INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 12(5), 599-611, October 2017.

Proportional-Integral-Derivative Gain-Scheduling Control
of a Magnetic Levitation System

C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,
S. Preitl, O.-M. Tanasoiu, S. Hergane

Claudia-Adina Bojan-Dragos, Radu-Emil Precup,
Stefan Preitl, Oana-Maria Tanasoiu, Stefania Hergane
Politehnica University of Timisoara
Department of Automation and Applied Informatics
Bd. V. Parvan 2, 300223 Timisoara, Romania
claudia.dragos@aut.upt.ro, radu.precup@upt.ro, stefan.preitl@upt.ro,
oanatanasoiu96@gmail.com, stefania.hergane@yahoo.com

Marius L. Tomescu*
Aurel Vlaicu University of Arad
Romania, 310330 Arad, Elena Dragoi, 2
*Corresponding author: tom_uav@yahoo.com

Abstract: The paper presents a gain-scheduling control design procedure for clas-
sical Proportional-Integral-Derivative controllers (PID-GS-C) for positioning system.
The method is applied to a Magnetic Levitation System with Two Electromagnets
(MLS2EM) laboratory equipment, which allows several experimental verifications of
the proposed solution. The nonlinear model of MLS2EM is linearized at seven operat-
ing points. A state feedback control structure is first designed to stabilize the process.
PID control and PID-GS-C structures are next designed to ensure zero steady-state
control error and bumpless switching between PID controllers for the linearized mod-
els. Real-time experimental results are presented for validation.
Keywords: gain-scheduling, magnetic levitation system, Proportional-Integral-
Derivative control; real-time experiments.

1 Introduction

During the last years several classical and adaptive control structures have been proposed
for positioning control system and applications for magnetic levitation systems. Some of these
structures are presented in [49], [52], [1], [35], [47], [56], [11], [27], [23], [19], [55], [5]. For example,
a high gain adaptive output feedback controller is designed in [5] by introducing two different
virtual filters and using back-stepping. Another adaptive control scheme that copes with the
modifications of the structural parameters of magnetic levitation systems is suggested in [1].

Gain-scheduling control solutions are popular nowadays, and they are briefly analyzed as
follows: fuzzy-based gain scheduling of exact feed-forward linearization control and sliding mode
controllers for magnetic ball levitation system are proposed in [28]. A high gain adaptive output
feedback control to a magnetic levitation system is discussed in [29]. A Proportional-Integral-
Derivative (PID) gain-scheduling controller for second order linear parameter varying, which
exclude time varying delay using a Smith predictor is given in [44]. Other interesting adaptive
gain scheduling control techniques for real practical applications are given in [6], [54], [7], [17], [53].

The paper is dealing with the position control of a ferromagnetic sphere in a Magnetic Lev-
itation System with Two Electromagnets (MLS2EM) laboratory equipment. A state feedback
control structure (SFCS) is first designed in order to stabilize the system by applying the control
signal only to the top electromagnet [8]. The simulated external disturbance can be applied

Copyright © 2006-2017 by CCC Publications



600
C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,

S. Preitl, O.-M. Tanasoiu, S. Hergane

to the bottom electromagnet. Starting with [9] the SFCS is next controlled by a Proportional-
Integral-Derivative Gain-Scheduling Control (PID-GS-C) structure. In the presented scheme,
the proportional, derivative and integral gains are adapted to the modifications of the operating
points. The paper proposes relatively simple classical and adaptive control structures which
belongs to the general case of linear and nonlinear control system structures [43], [38], [50], [20],
[42], [51], [32], [18], [36].

The paper is organized as follows: the nonlinear model and the linearized mathematical model
(MM) of MLS2EM are given in Section 2. The proposed control structure is next developed in
Section 3. The real-time experimental results are presented in Section 4 and the conclusions are
highlighted in Section 5.

