Journal of Robotics, Networking and Artificial Life, Vol. 2, No. 4 (March 2016), 263-267

Adaptive Sliding Mode Control for A 2 DOF Magnetic Levitation Systemwith Uncertain Parameters
Meng Duan

The Seventh Research Division and the Center for Information and Control, Beihang University (BUAA)
Beijing 100191, China
Yingmin Jia

The Seventh Research Division and the Center for Information and Control, Beihang University (BUAA)
Beijing 100191, China
E-mail: duanmeng90@126.com; ymjia@buaa.edu.cn
www.buaa.edu.cn

Abstract
This paper investigates the stability control for a 2 DOF magnetic levitation system with uncertain parameters andexternal disturbance. The electromagnetic forces between the magnets and coils are obtained by numericalcalculation. Then, an adaptive sliding mode controller is proposed to deal with the parameter uncertainties in thesystem matrices. The robust stability of the closed-loop system is proved by Lyapunov stability theory. Simulation
results are presented to verify the effectiveness of the proposed control strategy.
Keywords: adaptive sliding mode control, magnetic levitation, numerical calculation, uncertain input

1. Introduction
In recent years, magnetic levitation technologybecomes more and more important in the modern
industry, like high precision transportation, because
magnetic levitation technology can eliminate thefriction, vibrations and wear caused by contact. And
these machines have to move in multiple degrees offreed-om(DOF). In Ref. 1-2, two kinds of planar motors
using Halbach permanent magnet arrays which canmove in a large horizontal plane have been proposed.
Ref. 3 design-ed a long stroke 6 DOFs direct drivemachine, the mover was constructed by symmetriclinear magnet arrays and control currents generated bymanufactured PCBs.
However, large rotation ranges are limited by thedistribution of magnetic field in the above mentioned

technologies, forces and torques in six DOFs can not becalculated sufficiently accurately in real time.4 To thisend, Ref. 5 proposed a novel single-deck planar maglev
stage, its electromagnetic forces and torques areanalyzed by the ANSOFT software. Refs. 6 also usednumerical calculation method to obtain accurate
electromagnetic forces and torques, and large rotation
ranges have been realized.
Inspired by those works, this paper mainly focus on
the problem of stability control for a 2-DOF magnetic
levitation system subjects to parameter uncertainties.
The contributions include that(i) In the modeling process of electromagnetic forces,
the errors between the real position and preset one
are described as the uncertainties of control matrix.
(ii) Based on the sliding mode method, an adaptive
controller is designed to guarantee the robust

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M. Duan and Y. Jia

stability of the closed-loop system which has
uncertain parameters and external disturbances.
This paper is organized in six sections. Section 2
describes electromagnetic forces numerical calculation
model; Section 3 analyses the uncertain parameters ofsystem dynamic model; Section 4 presents the controller
design. Simulation results are presented in section 5.
And the conclusions are given in section 6.

Fig. 1. Magnetic levitation system

2. Electromagnetic Force ModelIn this paper, we investigate a 2 DOFs magnetic
levitation system that consists of one disk magnet andtwo cylindrical coils shown in Fig.1, Assume that thelevitated magnet can only move through x-axis and zaxis but can not rotate.
Generally, electromagnetic forces are calculated by
Lorentz’s formula at a fixed relative position between
the magnet and coils. The total forces are calculated as
the volume integrals:

IF =∫∫∫ J × Bm dV = ∫∫∫ a × Bm dV
A(1)

where J is the vector of the current density of coils, I is
the coil current, A is the cross-sectional area of thecoil, a is the direction of the current, Bm is the vector ofmagnetic field density generated by the magnet and it is
the function of the relative position of the magnet andthe coils, V is the volume of coils. From Eq. (1) we canget that the electromagnetic forces are proportional tothe magnitude of coil currents at a fixed relativeposition. So we can write Eq. (1) as follow:

