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Increasing the Robustness of Magnetic Levitation
System by Using PID-Sliding Mode Control
Nguyen Ho Si Hung, Le Thanh Bac, and Nguyen Huu Hieu Faculty of electrical engineering, University of Science and Technology – the University of Danang, Vietnam
Email: nguyenhosihung@gmail.com, lethanhbac@ac.udn.vn, nhhieu@dut.udn.vn
Abstract—This paper presents regulation and tracking
control design for a magnetic levitation system (Maglev).
First, the nonlinear dynamic model of magnetic levitation
system was built. Second, a sliding mode control (SMC) is
constructed to compensate the uncertainties occurring in the
magnetic levitation system. The control gains were
generated mainly by experimental method. Next, a
composite controller consisting of a PID plus a SMC
algorithm was proposed to enhance overall tracking
performance. The effectiveness of controllers was verified
through experiment results.
Index Terms—magnetic levitation system, sliding mode
control, composite controller
I. INTRODUCTION
Maglev was widely applied in many industrial fields
such as frictionless bearings in high speed trains, and
magnetically suspended wind tunnels [1]-[4]. Therefore,
issues of regulation and tracking control are a great deal
of importance. However, it is not easy at all for the
Maglev being unstable in the open-loop form and the
highly nonlinear feature of the system dynamics.
Among others, PID controller is widely used in
industrial applications for its ease of implementation.
However, it is not robust to variation of parameter and
disturbances [5].
To alleviate such difficulty, a SMC is proposed to
increase the robustness of system. SMC is a nonlinear
control method being robust to parameter variation and
external disturbances. However, the SMC gain must be
large enough to satisfy requirement of uncertainty bound
and guarantee closed-loop stability over the entire
operating space [6, 7]. On the other hand, a larger control
gain is more possible to ignite chattering behaviors.
Therefore, the SMC gain must be chosen to bargain the
robustness of the controller and the chattering behaviors.
Regarding this, it is then natural to formulate a
composite controller possessing the advantages of the
above-mentioned two controllers while avoiding their
disadvantages at the same time. Basically the SMC
dominates when the tracking errors are large while in the
region with smaller tracking errors the control authority is
switched to the PID controller to avoid possible
Manuscript received January 20, 2016; revised June 17, 2016.
chattering behaviors. Experimental results demonstrate its
validity of the proposed control algorithm.
The remainder of the paper is organized as follows: a
derivation of the system's dynamical model based on the
Newton's method is presented next. Following is central
part of this paper, namely, the control design. To
demonstrate the usefulness of the proposed designs,
simulation and experimental results done on Magnetic
Levitator - Model 730 of ECP are given in experiment
section. Conclusion is drawn in final section.
II. DYNAMICS OF THE MAGNETIC LEVITATION
SYSTEM
The physical structure of a typical Maglev is shown in
Fig. 1. The plant consists of a drive coil that generates a
magnetic field; a magnetic levitated permanent magnet
that can be moved along a grounded glass rod; and a
laser-based position sensor. The forces from coil, gravity,
and friction act upon the magnet. From Newton’s second
law of motion, the system dynamics can be written as:
𝐹𝑚 − 𝑚𝑔 − 𝑐��𝑚 = 𝑚��𝑚. (1)
where xr is the distance between the coil and the magnet,
m is the weight of the magnet, Fm is the magnetic force,
c is the friction constant, and g is the gravitational
constant. The magnetic force can be calculate as [8]:
𝐹𝑚 =𝑈
𝑎(𝑥𝑟+𝑏)𝑁 . (2)
where U is the control effort. N, a and b can be
determined by experimental methods (typically 3<N<4.5)
[9].
Figure 1. Magnetic plant.
