Fuzzy Sliding Control with Non-linear Observer
for Magnetic Levitation Systems
A. M. Benomair*, A. R. Firdaus, M. O. Tokhi
Department of Automatic Control and Systems Engineering
The University of Sheffield
Sheffield, UK
ambenomair1@sheffield.ac.uk
Abstract—Magnetic levitation (Maglev) systems make signif-
icant contribution to industrial applications due their reduced
power consumption, increased power efficiency and reduced
cost of maintenance. Common applications include Maglev
power generation (e.g. wind turbine), Maglev trains and medical
devices (e.g. magnetically suspended artificial heart pump). This
paper proposes fuzzy sliding-mode controller ’FSMC’ with a
nonlinear observer been used to estimate the unmeasured states.
Simulations are performed with nonlinear mathematical model
of the Maglev system, and the results show that the proposed
observer and control strategy perform well.
Index Terms—Magnetic levitation system; Nonlinear ob-
server; Sliding-mode control; Fuzzy sliding-mode control;
I. INTRODUCTION
Magnetically levitated train constitutes one of the signifi-
cant advance made due to Maglev technologies. The Japan’s
Maglev train has established world record train speeds of 581
km/h in 2003 and 603 km/h in 2014. Similarly, vertical axis
wind turbine (VAWT) using magnetic bearing is able to start
producing power with wind speeds as low as 1.5 m/s. Exper-
imental results of Maglev (VAWT) in [1] show that system
vibration can be reduced by 37.5%, and power generating
capability of the system is increased by 12% using Maglev
concept. A combination of sliding mode control (SMC) with
fuzzy logic is used in this work as, it is difficult to prove
the stability of fuzzy logic controller (FSMC). Moreover,
SMC is insensitive to external disturbance and to parameter
variations, and able to decouple systems high-dimensional
system into lower-dimensional sub-systems. This feature can
help reduce the size of rule base of an FLC. Finally this
combination can effectively alleviate the prevalent chattering
problem of SMC [2].
Sliding-mode control is used in [3] to control the linearised
magnetic ball levitation system over a fixed set of operating
points using singular perturbation method. In this paper two
kinds of traditional fuzzy sliding-mode control (FSMC) are
first introduced, based on calculation of the control action
using error / change of error e/ ˙e in the first instance and
secondly the sliding function S. Then a novel fuzzy sliding-
mode control (NFSMC) is proposed in which the fuzzy
natural control u
nfs
is calculated using sliding function with
integration of an adjustable gain. The simulation results show
the superior performance can be achieved to the NFSMC in
comparison to the traditional FSMC.
A modified dynamic sliding-mode control has been re-
ported in [4] to overcome the chattering problem of SMC,
where the proposed controller is used to stabilize the dynamic
of Maglev system in the new coordinates using feedback
linearisation control. The results show that the modified
dynamic SMC can provide smoother control action compared
to classic dynamic SMC and up to 25% more robustness to
parameter variations.
II. N
ONLINEAR DYNAMIC MODEL OF MAGLEV SYSTEM
The Maglev system consider here serves to keep a small
steel ball in stable levitation at some steady-state operating
position. An electromagnet is used to produce forces to
support the ball (see Fig. 1). The electromagnetic forces
are related to the electrical current passing through the
electromagnet coil;
f
em
= −
1
2
L
0
x
0
i
x
2
(1)
where f
em
is the electromagnetic forces, L
0
is the addi-
tional inductance, x
0
is an arbitrary of the object and i is the
coil current.
Fig. 1: Active magnetic levitation system (MLS)
24th Mediterranean Conference on Control and Automation (MED)
June 21-24, 2016, Athens, Greece
978-1-4673-8345-5/16/$31.00 ©2016 IEEE 256
The object is suspended by balancing between the force
of gravity and electromagnetic force. Applying Newtons 3rd
law of motion, the dynamic form of the mechanical system
can be written as
f
net
=f
g
− f
em
(2a)
m ¨x = mg+ c
i
x
2
(2b)
where, f
net
is the net force and f
g
is force due to gravity.