2 Process modeling

The controlled MLS2EM laboratory equipment includes: two electromagnets (EM1 - the top
electromagnet and EM2 - the bottom electromagnet), the ferromagnetic sphere, sensors to detect
position of the sphere, computer interface, drivers, power supply unit, connection cables and an
acquisition board. The nonlinear state-space MM of ML2SEM is [24]:

ṗ(t) = v(t),

v̇(t) = −i
2
EM1

(t)·FemP 1·exp(−p(t)/FemP 2)
m·FemP 2

+ g +
i2
EM2

(t)·FemP 1·exp(−(xd−p(t))/FemP 2)
m·FemP 2

,

i̇EM1(t) =
ki·uEM1(t)+ci−iEM1(t)

fiP 1
fiP 2

·exp(−p(t)/fiP 2)
,

i̇EM2(t) =
ki·uEM2(t)+ci−iEM2(t)

fiP 1
fiP 2

·exp(−(xd−p(t))/fiP 2)
,

y(t) = km ·p(t),

(1)

where: p ∈ [0, 0.0016] - the sphere position (m), v ∈ R- the sphere speed (m/s), iEM1, iEM2 ∈
[0.03884, 2.38] - the currents in the top electromagnet (EM1) and bottom electromagnet (EM2)
(A), uEM1,uEM2 ∈ [0.005, 1] - the signals applied to EM1 and EM2, respectively (V), and y -
the process output (m), i.e., the measured sphere position. The parameters of the process are
determined analytically and experimentally [24], [16].

The model (1) is linearized around seven operating points (o.p.s) with the coordinates
P (j)(p(j),v(j), iEM1

(j), iEM1
(j)) where j is the index of the current operating point, j = 1...7.

The linearized state-space models and their matrices are:

∆ẋ(j) = A(j)∆x(j) + b(j)∆u(j),

∆y(j) = cT
(j)

∆x(j),

∆x(j) = [∆p(j) ∆v(j) ∆i
(j)
EM1 ∆i

(j)
EM2]

T , ∆u(j) = [∆u
(j)
EM1 ∆u

(j)
EM2]

T ,

A(j) =




0 1 0 0

a
(j)

21 0 a
(j)

23 a
(j)

24

a
(j)

31 0 a
(j)

33 0

a
(j)

41 0 0 a
(j)

44


 ,b(j) = [ b(j)uEM1 b(j)uEM2 ] =




0 0
0 0

b
(j)

31 0

0 b
(j)

42


 ,cT (j) = [1 0 0 0],

(2)
where T stands for matrix transposition, with the following elements of the matrices A(j)

and b(j), which depend on P (j) :



Proportional-Integral-Derivative Gain-Scheduling Control
of a Magnetic Levitation System 601

a
(j)

21 =
i
(j)
EM1

2

m
FemP 1
F 2
emP 2

exp(−p(j)/FemP2) +
i
(j)
EM2

2

m
FemP 1
F 2
emP 2

exp(−(xd −p(j))/FemP2),

a
(j)

23 = −
2i

(j)
EM1
m

FemP 1
FemP 2

exp(−p(j)/FemP2), a
(j)

24 =
2i

(j)
EM2
m

FemP 1
FemP 2

exp(−(xd −p(j))/FemP2),
a

(j)

31 = −(kiu
(j)
EM1 + ci − i

(j)
EM1)

p(j)

fiP 1
exp(p(j)/fiP2), a

(j)

33 = −
fiP 2
fiP 1
·e

x10
fiP 2 ,

a
(j)

41 = −(kiu
(j)
EM2 + ci − i

(j)
EM2)

p(j)

fiP 1
exp((xd −p(j))/fiP2), a

(j)

44 = −
fiP 2
fiP 1

exp((xd −p(j))/fiP2),
b

(j)

31 = ki
fiP 2
fiP 1

exp(p(j)/fiP2), b
(j)

42 = ki
fiP 2
fiP 1

exp((xd −p(j))/fiP2).
(3)

The operating points were chosen as follows such that to belong to the steady-state zone of
the sphere position sensor input-output map [33], to cover the usual operating regimes and to
avoid the extremities of the input-output map due to instability that may occur:

P (1)(0.0063, 0, 1.22, 0.39),P (2)(0.007, 0, 1.145, 0.39),P (3)(0.0077, 0, 1.07, 0.39),

P (4)(0.0084, 0, 1, 0.39),P (5)(0.009, 0, 0.9345, 0.39),P (6)(0.0098, 0, 0.89, 0.39),

P (7)(0.0105, 0, 0.83, 0.39).