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Journal of Automation and Control Engineering Vol. 4, No. 6, December 2016
©2016 Journal of Automation and Control Engineeringdoi: 10.18178/joace.4.6.389-393
III. PROPOSED CONTROL SYSTEM.
By substituting Eq. 2 into Eq. 1, we get
��𝑟 = −𝑐
𝑚��𝑟 +
𝑈
𝑚𝑎(𝑥𝑟+𝑏)𝑁 − 𝑔 (3)
Define:
G(X; t) =1
ma(xr + b)N ; f(X; t) = −
c
mxr X = [xr xr]T
Eq. 3 can be rewritten as:
��𝑟 = 𝑓(𝑋; 𝑡) + 𝐺(𝑋; 𝑡)𝑈(𝑡) − 𝑔 (4)
Let f(X;t) = fn(X;t) + ∆f and G(X;t) = Gn(X; t) + ∆G,
with fn(X;t) and Gn(X;t) being the nominal known while
∆f and ∆G the unknown deviations. It follows that:
��𝑟(𝑡) = 𝑓𝑛((𝑋; 𝑡) + 𝐺𝑛(𝑋; 𝑡)𝑈(𝑡) − 𝑔 + 𝐿(𝑋; 𝑡)(5)
where L(X;t) =Δf + ΔGU(t) is the lumped uncertainty. It
is assumed that |L(X;t)|<δ with δ being a known positive
constant.
A. SMC Control Define the tracking error e = xr − xm and the sliding
surface:
𝑆 = ��(𝑡) + 𝜆1𝑒(𝑡) + 𝜆2 ∫ 𝑒(𝜏)𝑑𝜏
𝑡
0
(6)
where λ1
and λ2
are positive constants. The SMC
algorithm, shown in Fig. 2, has the following form
𝑈𝑆𝑀𝐶(𝑡) = 𝐺𝑛(𝑋; 𝑡)−1 [− 𝑓𝑛((𝑋; 𝑡) + 𝑔 + ��𝑚(𝑡) −
𝜆1��(𝑡) − 𝜆2𝑒(𝑡) − 𝛿𝑠𝑔𝑛(𝑆(𝑡))]
(7)
where sgn is the sign function.
Lyapunov function used to prove stability of system is:
𝑉 =1
2𝑆2 (8)
Differentiating V with respect to time and using (8),
we get:
�� = 𝑆�� = 𝑆[ 𝑓𝑛((𝑋; 𝑡) + 𝐺𝑛(𝑋; 𝑡)𝑈(𝑡) − 𝑔 + 𝐿(𝑋; 𝑡) −��𝑚(𝑡) + 𝜆1��(𝑡) + 𝜆2𝑒(𝑡) ] (9)
Replacing control law from Eq. 7 into Eq.9. The
result is exhibited following:
�� = 𝑆{𝑓𝑛((𝑋; 𝑡) + 𝐺𝑛(𝑋; 𝑡)𝐺𝑛(𝑋; 𝑡)−1 [− 𝑓𝑛((𝑋; 𝑡) + 𝑔 + ��𝑚(𝑡) − 𝜆1��(𝑡) − 𝜆2𝑒(𝑡) − δ𝑠𝑔𝑛(𝑆(𝑡)] − 𝑔 +
𝐿(𝑋; 𝑡) − ��𝑚(𝑡) + 𝜆1��(𝑡) + 𝜆2𝑒(𝑡) } = 𝑆(𝐿(𝑋; 𝑡) −
δ𝑠𝑔𝑛(𝑆))
(10)
The time derivative
of the candidate Lyapunov
function can be separated as:
𝑆 < 0 → 𝑠𝑔𝑛(𝑆) = −1 → 𝐿(𝑋; 𝑡) − δ𝑠𝑔𝑛(𝑆) > 0
→ �� = 𝑆(𝐿(𝑋; 𝑡) − 𝛿𝑠𝑔𝑛(𝑆)) < 0
𝑆 = 0 → �� = 0
𝑆 > 0 → 𝑠𝑔𝑛(𝑆) = +1 → 𝐿(𝑋; 𝑡) − δ𝑠𝑔𝑛(𝑆) < 0
→ �� = 𝑆(𝐿(𝑋; 𝑡) − δ𝑠𝑔𝑛(𝑆)) < 0
→ �� ≤ 0
in all case.
Thus, the designed control law is completely satisfied
the asymptotic stability.
Figure 2. SMC control.
Figure 3. PID-SMC controller.
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Journal of Automation and Control Engineering Vol. 4, No. 6, December 2016
©2016 Journal of Automation and Control Engineering
B. Combined PID and SMC.
In practice, the control gain δ might be too
conservative which might ignite chattering behavior.