Define the states x
1
= x (position), x
2
= V (velocity),
x
3
= i (current), the system input u = v (applied voltage)
and c =
1
2
L
0
x
0
. The nonlinear state space model of Maglev
system can be expressed as
⎡
⎢
⎢
⎢
⎢
⎣
˙x
1
˙x
2
˙x
3
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
x
2
g −
c
m
x
3
x
1
2
R
L
x
3
+
2c
L
x
2
x
3
x
2
1
+
1
L
u
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(3)
III. S
LIDING-MODE CONTROL
The state variables for the magnetic ball levitation system
are described as x
=
x
1
x
2
x
3
T
, the position of
the object, velocity and coil current. Define the vector of
the desired values as x
d
=
r
1
r
2
r
3
T
, where r
1
represents the desired position and r
2
& r
3
equal to zero.
Then the vector of tracking error is defined as
E =
e ˙e ¨e
=
e
1
e
2
e
3
(4)
A sliding mode controller can be effectively applied to
a nonlinear system in spite of parameter uncertainties and
external disturbances. The dynamics of a nonlinear system
can be described in the state space form as follows
˙x(t)=f(x; t)+g(x; t) u(t) (5)
For a control system, generally the sliding surface is
selected as
σ(x; t)=Sx(t) (6)
where S is a matrix of positive constant elements with
a dimension of [m × n]. For a 3rd order system, the time
varying surface σ(t) can be defined as
σ(t)=
d
dt
+ s
3−1
E (7)
where s is strictly positive constant and indicates the slope
of sliding surface. Thus from equation (7), the sliding surface
can be written as
σ(t)=
s
2
2s 1
⎡
⎣
e
1
e
2
e
3
⎤
⎦
(8)
or it can be simplified as follows
σ(t)=
s
1
s
2
1
⎡
⎣
e
1
e
2
e
3
⎤
⎦
(9)
where s
2
=2
√
s
1
considering the dynamical system in equation (5), the
developed control law is required to drive the system’s
trajectories towards the sliding surface in finite time t ≤ t
r
and maintain motion on the surface “ σ =0” thereafter, in the
presence of disturbance. A stability control law is designed
using Lyapunov stability condition σ
T
˙σ<0. The general
form of the control law for the sliding mode controller can
be written as
u = u
eq
+ u
n
(10)
where u
eq
and u
n
are the equivalent control and natural
control receptively. u
eq
can be interpreted as a continuous
control law to maintain the dynamics of the system on both
sides of the sliding surface “ i.e ˙σ =0” [5], while the
natural control u
n
is discontinuous control law to be designed
to account for nonzero uncertainties [6].
Substituting for u(t) from equation (10) into equation (5),
yields the closed-loop dynamics of the system as
˙x(t)=f(x; t)+g(x)(u
eq
+ u
n
) (11)
Since the necessary condition for the output trajectory to
remain on the sliding surface σ is that ˙σ =0which expressed
as ˙σ = S ˙x(t)=0. The equivalent control is augmented by
auxiliary control effort termed as hitting the sliding surface
and determined as
u
eq
= −(S ˆg(x))
−1
S
ˆ
f(x, t) (12)
The natural control is used to maintain the status trajectory
on sliding surfaces using signum function which requires
infinite switching on the part of actuator and the control
signal, and this is expressed [7] as
u
n
= −K(S ˆg(x))
−1
sign(σ) (13)
IV. F
UZZY SLIDING-MODE CONTROL
The general structure of FSMC’ is shown in Fig. 2.
The essential purpose of using this technique is to im-
prove the switching function of natural control (u
n
) with
fuzzy logic system. As the proposed method endeavours to
eliminate the chattering phenomenon during sliding mode
condition in the sliding surface.
The membership functions of fuzzy input (U
n
) and output
(FU
n
) are created through the the values of sliding surface
(σ) as seen in Fig. 3, making use of fuzzy sets of seven
257
Fig. 2: Block diagram of Maglev system with SMC controller
Fig. 3: Membership functions for fuzzy input ’U
n
’ and output
’FU
n
’
membership functions, namely: Negative Big (NB), Negative
Medium (NM), Negative Small (NS), Zero (Z), Positive
Small (PS), Positive Medium (PM), and Positive Big (PB),
with the range of [-1, 1] for both input and output.
Considering the fuzzy input and output, the rules-base to
produce the desired natural control (u
n
) can be set up as in
Table I
The natural control is obtained after defuzzification process
can be expressed as
u
n
= G
u
u
fn
(14)
where G
n
is a gain to factorise the output signal, while
u
fn
represents the crisp values obtained from the FLC. Thus
the overall control signal of FSMC can be written as
u = −(S ˆg(x))
−1
S
ˆ
f(x, t)+G
u
u
fn
(15)
V. N
ONLINEAR OBSERVER DESIGN FOR MLS
Let, the nonlinear system be represented as
˙x = f(x)+g(x) u
y = h(x)
(16)
where h(x) is a differentiable vector field of x. To design
an optimal controller based on state feedback for such a
nonlinear system; all states need to be measured or estimated.
