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Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control



M. Aliasghary, M. Teshnehlab, A. Jalilvand, M. Aliyari Shoorehdeli and M.A. Nekoui
 
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ABSTRACT

This study has developed classical and hybrid controllers for control of magnetic levitation system. Sliding mode and PID controllers are proposed as a classical controllers and neural network based controller is used for controlling a magnetic levitation system. Adaptive neural networks controller needs plant`s Jacobain, but here this problem solved by sliding surface and generalized learning rule in case to eliminate Jacobain problem. The simulation results show that these methods are feasible and more effective for magnetic levitation system control.

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  How to cite this article:

M. Aliasghary, M. Teshnehlab, A. Jalilvand, M. Aliyari Shoorehdeli and M.A. Nekoui, 2008. Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control. Journal of Applied Sciences, 8: 2561-2568.

DOI: 10.3923/jas.2008.2561.2568

URL: https://scialert.net/abstract/?doi=jas.2008.2561.2568
 

INTRODUCTION

One of the ways that non-contact surfaces are maintained is via magnetic suspension. This system is commonly referred to as Magnetic Levitation system (Maglev) which has been used in the vehicle suspension system and magnetic bearing system by B.A. Holes of University of Virginia in 1937 for the first time. In 1954 this system was utilized by Laurencean and Tournier at ORENA in France for the purpose of aerodynamic testing in wind tunnels (Covert, 1988). Furthermore, the system have practical uses in many industrial systems such as in high-speed maglev passenger trains, levitation of wind tunnel models, vibration isolation of sensitive machinery, levitation of molten metal in induction furnaces and levitation of metal slabs during manufacturing.

In recent years, a lot of works have been reported in the literature for controlling magnetic levitation systems. The feedback linearization technique has been used to design control laws for magnetic levitation systems (Jing-Chung, 2002; Trumper et al., 1997; Hajjaji and Ouladsine, 2001). Other types of nonlinear controllers based on nonlinear methods have been reported by Yang and Tateishi (1998), Tanaka and Torii (2004), Huang et al. (2000) and Teng and Qiao (2008). Control laws based on phase space (Zhao et al., 1999), linear controller design (Rifai and Youcef-Toumi, 1998) and neural network techniques and fuzzy controllers have also been used to control magnetic levitation systems (Lairi and Bloch, 1999; Anh and Timothy, 2008; Muñoz-Gómez et al., 2006). One of the first applications of Sliding Mode Control (SMC) to magnetic levitation systems was carried out by Cho et al. (1993). Chen et al. (2001) designed an adaptive sliding mode controller for a rather different type of magnetic levitation systems called dual-axis maglev positioning system. Muthairi and Zribi (2004) designed static and dynamic sliding mode controller for the magnetic levitation system and Chao-Lin et al. (2005) designed a novel Fuzzy sliding-mode controller for magnetic ball levitation system.

In this study, we consider a magnetic levitation system and propose classic controllers like sliding mode, PID controller and new hybrid controllers. In proposed hybrid controllers the Feedback Error Learning (FEL) based-on sliding mode is used to train the neural network with Radial Basis Function (RBF). The proposed methods demonstrate the advantages of the adaptive neural network and sliding mode control strategies with achieving better performance.

MATHEMATICAL MODEL OF THE SYSTEM

Figure 1 is shown a diagram of the magnetic levitation system. Note that only the vertical motion is considered. The dynamic model of the system can be written as (Barie and Chiasson, 1996; Muthairi and Zribi, 2004):

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(1)

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 1: Diagram of the magnetic levitation system

where, p denotes the ball`s position, v is the ball`s velocity, i is the current in the coil of the electromagnet, e is the applied voltage, R is the coil`s resistance, L is the coil`s inductance, g is the gravitational constant, C is the magnetic force constant and m is the mass of the levitated ball. The inductance L is a nonlinear function of ball`s position p. The approximation of L is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(2)

where, L1 is a parameter of the system.