Regarding this, we propose a combined PID and SMC
controller to reduce chattering as well as preserve
robustness at the same time. The block diagram of the
proposed controller is shown in Fig. 3, where the
combined PID and SMC is given by:
𝑈𝑃𝐼𝐷−𝑆𝑀𝐶 = 𝐾1𝑈𝑃𝐼𝐷 + 𝐾2𝑈𝑆𝑀𝐶 (11)
With K1
and K2
being positive constants chosen
empirically.
IV.
EXPERIMENT
Experimental works for verifying the validity of the
proposed controller are conducted here. Parameter
identification using curve-fitting technique is done first.
The results are m=0.121 (kg); c=2.69; a=1.65; b=6.2;
N=4.
Initial conditions of this experiment are that the initial
magnet position (xr) is 20 mm in all experiments and the
controlled stroke of the disk (Δx) is 10 mm. The chosen
PID gain are Kp=1.72, Kd=0.065, Ki=0.5, the chosen
SMC gains are λ1=30; λ2=10; δ=8 and the chosen PID-
SMC constants are K1=0.5; K2=0.5.
The errors were calculated by the sum of squared
tracking errors (SSTE), unit of actuator is Count (10000
couts = 1 cm).
SSTE = ∑ (error(kT))2n
k=1 (12)
where t=kT is time from 0 to 4s, and T=0.002562. To explore the adaptability of the proposed design to
variation of parameters, two case studies are considered
in the following. Case 1 (m=0.121kg): Tests were implemented with
sinusoidal command and their experimental results were
shown in Fig. 4. The responses of magnet position of PID,
SMC and PID SMC are displayed in Fig. 4(a), Fig. 4(b)
and Fig. 4(c), respectively. Their errors are illustrated in
Fig. 4(d) and Fig. 4(e), and Fig. 4(f), respectively. The
error measures were calculated by SSTE method and
shown in Table I. The response of magnet position of
SMC is better than PID and error of SMC is also less than
PID error. However, there are always chattering in
operation process of SMC. This is a common problem in
general SMC controllers. Meanwhile, using PID-SMC
helps to solve the above problems. Experimental results
shown in Fig. 4 and Table I confirm that the proposed
PID-SMC controller improves the tracking performance
as well as reduces chattering error. Besides, Fig. 5 shows
that sliding surface of SMC is stronger oscillation than
sliding surface of PID-SMC.
TABLE I.
ERROR
MEASURES OF PID,
SMC,
PID-SMC
IN CASE 1
Performance
Reference
Trajectory
Sum of squared Tracking error [mm2] (SSTE)
PID
SMC
PID_SMC
Sinusoidal trajectory
7.6x102
3.4x102
1.3x102
Figure 4. Performance and tracking error of PID, SMC, PID-SMC in case 1.
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Journal of Automation and Control Engineering Vol. 4, No. 6, December 2016
©2016 Journal of Automation and Control Engineering
Figure 5. Sliding surface of SMC (a), Sliding surface of PID-SMC (b) in case 1.
Case 2 (m=0.151kg): Tests were implemented with
sinusoidal command and their experimental results were
shown in Fig. 6. Responses of magnet position of PID,
SMC, PID-SMC are displayed in Fig. 6(a), Fig. 6(b) and
Fig. 6(c), respectively. Their errors are illustrated in Fig.
6(d), Fig. 6(e), and Fig. 6(f), respectively. The error
measures were calculated by SSTE method and shown in
Table II. In this case, the error of PID increases
drastically so its tracking performance is poor. In contrast,
SMC errors do not grow up significantly due to the
robustness of SMC to the variation of system parameters
and disturbances. Similarly, the PID-SMC controller has
same characteristics but without igniting chattering
behaviors.As can be seen in the Fig. 7, the sliding surface
of PID-SMC is smaller oscillation than sliding surface of
SMC.
TABLE II. ERROR MEASURES OF PID, SMC, PID-SMC IN CASE 2
Performance
Reference Trajectory
Sum of squared Tracking error [mm2] (SSTE)
PID SMC PID_SMC
Sinusoidal trajectory 1.2x103 4.2x102 1.7x102
Figure 6. Performance and tracking error of PID, SMC, PID-SMC in case 2.
Figure 7. Sliding surface of SMC (a), Sliding surface of PID-SMC (b) in case 2.
