop in
a nonintrusive way, following the retuning method in [11].
Controller settings are verified on the real-life laboratory model
of the MLS.
The paper is organized as follows. In Section II the reader
is introduced to the mathematical tools of fractional-order
calculus used throughout this paper. In Section III the nonlinear
model of the MLS is presented. In Section IV the control
design method, forming the main contribution of this paper, is
described. Experimental results that verify the proposed control
design approach follow in Section V. Finally, in Section VI
conclusions are drawn.
II. MATHEMATIC AL TO OL S
First, we consider fractional-order modeling. Fractional-
order calculus is a generalization of integration and differenti-
ation operations to the non-integer order operator aDα
t, where
aand tare the lower and upper terminals of the operation,
and αis the fractional order, such that
aDα
t=




dα
dtα<(α)>0,
1<(α)=0,
´t
a(dt)−α<(α)<0,
(1)
where α∈R+. The Laplace transform of Dαof a signal x(t)
with zero initial conditions is given by
L{Dαx(t)}=sαX(s).(2)

A transfer function representation of a fractional dynamical
model may be given by
G(s) = bmsβm+bm−1sβm−1+· · · +b0sβ0
ansαn+an−1sαn−1+· · · +a0sα0,(3)
where usually β0=α0= 0. The system in (3) has a
commensurate order γ, such that λ=sγ, if it can be
represented in the following way:
H(λ) =
m
P
k=0
bkλk
n
P
k=0
akλk
,(4)
where nis called the pseudo-order of the system. The form
(4) can also be used to determine the stability of the system by
means of, e.g., Matignon’s theorem [12], which is given next.
Theorem 1. (Matignon’s stability theorem) The fractional
transfer function G(s) = Z(s)/P (s)is stable if and only if
the following condition is satisfied in σ-plane:
|arg(σ)|> q π
2,∀σ∈C, P (σ) = 0,(5)
where σ:= sq. When σ= 0 is a single root of P(s), the system
cannot be stable. For q= 1, this is the classical theorem of
pole location in the complex plane: no pole is in the closed
right plane of the first Riemann sheet.
It can be seen, that fractional-order systems offer a larger
region of stability than conventional linear systems—roots of
the characteristic polynomial P(σ)may be located in the right
half of the complex plane, as long as the condition (5) is
satisfied. This theorem works for commensurate-order systems,
where the commensurate order is given by q.
We now turn to fractional-order control. The parallel form
of the PIλDµcontroller is given by
C(s) = Kp+Kis−λ+Kdsµ.(6)
In this work, we consider the negative unity feedback closed
loop system of the form
W(s) = C(s)G(s)
1 + C(s)G(s),(7)
where C(s)is the PIλDµcontroller, and G(s)is the plant
under control.
Finally, in terms of implementation of fractional-order
controllers we consider Oustaloup’s approximation method,
described in [13], which allows to obtain a band-limited
approximation of a fractional-order operator in the form sα≈
H(s), where α∈(−1,1) ⊂Rand H(s)is a conventional
linear, time-invariant system.
III. MODEL OF THE MAGNETIC LEVITATION SYSTEM
The MLS consists of an electromagnet, a light source and
sensor to measure the position of the levitated sphere, and a
sphere rest, the height of which is variable and determines
the initial position xmax of the sphere in control experiments.
Electromagnet
Sphere
Sphere rest
Light source
Light sensor
x
0
Fig. 1. Physical model of the MLS
The position of the sphere is deremined relative to the elec-
tromagnet and has an effective range of x∈[0, xmax]mm.
A schematic drawing depicting this configuration is given in
Fig. 1. The basic principle of MLS operation is to apply voltage
to the electromagnet to keep the sphere levitated [5].
In [6] and [10] the following dynamical model for the MLS
is used:
m¨x=mg −ci2(u)
x2,(8)
where mis the mass of the sphere, xis the position of the
sphere, gis gravitational acceleration, i(u)is a function of
voltage corresponding to the electrical current running through
the coil of the electromagnet under input u, and cis some
constant. However, the following practical observations can be
made:
•It is essential to model the dynamics of the electrical
current running through the coil;
•The parameter cis, in fact, not constant and depends on
the position of the sphere x.
Therefore, we use the model description provided by INTECO,
which takes into account the dynamics of the coil current. In
addition, we model the parameter cby a polynomial c(x). The
following model is finally established:
˙x1=x2,
˙x2=−c(x1)
m
x2
3
x2
1
+g, (9)
˙x3=fip2
fip1
i(u)−x3
e−x1/fip2,
where x1is the position of the sphere, x2is the velocity of the
sphere, and x3is the coil current, fip1and fip2are constants.
