INTRODUCTION
Today, express trains have significantly eased traffic congestions in
urban areas. The basic idea in designing such trains is the magnetic suspension.
As shown in Fig. 1, in a single suspension system, the
mass is under the influence of two forces: the force of gravity and the
magnetic force. In the case of passenger trains, a different type of suspension
system as shown in Fig. 2 is used.
In fact, the two groups of electromagnets embedded in a module are connected
with a rigid body: therefore their motion states are coupled. Currently,
the general control method controls the two groups of electromagnets separately
using two independent controllers, each of which acts according to respective
controlled object: the coupling between the two groups of electromagnets
is regarded as disturbance and suppressed by enhancing the robustness
of individual controllers. However, this method cannot actively overcome
the uncertainty issues and the control performance is not desirable especially
in the presence of external disturbances.
In recent years so many studies have been carried out on electromagnet suspension
systems. Chen et al. (2003) presented a larger
moving range dualaxis magneticlevitation (maglev) system. A repulsive maglev
system with four active guiding tracks was adopted.

Fig. 1: 
Magnetic suspension system 

Fig. 2: 
The structure of the suspension system 
The system was treated as
a multiinput multioutput system and an adaptive controller was designed. Huang
and Lin (2003) proposed a novel adaptive fuzzy sliding mode controller and
successfully employed it to control an active hydraulic suspension system. A
maglev transportation system including levitation and propulsion control is
the subject of considerable scientific interest because of highly nonlinear
and unstable behaviors. Wai and Lee (2008a), based on
the concepts of mechanical geometry and motion dynamics, developed the dynamic
model of a maglev transportation system including levitated electromagnets and
a propulsive Linear Induction Motor (LIM). Then, a modelbased Sliding Mode
Control (SMC) strategy was introduced. Moreover, Wai and
Lee (2008b) focused on the sequential developments of backsteppingbased
control systems including a Backstepping Control (BSC), an adaptive BSC (ABSC)
and an adaptive dynamic surface control (ADSC) for the levitated positioning
of the linear maglev rail system. Yang et al. (2008) proposed an adaptive robust output feedback controller for the position
tracking problem of a magnetic levitation system while using a noisy position
sensor. Bonivento et al. (2005) addressed the
problem of positioning a ball in a vertical magnetic field created by a pair
of electromagnets while rejecting some external disturbances. The presence of
uncertainties on the physical parameters characterizing the system has been
taken into account, designing a control law capable of solving the problem robustly. De Queiroz and Pradhananga (2007) presented a general
formulation for constructing stabilizing control laws with a timevarying bias
flux. The design of a bias flux function of the mechanical states, general conditions
were established on the bias function that ensures a singularityfree controller
and power losses that converge to zero as the mechanical states converge to
zero, without affecting the system stability. Position regulation of a magnetic
levitation device is achieved through a Control Lyapunov Function (CLF) feedback
design. Peterson et al. (2006) showed experimentally
that, by selecting the CLF based on the solution to an algebraic Riccati equation,
it is possible to tune the performance of the controller using intuition from
classical LQR control. Lin et al. (2005) proposed
a hybrid controller using a Recurrent Neural Network (RNN) to control a levitated
object in a magnetic levitation system. A nonlinear dynamic model of the system
is described and a computed force controller, based on feedback linearization,
is proposed, based on feedback linearization, to control the position of the
levitated object. Fialho and Balas (2002) presented a
framework for designing road adaptive suspension controllers. Linear parametervarying
techniques were used in combination with nonlinear backstepping to achieve the
desired nonlinear response of the vehicle suspension. Hassanzadeh et al. (2008) defined an objective function based (on) transient
response of step response. Then considered a maglev system as a case study in
which by using genetic algorithm, the optimal controller parameters could be
assigned. Aliasghary et al. (2008) introduced
classical and hybrid methods for controling the Maglev system. A new RBFsliding
mode control method, a new FELsliding mode control for Maglev were proposed,
which combines the merits of adaptive neural network and sliding mode control.
The goal of designing an adaptive controller for a double magnetic suspension
system is to suspend an object in a certain distance from a magnetic rail by
using two electromagnets. As mentioned before, one of the most significant applications
of magnetic suspension is in express trains. The advantage of the suggested
method over the work done by DeSheng et al. (2006)
is that their presented decoupled controller shows no robustness to mass changes,
parameters changes, noise and outer disturbance; so that for example, with a
small variation into the train, the performance of their proposed method will
decrease. However the current study will solve the mentioned problems and improves
their method.
MATERIALS AND METHODS
Double ElectroMagnet (DEM) has five degrees of freedom in movements:
heave, sway, pitch, roll and yaw; Among them, only heave and pitch movements
are to be controlled: hence the system in this case has two degrees of
freedom.
Variable parameters: The in hand variable parameters in the problem
are: the mass of the object (m) and the inductance of the coil of the
electromagnets (K).
System modeling: Structure of the system: The structure of DEM shown
in Fig. 2 includes two identical electromagnets which are
connected by a rigid bracket. Magnets 1 and 2 provide the suspension forces
needed for points 1 and 2, respectively. The suspended object can be considered
as a solid object with two electromagnets. In addition, two sensors are used
to measure the states of the suspension and two poises can be used to provide
load forces.
Double ElectroMagnet can be simplified as shown Fig. 3.
Parameters of the system are defined as follows: m is the mass of the
suspended object, I is the spinning inertia in the center of the object
O, F_{1} and F_{2} are magnetic forces, N_{1}
and N_{2} are load forces on the two ends of the bracket, d is
the distance of the center of the object from the rail, δ_{1}
and δ_{2} are the distances from the points where, F_{1}
and F_{2} are applied, respectively.

