Proportional-integral-derivative control of rigid rotor-active magnetic bearing system via eigenvalue assignment for decoupled translational and conical modes

Abstract

Proportional-integral-derivative (PID) control has been widely adopted for stable and reliable operation of rigid rotors supported by a pair of active magnetic bearing systems. Conventional centralized PID control methods manipulate the feedback gains in prescribed forms until they achieve the desired control performance. In this study, an eigenvalue assignment for decoupled translational and conical modes is proposed in the complex domain to yield a unique PID controller in a closed form, preserving the isotropic bearing characteristics. The eigenvalue assignment necessitates the constraints required for decoupling of the translational and conical whirl motions from the complex equation of motion written in the center of gravity coordinates of the rigid rotor. The complex equation of motion integrates the rigid rotor and electro-magnetic control force models defined in two different coordinate systems by utilizing complex coordinate transformation relations. A flywheel energy storage system is taken as a simulation example in order to demonstrate the validity and effectiveness of the proposed eigenvalue assignment. The simulation results show that the eigenvalue assignment algorithm is superior to conventional control methods at systematically ensuring sufficient stability margins for the lightly-damped and unstable conical modes.

Keywords

Cross feedback decoupled control eigenvalue assignment magnetic bearing PID control rigid rotor rigid body mode translational mode conical mode noncollocation

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References

Ahrens

Kucera

Larsonneur

(1996) Performance of a magnetically suspended flywheel energy storage device. IEEE Control Systems Technology 4(5): 494–502.

Beatty R (1985) Notch Filter Control of Magnetic Bearings to Improve Rotor Synchronous Response. MS thesis, MIT, USA.

Bleuler

Gähler

Herzog

(1994) Application of digital signal processors for industrial magnetic bearings. IEEE Control Systems Technology 2(4): 280–288.

Burrows

Sahinkaya

(1983) Vibration control of multi-mode rotor-bearing systems. In Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences 386: 77–94.

Byun

Lee

(1988) Pole assignment in rotating disk vibration control using complex modal state feedback. Mechanical Systems and Signal Processing 2(3): 225–241.

Dimond T, Allaire P, Mushi S, et al. (2012) Modal tilt/translate control and stability of a rigid rotor with gyroscopics on active magnetic bearings. International Journal of Rotating Machinery 2012: Article ID 567670.

Genta

(2005) Dynamics of Rotating Systems, New York: Springer-Verlag.

Horiuchi Y, Inoue M, Sato N, et al. (2000) Development of magnetic bearing momentum wheel for ultra-precision spacecraft attitude control. In Proceedings of the 7^th International Symposium on Magnetic Bearings, Zurich, Switzerland, 23–25 August, pp. 525–530.

Humphris

Kelm

Lewis

(1986) Effect of control algorithms on magnetic journal bearing properties. ASME Journal of Engineering for Gas Turbines and Power 108(4): 624–632.

10.

Inoue

Niimi

Ishida

(2012) Vibration suppression of the rotor system using both a ball balancer and axial control of the repulsive magnetic bearing. Journal of Vibration and Control 18(4): 484–498.

11.

Jei

Lee

(1992) Does curve veering occur in the eigenvalue problem of rotors? ASME Journal of Vibration and Acoustics 114(1): 32–36.

12.

Junkins

Kim

(1993) Introduction to Dynamics and Control of Flexible Structures, Washington, DC: American Institute of Aeronautics and Astronautics.

13.

Kim

Lee

(1996) Isotropic optimal control of active magnetic bearing system. ASME Journal of Dynamic Systems, Measurement, and Control 118(4): 721–726.

14.

Kim

Lee

(1990) Optimal pole assignment into specified regions and its application to rotating mechanical systems. Optimal Control Applications & Methods 11: 197–210.

15.

Kim WY, Lee JM, Kim SJ, et al. (2010) Design and control of hybrid magnetic bearing in a flywheel energy storage system. In Proceedings of the 8^th IFToMM-International Conference on Rotor Dynamics, Seoul, Korea, 12–15 September, pp. 496–501.

16.

Knospe

Hope

Tamer

(1996) Robustness of adaptive unbalance control of rotors with magnetic bearings. Journal of Vibration and Control 2(1): 33–52.

17.

Lancaster

(1966) Lambda-matrices and Vibrating Systems, Oxford; New York: Pergamon Press.

18.

Lee

(1993) Vibration Analysis of Rotors, Dordrecht; Boston: Kluwer Academic Publishers.

19.

Lee CW (2006) Mechatronics in rotating machinery. In Proceedings of the 7^th IFToMM-International Conference on Rotor Dynamics, Vienna, Austria, 25–28 September 2006, Keynote paper.

20.

Okada

Nagai

Shimane

(1992) Cross feedback stabilization of the digitally controlled magnetic bearing. ASME Journal of Vibration and Acoustics 114(1): 54–59.

21.

Schweitzer

Maslen

(2009) Magnetic Bearings, Dordrecht; New York: Springer.

22.

Sivrioglu

Nonami

Saigo

(2004) Low power consumption nonlinear control with H∞ compensator for a zero-bias flywheel AMB System. Journal of Vibration and Control 10(8): 1151–1166.

23.

Youcef-Toumi

Reddy

(1992) Dynamic analysis and control of high speed and high precision active magnetic bearings. ASME Journal of Dynamic Systems, Measurement, and Control 114(4): 623–633.