By means of a series of experiments, we have found, that it
is sufficient to model c(x1)as a 4th order polynomial of the
form
c(x1) = c4x4
1+c3x3
1+c2x2
1+c1x1+c0,(10)
and i(u)as a 2nd order polynomial of the form
i(u) = k2u2+k1u+k0.(11)
Note, that the voltage control signal is normalized and has the
range u∈[0,1] corresponding to the pulse-width modulation
duty cycle 0. . . 100%.

+
+
+
+
PID Plant
−
−
CR
r
e
y
u
Original PID control loop
Fig. 2. Retuning method for existing closed-loop control systems
IV. DES IG N AN D IMPLEMENTATION OF SUBOPTIMAL
STABILIZING FRACTIONAL-ORDER PID CONTROLLERS
In the following, we summarize the method, that shall be
used to design FOPID controllers for the MLS.
A. Model Linearization and Stability Analysis
We will analyze the stability of linear approximation around
a working point (u0, x10). We linearize the model in (9) and
obtain the following transfer function of the MLS:
GM(s) = b3a23
s3−a33s2−a21 s+a21a33
,(12)
where
a21 =(−2c4x4
10 −c3x3
10 +c1x10 + 2c0)x2
30
mx3
10
,(13)
a23 =−2c(x10)x30
mx2
10
, a33 =i(u0)−x30
fip1
ex10/fip2,(14)
b3=fip2
fip1
(k1+ 2k2u0)ex10/fip2.(15)
To analyze the stability of the closed-loop fractional-order
control system in (7) we shall use Matignon’s theorem. The
characteristic polynomial is given by
Q(s) = s3+λ−a33s2+λ−a21 s(16)
+(b3a23Kp+a21 a33)sλ
+b3a23Kdsλ+µ+b3a23 Ki.
Thus, a point of the form (Kp, Ki, Kd, λ, µ)in the PIλDµ
parameter space can be selected and the stability of the closed-
loop control system can be verified.
B. PID Controller Retuning Method
The main idea of the retuning method is illustrated in Fig. 2.
The method allows to incorporate fractional-order dynamics
into a conventional PID control loop without making changes
to the loop itself, but rather adding a second loop with the re-
tuning FOPID controller. The following proposition establishes
the relations between the parameters of the controllers [11].
Proposition 2. Consider the original integer-order PID con-
troller of the form
CP ID (s) = KP+KIs−1+KDs. (17)
Let CR(s)be a controller of the form
CR(s) = K2sβ+K1sα−KDs2+ (K0−KP)s−KI
KDs2+KPs+KI
,
(18)
where the orders αand βare such, that −1< α < 1and
1< β < 2. The PIλDµcontroller resulting from a classical
PID controller will have the following coefficients
K?
P=K0, K?
I=K1, K?
D=K2,(19)
and the orders will be
λ= 1 −α, µ =β−1.(20)
It can be shown, that the structure in Fig. 2 may be replaced
by a feedback of the form (7), where
C(s)=(CR(s) + 1) ·CP ID (s)(21)
and G(s)corresponds to the plant. Therefore, the parameters
of the retuning controller CR(s)in (18) may be computed from
those of the FOPID controller C(s).
The application of the retuning method to the problem
of control of the MLS is motivated by that we shall make
use of closed-loop identification which may lead to a model
that is sensitive to changes in parameters of the original PID
controller. With the retuning method, a suitable controller is
added into an external loop, and its control law is regulating
the reference signal. Therefore, the underlying closed-loop
system continues to operate as before, but the dynamics
introduced to the reference signal allow to potentially enhance
its performance.
C. Determination and Optimization of Stabilizing FOPID
Controllers
To determine stabilizing controllers a randomized method
may be used, where FOPID controller parameters are randomly
selected from Kp∈[Kl
p, Ku
p], Ki∈[Kl
i, Ku
i], Kd∈
[Kl
d, Ku
d], λ ∈[λl, λu],µ∈[µl, µu]. Note, that the choice
of λand µmust lead to a commensurate-order system, since
only then the results of the stability test are reliable, otherwise
they are only approximate [14]. For example, one can choose
a minimum commensurate order q= 0.01.