The object position can be measured by an appropriate sensor
and coil current. The velocity measurement, however, is not
TABLE I: The Fuzzy rule base for FSMC
P
P
P
P
P
P
u
fn
σ
NB NM NS Z PS PM PB
PB PM PS Z NS NM NB
straight forward. The difficulties associated with unmeasured
states can be solved through a state estimation process.
A nonlinear observer can be set up as
˙
ˆx = f (ˆx)+g(ˆx)u + l(ˆx)(y − h(ˆx)) (17)
where g(ˆx)=
001/L
T
, h(ˆx)=ˆx
1
and l(x) is
nonlinear gain which can be written as
l(ˆx)=(J(ˆx))
−1
G (18)
Here, J is the Jacobian matrix of coordinates obtained from
nonlinear coordinate transformation as follows
J(ˆx)=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
10 0
01 0
2c
m
ˆx
2
3
ˆx
3
1
0 −
2c
m
ˆx
3
ˆx
2
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(19)
In general, the nonlinear gain l can be written as
l =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
G
1
G
2
ˆx
3
ˆx
1
G1 −
m
2c
ˆx
2
1
ˆx
3
G
3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(20)
Therefore, the nonlinear observer for MLS system can ex-
pressed as
⎡
⎢
⎢
⎢
⎢
⎣
˙
ˆx
1
˙
ˆx
2
˙
ˆx
3
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ˆx
2
g −
c
m
ˆx
3
ˆx
1
2
−
R
L
ˆx
3
+
2c
L
ˆx
2
ˆx
3
ˆx
2
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
0
1
L
⎤
⎥
⎥
⎥
⎥
⎥
⎦
u
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
G
1
G
2
ˆx
3
ˆx
1
G1 −
m
2c
ˆx
2
1
ˆx
3
G
3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(x
1
− ˆx
1
)
(21)
Verification of this kind of observer is considered as a
difficult task as it requires that the right side of equation
(3) should ascertain the global Lipschitz condition. For this
reason this type of observer is considered only as local stable
observer, which means that the estimation error dynamics of
equation (17) should have a finite escape time (observer error
converges to zero within a finite time). Thus in this paper the
gains G were constructed in such a way, that the observer
dynamics (High-Gain Observer HGO) are much faster than
the system dynamics (at least five-times).
258
The estimated states which are obtained from the nonlinear
observer are used to implement the control law of the exact
linearizing controller (see Fig. 4).
VI. S
IMULATION RESULTS
A block diagram representation of active magnetic lev-
itation with the proposed controller is shown in Fig. 3.
Simulations were carried out with the following values for
the system parameters: M = 21.2e-3 kg, C = 8.248*1e-5,
N*m/Aˆ2,R=4.2Ω, L = 0.02 H. The linear gain G was
calculated according to the formula with the value of a=50,
that was used in [8] the form of
G =
⎡
⎢
⎢
⎢
⎢
⎣
6a
12a
2
8a
3
⎤
⎥
⎥
⎥
⎥
⎦
(22)
Initially, the set point control for the system was located
at 15 mm with initial position of x
1e
=18× 10
−3
m,
x
2e
=0m/s and the obtained initial current as x
3e
=
m × g × 18 × 10
−3
= 0.9039 A. Fig. 5 shows the perfor-
mances of ’ SMC and FSMC. It is noted that both controllers
could stabilise the object and track the set point control and
the response rise time was shorter with FSMC. Moreover,
FSMC generated much smoother control signal than SMC
as shown in Fig. 6
Table II explores the effect of uncertainty in the mass of
object in the system with set-point of 15 mm, as the mass
was decreased to the limit of 70% and increased up to around
255%. It is noted that FSMC achieved the minimum Integral
Absolute Error (IAE) as well as minimum Mean Square Error
(MSE).
Step responses of the magnetic ball system with these two
controllers are shown in Figures 7-10 in presence of different
percentage mass uncertainties.
Figures 11 and 12 show the control efforts of SMC and
FSMC with mass percentages of 80% and 120% respectively.
