increasing the robustness of magnetic levitation system … · increasing the robustness of...

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: [email protected], [email protected], [email protected]

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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©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.

391



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.

392



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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