Once a stable point is found, the following procedure is
carried out. Two of the controller parameters are parametrized
as (p1, p2), all other parameters are fixed. A limited number
of steps Nis selected and a sweep with step sizes ∆p1and
∆p2is done from the initial stable point. Four directions are
considered. The main idea is illustrated in Fig. 3. Each time
only a single parameter is changed. If, at any step, an unstable
control loop is obtained, then the previous parameter value
shall determine the approximate stability boundary for the
corresponding direction. Otherwise, all points will be tested
within the range ∆p1·Nand ∆p2·N. Testing is done by
means of the characteristic polynomial in (16) and Matignon’s
theorem. Finally, the stability region will not always have
a rectangular shape. Thus, it is possible to determine the
shape by testing every point within the approximate rectangular
stability boundary. This is a heuristic method similar to [15]
and [16].
Once the procedure is complete, stable parameter ranges are
obtained for all controller parameter pairs and may be used
in FOPID controller optimization as lower and upper bounds

Approximate rectangular stability boundary
Fig. 3. Determination of the approximate rectangular stability boundary in the
(p1, p2)plane
for corresponding controller parameters. Optimizing only two
parameters at a time can be beneficial from the perspective
of conditioning the problem, albeit in this case it will not be
possible to satisfy several design constraints. Yet it poses great
difficulty to impose feasible robustness specifications in case of
the MLS. Thus, suboptimal controllers may be designed. The
performance of the system will be evaluated experimentally,
settling time τs, percent overshoot θ, and percent maximum
deviation from reference due to disturbance θdare used as
performance measures. In essence, we consider time-domain
simulations of the nonlinear model in (9) and minimize a cost
defined by
ISE =ˆt
0
|e(τ)|dτ, (22)
where e(τ)is the error signal. The choice of this particular
performance index is dictated by the necessity to minimize
the overshoot [6]. The optimization procedure is carried out
by means of the method described in [17], [18].
In what follows, we illustrate the proposed method on the
basis of experimental results.
V. EXPERIMENTAL RESU LTS
For the purpose of validating our control design approach
we use a real-life MLS provided by INTECO [5] and de-
picted in Fig. 4. It is connected to a computer running
MATLAB/Simulink thereby allowing to conduct real-time ex-
periments. The specific parameters of the model in (9) are as
follows: m= 0.0585kg, xmax = 0.0155m, g= 9.81m/s2.
Other parameters need to be identified. The corresponding
procedure is detailed in the following subsection.
A. Identification of the Nonlinear Model
Our task is to identify two functions i(u)and c(x), as well
as parameters fip1and fip2of the nonlinear model in (9).
Identification of i(u)is relatively simple and straightfor-
ward is done with the sphere removed from the MLS, since
only the coil current is measured. We obtain the following
polynomial:
i(u) = −0.3u2+ 2.6u−0.047.(23)
In addition, the deadzone in control is found to be udz =
[0,0.0182].
Fig. 4. Real-life laboratory model of the MLS
0 5 10 15
4
6
8
10
12
14
16 x 10−3
Time [s]
Position [m]
Original system
Identified model
Fig. 5. Results of nonlinear model parameters identification
Determination of c(x1)and parameters fip1and fip2, on
the other hand, is more involved. Because MLS is open-
loop unstable, only closed-loop identification is applicable. Our
approach is to use the existing PID control loop with
KP=−39, KI=−10, KD=−2.05 (24)
provided by INTECO. It should be noted, that a constant input
uc= 0.38 is added to the control law uP ID (t)in (24), that is
the full control law u(t)is such that
u(t) = uP ID (t) + uc.(25)
In order to determine the values of the parameters, we
employ time-domain simulations and minimize the model
output error by means of the least-squares method. The results
are as follows:
c(x1)=3.9996x4
1+ 3.9248x3
1−0.34183x2
1
+ 0.007058x1+ 2.9682 ·10−5(26)
and
fip1= 1.1165 ·10−3m/s, fip2= 26.841 ·10−3m.(27)
The results of the identification are presented in Fig. 5. It can
be seen, that a close fit to the response of the original response
of the system is achieved.

B. Design of FOPID Controllers
We first obtain a linear model as discussed in Section IV-A.
We choose a working point u0= 0.3726, x10 = 9.84 ·10−3
and obtain
GM(s) = −1788
s3+ 34.69s2−1737s−60240.(28)
Next, we apply the method detailed in Section IV-C. First,
we randomly generate FOPID controllers using the ranges
Kp∈[−100,0], Ki∈[−50,0], Kd∈[−25,0], λ ∈[0.8,1.2],
µ∈[0.5,1.0]. On the average, about 20 out of 100 tested
controllers are found to produce a stable closed-loop system.
