Elsevier

Neurocomputing

Volume 72, Issues 13–15, August 2009, Pages 3220-3230
Neurocomputing

Direct decentralized neural control for nonlinear MIMO magnetic levitation system

https://doi.org/10.1016/j.neucom.2009.02.009Get rights and content

Abstract

A direct modified Elman neural networks (MENNs)-based decentralized controller is proposed to control the magnets of a nonlinear and unstable multi-input multi-output (MIMO) levitation system for the tracking of reference trajectories. First, the operating principles of a magnetic levitation system with two moving magnets are introduced. Then, due to the exact dynamic model of the MIMO magnetic levitation system is not clear, two MENNs are combined to be a direct MENN-based decentralized controller to deal with the highly nonlinear and unstable MIMO magnetic levitation system. Moreover, the connective weights of the MENNs are trained online by back-propagation (BP) methodology and the convergence analysis of the tracking error using discrete-type Lyapunov function is provided. Based on the direct and decentralized concepts, the computational burden is reduced and the controller design is simplified. Furthermore, the experimental results show that the proposed control scheme can control the magnets to track various periodic reference trajectories simultaneously in different operating conditions effectively.

Introduction

Magnetic levitation systems have been successfully and widely implemented for many engineering applications, such as vibration isolation, frictionless bearings, high-speed maglev passenger trains, and fast-tool servo systems [1], [2], [3], [4], [5]. Due to the features of the open-loop instability and highly inherent nonlinearities in electromechanical dynamics of the magnetic levitation systems, the development of a high performance control design for the position control of the levitated magnets is very important. Since it is very difficult to acquire an exact mathematical model to describe the electromechanical dynamics of the magnetic levitation systems, in general, only the approximated dynamic model, which consists of the state variables of position, velocity, and coil current signals are used. Therefore, many studies have been reported and discussed in recent years [6], [7], [8], [9], [10], [11] based on the approximated dynamic models. For this reason, additional compensator or estimator is required to obtain good control performance, thus the controller designs in the above literatures become complicated. In [6], an integral variable-structure grey control for a magnetic levitation system for position tracking was presented. The grey prediction compensator was used as a disturbance boundary estimator to reduce chattering or steady-state error when uncertainty values are overestimated or underestimated, respectively. However, the sign function in the grey prediction compensator and discontinuous switched type will cause chattering unavoidably. Moreover, in [7], [8], the feedback linearization techniques were adopted to transform the system model into an equivalent model with simple form. Although the dynamic model of the magnetic levitation system was simplified, only the nominal system parameters were considered in feedback linearization design. This usually leads to problems with deteriorated performance and instability since the system parameters are usually varied with thermal drift. Furthermore, in [9], the nonlinear dynamics was approximated by the use of the Taylor's series expansion. However, the high-order terms in Taylor's series expansion were neglected for the simplification of the original dynamic model to a second-order differential equation. As a result, the development of high performance controller based on this simplified model was very difficult to complete. In addition, a robust feedback linearization controller for an electromagnetic suspension system was presented in [10]. Although the stability of the control system was guaranteed theoretically, it seems that the relatively large overshoots and oscillation in transients were existed in their experimental results. This deteriorated the transient performance and resulted in impractical applications of the levitation systems. Therefore, some adaptation laws should be applied to solve the mentioned difficulty. On the other hand, an adaptive robust nonlinear controller was proposed in [11] for a magnetic levitation system via the backstepping design approach. First, a proportional-integral (PI) controller was designed to stabilize the position error of the levitated object. Then, an adaptive robust nonlinear controller was designed to attenuate the effects of parameter uncertainties.

In the studies mentioned above, the single-input-single-output (SISO) magnetic ball-like system is always turned into a magnetic levitation system for researching. However, in the most practical applications, complicated multi-input multi-output (MIMO) magnetic levitation systems are used more extensively than SISO magnetic levitation systems such as two-degrees-of-freedom mathematical model in [12] and five-degrees-of-freedom active magnetic bearing in [13]. Moreover, in MIMO magnetic levitation systems, due to not only highly nonlinear and unstable characteristics but also suffered uncertainties including the external disturbance, gravity, variation of the magnetic strength, the effect from the temperature, repulsive or attractive force among different magnets, coupling effect among various inputs and outputs, and so on are very serious, it is more difficult to obtain an exact dynamic model than SISO magnetic levitation systems. Therefore, the MIMO magnetic levitation system control problem is becoming important and challenging to the control engineers and researchers.

