About the observability and controllability of a levitation system.

Teodorescu, Adriana ; Dolga, Valer

Abstract: The magnetic levitation system (MLS) is a multiple application system. The problem in analyzing this type of system includes a synergic integration of sensorial elements and control. This paper describes the state system model, the linearization over the equilibrium point and analyzes the observability and controllability of the system.

Key words: magnetic levitation system, feedback, nonlinear system, system linearization.

1. INTRODUCTION

The magnetic levitation system (MLS) can be assimilated with a mechatronic system (Shiao, 2001). The applicability of a MLS is wide: industrial technology--transportation, kinetic energy storage, special actuators, haptic magnetic levitation, etc. (Lilienkamp & Lundberg, 2004; Morita et al., 2002). The magnetic levitation constitutes a classical control task for which various solutions were admitted (Kemin & Tekkouk, 2006; Li, 2005; Sinha & Nagurka, 2005; Wai et al., 2005).

The analysis of the mathematical models for the studied system outlined several variants formulated by different experts. For this reason, the authors of this paper intend to develop an own model and to investigate the observability and the controllability of this system model.

2. SYSTEM DESCRIPTION

The considered magnetic levitation system consists of an iron magnetic ball hanged in a magnetic field with a controlled tension. The scheme is showed in Figure 1 (Dolga 2007).

The levitation system includes the following subsystems: the iron mass core coil number 1, the position sensor in order to determine the metallic ball's number 2 position, which is in sustentation with the coil and various circuits used for alimentation, amplification, control etc.

3. MATHEMATICAL MODEL

3.1. The state model of the system

In the given context, the dynamic model of the levitation system can be individualized for each variant of approach of induction calculation and it is described by the equations (1-3):

[FIGURE 1 OMITTED]

dx / dt = v (1)

e = Ri + d[L(x)i] / dt (2)

m dv / dt = mg - [F.sub.em] (3)

where: "x"- is the ball's position towards the reference position; "v"- represent the ball's velocity; "i"- represents the power in the electromagnetic coiling; "e"- represents the supplied tension of the coil; "R"- represents the strength of electromagnetic coiling; "L"- represents the coiling inductivity; "g"- represents gravity acceleration who is constant; "m"-represents ball's mass.

The induction in the coil for the presented circuit is:

L = [[mu].sub.0]S[N.sup.2] / l / [[mu].sub.r] + x (4)

where: "N"- represents the coil number of whirls and "S"- the closing cross section of the magnetic flux.

The magnetic levitation force can be determined based on the previous equation:

[F.sub.em] = -[[[partial derivative][W.sub.m] / [partial derivative]].sub.i=ct] = [i.sup.2]/2 x [partial derivative]L(x) / [partial derivative]x (5)

The mathematical model can be developed using the state model of the system and the state variables

X = [[[x.sub.1] [x.sub.2] [x.sub.3].sup.T] = [[x v I].sup.T] and u = e (k = [[mu].sub.0]S[N.sup.2]):

[dx.sub.1] / dt = [x.sub.2] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The nonlinear system can be linearized with the classical principle used to linearize action systems (Fourier system development and holding on the first order terms).

3.2. The linearization of the mathematical model

The basic idea starts from the serial Taylor development. Having the f(x) nonlinear function, in a Taylor serial development around an equilibrium point x0 the relationship (9) is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Taking into consideration [DELTA]x = x - [x.sub.0], [y.sub.0] = f ([x.sub.0]) and y = f(x) and neglecting higher rank terms than 2 ([DELTA]x < 1 and so [([DELTA]x).sup.2] << 1, it results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This function linearization can be imposed even for two or more independent variables.

Based on the equations (6) - (8), one can determine the equilibrium point [x.sub.0]:

[x.sub.10] = l / [[mu].sub.r] + e / R. [square root of k / 2 mg] (11)

[x.sub.20] = 0 and [x.sub.30] = e / R. (12)

The linearized equation system (6)-(8) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The general expression for the state model is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Where (based on the equations (13)), the definition matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

B = [[0 0 e / R [square root of 2mgk].sup.T] (16)

C = [1 0 0] (17)

Based on the previous matrices, one can define the observability matrix (Salgado & Yuz, 2002; Sinha & Nagurka, 2005) and the controllability matrix:

[[GAMMA].sub.0][A, C] = [[C CA ... C[A.sup.n-1]].sup.T] (18)

[[GAMMA].sub.C][A, B] = [B AB ... [A.sup.n-1]B] (19)

A system is observable if [[GAMMA].sub.0] has full column rank n and, in turn, a system is controllable if [[GAMMA].sub.c] has full row rank. The determinations applied from the relationships (15)-(19) show that the system is both observable and controllable.

4. CONCLUSIONS

The authors developed an own state model for the magnetic levitation system starting from a set of own theoretical considerations. The model was initially nonlinear but it was linearized by classical mathematical methods.

The obtained model endorsed to analyze the observability and the controllability of the system and to conclude that the magnetic levitation system is both observable and controllable. This fact allows the realization of a controller that is appropriate for the magnetic levitation system.

Consequently, a rapid prototyping system can be analyzed within the combined environment Matlab / Simulink--dSPACE.

5. REFERENCES

Dolga, V. (2007). Modelarea si simularea unui sistem de levitatie magnetica. Annals of the Oradea University, fascicle of Management and Technological Engineering, vol.v(xv) 2007, ISSN 1583-0691, p.35-37

Kemin, K. & Tekkouk, O. (2006). Constrained generalised predictive control with estimation by genetic algorithm for a magnetic levitation system. Intern. J. of Innovative Comp., Inform. and Control, vol.2, no.3, june 2006, p.543-552, ISSN

Li, J.H. (2005). DSP--Based Control of a PWM-driven Magnetic Levitation System, Proceedings IEEE ICSS2005 Intern. Conf. on Systems & Signals, Kaohsiung, Taiwan, pp.483-487, NSC 90-2213-E-168-003

Lilienkamp, L.A. & Lundberg, K. (2004). Low-cost magnetic levitation project kits for teaching feedback system design, Available from: http://web.mit.edu/klund/www/papers/ ACC04_maglev.pdf Accessed: 2007-04-27

Morita, T.; Shimizu, K.; Hasegawa, M; Oka, M.K. & Higuchi, T. (2002). A Miniaturized Levitation System with Motion Control using a Piezoelectric Actuator, IEEE Trans. On Control Systems Tech., vol.10, no.5, p.666-670

Salgado, M. & Yuz, J. (2002) State space Analysis and System Properties, in Mechatronics, CRC Press LLC, ISBN 0849300665

Shiao, Y.S. (2001). Design and Implementation of a Controller for a Magnetic Levitation System. Proc. Natl. Sci. Counc., vol.11, no.2, pp.88-94

Sinha, R. & Nagurka, M. (2005). Analog and LabView based control of a MAGLEV system with NI_ELVIS, Proceedings of 2005 ASME Intern. Mech. Eng. Congress Exp., november 5-11, Orlando, Florida (USA), paper IMECE 2005-81600

Wai, R.J.; Lee, J.D. & Liao, C.C. (2005). Model-free Control Design for Hybrid Magnetic Levitation System, Available from: ieeexplore.ieee.org/iel5/9859/31200/01452519.pdf Accessed: 2007-04-27