ˆF = bI
where=bˆ

I
a ×Bm dV .
A ∫∫∫In practice, the impact between coils can be ignored,
so the forces generated by each coil are independent, thetotal forces can be calculated simply by adding theforces generated by each coil. The forces can be written
as:=Fˆ ==bˆ I BIi 1, 2
∑

(3)

i i

where I = [ I1 , I 2 ]T , Bˆ = [bˆ1 , bˆ2 ]T .
Because the forces generated by each coil current
are proportional and independent, a numerical model
can be established that only includes one magnet andone coil, the electromagnetic forces are calculated inthis model on each preset position along x-axis and zaxis with a unit ampere of current to get bˆi , then use Eq.
(3) to get F .
But the preset positions in numerical calculation
model are discrete, when the magnet moves on thepositions that have not been numerical calculated, wecan not find the matrix B̂ . In Ref. 6, B. Peter uses
nearest-neighbor positions to replace real positions, sothe B̂ is not certain at some positions, in this paper, theuncertainties of B̂ are introduced as ∆B̂ , obviously, it isbounded, so the electromagnetic forces can be writtenas:=F ( Bˆo + ∆B ) I(4)

where Bˆo is the nominal part of B̂ .
3. Model Analysis
The dynamic model of the magnetic levitation
system can be derived by using Newton’s Law asfollow:=Fx ∑=f xi mx(5)
1, 2Fz =mz + mgi=
∑ f zi =where m is the mass of magnet, g is acceleration ofgravity. =Choose X [ =x1 , x2 , x3 , x4 ] [ x, x , z , z ] , apply
Eq. (4) to Eq. (5), the dynamic model can be expressed
as follow:=X f ( X ) + ( B + ∆B ) I(6)

where f ( X ) = [ x2 , 0, x4 , g ]T , I = [ I1 , I 2 ]T , ∆B =∆Bˆ / m ,(2)T 0 bˆ11 0 bˆ21 
 0 b11=B = /m ˆ
ˆ
0 b12 0 b22 0 b12Published by Atlantis Press
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T0 b21 .
0 b22 

Adaptive Sliding Mode Control

Simultaneously consider the uncertain parameters ofsystem dynamic model and external disturbances, write
dynamical model as follow:

1 1Pˆ = s1 ,s2 
ρ2
 ρ1


=X f o ( X ) + ∆f ( X ) + ( B + ∆B ) I + w

where r 1 and r 2 are the adaptation gain and both are
positive. Assume that Pd 1 and Pd 2 are the terminal
solution of P̂1 and P̂2 which satisf W1 ≤ Pd 1 , W2 ≤ Pd 2 ,respectively.
Choose the adaption error as:(7)

where f o ( X ) is the nominal part of f ( X ) , ∆f ( X ) is theuncertain part of f ( X ) , w is external disturbance,
assume that both of them are bounded.

 Pˆ1 − Pd 1   P1 =P =   .
ˆ P2 − Pd 2   P2 4. Adaptive Sliding Mode Controller Design
Define the desired state as X d = [ xd 1 , xd 2 , xd 3 , xd 4 ] ，the tracking error can be written as:
TT
1T T2]
E [e , e =Xd − X=(9)where c1 = [c11 , c12 ]T , c2 = [c21 , c22 ]T .The constant c1 andc2 are chosen to be positive to make sure that thepolynomial c11λ + c12 and c11λ + c12 is Hurwitz.
Differentiating s yields:s= c f ( X ) − c X d + c Bo I + W ( X , I )TTT(10)c, W ( X , I ) is the lumped uncertain
where c =  1c2 parameters and external disturbance:W ( X , I ) = [W1 , W2 ] = cT ∆f ( X ) + cT ∆BI + cT w
T(11)in practice, the input coil currents are bounded, we canguarantee that W ( X , I ) is bounded.
We design the control input I to be:=I Io + I s(12)ρ
1 T
1s s + P T  122
0=V

where e1 = [ E (1), E (2)]T , e2 = [ E (3), E (4)]T .