Fig. 3: 
Details of suspension system 
d_{1} and d_{2} are the distances between the solid object and the corresponding points
on the rail. l is the distance between the center of the object
O and the point where magnetic forces are applied and finally, L is the
distance between the center of the object and the point where load forces
are applied.
Simplifying assumptions
• 
The stiffness of the rail is assumed to be infinite
and only the movement of the DEM relative to the rail is significant 
• 
It is assumed that leakage flux, edge effect of the magnetic force,
magnetic resistance of the core and the rail are negligible 
• 
A weight center is considered for the weight of the object. The
weight of the two magnets is assumed to be identical: therefore, the
total weight of the bracket and the magnets can be represented by
a weight center, O, shown in Fig. 3 
• 
The lengths of the magnets are assumed to be very small and the
points where the forces are applied are fixed 
• 
Load forces, created by the poises, have only one downward component 
Double ElectroMagnet (DEM) suspension system is a complex system including
mechanical dynamics of DEM, the relation between the current and the electromagnetic
force and the relation between the voltage and the current.
Mechanical dynamic equations: Double ElectroMagnet (DEM) has
five degrees of movement freedom: heave, sway, pitch, roll and yaw. For
the controller, only heave and pitch movements are taken into account.
Vertical movement and the movements of the weight center, O and the twisting
are movements around the main rotation pivot. The positive directions
of movement and rotation are downward and counterclockwise, respectively.
According to the principle of force transfer and the second law of Newton,
the mechanical dynamics equations are obtained as follows:
According to Fig. 3, we have:
Using Eq. 1 and 2, we have:
where, the parameters are defined as:
Magnetic force equation: The magnetic force between the rail and
the magnets can be represented as:
where, K is the inductance coefficient, δ is the distance between
the rail and magnets and i is the electrical current of the coil of the
magnet. According to Fig. 3 we have:
Where:
Hence, the magnetic force in Fig. 3 can be represented
as follows:
Electricaldynamic equations: The electricaldynamic equations
of the electromagnetic suspension system can be considered as a resistanceinductance
circuit and this can be modeled as follows:
where, u (t) is the control voltage of the magnet and R is the total
resistance of the whole circuit. Using Eq. 57, we have
the electrical equations of the magnet 2 as follows:
where, u_{1}(t) and u_{2}(t) are the control voltages
of the two magnets.
The state space representation: Let us select the state of the
system as follows:
The one may come up with the following equations as the statespace representation
of the system:
Where:
In the following the case where all system parameters are not exactly
known is to be investigated.
The case in which the exact values of some parameters of the system are
not known is to be investigated. It is inevitable to apply adaptive methods
for controlling such systems. Designing an adaptive controller includes
two major steps:
• 
The control law 
• 
The adaptation law 
In this study a linearizing feedback is proposed. The introduced feedback
is very useful in designing the control law.
ADAPTIVE CONTROL
Assume the nonlinear system can be represented as the following companion
form,
Then designing the control law and the adaptation law can be done by
using the exact feedback linearization: the dynamic equation of the system
can be expresses as:
In case where all system parameters are known, the control law is given
by:
However, in this study it is assumed that the parameters of the system are
not exactly known. Hence the exact values of the functions
and
are not known either and therefore, the Eq. 16 cannot be
realized to obtain the control law. In this study, the idea of substituting
the parameters by their estimation is introduced, which lead us to the following
control law (Khalil, 2002):
Now, the necessary conditions for implementation of this method should
be discussed: The first step is to represent the functions f (x) and g
(x) linear in parameters. Hence, the coefficients are considered as some
unknown parameters that are to be estimated. Hence, the estimated values
of
and
can be obtained and an adaptation law for the estimation of coefficients
can be presented. The second step is to find an estimate for each ,
which is the derivative of the output.
Suppose that f (x) and g (x) are linear in parameters (as will be shown,
this is possible for the discussed system):
where, f_{i} and g_{i} are some known functions and α_{i}
and β_{i} are unknown parameters that are to be estimated.
Now, by considering the discussed form for f (x) and g (x), an expression
for should
be found. Where, r is the system relative degree.
Having the equations, some new unknown above parameters emerge which
are the combinations of previous unknown parameters. By estimating the
new unknown parameters, the estimated values of can
be obtained. Now, we return to the case in which the exact parameter values
are known. It was mentioned that for such cases the control parameter
υ is chosen as follows so the trace error converges to zero.
We use the derivatives of the output which are not available. Hence their
estimates are used. In such case, the estimated value of υ is defined
as follows:
It is known that the relative degree of the system is three. Hence, the
control law is given by:
Now, it is time to discuss the necessary adaptation law for determining
the values of the unknown parameters. In order to do so, first the control
input defined in Eq. 21 is substituted in Eq.
16, hence we have:
Subtracting υ from the above equation results in:
which can be written as:
Substituting the functions in Eq. 25 with their estimates
in Eq. 19, the right hand side of the above equation
can be written in standard regressor form.
where, Φ is the difference between estimated (Φ = θ_{e}–θ),
W is the socalled nonlinear regressor.
This represents a relation between the tracking error and the estimations
error of the parameters which can be written as:
Now, we have:
The transfer function F(s) is not an SPR transfer function: therefore, it should
be converted to a SPR transfer function by defining a new filtered error variable
as (Slotine and Li, 1991):
where, the values of the coefficients p_{1} and p_{2}
are found such that
is an SPR transfer function. However, in order to obtain the adaptation
law, the value of e_{1} is needed which in turn requires the first
and the second order derivatives of the tracking error (e). If direct
differentiation is used to get these derivatives, the system becomes noisesensitive:
therefore this method is not applicable.
To obtain the adaptation law, first, a combination is defined as the
error:
Since, the constant coefficients are not affected by the filter F(s),
the above equation can be written as:
Now, by using Eq. 27 and Eq. 31,
we have:
where, the variable ξ is defined as:
Now, by using Eq. 32 and 33, the
adaptation law can be obtained by using either a gradient algorithm or
a least square algorithm. In case of SISO system, the adaptation law obtained
by the normalized least square algorithm is:
For the Multi Input Multi Output (MIMO) system ξ has a matrix form
and the above equation changes to:
The problem that may raise here is that the term ξ^{T} pξ
may be very large and therefore
and
become singularlike point and cause the parameters to diverge. To solve
this issue, the adaptation law should be modified as follows:
This can be considered as a new variation of the normalized least square
algorithm.
Implementation of the proposed model: In present case , f (x)
and g (x) can be written as:
According to Eq. 18 and 38, we
have n_{1} = 6 and n_{2} = 1. Now, the derivatives of
the output can be calculated as follows:
Hence:
And finally
can be expressed as:
As can be seen from the above equation, the function
has four terms. The function
can be obtained in a similar way:
To obtain the matrixes W and Φ, one can use Eq. 25.
According to Eq. 38 and 41 we have:
and
In Eq. 45, k_{1} is the coefficient of the
dynamic of the error. Here, for the parameters to converge to their true
values, matrix W must satisfy Persistently Excitation (PE) condition.
The continuous stimulation condition of the signal should be discussed
for the matrix W. The adaptation algorithm is given by Eq.
25.
Present proposed method can now be shown in Table 1.
The proposed control method is simulated by using MATLAB software. It
is worth noting that that the current system is a coupled 2input2output
system.
Table 2: 
Parameters of the system 