(4)

The transfer function (t.f) of the state-space linearized MM (2) used in the control system
design is:

H
(j)
PC(s) = c

T (j)(sI−A(j))−1b(j)uEM1 =
k(j)

3∏
k=1

(s−p(j)k )
=

k
(j)
p

3∏
k=1

(1 + T
(j)
k s)

, (5)

where k(j)p = k(j)/
3∏

k=1

p
(j)
k , I is the third-order identity matrix and the time constants are T

(j)
k =

−1/k(j)p , k = 1...3, j = 1...7. The expressions of the t.f.s H
(j)
PC(s) are given in [22], [23].

3 Control solutions design

In order to support the development of next control solutions, the SFCS is designed for
the reduced third-order linear mathematical model (u2 = uEM2 = 0) of unstable magnetic
levitation system [22]. The first three state variables are kept and they lead to the state vector
x = [ p v iEM1 ]

T .
The pole placement method is applied to compute the state feedback gain matrix, kT

(j)

c =

[k
(j)
c1 k

(j)
c2 k

(j)
c3 ]

T , j = 1...7 . Therefore, for each linearized MM, with the t.f. H(j)PC(s) (5), the
closed-loop system poles p∗(j)k , k = 1...3, j = 1...7, [22], [23], have been imposed in order to
guarantee the stability of the linearized system. With the obtained state feedback gain matrix,
kTcbest = k

T
c_5 = [ 66.63 1.62 −0.15 ], two types of closed-loop t.f.s of the new state feedback

control structure (nSFCS), H(j)SFCS_5(s) result as:

H
(j)
SFCS_5(s) = H

(j)
x_5(s) = c

T (j)(sI−A(j)x_5)
−1b

(j)
uEM1 = c

T (j)[sI− (A(j) −b(j)uEM1k
T
c_5kAS)]

−1b
(j)
uEM1

=




k
(j)
SF CS_5

(1+T
(j)
1x_5s)(1+2ζ

(j)
5 T

(j)
2x_5s+T

(j)
2x_5

2
s2)
, j = 1...3, j ∈{6, 7},

k
(j)
SF CS_5

(1+T
(j)
1x_5s)(1+T

(j)
2x_5s)(1+T

(j)
3x_5s)

, j ∈{4, 5}.

(6)



602
C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,

S. Preitl, O.-M. Tanasoiu, S. Hergane

Table 1: nSFCS poles and parameters

nSFCS poles nSFCS parameters
O.p. p∗(j)1_5 p

∗(j)
2_5 p

∗(j)
3_5 k

(j)
SFCS_5 T

(j)
1x_5 T

(j)
2x_5 ζ

(j)
5 T

(j)
etax_5

(1) -0.79+1.02i -0.79-1.02i -0.10 0.084 0.0988 - 0.6 0.0077
(2) -0.92+0.88i -0.92-0.88i -0.13 0.065 0.0778 - 0.7 0.0078
(3) -1.07+0.62i -1.07-0.62i -0.16 0.054 0.0618 - 0.9 0.0081
(4) -1.65 -0.82 -0.21 0.046 0.0485 0.0123 - -
(5) -2.32 -0.41 -0.32 0.041 0.0314 0.0244 - -
(6) -3.07 -0.28+0.16i -0.28-0.16i 0.038 0.0033 - 0.9 0.0308
(7) -3.78 -0.23+0.20i -0.23-0.20i 0.034 0.0026 - 0.7 0.0332

The new SFCS poles obtained with the state feedback gain matrix kTc_5 and the nSFCS
parameters used next in the PID-GS-C structure are given in Table 1.