After inspection, three of them are selected for the optimization
phase:
C1(s) = −42.8642 −18.5653
s1.06 −3.0559s0.94,(29)
C2(s) = −54.3649 −47.6078
s0.82 −6.5436s0.98,(30)
C3(s) = −45.3118 −4.24932
s0.86 −3.51115s0.98.(31)
For each controller in this set, we find stability boundaries
in different parameter planes, that is in (Kp, Ki),(Kp, Kd),
and (Ki, Kd), so that we can obtain a wider set of results.
Using the method in Section IV-C, with a step of ∆p= 1
and considering a maximum of N= 20 steps we locate the
following bounds:
KC1
p∈[−62,−34], KC1
i∈[−38,−1],(32)
KC2
p∈[−74,−35], KC2
d∈[−26,−3],(33)
KC3
i∈[−24,−1], KC3
d∈[−23,−2].(34)
We then proceed directly to the optimization procedure.
The FOMCON toolbox FOPID optimization tool is used [17],
[19]. We set the bounds of controller gains as in (32)–(34)
for each controller and optimize only the corresponding gains.
The number of iterations is, in general, limited to Niter = 5.
After optimization, the following controllers are obtained:
C∗
1(s) = −45.839 −18.504
s1.06 −3.0559s0.94,(35)
C∗
2(s) = −54.444 −47.6078
s0.82 −3.7773s0.98,(36)
C∗
3(s) = −45.3118 −4.916
s0.86 −2.9074s0.98.(37)
In the following, we provide the results of performance evalu-
ation of both the randomly generated FOPID controllers, and
the suboptimal ones. The controllers are evaluated in a two-
cascade closed control loop as detailed in Section IV-B. The
parameters of the retuning controllers are computed by means
of (19) and (20). The performance of FOPID controllers is
compared to the performance of the original PID control loop,
where the parameters of the PID controller are equal to those in
(24). The reference set point is xr= 0.010m, and a disturbance
impulse is considered, appearing for 200ms on the 10th second
Table I. Comparison of FOPID controller performance
FOPID τs[s] θ[%]θd[%] FOPID∗τs[s] θ[%]θd[%]
C1(s)1.85 24.0 60.3 C∗
1(s)1.68 14.8 56.4
C2(s)1.39 19.4 37.5 C∗
2(s)0.86 11.6 34.6
C3(s)4.68 14.6 55.7 C∗
3(s)3.84 15.0 58.3
0 5 10 15
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Position [m]
Reference
Original PID Control
Retuning FOPID Control
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Control law u(t)
Time [s]
Original PID Control
Retuning FOPID Control
Fig. 6. Step experiment: Original PID control vs. Retuning FOPID controller
of the simulation. With the conventional PID controller the
following results are achieved:
τs= 3.34 s, θ = 66.0%, θd= 60.6%.
In Table I the performance evaluation of the FOPID con-
trollers working in the retuning control loop is presented. It can
be seen, that the best performance is achieved, when controller
C∗
2(s)is used. The result of real-time simulation of this
controller versus the original PID control loop is provided in
Fig. 6. It can be seen that a significant improvement in control
system response is obtained. The controller C3(s)outperforms
the original PID only in terms of overshoot, while C∗
3(s)offers
similar settling time with a much smaller overshoot.
In addition, we consider a reference tracking experiment to
illustrate the ability of the controllers to provide appropriate
regulation across a wider operating range. The comparison
of the performance of the C∗
2(s)controller and the original
control loop is presented in Fig. 7. Once again, improvements
in the control loop performance can be observed.
VI. CONCLUSIONS
In this paper, we have presented a method for FOPID
controller design that allows incorporating fractional-order
dynamics into existing PID control loops. An unstable plant,
namely the MLS system was considered. A nonlinear model of
this plant was identified from a closed-loop experiment. Lin-
ear analysis methods were employed to determine stabilizing

0 5 10 15 20 25 30
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Position [m]
Reference
Original PID Control
Retuning FOPID Control
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Control law u(t)
Time [s]
Original PID Control
Retuning FOPID Control
Fig. 7. Reference tracking: Original PID control vs. Retuning FOPID controller
FOPID controllers and stability boundaries in two-dimensional
parameter planes thereof. The controllers were then evaluated,
and those with best performance were optimized. In all cases,
the optimization procedure enhanced the performance of the
control loop. Virtually all retuning controllers offer superior
performance compared to the original control loop, thereby
establishing the validity of the proposed approach.
ACKNOWLEDGMENT
This work was partially supported by the Estonian Doctoral
School in Information and Communication Technology.
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