In the general control problems, MIMO systems have more practical and wider applications than SISO systems. However, in comparison with the vast amount of results on controller design for SISO systems in the control literatures, there are relatively fewer results available for MIMO systems due to its complication. In [14], a decentralized PID controller design procedure based on backstepping principles was presented to operate MIMO dynamic processes. By analyzing the regulation dynamics, external disturbance can be rejected via appropriate tuning of the backstepping design parameters. Moreover, in [15] a systematic procedure for constructing a MIMO fuzzy controller that guarantees identical performance to an existing stabilizing linear controller was presented. Furthermore, [16] proposed a design of adaptive sliding surfaces for a MIMO system with matched and mismatched perturbations. By utilizing some adaptive gains designed both in the sliding surface function and in the controllers, the property of asymptotical stability of controlled systems is guaranteed as well as the upper bound of partial perturbations is not required. In addition, a robust adaptive controller using a feedforward Takagi–Sugeno (T–S) fuzzy approximator for a MIMO nonlinear plants was proposed in [17]. Different to typical fuzzy approximation approaches, the desired commands are taken as the input variables of a T–S fuzzy system. Meanwhile, the unknown feedforward terms required during steady state are adaptively approximated and compensated. Additionally, some other works such as model reference adaptive control [18], PID control [19], sliding mode control [20], and neural network control [21] for various MIMO systems were reported in recent years. However, in [15], [16], [17], [18], [19], [20], [21], only simulations were demonstrated because the complicated theories are very difficult to implement on MIMO systems practically.

Neural network control is one of the intelligent control approaches which does not require mathematical model and has the ability to approximate nonlinear systems effectively. Among the several kinds of the neural network structures, the Elman neural network (ENN) used in this study was first proposed for speech processing [22]. Due to its distinguished dynamical characteristics, it has been widely applied for the identification and control of dynamical systems [23], [24]. Generally, the ENN can be considered as a special kind of feed-forward neural network with additional memory neurons and local feedback [22]. Due to the context neurons and local recurrent connections between the context layer and the hidden layer, it has certain dynamical advantages over static neural network, such as multi-layer perceptrons and radial-basis function networks. However, the typical ENN cannot approximate high-order dynamic systems precisely and its convergence speed is usually slow and not suitable for some time critical applications. Therefore, several kinds of modified ENNs were proposed to overcome such issue and to improve the dynamic characteristics and convergence speed of the original ENN [25], [26]. In [25], a modified ENN (MENN) approximation-based computed-torque controller is proposed to deal with unmodeled bounded disturbances and unstructured unmodeled dynamics of the robot arm. An improved ENN was developed to realize failure detection in a hydraulic servo system in [26]. Moreover, in [27], a MENN with self-feedback links of the context neurons is proposed which provide a dynamic trace of the gradients in the parameter space and enable the network to model dynamic systems of orders higher than one efficiently.

Since the exact dynamic model of the MIMO magnetic levitation systems is vague, a direct decentralized neural network control strategy is proposed in this study to control the magnets to track various periodic reference trajectories simultaneously. Decentralized control strategy has fewer tuning parameters than centralized control and is simple to design and implement due to single control loop [28]. In this study, first, the dynamic of the MIMO magnetic levitation system is analyzed briefly. Then, two MENNs with online tuning ability are combined as a direct MENN-based decentralized controller to control two levitated magnets of the magnetic levitation system to track periodic sinusoidal and trapezoidal reference trajectories. Moreover, the structure of adopted MENN [27] and its online tuning algorithm are described. Furthermore, to guarantee the convergence of the MENN, an analytical method based on a discrete-type Lyapunov function is provided. Finally, some experimental results illustrating the validity of the proposed control scheme for the MIMO magnetic levitation system are demonstrated. Using the proposed direct MENN-based decentralized controller, good control performance of the transient response for the position tracking of the levitated magnets is guaranteed, and robustness to the system uncertainties and external disturbance is obtained as well.