Due to the fact that the natural SFCS does not contain an I component, so it cannot ensure
the zero steady-state control error, the SFCS, as controlled plant, is included in a cascade control
structure (CCS) with PID controller in the outer loop. Depending on the operating points, seven
control solutions with PID controllers have been designed using pole-zero cancellation. The t.f.s
of the designed PID controllers extended with a first order lag filter can be expressed as:

H
(j)
PID_5(s) =




k
(j)
c_5(1+2ζ

(j)
c_5T

(j)
c_5s+T

(j)
c_5

2
s2)

s(1+T
(j)
fd_5s)

,j ∈{1, 2, 3, 6, 7},

k
(j)
c_5(1+T

(j)
c1_5s)(1+T

(j)
c2_5s)

s(1+T
(j)
fd_5s)

,j ∈{4, 5},
(7)

with the tuning parameters k(j)c_5, T
(j)
c_5, T

(j)
c1_5, T

(j)
c2_5 and T

(j)
fd_5:

k
(j)
c_5 =




1/(2 ·k(j)SFCS_5 ·T
(j)
etax_5), j = 1...3

0.05/(2 ·k(j)SFCS_5 ·T
(j)
Σx_5), j ∈{4, 5}

0.01/(2 ·k(j)SFCS_5 ·T
(j)
1x_5), j ∈{6, 7},

,




T
(j)
c_5 = T

(j)
etax_5, j ∈{1, 2, 3, 6, 7},

T
(j)
c1_5 = T

(j)
1x_5, j ∈{4, 5},

T
(j)
c2_5 = T

(j)
2x_5, j ∈{4, 5},

T
(j)
fd_5 = 0.1 ·T

(j)
1x_5, j = 1...7,

ζ
(j)
c_5 = ζ

(j)
5 .

(8)

Due to the oscillatory regime, the t.f.s coefficients k(j)c_5 must be adjusted. The numerical
values of PID-C parameters and the system performance indices – overshoot and settling time –
are synthesized in Table 2.

Due to the process nonlinearities, which also depend on the operating point, the switching
from one PID controller to another one is a useful solution to ensure improved performance
according to [48]. Therefore, a PID-GS-C structure illustrated in Figure 1 is designed. The
PID-GS-C structure is developed on the basis of the PID controller (with the notation PID-C in
Figure 1), with the t.f.:

∆u1x(t) = kp(t)e(t) + ki(t)

∫
e(t)dt + kd(t)ė(t), (9)

where kp is the proportional gain, kd is the derivative gain and ki is the integral gain.



Proportional-Integral-Derivative Gain-Scheduling Control
of a Magnetic Levitation System 603

Table 2: PID controller parameters

O.p. PID controller tuning PID control system
parameters performance indices

k
(j)
c_5 T

(j)
c_5 t

(j)
r_5 σ

(j)
1_5

(1) 60.65 0.009 4,25 0,24
(2) 98.23 0.008 4,25 0,24
(3) 150.83 0.006 4,5 0,24
(4) 89.88 0.005 5,0 0,23
(5) 28.67 0.003 4,0 0,23
(6) 40.98 0.003 3,5 0,24
(7) 111.24 0.003 4,0 0,24

Figure 1: Block diagram of PID-GS-C system structure

As shown in Figure 1, the gain-scheduling (GS) block uses the reference input r(t) and the
control error e(t) as input variables, and the tuning parameters of the PID-C, namely kp, kd and
ki, as output variables. The PID tuning parameters are obtained as follows:

kp(t) = kp max − (kp max −kp min) exp[−(α(t)|e(t)|)],
kd(t) = kd max − (kd max −kd min) exp[−(α(t)|e(t)|)],
ki(t) = (1 −α(t))ki max.