Here, it is assumed that some of the parameters of the system
(m (the mass of the plate), R (the resistance of the coils) and K (the
inductance coefficient of the coils)) are not known Since the unknown
parameters are combination of the above parameters, by assigning values
to the parameters m, R and K, initial values for the parameters that should
be estimated can be easily obtained. The point here is that in this case,
initial values of m, R and K should be known: these initial values cannot
be set to zero because by doing so, the convergence of the parameters
are jeopardized. In simulation, an unknown limited poise is assumed to
be placed on the plate and the performance of the proposed adaptation
algorithm is evaluated. It is noticeable that the value of ÿ is estimated
in every time step. The initial values for m, R and K are 10, 15 and 0.1,
respectively. The values of the Parameters are shown in Table
2.
The reference model: For the suspended plate to have a smooth
motion, a reference model is used to provide the plate with appropriate
motion conditions. The point is that the relative degree of the reference
model should be greater or equal to the relative degree of the system.
Since the relative degree of the system is equal to 3, the reference model
is assumed as:
where, k_{1}, k_{2} and k_{3} are 70, 1600 and
12000, respectively.
RESULTS AND DISCUSSION
The advantage of the proposed algorithm in comparison to nonlinear controllers
(DeSheng et al., 2006; Li
and Chang, 1999: Liu et al., 2005) is that
knowing the mass changes is not necessary, it was also shown that in the presence
of input and output perturbation, the modified proposed algorithm ends to the
satisfying results (Namerikawa and Fujita, 2001).
In Fig. 4, the reference path that is to be followed
is depicted. In addition, Fig. 5 shows the output of
the system. As it can be shown, a smooth path for the motion of the plate
is produced. In Fig. 6, the output of the system tracking
the reference model path is shown: this figure shows the satisfactory
performance of the control system tracking the desired path, but estimated
parameters do not converge to the real values (Fig. 7,
8) because of the matrix W is not PE.