Let the states and control input be chosen such. Thus, the state-space model of the magnetic levitation system can be written as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(3)

Consider the following nonlinear change of coordinates:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(4)

The dynamic model of the magnetic levitation system in the new coordinates system can be written as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(5)

Where:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(6)

The function f (z) and g (z) correspond in the original coordinates to the following functions, respectively:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(7)

The relationship between the input and the output of the system can be found as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(8)

The parameters of the magnetic levitation system are as follows (Barie and Chiasson, 1996). The coil`s resistance R = 28.7 Ω, the inductance L1 = 0.65 H, the gravitational constant gc = 9.81 msec-2, the magnetic force constant C = 1.24x10-4 and the mass of the ball m = 11.87 g and x1d = 0.01 is the desired value of x1.

MATERIALS AND METHODS

Sliding mode controller: Sliding mode control is a variable structure control utilizing a high-speed switching control law to drive a system state trajectory onto a specified and user chosen surface, so called sliding surface and to maintain the system state trajectory on the sliding surface at subsequent times (Slotin and Li, 1991). In this paper, the sliding surface on the phase plane can be defined as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(9)

In case n = 2:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(10)

Based on the Lyapunov theorem, the sliding surface reaching condition is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(11)

Using Eq. 10 and 3, the switching surface can be written as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(12)

Note that the choice of the switching surface guarantees that x1–x1d converges to 0 as t → ∞, when we have S = 0.

The following proposition gives the U :

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(13)

For eliminating chattering in control signal, we use saturation function instead of sign. So the following proposition gives the Us:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(14)

Radial basis function neural networks: In this study, we use a type of neural networks which is called the Radial Basis Function (RBF) networks (Powell, 1987). These networks have the advantage of being much simpler than the perceptrons while keeping the major property of universal approximation of functions (Poggio and Girosi, 1987). RBF networks are embedded in a two layer neural networks, where each hidden unit implements a radial activated function. The output units implemented a weighted sum of hidden unit outputs. The input into an RBF network is nonlinear while the output is linear. Their excellent approximation capabilities have been studied in (Park and Sandberg, 1991). The output of the first layer for a RBF network is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(15)

The output of the linear layer is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(16)

where, xεRn and yεRm are input vector and output vector of the network, respectively and φ = [φ1, Λ, φn]T is the hidden output vector. n is the number of hidden neurons, Wj = [wj1, Λ, wjn]T is the weights vector of the network, parameters ci and σi are centers and radii of the basis functions, respectively. The adjustable parameters of RBF networks are W, ci and σi. Since the network`s output is linear in the weights, these weights can be established by least-square methods. The adaptation of the RBF parameters ci and σi is a non-linear optimization problem that can be solved by gradient-descent method.

RBF sliding mode controller: The sliding variable, S will be used as the single-input signal for establishing a RBF neural network model to calculate the control law, u. Then for the single-input and single-output case in this paper, the output of the controller based on RBF networks is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(17)

where, n is the number of hidden layer neurons and u is the final closed-loop control input signal. In order to combine the advantages of sliding mode and adaptive control schemes into the RBF neural network, an adaptive rule is introduced to adjust the weightings between hidden and output layers.

If a control input u can be chosen to satisfy this reaching condition (11), the control system will converge to the origin of the phase plane. Adaptive law is used to adjust the weightings for searching the optimal weighting values and obtaining the stable convergence property. The adaptive law is derived from the steep descent rule to minimize the value of Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control with respect to W. Then the updated equation of the weighting parameters is:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(18)

Or

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(19)

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(20)

And from Eq. 12, we have:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(21)

Form Eq. 3, we can find that:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 2: Block diagram of the modified RBF-sliding mode

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(22)

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(23)

Finally we can find updating rule as follow:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(24)

From Eq. 16, we have:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(25)

It is clear that we do not need any identifier for magnetic levitation system.

For improve control signal, a modified RBF-sliding mode controller is now designed for the magnetic levitation system. Figure 2 is a block diagram of the modified RBF-sliding mode controller.

Feedback error learning architecture: The structure of FEL is shown in Fig. 3, which is proposed by Kawato et al. (1988). In this architecture, the neural network is used as a feed forward controller and trained by using the output of a feedback controller as error signal. The total control input U to the plant is equal to:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(26)

where, UC(t) and UN(t) are outputs of CFC and NN, respectively.