(10)

The parameters kp max, kp min, kd max, kd min, ki max and ki min are determined from the PID-
GS-C tuning parameters:

kp max =

{
max[k

(j)
c_5 · (T

(j)
c1_5 + T

(j)
c2_5)],j ∈{4, 5},

max[2 ·k(j)c_5 · ς
(j)
c_5 ·T

(j)
c_5],j ∈{1, 2, 3, 6, 7},

kp min =

{
min[k

(j)
c_5 · (T

(j)
c1_5 + T

(j)
c2_5)],j ∈{4, 5},

min[2 ·k(j)c_5 · ς
(j)
c_5 ·T

(j)
c_5],j ∈{1, 2, 3, 6, 7},

kd max =

{
max(k

(j)
c_5 ·T

(j)
c1_5 ·T

(j)
c2_5),j ∈{4, 5},

max(k
(j)
c_5 ·T

(j)2

c_5),j ∈{1, 2, 3, 6, 7},

kd min =

{
min(k

(j)
c_5 ·T

(j)
c1_5 ·T

(j)
c2_5),j ∈{4, 5},

min(k
(j)
c_5 ·T

(j)2

c_5),j ∈{1, 2, 3, 6, 7},
ki max = max(k

(j)
c_5),ki min = min(k

(j)
c_5),j = 1...7.

(11)

The parameter 0 ≤ α(t) ≤ 1 is included in order to have a smooth and continuous variation
of the switching from one PID controller to another one. The following equation is used to get
the value of this parameter:

α(t) = tanh(ηβ(t)) = [exp(2ηβ(t)) − 1]/[exp(2ηβ(t)) + 1], (12)



604
C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,

S. Preitl, O.-M. Tanasoiu, S. Hergane

where the parameter η determines the rate at which α(t) changes between 0 and 1 and chosen in
order to ensure a certain dynamics of the variation of α(t). The parameter β(t) is set in terms
of [35]:

β(t) =

{
1, |e(t)| > ξ,
0, |e(t)| < ξ,

, ξ = 0.9 ·r(t). (13)

As shown in [35], to design the PID-GS-C structure, the following conditions can be taken into
account: when the system is in steady-state error (i.e., |e(t)| is large), kp max and ki min are
activated in order to produce a large control signal and to overcome the undesirable oscillation
and overshoot; during the steady-state regime, ki max and kp min are activated to obtain a small
value of |e(t)| and to overcome the undesirable problem of overshoot.

4 Experimental results

All control structures, namely, SFCS, PID-C and PID-GS-C, were tested on the nonlinear
laboratory system and validated by real-time experiments. In all cases, the reference input was
set to 0.007 m from the top electromagnet and the control structures responses were tested on
the time frame of 20 s. The responses of the sphere position, the current and the control signal
in the top electromagnet were plotted.

The values of the parameters of the PID-GS-C block in Figure 1 are kp max = 5516, kp min =
0.6, kd max = 0.1225, kd min = 0.0036, ki max = 151, ki min = 0 and η ∈ {0.001, 0.1}. They have
been obtained using several experiments such that to get the best values in the context of a
compromise to tradeoff to overshoot. But other empirical performance indices can considered in
this tuning.

The following experimental scenarios were considered and performed:
(a) The state feedback control system structure with the best state feedback gain matrix was

tested on the laboratory equipment and the results are presented in Figure 2.
(b) The PID controller designed at seven operating points was tested on laboratory equip-

ment. The experimental results of the control system with PID controller designed only for three
o.p.s defined in Tables 1 and 2, i.e., (1), (5) and (7) are presented as responses of several variables
measured in the laboratory setup in Figures 3 to 5 as follows: the control signal versus time in
Figure 3, he current through EM1 versus time in Figure 4 and the controlled output versus
time in Figure 5. These results are better in comparison with the results presented in Figure 2
because of the PID-C, which ensures the zero steady-state control error and the reference input
is tracked. The pair of complex conjugated poles from the cases that correspond to the o.p.s (1)
to (3) (similarly to the o.p.s (6) and (7)) and the nonlinearities of the process lead to oscillations
at the beginning of transient responses and during the real-time experiments.