T[c1T e1 , c2T e2 ]T=s [ s=1 , s2 ]0P.
ρ 2 (13) Pˆ sgn( s1 ) Is =−(cT Bo ) −1 Pˆ sgn( s ) =−(cT Bo ) −1  1ˆ P2 sgn( s2 ) (14)

(16)

(17)

The time derivative of the Lyapunov candidate
function can be found to be:ρV sT s + P T  1=
00  P

ρ 2 T ρ ( Pˆ − P ) = s (c f ( X ) − c X d + cT Bo I + W ( X , I )) +  1 1 d 1  Pˆ ρ 2 ( P2 − Pd 2 ) TTTT Pˆ sgn( s1 )   Pˆ1 − Pd 1   s1 sgn( s1 ) =sT (W ( X , I ) −  1 ) + ˆˆ P2 sgn( s2 )   P2 − Pd 2   s2 sgn( s2 ) =s1W1 + s2W2 − s1 Pd 1 − s2 Pd 2 ≤ 0.

Because V ∈ L2 , V is bounded, according to Barbalat Lemma, V would converge to zero, so the tracking
error E would converge to zero. In order to alleviate theinput chattering caused by the sgn( si ) in Eq. (14), thesaturation function sat ( si / φ ) is used to replace sgn( si ) ,7
where φ is constant.
5. Simulation Results

whereIo =−(cT Bo ) −1 (cT f ( X ) − cT X d )(15)

Define a Lyapunov candidate function as:(8)

Because the levitated magnet moves along two axes,
the sliding-surface function can be introduced as:

Tand P̂1 and P̂2 are adjustable parameters, and theadaptive law is:

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Fig. 2. Numerical calculation model.

M. Duan and Y. Jia

designed as [0,sin t , 0,sin t ]T , and the parameters of thecontroller are designed as: c=ρ=0.5 ,
c=[0.5,1] , ρ=12
12Fig. 3. Vibration of electromagnetic forces along x-axis withdifferent air gaps

φ = 0.1 , Pˆ1 (0) = 30 , Pˆ2 (0) = 70 .
The position tracking response performance of the2-DOF magnetic levitation system is shown in Fig. 4,
where the levitated magnet can track the desire positionx = 32 mm and z = 15 mm. From the tracking curve
shown in Fig. 4, we can see that the magnet can reachand keep balance at the desired position with smallcurrents.
The trajectory tracking response performance isshown in Fig. 5, from the simulation results, we can seethat the levitated magnet can track the desired trajectories x = 6 sin t + 22 mm and z = 6sin(0.6t ) + 9 mm.
6. Conclusions

Fig. 4. Position tracking response performance.

Fig. 5. Trajectory tracking response performance

In this paper, the electromagnetic force model iscalculated by Radia which is a free 3D magnetostatics
computation software package developed by ESRF andrunning in Mathematica. The numerical calculation
model established in Radia is shown in Fig. 2. The coilin the numerical calculation model is 26mm in height,
20mm in the inside diameter and 44mm in the outside
diameter. The current density in the coil is 0.97A/mm2.
The magnet is cylindrical with 50mm in diameter and10mm in height, its material is NdFeB and the remnant
magnetization is 1.22T. The calculation sampling interval is 1mm along z-axis and x-axis. Figure. 3 shows thatvibration of electromagnetic forces when the magnet
moves along x-axis with different air gaps.
In this section, two typical simulation results which
including the trajectory and position tracking arepresented. In both two simulations, ∆f ( X ) is supposed
to be [0.3sin t ( x1 + x2 ), 0, 0.2 cos t ( x1 + x2 ), 0]T , w is

In this paper, electromagnetic forces obtained by
the method of numerical calculation, which brings
the uncertainties to the control matrix, to deal with
this, an adaptive sliding model controller has beendeveloped and applied to the 2-D maglev system,
the simulations confirmed the efficacy of theproposed controllers.
Acknowledgements
This work was supported by the National Basic
Research Program of China (973 Program:2012CB821200, 2012CB821201) and the NSFC
(61134005, 61327807, 61521091, 61520106010).
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Adaptive Sliding Mode Control

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