Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 2 (September 2014), 169-173

Adaptive Sliding Mode Control for Magnetic Levitation Vehicles *Juanjuan He and Yingmin Jia

The Senventh Research Division, Beihang University (BUAA),

Beijing, 100191, China
E-mail: hejuanjuan@smss.buaa.edu.cn; ymjia@buaa.edu.cn

Abstract
This paper focuses on stability control for the levitated positioning of the magnetic levitation vehicle system. For
the nonlinear magnetic levitation system model, the output feedback linearization method is employed to derive aglobal linearization error model. However, there exists uncertain item in the error model. To stabilize this error
model, the adaptive sliding mode control method is used here. Simulations show that the magnetic levitation system
can be stability quickly under controlled by the proposed control scheme.
Keywords: Magnetic levitation vehicle system; nonlinear model; output feedback linearization; adaptive sliding
mode control

1. Introduction
As the smart transportation establishing becomes a hot
research issue in recent years, how to improve theoperating efficiency and convenience of the entire city
is becoming increasingly important. Therefore, it isnecessary to establish a new high-speed and efficientmeans of transport. Up to 500km/h of speed, the maglev
train can greatly improve the transport capability.
Furthermore, magnetic levitation technology has seen arapid development since the 20th century, and made theinteresting in the field of real-life applications [1], suchas transportation systems [2,3], wind tunnel levitation[4],magnetic bearing systems and antivibration table.
However, the design of maglev system controller stillpresents plenty of formidable challenges [5] including
the nonlinear solenoid model, instability, and inevitably
uncertain parameters. Hence, a proper controlling mode

should be designed for the suspension system toguarantee the robust stability of the entire system. For
traditional methods, nonlinear magnetic levitation
vehicle system model is linearized at one equilibrium
point [6], which leads to the fact that the design of thecontroller seriously depends on the selected equilibrium
point. So this type of linearization cannot be applied tothe situation that the gap is a large range of variation.
Thus if the traditional linearization method is applied tothis situation, it may lead to the performancedegradation or instability of the system. Hence, a newsolution should be proposed.
In this paper, latest control method is applied in allusion
to the magnetic levitation planner. Firstly, output
feedback linearization technique is applied to thenonlinear magnetic levitation vehicle system model,
which derives a global linearization error model.
However, the simplified model is unstable and includes
uncertain factors, so the adaptive sliding mode

*The Seventh Research Division, Beihang University,No.37, Xueyuan Road, Haidian Dist., Beijing 100191, China.

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J. He, Y. Jia

controller design method is used here, which makes thelinearization error system stable. Finally, a series ofsimulation experiments are shown in this paper, which
is used to demonstrate the effects of output feedback
linearization method and adaptive sliding mode
controller.
This paper is organized in five sections. In the next
section, model analysis of magnetic levitation vehicle
system is formulated, and output feedback linearization
is proposed. Section 3 is related to the controller design.
Simulation results are presented in section 4. Finally, insection 5, some concluding remarks are outlined.
2. Model Analysis of The Maglev Systemdψ(x,i)(2)
dtwhere u is control voltage, and R , Ψ are circuitous
resistance and flux linkage respectively.
u(t)=Ri(t)+2.2. System Dynamics

The complete model of the maglev system can bederived by using Newton’s Law as follow
(3)
 F=mx(t)Firstly, to obtain the system dynamics, we apply Eq.(1)and Eq.(2) to Eq.(3). As a result, the dynamic equations
can be rewritten as

μ0 n2σi2mx(t)=-mg4x(t)2

2.1. Nonlinear Solenoid Model Analysis
The schematic diagram of the magnetic levitation
system is depicted in Fig.1. For the sake of achieving
the goal of high-precision positioning, a complex
magnetic levitation system vehicle model needs to beanalyzed thoroughly.

where m is mass of the suspension; g is acceleration ofgravity; and σ is magnetic area .
2.3. Analysis of Nonlinear Model