Fig. 4: 
The reference path 

Fig. 6: 
Following the reference model path 
Difficult paths are selected to show the abilities of the proposed control
method in path following. In the path shown above, first one side of the
plate approaches the reference value and then the other side does the
same. Of course, such paths have the advantage of protecting the system
from a sudden large amount of error.

Fig. 7: 
Estimated parameters curves model path 

Fig. 8: 
Estimated parameters curves 

Fig. 9: 
Changes in the mass of the plate caused by the additional
poises 
For this part, the case in which the parameters of the system are not
completely identified is studied and the performance of the controller
is shown when an unknown poise is placed on the plate. Figure
9 shows the change in the mass of the plate caused by the additional
poises.

Fig. 10: 
Reference model path tracking in case of changes in
the mass 

Fig. 11: 
The effects of the changes in mass on the output of
the system 
In Fig. 10, reference model path tracking, in the
case of a changing mass for the plate, is depicted. As shown in Fig.
10, the proposed control algorithm is capable of producing a good
control input for tracking of the reference model path. In order to visualize
the effects of changes in mass on the output of the system, a part of
Fig. 10 is magnified and shown in Fig.
11 and as can be shown, although the mass is changed, the tracking
is well done. This is due to the effect of changes in mass that is compensated
by the changes on the estimated parameters and the adaptation algorithm.
The nature and the magnitude of the changes in mass should be known, otherwise
a complete tracking would be impossible and the output of the system would
have a steady state error. The proposed adaptive control method provides
the capability of complete tracking with the mass changes being unknown.
In Fig. 11, the effect of the changes in mass is shown
at 20th to 80th second time interval.

Fig. 12: 
Changes in the estimated parameters in case of changes
in mass 

Fig. 13: 
Changes in the estimated parameters in case of changes
in mass 
Looking at Fig. 8,
one can realize that the 40th second is the time that the mass stops changing
and the curve has its peek. Figure 12 and 13 show the curves of changes in parameters in such case. In this case, the
parameters do not converge to their actual values because the signal that
is used for estimation of the parameters is not a PE signal.
Noise: Noiserobustness is an important issue in a control system
because all sensors collect noise from the environment. On the other hand,
noise exists in all environments. In the following First, the effects
of noise are discussed and then perturbation in the output of the control
input is investigated.
To investigate the effects of noise, the adaptive algorithm and its noiserobustness
capability is discussed and if the system is robust to a large domain
of noise, the proposed adaptive control algorithm should be corrected.
To do so toward this, it is started with the small value of 1e–5
for the noise domain and gradually increase the noise domain.

