INTRODUCTION
One of the ways that noncontact surfaces are maintained is via
magnetic suspension. This system is commonly referred to as Magnetic Levitation
system (Maglev) which has been used in the vehicle suspension system and
magnetic bearing system by B.A. Holes of University of Virginia in 1937
for the first time. In 1954 this system was utilized by Laurencean and
Tournier at ORENA in France for the purpose of aerodynamic testing in
wind tunnels (Covert, 1988). Furthermore, the system have practical uses
in many industrial systems such as in highspeed maglev passenger trains,
levitation of wind tunnel models, vibration isolation of sensitive machinery,
levitation of molten metal in induction furnaces and levitation of metal
slabs during manufacturing.
In recent years, a lot of works have been reported in the literature
for controlling magnetic levitation systems. The feedback linearization
technique has been used to design control laws for magnetic levitation
systems (JingChung, 2002; Trumper et al., 1997; Hajjaji and Ouladsine,
2001). Other types of nonlinear controllers based on nonlinear methods
have been reported by Yang and Tateishi (1998), Tanaka and Torii (2004),
Huang et al. (2000) and Teng and Qiao (2008). Control laws based
on phase space (Zhao et al., 1999), linear controller design (Rifai
and YoucefToumi, 1998) and neural network techniques and fuzzy controllers
have also been used to control magnetic levitation systems (Lairi and
Bloch, 1999; Anh and Timothy, 2008; MuñozGómez et al.,
2006). One of the first applications of Sliding Mode Control (SMC) to
magnetic levitation systems was carried out by Cho et al. (1993).
Chen et al. (2001) designed an adaptive sliding mode controller
for a rather different type of magnetic levitation systems called dualaxis
maglev positioning system. Muthairi and Zribi (2004) designed static and
dynamic sliding mode controller for the magnetic levitation system and
ChaoLin et al. (2005) designed a novel Fuzzy slidingmode controller
for magnetic ball levitation system.
In this study, we consider a magnetic levitation system and propose classic
controllers like sliding mode, PID controller and new hybrid controllers.
In proposed hybrid controllers the Feedback Error Learning (FEL) basedon
sliding mode is used to train the neural network with Radial Basis Function
(RBF). The proposed methods demonstrate the advantages of the adaptive
neural network and sliding mode control strategies with achieving better
performance.
MATHEMATICAL MODEL OF THE SYSTEM
Figure 1 is shown a diagram of the magnetic levitation
system. Note that only the vertical motion is considered. The dynamic
model of the system can be written as (Barie and Chiasson, 1996; Muthairi
and Zribi, 2004):