As it can FSMC delivered a smooth the control signal and
overcome the phenomena of chartering.
VII. CONCLUSION
A nonlinear full-order observer-based controller with fuzzy
sliding-mode controller has been proposed FSMC to stabilise
an active magnetic levitation system. The performance of
Fig. 4: Overall Maglev system
Fig. 5: System response using SMC and FSMC with set-point
of 15 mm
Fig. 6: The control effort of SMC and FSMC controllers with
set-point of 15 mm
TABLE II: The performance indices for the control schemes
with set-point of 15 mm
mass uncertainty SMC FSMC
(m %) IAE MSE IAE MSE
70 - - 16.405 8.03 × 10
−3
78* 27.398 4.46 × 10
−3
--
80 27.478 4.24 × 10
−4
10.959 5.56 × 10
−4
90 13.933 3.23 × 10
−4
5.631 3.32 × 10
−4
105 5.284 3.94 × 10
−4
2.414 2.85 × 10
−4
110 9.157 5.11 × 10
−4
6.168 3.81 × 10
−4
120 11.095 9.33 × 10
−4
10.966 6.21 × 10
−4
122.3* 28.344 2.70 × 10
−3
--
130 - - 15.848 2.81e−1
150 - - 25.848 3.61
180 - - 41.334 8.03
200 - - 51.805 9.98
250 - - 78.624 12.95
255.5* - - 81.377 13.17.
proposed FSMC has been assessed in comparison to that
of conventional, and it has been demonstrated that FSMC
has stabilised the system in the presence of high mass
uncertainty of up to 155% ’that is mass percentage of 255%’,
as compared to SMC which could stabilise the system only
up to the limit of mass uncertainty of around 23%. The results
show that the generated control signals using FSMC are much
smoother than those with SMC, and this has improved the
regulation and quality of control provided by FSMC with
259
Fig. 7: System response in the presence of mass uncertainty,
m=80%
Fig. 8: System response in the presence of mass uncertainty,
m=90%
Fig. 9: System response in the presence of mass uncertainty,
m=110%
nonlinear observer. Future work will address experimental
implementation using the proposed control schemes.
R
EFERENCES
[1] C. Aravind, S. Tay, M. Norhisam, I. Aris, and J. Kumar, “Reduced
mechanical oscillations using the maglev concept in vertical axis wind
turbine,” in Renewable Energy and Sustainable Energy (ICRESE), 2013
International Conference on. IEEE, 2013, pp. 248–252.
[2] F. Song and S. M. Smith, “A comparison of sliding mode fuzzy
controller and fuzzy sliding mode controller,” in Fuzzy Information
Processing Society, 2000. NAFIPS. 19th International Conference of
the North American. IEEE, 2000, pp. 480–484.
[3] C.-L. Kuo, T.-H. S. Li, and N. R. Guo, “Design of a novel fuzzy
sliding-mode control for magnetic ball levitation system,” Journal of
Intelligent and Robotic Systems, vol. 42, no. 3, pp. 295–316, 2005.
Fig. 10: System response in the presence of mass uncertainty,
m=120%
Fig. 11: The control effort of SMC & FSMC with mass
percentage of 80%
Fig. 12: The control effort of SMC & FSMC with mass
percentage of 120%
[4] N. Al-Muthairi and M. Zribi, “Sliding mode control of a magnetic
levitation system,” Mathematical Problems in Engineering, vol. 2004,
no. 2, pp. 93–107, 2004.
[5] J.-J. E. Slotine, W. Li et al., Applied nonlinear control. Prentice-hall
Englewood Cliffs, NJ, 1991, vol. 199, no. 1.
[6] R. DeCarlo, S. H. Zak, G. P. Matthews et al., “Variable structure
control of nonlinear multivariable systems: a tutorial,” Proceedings of
the IEEE, vol. 76, no. 3, pp. 212–232, 1988.
[7] A. R. FIRDAUS and M. TOKHI, “Sliding mode control–based interval
type-2 fuzzy logic controller for quadcopter uavs,” in ASSISTIVE
ROBOTICS: Proceedings of the 18th International Conference on
CLAWAR 2015. World Scientific, 2016, pp. 555–563.
[8] A. M. Benomair and M. O. Tokhi, “Nonlinear full-order observer-
based controller design for active magnetic levitation via lqr-feedback
linearization,” in press.
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