Fig. 14: 
The output of the system in presence of noise 
The output
of the system for the following cases of measuring of the domain of the
noise signals is shown in Fig. 14.
As it can be seen in the Eq. 47, the noise domain
for x_{5} and x_{6} is 10 times larger than that of other
variables. Because these two variables represent electrical current which
have a domain 10 to 20 times larger than other state variables. Considering
the domain of the noise for the state variables can be translated to inaccurate
measuring instruments of electrical current in comparison with the sensors
of measuring the position and velocity. In other simulations these relations
are preserved. Figure 14 shows that the proposed system
is robust to the noise in Eq. 47.
The way in which the parameters change is shown in Fig. 14.
In the next experiment, the domain of the noise is multiplied by 10 and
the estimated parameters do not converge to the real values (Fig.
15, 16) because the signal that is used for the estimation
of the parameters is not a PE signal. In such case, the system diverges
in the 4th time step as shown in Fig. 17. To solve this
problem, the adaptation rule is changed to what follows:

Fig. 15: 
The changes of the parameters 

Fig. 16: 
The changes of the parameters 
As it is shown in the Eq. 48b, in order to implement
the algorithm, estimation of the parameters is needed. In Eq.
48, w and R_{0} are design parameters. In the simulation, these
values are considered as follows:
Once again, the domain of noise in Eq. 47 is multiplied
by 10. The output of the system is shown in Fig. 18.
As shown in this figure, despite the presence of noise with large domain,
the adaptive algorithm has a satisfying performance and the output of
the system follows the reference value and its fluctuations.

Fig. 17: 
The output of the proposed algorithm presence of noise 

Fig. 18: 
The output of the system for the enhanced adaptive algorithm
in presence of noise 
The proposed
algorithm here is the enhanced version of the algorithm for which the
leakage and the dead zone is taken into account. The term p (wxnorm (e_{1})xθ_{e})/I+norm
(wxpxw) is related to the leakage and R_{0} represents the dead
zone. The term norm (e_{1}) appears in the leakage term because
the influence of this term decreases with the combinational error and
the algorithm becomes more and more similar to the adaptive algorithm
that is proposed in the past.
Figure 19 and 20 show the changes
in the estimated parameters and Fig. 21 shows the norm
fluctuations of .
This value can be considered as a measure for the convergence of the parameters.
In addition, this norm can be used to determine the dead zone.
Perturbation in the input: The simulation results indicate that
perturbation in the input hardly affects the algorithm. Since, the control
input of the system in absence of noise and perturbation is about 50 volts.

Fig. 19: 
The change in estimated parameters for the enhanced
adaptive algorithm in presence of noise 

Fig. 20: 
The change in estimated parameters for the enhanced
adaptive algorithm in presence of noise 
To show the satisfying performance of the proposed algorithm in the past
in presence of perturbation and absence of noise, the perturbation domain
is set to 20. As shown in Fig. 22, in this case, the
system is capable of following the reference value satisfactorily. In Fig. 23, 24, the change in the estimated
parameters is shown. Since noise avoidance cannot be done completely,
the same perturbation is considered for the adaptive algorithm in absence
of noise. The simulation results show that in such case, complete trajectory
following is not possible but if perturbation is added to the system,
complete following is reachable by fluctuating around the reference value.

Fig. 21: 
The fluctuations of norm of 

Fig. 22: 
The output of the control system by using the algorithm
in presence of perturbation 

Fig. 23: 
The change in estimated parameters by using the algorithm
in presence of perturbation 

Fig. 24: 
The change in estimated parameters by using the algorithm
in presence of perturbation 
CONCLUSION
As mentioned earlier, this study is presented in order to improve the study
of DeSheng and Kun (2006) for overcoming the problems
such as noise and disturbance, mass changes of the train and uncertainties of
parameters which decrease the efficiency of their method. As the simulation
shows the current proposed method will overcome all of these problems. In this
study, an enhanced adaptation algorithm based on the normalized least square
algorithm is implemented. As the simulation results show, the proposed algorithm
has a satisfying performance in tracking in presence of unknown changes in the
mass. The advantage of the proposed algorithm in comparison to nonlinear controllers
is that knowing the mass changes is not necessary. It is also noticeable that
the initial values of the parameters cannot be set to zero because the parameters
diverge time goes by due to the structure of the system. As shown, the system
is sensitive to noise in measurement and perturbation in output shows less sensitivity
to perturbation in the input. Generally, it was also shown that in the presence
of input and output perturbation, the modified proposed algorithm ends to the
satisfying results.