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Journal of Automation and Control Engineering Vol. 4, No. 6, December 2016
©2016 Journal of Automation and Control Engineering
V. SUMMARY
This paper has successfully demonstrated the
effectiveness of SMC and PID-SMC to control the
position of a magnetic levitated object. As expected, the
SMC exhibits good tracking performances robustness to
parameter variation and disturbances. However, it creates
larger chattering behaviors. The proposed PID-SMC
algorithm retains the advantages of SMC algorithm while
avoids chattering at the same time. The experimental
results confirm these features clearly.
REFERENCES
[1] M. Ono, S. Koga, and H. Ohtsuki, “Japan’s superconducting
Maglev train,” IEEE Instrum, Meas. Mag, vol. 5, no. 1, pp. 9–15,
Mar. 2002.
[2] D. M. Rote and Y. Cai, “Review of dynamic stability of repulsive-
force Maglev suspension systems,” IEEE Trans. Mag., vol. 38, no. 2, pp. 1383–1390, Mar. 2002.
[3] M. Y. Chen, M. J. Wang, and C. L. Fu, “A novel dual-axis
repulsive Maglev guiding system with permanent magnet: Modeling and controller design,” IEEE/ASME Trans,Mechatronics,
vol. 8, no. 1, pp. 77 – 86, Mar. 2003.
[4] H. M. Gutierrez and P. I. Ro, “Magnetic servo levitation by sliding-mode control of nonaffine systems with algebraic input
invertibility,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1449–
1455, Oct. 2005. [5] H. Liu, X. Zhang, and W. Chang, “PID control to Maglev train
system,” International Conference, Industrial and Information
Systems, pp. 341–342, 2009. [6] J. J. E. Slotine and W. Li, Applied Nonlinear Control, edited by
Englewood Cliffs, NJ: Prentice-Hall, 1991.
[7] W. Perruquetti and J. P. Barbot, Sliding Mode Control in Engineering, Marcel Dekker, Ed. New York, , Inc 2002.
[8] J. D. Kraus, Electromagnetics, New York: McGraw-Hill, 1992.
[9] F. J. Lin and L. T. Teng, “Intelligent sliding mode control using RBFN for magnetic levitation system,” IEEE Trans. Insdistrial
Electronics, vol. 54, no. 3, pp. 1752–1762, 2007. [10] E. T. Moghaddam, “Sliding mode control for magnetic levitation
system using hybrid extended kalman filter,” CSCanada, vol. 2,
no. 2, pp. 35–42, 2011.
Nguyen Ho Si Hung
was born in Danang,
Vietnam in 1986. He received the B.S degrees in electrical engineering from
University of Science and Technology –
The
University of Danang, Vietnam in 2010 and received the M.S degrees in digital
mechatronic technology from Institute of
Digital mechatronic technology of Culture University, Taiwan in 2014.
His research
interests including nonlinear control theories,
artificial intelligence control theories
and magnetic levitation system.
Le Thanh Bac
was born in Bac Giang, Vietnam in 1966. He received the B.S
degrees from Thai Nguyen University of
Technology, Vietnam in 1987. He received the M.S degrees and PhD
degrees in
Electrical devices and Electric power system of
Peter the Great St. Petersburg Polytechnic
University, Russia
in 2005 and 2007,
respectively. He was an Associate Professor
in 2010. From 2013 to now, he was Chief of Office Manager
in The University of Danang, Vietnam
His research
interests including electrical devices and electric power system,
nonlinear control theories, intelligence control theories, and magnetic levitation system.
Nguyen Huu Hieu was born in Danang, Vietnam in 1981. He received the B.S
degrees from École centrale de Lyon, France
in 2004 He received the M.S degrees and PhD degrees in Electric of Joseph Fourier
University, Grenoble, France in 2005 and
2008, respectively. From 2011 to 2014, He was a Vice Dean of Faculty of Electrical
Engineering of Danang University of Science
and Technology (DUT) - The University of
Danang, Vietnam. From 2014 to now, he was Dean of Faculty of
Electrical Engineering of DUT. His research interests including
modeling of electrical devices and electric power system, embedded system, nonlinear control theories, intelligence control theories, and
magnetic levitation system.
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Journal of Automation and Control Engineering Vol. 4, No. 6, December 2016
©2016 Journal of Automation and Control Engineering
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