Fig. 1: 
Diagram of the magnetic levitation system 
where, p denotes the ball`s position, v is the ball`s velocity, i is
the current in the coil of the electromagnet, e is the applied voltage,
R is the coil`s resistance, L is the coil`s inductance, g is the gravitational
constant, C is the magnetic force constant and m is the mass of the levitated
ball. The inductance L is a nonlinear function of ball`s position p. The
approximation of L is:
where, L_{1} is a parameter of the system.
Let the states and control input be chosen such. Thus, the statespace
model of the magnetic levitation system can be written as:
Consider the following nonlinear change of coordinates:
The dynamic model of the magnetic levitation system in the new coordinates
system can be written as:
Where:
The function f (z) and g (z) correspond in the original coordinates to
the following functions, respectively:
The relationship between the input and the output of the system can be
found as:
The parameters of the magnetic levitation system are as follows (Barie
and Chiasson, 1996). The coil`s resistance R = 28.7 Ω, the inductance
L_{1} = 0.65 H, the gravitational constant g_{c} = 9.81
msec^{2}, the magnetic force constant C = 1.24x10^{4}
and the mass of the ball m = 11.87 g and x_{1d} = 0.01 is the
desired value of x_{1}.
MATERIALS AND METHODS
Sliding mode controller: Sliding mode control is a variable
structure control utilizing a highspeed switching control law to drive
a system state trajectory onto a specified and user chosen surface, so
called sliding surface and to maintain the system state trajectory on
the sliding surface at subsequent times (Slotin and Li, 1991). In this
paper, the sliding surface on the phase plane can be defined as:
In case n = 2:
Based on the Lyapunov theorem, the sliding surface reaching condition
is:
Using Eq. 10 and 3, the switching surface can be written as:
Note that the choice of the switching surface guarantees that x_{1}–x_{1d}
converges to 0 as t → ∞, when we have S = 0.
The following proposition gives the U :
For eliminating chattering in control signal, we use saturation function
instead of sign. So the following proposition gives the U_{s}:
Radial basis function neural networks: In this study, we use a
type of neural networks which is called the Radial Basis Function (RBF)
networks (Powell, 1987). These networks have the advantage of being much
simpler than the perceptrons while keeping the major property of universal
approximation of functions (Poggio and Girosi, 1987). RBF networks are
embedded in a two layer neural networks, where each hidden unit implements
a radial activated function. The output units implemented a weighted sum
of hidden unit outputs. The input into an RBF network is nonlinear while
the output is linear. Their excellent approximation capabilities have
been studied in (Park and Sandberg, 1991). The output of the first layer
for a RBF network is:
The output of the linear layer is:
where, xεR^{n} and yεR^{m} are input
vector and output vector of the network, respectively and φ = [φ_{1},
Λ, φ_{n}]^{T} is the hidden output vector. n
is the number of hidden neurons, W_{j} = [w_{j1}, Λ,
w_{jn}]^{T} is the weights vector of the network, parameters
c_{i} and σ_{i} are centers and radii of the basis
functions, respectively. The adjustable parameters of RBF networks are
W, c_{i} and σ_{i}. Since the network`s output is
linear in the weights, these weights can be established by leastsquare
methods. The adaptation of the RBF parameters c_{i} and σ_{i}
is a nonlinear optimization problem that can be solved by gradientdescent
method.
RBF sliding mode controller: The sliding variable, S will be used
as the singleinput signal for establishing a RBF neural network model
to calculate the control law, u. Then for the singleinput and singleoutput
case in this paper, the output of the controller based on RBF networks
is:
where, n is the number of hidden layer neurons and u is the final closedloop
control input signal. In order to combine the advantages of sliding mode
and adaptive control schemes into the RBF neural network, an adaptive
rule is introduced to adjust the weightings between hidden and output
layers.
If a control input u can be chosen to satisfy this reaching condition
(11), the control system will converge to the origin of the phase plane.
Adaptive law is used to adjust the weightings for searching the optimal
weighting values and obtaining the stable convergence property. The adaptive
law is derived from the steep descent rule to minimize the value of with respect to W. Then the updated equation of the weighting parameters
is:
Or
And from Eq. 12, we have:
Form Eq. 3, we can find that:

Fig. 2: 
Block diagram of the modified RBFsliding mode 
Finally we can find updating rule as follow:
From Eq. 16, we have:
It is clear that we do not need any identifier for magnetic levitation
system.
For improve control signal, a modified RBFsliding mode controller is
now designed for the magnetic levitation system. Figure
2 is a block diagram of the modified RBFsliding mode controller.
Feedback error learning architecture: The structure of FEL is
shown in Fig. 3, which is proposed by Kawato et al.
(1988). In this architecture, the neural network is used as a feed forward
controller and trained by using the output of a feedback controller as
error signal. The total control input U to the plant is equal to:
where, U_{C}(t) and U_{N}(t) are outputs of CFC and NN,
respectively.
The feedbackerror learning scheme has the following advantages: (1)
the teaching signal is not required to train the neural network. Instead,
the error signal is used as the training signal, (2) the learning and
control are performed simultaneously in sharp contrast to the conventional
learnthencontrol approach and (3) backpropagation of the error sign
through the model of the controlled object or through the model of the
controlled object is not necessary.
In this control procedure, the CFC controller is a sliding mode controller
and in Fig. 4, modified FELsliding mode controller,
the CFC controller is a proportionalplusderivative (PD) controller.
The total control input U to the plant is equal to:

Fig. 3: 
General structure of FEL 

Fig. 4: 
The modified hybrid control approach 
where, U_{P}(t), U_{N}(t) and U_{S}(t) are PD
controller, NN output and sliding mode outputs, respectively.
The PD controller and sliding mode controller guarantees the stability
of the overall system and ensures adequate performance prior to convergence
of the neural network weights and reduces the steadystate output errors
due to disturbance inputs.
PD controller: Considering the following PD control for the Fig.
4 block diagram for eliminate error between x_{1d} and x_{1}:
PID controller: The continuous form of a PID controller, with
input e (0) and output u_{PID} (0), is generally given as:
where, K_{P} is the proportional gain, T_{i} is the integral
time constant and T_{d} is the derivative time constant. We can
also rewrite (28) as:
where, is
the integral gain and K_{d} = K_{P}T_{d} is the
derivative gain.
RESULTS AND DISCUSSION
Here, simulation results are presented. The figures show the position
versus time (millisec) and the control (the applied voltage) versus time
for the system.
First, the sliding mode controller is applied to the magnetic levitation system.
The parameters of controllers are chosen such that W = 350, λ_{1}
in sliding surface is set as 61 and λ_{2} = 930. The simulation
results are shown in Fig. 5. It can be seen from the Fig.
5 that there is a small steadystate error in the position and some chattering
can be seen due to this controller. Figure 6 shows sliding
mode controller that we eliminate steadystate and chattering by using saturation
function instead of sign function (14).
Second, the RBFSliding mode controller of system results are shown in
Fig. 7, the control signal has chattering and it is
not suitable for voltage source so we proposed modified RBFSliding mode
controller of magnetic levitation system. The results in Fig.
8 shows that we improve control signal by proposing this new methods.
Third, Fig. 9 shows the FELSliding mode controller
of system that there is a small steadystate error in the position and
we eliminate it by modified FELsliding mode controller. Figure
10 shows the modified FELsliding mode controller of magnetic levitation
system. For PD controller k_{p} and k_{d} are set as 1.4
and 0.5.
Fourth, Fig. 11 shows the PID controller of plant.
The parameters of controllers are chosen such that K_{P}, K_{i}
and K_{d} are 1990, 9996 and 88.5, respectively.
Finally, in Fig. 12, we compare all results with together.
Therefore, the simulation results indicate that the proposed control schemes
work well when applied to the magnetic levitation system.

Fig. 5: 
Classical sliding mode control (a) position of ball
and (b) control signal) 

Fig. 6: 
Sliding mode without chattering (a) position of ball
and (b) control signal 

Fig. 7: 
RBFSliding mode controller without low pass filter
(a) position of ball and (b) control signal 

Fig. 8: 
Modified RBFSliding mode controller (a) position of
ball and (b) control signal 

Fig. 9: 
FELSliding mode controller without PD controller (a)
position of ball and (b) control signal 

Fig. 10: 
Modified FELsliding mode controller (a) position of
ball and (b) control signal 

Fig. 11: 
PID controller (a) position of ball and (b) control
signal 

Fig. 12: 
Comparison of all methods results (Modified RBFSliding
mode, Sliding mode without chattering, RBFSliding mode, Modified
FELSliding mode and PID) 
CONCLUSION
This study introduced classical and hybrid methods for control the
magnetic levitation system which has practical uses in many industrial
systems.
In this research, a new RBFsliding mode control method, a new FELsliding
mode control method and modified FELsliding controller for magnetic levitation
is proposed, which combines the merits of adaptive neural network and
sliding mode control. Based on the Lyapunov stability theory, a RBFsliding
mode controller is designed for stabilization of magnetic levitation system
to the desired point in the state space. The adaptive neural network controller
in this new approaches uses generalized learning rule therefore does not
need to compute Jacobain of plant which comes more simplicity. Simulation
results show that the proposed hybrid controllers are able to control
magnetic levitation and the chattering phenomenon of conventional switching
type sliding control does not occur any more in control process.