Feedback Linearization and Model Reference Adaptive Control of a Magnetic Levitation System

Luiz H. S. Torres, Leizer Schnitman, Carlos A. V. V. Júnior
Centro de Capacitação Tecnológica em Automação Industrial (CTAI), Universidade Federal da Bahia
Rua Aristides Novis, no 02, Escola Politécnica, 2^o andar, 40.210-630,
Salvador, Bahia, Brasil

J. A. M. Felippe de Souza
Electromechanical Engineering Dept, University of Beira Interior
Covilhã, Portugal

Abstract:

The aim of this paper is to combine two techniques to control a nonlinear Magnetic Levitation System (MLS). Firstly, a feedback linearization technique (here, exact linearization with state feedback) is applied to obtain a linear system. Secondly, the linearization is made via direct cancellation of nonlinear functions, which represent the phenomenological model of the system. Finally, to deal with the presence of uncertainty in the system model, an adaptive controller is used. The controller is based on model reference adaptive control to estimate the functions that contain the nonlinearities of the system. The exact linearization and the adaptive controller were implemented in a simulated environment (Matlab Simulink ©). The linear adaptive controller structure guarantees the parameters adaptation and the overall stability of the system. The results show that the controller output signal tracks a reference input signal with a small error.

Keywords:

Adaptive Control; Direct Approach; Exact Linearization; Magnetic Levitation; Model Reference.

>>Full text

CITE THIS PAPER AS: Luiz H. S. TORRES, Leizer SCHNITMAN, Carlos A. V. V. JUNIOR, J. A. M. FELIPPE DE SOUZA, Feedback Linearization and Model Reference Adaptive Control of a Magnetic Levitation System, Studies in Informatics and Control, ISSN 1220-1766, vol. 21 (1), pp. 67-74, 2012. https://doi.org/10.24846/v21i1y201208

1. Introduction

In recent years, due to computational developments that have enabled more complex applications of nonlinear problems, the area of nonlinear control systems has been the subject of many studies (Soltanpour and Shafie, 2010). The present paper shows a combination of an adaptive controller and a feedback linearization technique to control a Magnetic Levitation System (MLS). This system was chosen since it has nonlinear dynamics and a didactic kit of the physical system is available to continue with future work.

The MLS used is manufactured by ECP – Educational Control Products (www.ecp.com) and will be described in more detail in section II. Here one desires to control a magnet displacement over a glass stick as a result of the application of an electrical current on a coil (ECP, 1999).

The relationship between the electrical flow and the magnetic disc movement is given by a second order nonlinear ordinary differential equation. This nonlinear relationship belongs to a class of engineering systems of the type .

Several nonlinear control strategies can be used to control the disc position, such as, for example: fuzzy, neural network, adaptive control, feedback linearization (Khalil, 1996; Abdel-Hady and Abuelenin, 2008; Torres et al., 2010a). Here both the exact feedback linearization technique and adaptive control will be used.

Exact feedback linearization can enable a transformation from a nonlinear system to a linear through the addition of nonlinear compensators. Thus, this transformation allows designing a linear controller for the system linearized. However, the exact linearization technique with state feedback requires a mathematic model that represents the dynamics of the real plant (Slotine, 1991).

Furthermore, the uncertainties in the phenomenological model can compromise better results. To deal with some uncertainties in the system’s model, an adaptive controller is used. The controller is based on direct model reference approach (Narendra and Valavani, 1978; Ioannou and Sun, 1996) to provide a control law that will be used to the system after the linearization.

The aim of this paper is to use the combination of two techniques to control the MLS: exact linearization with state feedback and Model Reference Adaptive Control (MRAC). These two techniques combined will enable a linear controller structure to deal with some uncertainties in the system model to MSL?s control disk position (a typical nonlinear dynamical system).

In section 2 the MLS is briefly explained .The proposal of the exact linearization with state feedback and its application over the MLS are presented in section 3. In section 4, the adaptive controller structure with a linear control law is presented. The simulation and analysis results are discussed in section 5. Finally, some remarks about the controller performance are presented in section 6. It is important to mention that a first version of this paper was published in (Torres et al., 2010b).

References:

ASTRÖM, K., B. WITTENMARK, Adaptive control, 573p, Dovler: New York, USA, 2008.
ABDEL-HADY, F., S. ABUELENIN, Design and Simulation of a Fuzzy-Supervised PID Controller for a Magnetic Levitation System. Studies in Informatics and Control, Vol. 17, No. 3, 2008, pp. 315-328.
ECP, 1999. Manual for Model 730: Magnetic Levitation System. Bell Canyon, USA.
GUADARBASSI, G., S. SAVARESI, Approximate Linearization via Feedback – An Overview, Automatica, Vol. 37, 2001, pp. 1-15.
IOANNOU, P. A., J. SUN, Robust Adaptive Control. Prentice Hall: New York, 1996, 833p.
KHALIL, H., Nonlinear Systems, 2nd ed., Prentice Hall: Michigan, 1996.
LAITHWAITE, E. R., Electromagnetic Levitation, in Proceedings of Institution of Electrical Engineers, Vol. 112, no. 12, 1965, pp. 2361-2375.
NAM, K., A. ARAPOSTATHIS, K. K. LEE, Some Numerical Aspects of Approximate Linearization of Single Input Nonlinear Systems. IEEE Transactions on Automatic Control, Vol. 57, 1993, pp. 463-472.
NARENDRA, K. S., L. S. VALAVANI, Stable Adaptive Controller Design – Part I – Direct Control. IEEE Transactions on Automatic Control, Vol. AC-23, no. 4, 1978, pp. 570-583.
SILVA, E. B., Modeling and Controlling a Magnetic Levitation System, B. Sc. thesis, Dept. Elect. Eng., UFBA Univ., Salvador Bahia, Brazil, 2009.
SOLTANPOUR, M. R., E. S. SHAFIE, Design and Stability Analysis of a Robust Impedance Control System for a Robot Manipulator. Studies in Informatics and Control, Vol. 19, No. 1, pp. 5-16.
SLOTINE, J. J., W. LI, Applied Non-linear Control. Prentice Hall: Englewood Cliffs, New Jersey, 1991.
TORRES, L. H. S., L. SCHNITMAN, C. A. V. VASCONCELOS JÚNIOR, J. A. M. FELIPPE DE SOUZA, Exact Linearization and Fuzzy Logic Applied to the Control of a Magnetic Levitation System, in Proc. 2010 IEEE World Congress on Computational Intelligence, WCCI 2010, Barcelona-Spain, 2010, pp. 1347-1351.
TORRES, L. H. S., L. SCHNITMAN, C. A. V. VASCONCELOS JÚNIOR, J. A. M. FELIPPE DE SOUZA, Feedback Linearization and Model Reference Adaptive Control of a Magnetic Levitation System, in Proc. XIV Latin American Congress of Automatic Control / XIX Congress of the Chilean Association of Automatic Control – ACCA, Santiago-Chile, 2010.