Let the states be chosen as x1 =x , x 2 =x , x 3 =i . u is theinput voltage of the system changing from 0 to 5 volts,
Tand X=  x1 x 2 x 3  is the state vector. Thus, thestate-space model of the magnetic levitation system canbe written as
X=F(X)+G(X)U(t)(5)
Y(t)=x1

where S=Fig.1. Schematic diagram of magnetic levitation system

Before deriving, we make the reasonable assumptionthat the air-gap flux leakage is zero. Then, the idealized
nonlinear solenoid model can be calculated as
μ 0 n 2 σi 2 (t)(1)
4x 2 (t)where F is the electromagnetic force between the coiland suspension; the constant μ 0 =4π×10-7 H/m is theabsolute magnetic permeability; n is the coil turn; σ isthe magnetic cross section; i(t) is the coil current, andx(t) is the levitation height between coil and suspension.
The coil model is given by(4)μ 0 σn 2,and F(X),G(X),U(t) are2





x2

0 S  x 2  3F(X)=   -g  ,G(X)=  0  , U(t)=u(t).
 2m  x1  x1 S x x23
-Rx1 x 3  2 x1
F=

2.4. Linearization of System Model

For the purpose of the afterward controller design, thefeedback linearization is applied to the system model
shown as Eq.(5) in this subsection. As a result, the third
derivative of system outputs can be finally obtained asfollows

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Maglev Vehicle’s Suspension Control
Y=B(C+KU)(6)

will need to investigate the dynamics of the sliding
surface variable S as follows  +ς z +ς z =BU -Y +W+ς z +ς z
S=z
(12)

Obviously, when x1 ,x 3  0 ,then K  0 . In physics
system, x j ,j  (1,3) , which are Central levitation heightof the suspension and coil current respectively ， thus
their product could not be zero, so the decoupling
matrix K is invertible.
Thus, the decoupling control inputs can be obtained as
U=K -1 (U c -C)(7)
where U c =v .
Then the linearized system model can be arrived at byapplying Eq.(7) to Eq.(6) and finally obtained as follows

In addition to the SMC, an adaptive controller is applied
for estimating the parameters of the system online while
simultaneously controlling the system. After we havethe estimates of the system parameters, these estimates
can be adopted to the control command in Eq.(12) toform appropriate SMC with boundary layer asˆ -ς z -ς z ]U =B-1[-k S-Psgn(S)+Y(13)where K=

2
3

23Rx Sx 2 xx3
1.
, B= , and C=x1mx12x1331
Y=BU c = vm(8)(9)

where W denotes the system’s uncertainty and isassumed bounded W  P . Here, we assume that P isunknown.
In order to design a controller which possesses a better
ability to gain high robustness and self-tuning property,
two advanced control methods have been intergrated. In
this paper, we will introduce the controller design andprovide the stability analysis.
Based on the model Eq.(9), which is compactly reexpressed asz =E
13cc

1d1 3d1 32 22 2

where k1 is a positive constant, P̂ is the estimate of P ,and sgn() is the symbolic function.
Thus, substituting Eq.(13) into Eq.(12), we obtain +W)+B
S=(ςz +ς z -Y
1 32 2d(14)1The error output vector is defined as E=Y-Yd ,where Yd is the desired position and Y is the current
position. Then, Eq.(8) can be rewritten asz 2 =E +Wz =E=BU -Yc2 2ˆ -ς z -ς z )][B-1 (-k1S-Psgn(S)+Y
d 1 32 2
ˆ=-k S-Psgn(S)+W

3. Adaptive Sliding mode Controller Design1 
E= v-Yd +Wm

1 3(10)dFurthermore, we define the sliding surface variable S as
S=z 3 +ς1z 2 +ς 2 z1(11)Where ς1 ,ς 2 are positive constants. In this paper, we try
Tto regulate the state Z=  z1 z 2 z 2  to zero, in otherwords, the current outputs will achieve the goal ofprecise tracking of positioning and attitude. In thecontext of SMC, asymptotical convergence of thevariable S to zero will apparently imply asymptoticalconvergence of Z as well to zero. To validate this, weBy appropriate gains k1 , P̂ , ς1 and ς 2 , we can ensure theconvergence of S . Hence, the state Z converges to zero,
which means the tracking error E to zero, and theestimation error P converging to zero.
Stability analysis: we define a Lyapunov function
candidate V , which is a positive definite function
11
V= ST S+ ρ-1P 2
22(15)

