636 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011 Robust Nonsingular Terminal Sliding-Mode Control for Nonlinear Magnetic Bearing System Syuan-Yi Chen, Student Member, IEEE, and Faa-Jeng Lin, Senior Member, IEEE Abstract—This study presents a robust nonsingular terminal sliding-mode control (RNTSMC) system to achieve finite time tracking control (FTTC) for the rotor position in the axial di- rection of a nonlinear thrust active magnetic bearing (TAMB) system. Compared with conventional sliding-mode control (SMC) with linear sliding surface, terminal sliding-mode control (TSMC) with nonlinear terminal sliding surface provides faster, finite time convergence, and higher control precision. In this study, first, the operating principles and dynamic model of the TAMB system using a linearized electromagnetic force model are introduced. Then, the TSMC system is designed for the TAMB to achieve FTTC. Moreover, in order to overcome the singularity problem of the TSMC, a nonsingular terminal sliding-mode control (NTSMC) system is proposed. Furthermore, since the control characteristics of the TAMB are highly nonlinear and time-varying, the RNTSMC system with a recurrent Hermite neural network (RHNN) uncer- tainty estimator is proposed to improve the control performance and increase the robustness of the TAMB control system. Using the proposed RNTSMC system, the bound of the lumped uncertainty of the TAMB is not required to be known in advance. Finally, some experimental results for the tracking of various reference trajectories demonstrate the validity of the proposed RNTSMC for practical TAMB applications. Index Terms—Hermite polynomials, magnetic bearing system, nonsingular terminal sliding-mode, recurrent neural network, tracking control. I. INTRODUCTION S LIDING-mode control (SMC) is a well-known powerful control scheme which has been successfully and widely applied for both linear and nonlinear systems [1]. In general, the most commonly used sliding surface is the linear sliding surface, which can guarantee the asymptotic stability and de- sired performance of the closed-loop control system by using linear sliding mode [1]. Although the parameters of the linear sliding surface can be adjusted appropriately to obtain the ar- bitrary convergence rate, the system states can not reach the equilibrium point in finite time [2]. To overcome this draw- back, terminal sliding-mode control (TSMC) with nonlinear ter- minal sliding surface is recently proposed based on the con- cept of a terminal attractor [2], [3]. Compared with the con- ventional SMC with linear sliding surface, TSMC offers some superior properties such as faster, finite time convergence, and Manuscript received March 09, 2010; accepted May 04, 2010. Manuscript received in final form May 10, 2010. Date of publication June 10, 2010; date of current version April 15, 2011. Recommended by Associate Editor M. L. Corradini. This work was supported by the National Science Council of Taiwan under Grant NSC 98-2221-E-008-115-MY3. The authors are with the Department of Electrical Engineering, National Central University, Chungli 320, Taiwan (e-mail: 955401017@cc.ncu.edu.tw; linfj@ee.ncu.edu.tw). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2050484 higher control precision [2]. However, there are two disadvan- tages of TSMC which are the singularity point problem and the requirement of the bound of the uncertainty. Fortunately, the first problem has been overcame by nonsingular terminal sliding-mode control (NTSMC) [4], [5] and the second problem can be solved by well-designed uncertainty estimator [5]–[7]. In general, the hidden neurons in neural networks (NNs) usu- ally have the same activation functions such as sigmoid or radial basis functions. Moreover, the ideas of using different activation functions for different hidden neurons have been proposed in [8]–[11]. In [10], a novel one-hidden-layer NN was proposed in which each hidden neuron employs a different orthonormal Her- mite polynomial basis function (OHPBF) for its activation func- tion. Therefore, the weighting sum of the OHPBFs series ex- pansion yields an approximation to any function and may be re- garded as a Hermite neural network (HNN). On the other hand, since a recurrent neuron has an internal feedback loop to capture the dynamic response of a system, the recurrent neural networks (RNNs) can perform excellent dynamic mapping through their own natural temporal operation and demonstrate good control performance in the presence of unmodeled dynamics, param- eter variations, and external disturbances [12]–[14]. Based on the noncontact and frictionless characteristics, active magnetic bearing (AMB) offers many practical and promising advantages over conventional bearings such as longer life, lower rotating frictional losses, higher rotational speed, and elimination of the lubrication [15]–[17]. In most ap- plications, the controlled rotor should be positioned and moved precisely and functionally to deal with the different operation demands and environments. Therefore, the development of sophisticated controller, which is capable of tracking various reference trajectories precisely over a large fraction of the air gap, is imperative to deal with different operating demands and environments [18]–[20]. In this study, the controlled TAMB system is represented by a nonlinear dynamic model, in which both the system parameter variations and external disturbance are considered. Then, the TSMC is designed to control the rotor position in the axial direc- tion of the TAMB system to achieve FTTC. Moreover, the sin- gularity problem of the TSMC is further solved by the designed NTSMC. To improve the control performance and increase the robustness of the TAMB control system, the RNTSMC system, which combines the advantages of the NTSMC, RNNs and OH- PBFs, is proposed. In the RNTSMC, the RHNN uncertainty es- timator with superior approximated ability is employed to di- rectly estimate the lumped uncertainty of the TAMB. Thus, the exact value of the bound of the lumped uncertainty is unneces- sary. Finally, some experimentation illustrating the validities of the proposed RNTSMC for the TAMB control system are com- pared and discussed. 1063-6536/$26.00 © 2010 IEEE