(c) The experimental results of the control system with PID-GS controller are presented as
responses of several variables measured in the laboratory setup in Figures 6 to 8 as follows: the
control signal versus time in Figure 6, he current through EM1 versus time in Figure 7 and the
controlled output versus time in Figure 8. These results are better in comparison with the results
presented in Figures 3 to 5 because of the PID-GS controller that ensures improved dynamic
behavior characterized by smaller overshoot and settling time.



Proportional-Integral-Derivative Gain-Scheduling Control
of a Magnetic Levitation System 605

Figure 2: Real-time experimental results of state feedback control system structure with the best
state feedback gain matrix

Figure 3: Control signal u1 = u
(j)
EM1, j ∈ {3, 5, 6}, versus time for control systems with PID

controller for o.p.s. (3), (5) and (6)

Figure 4: Current iEM1 versus time for control systems with PID controller designed for o.p.s.
(3), (5) and (6)



606
C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,

S. Preitl, O.-M. Tanasoiu, S. Hergane

Figure 5: Sphere position p versus time for control systems with PID controller designed for
o.p.s. (3), (5) and (6)

Figure 6: Control signal u1 = u
(j)
EM1 versus time for control systems with PID-GS controller

Figure 7: Current iEM1 versus time for control systems with PID-GS controller



Proportional-Integral-Derivative Gain-Scheduling Control
of a Magnetic Levitation System 607

Figure 8: Sphere position p versus time for control systems with PID-GS controller

5 Conclusions

The paper has presented three control structures developed to control the position of the
sphere in an MLS2EM laboratory setup. In order to use relatively simple control structures, the
presented nonlinear model of the MLS2EM was linearized around seven operating points. To
stabilize the process, a state feedback control structures was designed and the best state feedback
gain matrix was found.

Since the SFCS does not ensure the zero steady-state control error, seven PID controllers were
designed. To ensure the switching between different PID controllers, a PID-GS controller was
developed and implemented. All control system structures were tested on the nonlinear model,
accepting the main values of the parameters given in [24]. This paper has considered a low-cost
implementation of the controllers but this must be viewed in connection with the complexity of
the control algorithms, and several approaches can be used [10], [12], [37], [13], [15], [31], [46].

The real-time experimental results prove that the PID-GS-C structure discussed in this paper
guarantees the improvement of control system performance regarding to step modifications of
reference input. They ensure zero steady-state control error, small overshoot and settling time.
As shown in the previous section, the choice of the parameters of the PID-GS-C block in Figure
1 has been carried out by conducting several experiments such that to aim the best values from
the point of view of the compromise to the achievement of best overshoot and settling time. This
is a limitation of the control system structure presented in this paper.

The systematic choice of the parameters of PID-GS-C will represent a direction of future
research by ensuring the optimal tuning using classical [4], [22], [39], [2], [25], [14] and modern
optimization algorithms [40], [41], [45], [26], [33], [30], [34], [21], [3] or their combinations. Future
research will also be focused on the design of control systems with PI(D) fuzzy gain-scheduling
controllers, Takagi-Sugeno fuzzy controllers and hybrid structures including sliding mode control
and gain-scheduling control for improved performance indices.

Acknowledgment

This work was supported by grants from the Partnerships in priority areas – PN II pro-
gram of the Romanian Ministry of National Education and Scientific Research (MENCS) –
the Executive Agency for Higher Education, Research, Development and Innovation Funding



608
C.-A. Bojan-Dragos, R.-E. Precup, M.L. Tomescu,

S. Preitl, O.-M. Tanasoiu, S. Hergane

(UEFISCDI), project numbers PN-II-PT-PCCA-2013-4-0544, PN-II-PT-PCCA-2013-4-0070 and
PN-II-RU-TE-2014-4-0207.

Bibliography

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