ˆ , P̂ is thewhere the estimation error is defined as P=P-P
estimates of P ,and ρ is positive.
The time derivative of the Lyapunov candidate
function V can be found to be 1 pp
 T S+ V=Sρ
1 ˆ - pp
ˆ=ST  -k1S-Psgn(S)+W(16) ρ1 ˆ
 -ST k1S-Pˆ S + S P- pp
ρUsing the adaptive control theory to establish bounds ofparameter estimates in the presence of modeling error
terms, the adaptive laws is devised as ˆ  S(17)p=-p=After substituting Eq.(17) into Eq.(16), we can get  -ST k S-Pˆ S + S P-P SV

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1=-S k1S-Pˆ S +Pˆ S
TT< -S k1S < 0(18)J. He, Y. Jia

where k1 >0 .
According to Lyapunov stability theory, the tracking
error E(t) will converge to zero. In other words, it
achieves the goal of precise tracking of positioning.

4. Simulation Results
In this section, a number of typical simulation results
are presented, including the transient and the steadystate responses in different situations. The simulation
results are provided to demonstrate the performance ofthe developed magnetic levitation system with
controller presented in section 3. Based on these results,
we will make some conclusions which are important for
the future work in this research.

Fig.3. Tracking response performance of central levitation
height along the x-axis with uncertainty boundFig.2. Block diagram of magnetic levitation system

In order to demonstrate the controlling performance ofthe magnetic levitation system more effectively, theblock diagram of the system as shown in Fig.2. isconstructed in this section. Base on the simulation block
diagram, serials experiments are performed.
The tracking response performance of magnetic
levitation vehicle system is shown in Fig.3., where thecentral levitation height can track the desire
signal x(t)=0.5sint+1 . From the position error curve
shown in Fig.3., we can see that the error can becontrolled within 0.03cm. The simulation result, asshown in Fig.4., demonstrates the ball can keep balanceat the equilibrium point. The simulation results for
repeating a 0.2cm step-train response in x-axis, as
shown in Fig.5., indicates that the effective tracking
performance of the proposed magnetic levitation vehicle
system.

Fig.4.Results for central levitation height holding at 2cm

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Maglev Vehicle’s Suspension Control

Maglev Design, Maglev 93 Conf. Argonne National
Laboratory, (1993) 19-21.
4. E. E. Covert, M. Vlajinac, T. Stephens, and M. Finston,
Magnetic balance and suspension systems for use withwind tunnels, Progress in Aerospace Sciences (14)(1973)27-107.
5. M.Y. Chen, M.J. Wang, and L.C. Fu, Modeling andcontroller design of a maglev guiding system forapplication
inprecision
positioning.Industrial
Electronics, IEEE Transactions on 50(3) (2003)493-506.
6. H. Chen, Z. Long, W. Chang, Fault tolerant control
research for high-speed maglev system with sensorfailure, Intelligent Control and Automation,WCICA. (TheSixth World Congress on. IEEE, 2006), pp. 2281-2285.

Fig.5.Results for repeating a 0.2cm step-train response

5. Conclusion
This paper does a research on the suspension control
problem of the magnetic levitation vehicle system,
which in nature, is served as strongly coupling,
nonlinear and instability. The related analysis had been
applied in to the maglev system. The linearization
method and adaptive sliding mode controller can stable
the system effectively.

Acknowledgments
This work was supported by the National Basic
Research Program of China(973 Program: 2012CB821
200,2012CB821201) and the NSFC(61134005, 609210
01,61327807).

References
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2. B. V. Jayawant, P. K. Sinha, and D. G. Aylwin, Feedback
control systems for dc electromagnets in passengercarrying vehicles. International Journal of Control 24(5)(1976) 627-639.
3. M. Proise, System Concept Definition of the Grumman
Superconducting Electromagnetics Suspension (EMS)

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