The feedback-error learning scheme has the following advantages: (1) the teaching signal is not required to train the neural network. Instead, the error signal is used as the training signal, (2) the learning and control are performed simultaneously in sharp contrast to the conventional learn-then-control approach and (3) back-propagation of the error sign through the model of the controlled object or through the model of the controlled object is not necessary.

In this control procedure, the CFC controller is a sliding mode controller and in Fig. 4, modified FEL-sliding mode controller, the CFC controller is a proportional-plus-derivative (PD) controller. The total control input U to the plant is equal to:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 3: General structure of FEL

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 4: The modified hybrid control approach

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(27)

where, UP(t), UN(t) and US(t) are PD controller, NN output and sliding mode outputs, respectively.

The PD controller and sliding mode controller guarantees the stability of the overall system and ensures adequate performance prior to convergence of the neural network weights and reduces the steady-state output errors due to disturbance inputs.

PD controller: Considering the following PD control for the Fig. 4 block diagram for eliminate error between x1d and x1:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(28)

PID controller: The continuous form of a PID controller, with input e (0) and output uPID (0), is generally given as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(29)

where, KP is the proportional gain, Ti is the integral time constant and Td is the derivative time constant. We can also rewrite (28) as:

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
(30)

where, Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Controlis the integral gain and Kd = KPTd is the derivative gain.

RESULTS AND DISCUSSION

Here, simulation results are presented. The figures show the position versus time (millisec) and the control (the applied voltage) versus time for the system.

First, the sliding mode controller is applied to the magnetic levitation system. The parameters of controllers are chosen such that W = 350, λ1 in sliding surface is set as 61 and λ2 = 930. The simulation results are shown in Fig. 5. It can be seen from the Fig. 5 that there is a small steady-state error in the position and some chattering can be seen due to this controller. Figure 6 shows sliding mode controller that we eliminate steady-state and chattering by using saturation function instead of sign function (14).

Second, the RBF-Sliding mode controller of system results are shown in Fig. 7, the control signal has chattering and it is not suitable for voltage source so we proposed modified RBF-Sliding mode controller of magnetic levitation system. The results in Fig. 8 shows that we improve control signal by proposing this new methods.

Third, Fig. 9 shows the FEL-Sliding mode controller of system that there is a small steady-state error in the position and we eliminate it by modified FEL-sliding mode controller. Figure 10 shows the modified FEL-sliding mode controller of magnetic levitation system. For PD controller kp and kd are set as 1.4 and 0.5.

Fourth, Fig. 11 shows the PID controller of plant. The parameters of controllers are chosen such that KP, Ki and Kd are 1990, 9996 and 88.5, respectively.

Finally, in Fig. 12, we compare all results with together. Therefore, the simulation results indicate that the proposed control schemes work well when applied to the magnetic levitation system.

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 5: Classical sliding mode control (a) position of ball and (b) control signal)

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 6: Sliding mode without chattering (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 7: RBF-Sliding mode controller without low pass filter (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 8: Modified RBF-Sliding mode controller (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 9: FEL-Sliding mode controller without PD controller (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 10: Modified FEL-sliding mode controller (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 11: PID controller (a) position of ball and (b) control signal

Image for - Hybrid Control of Magnetic Levitation System Based-on New Intelligent Sliding Mode Control
Fig. 12:
Comparison of all methods results (Modified RBF-Sliding mode, Sliding mode without chattering, RBF-Sliding mode, Modified FEL-Sliding mode and PID)

CONCLUSION

This study introduced classical and hybrid methods for control the magnetic levitation system which has practical uses in many industrial systems.

In this research, a new RBF-sliding mode control method, a new FEL-sliding mode control method and modified FEL-sliding controller for magnetic levitation is proposed, which combines the merits of adaptive neural network and sliding mode control. Based on the Lyapunov stability theory, a RBF-sliding mode controller is designed for stabilization of magnetic levitation system to the desired point in the state space. The adaptive neural network controller in this new approaches uses generalized learning rule therefore does not need to compute Jacobain of plant which comes more simplicity. Simulation results show that the proposed hybrid controllers are able to control magnetic levitation and the chattering phenomenon of conventional switching type sliding control does not occur any more in control process.

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