Section snippets

Dynamic analysis of MIMO magnetic levitation system

The side view and front view of the MIMO magnetic levitation system are shown in Fig. 1 [29]. The system consists of the upper and lower drive coils that produce magnetic fields in response to DC current. The magnetic field results in repulsive or attractive force depending on the same or distinct polarity between the coil and the magnet. Two levitated magnets travel along a precision ground glass guide rod in which the magnetic north pole in the lower magnet and the magnetic south pole in the

Proposed control scheme

In a typical ENN, the hidden layer neurons are fed by the outputs of the context neurons and the input neurons. Context neurons are known as memory units as they store the previous output of hidden neurons. In order to make the neurons sensitive to history of input data, self-connections of the context nodes are equipped in the adopted MENN [27]. Thus, the MENN possesses well recalling ability. Therefore, the MENN can improve the convergence error and reduce the learning time effectively. In

Experimental results

In this section, the tracking control of the MIMO magnetic levitation system is implemented using the proposed direct MENN-based decentralized controller. In order to reduce the computational burden, only small number of neurons is used. In each MENN, there are only two, four, four and one neurons at the input, hidden, context and output layers, respectively. The number of the hidden (context) neurons is obtained by trial and error, and is chosen to achieve the best transient control

Conclusions

This study has successfully demonstrated the development and application of the proposed direct MENN-based decentralized controller to control the positions of two magnets of the MIMO magnetic levitation system. First, the operating principles of the magnetic levitation system with two moving magnets were introduced. Then, the theoretical bases and the convergence analyzes of the proposed direct MENN-based decentralized controller were described in detail. Moreover, experimentations were

Acknowledgment

The author would like to acknowledge the financial support of the National Science Council of Taiwan, ROC through its Grant NSC 95-2221-E-008-177-MY3.

Syuan-Yi Chen was born in Changhua, Taiwan, in 1982. He received the B.S. degree in Electrical Engineering from the National Kaohsiung University of Applied Science, Kaohsiung, Taiwan, and M.S. degree in Industrial Education from the National Taiwan Normal University, Taipei, Taiwan, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree in Electrical Engineering, National Central University, Chungli, Taiwan. His research interests include magnetic levitation systems,

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  • Cited by (0)

    Syuan-Yi Chen was born in Changhua, Taiwan, in 1982. He received the B.S. degree in Electrical Engineering from the National Kaohsiung University of Applied Science, Kaohsiung, Taiwan, and M.S. degree in Industrial Education from the National Taiwan Normal University, Taipei, Taiwan, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree in Electrical Engineering, National Central University, Chungli, Taiwan. His research interests include magnetic levitation systems, active magnetic bearing system, nonlinear control theories, and intelligent control theories.

    Faa-Jeng Lin received the Ph.D. degree in electrical engineering from National Tsing Hua University, Taiwan, in 1993. From 1993 to 2001, he was an Associate Professor and then a Professor in the Department of Electrical Engineering, Chung Yuan Christian University, Taiwan. From 2001 to 2003, he was Chairperson and Professor in the Department of Electrical Engineering, National Dong Hwa University, Taiwan. He was Dean of Research and Development from 2003 to 2005 and Dean of Academic Affairs from 2006 to 2007 at the same University. Currently, he is Professor in the Department of Electrical Engineering, National Central University, Taiwan. He is also the Chair, Power System Division, National Science Council, Taiwan, 2007–2009. His research interests include AC and ultrasonic motor drives, DSP-based computer control systems, fuzzy and neural network control theories, nonlinear control theories, power electronics, and micro mechatronics.

    Prof. Lin received the Outstanding Research Professor Award from the Chung Yuan Christian University in 2000; the Crompton Premium Best Paper Award from the Institution of Electrical Engineers (IEE), United Kingdom, in 2002; the Outstanding Research Award from the National Science Council in 2004; the Outstanding Research Professor Award from the National Dong Hwa University in 2004; the Outstanding Professor of Electrical Engineering Award in 2005 from the Chinese Electrical Engineering Association, Taiwan. Moreover, he was the recipient of the Distinguished Professor Award from the National Central University in 2007. Furthermore, he is Fellow of the Institution of Engineering and Technology (IET, former IEE).

    Kuo-Kai Shyu received the B.S. degree from Tatung Institute of Technology, Taipei, Taiwan, in 1979, and M.S. and Ph.D. degrees from National Chung-Kung University, Tainan, Taiwan, in 1984 and 1987, respectively, all in Electrical Engineering. In 1988, he joined the National Central University, Chung-Li, Taiwan, where he is currently a Professor of the Department of Electrical Engineering. He had been the chairman of the department from 2004 to 2007. From 1988 to 1999, he was a Visiting Scholar with the Electrical and Computer Engineering Department, Auburn University, Auburn, AL. His teaching and research interests include variable-structure control systems and signal processing with applications in motor control, power electronics, and biomedical systems.

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