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To intuitively analyze the absolute and relative stability of the forward and backward whirling modes of the high-speed magnetically suspended flywheel rotor with significant gyroscopic effects and bending modes, this paper presents a whirling modes stability criterion based on complex coefficient frequency characteristics method. Through variable reconstruction, the multipleinput multiple-output magnetically suspended rotor system is converted into a single-input single-output (SISO) system with complex coefficients. The stability equivalence of the systems before and after variable reconstruction has been proven. Based on this, the inherent relationships between the distribution of the closedloop poles of the complex coefficient SISO system and the whirling modes stability are revealed, and the Nyquist stability criterion is further extended by applying the argument principle. All of these lay the foundation for the whirling modes stability theorem, and then, their stability criterion is further developed. Simulation and experimental results prove the effectiveness and correctness of the presented criterion.Index Terms-Bending mode, characteristic locus method, complex coefficient frequency characteristics, gyroscopic effects, magnetically suspended flywheel (MSFW), stability criterion, whirling mode.

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For a magnetically suspended flywheel (MSFW), the residual unbalance of rotor can cause synchronous vibration and reduce the attitude control precision of spacecrafts. To eliminate the vibration caused by the residual unbalance of rotor suspended by magnetic bearing, the theorem of unbalance vibration is researched, and it is pointed out that the vibration can be successfully suppressed through the elimination of synchronous component in magnetic suspension forces by a magnetic suspension force compensation method. Based on the general notch filter, this method is proposed to be closed-loop style when the rotor rotates beyond its critical stable speed and to be open-loop style when the speed of the rotor is less than the critical stable speed by additional displacement stiffness compensation segment, open and closed-loop control method. The stability of this method is analyzed and the ability to eliminate n octave synchronous is verified in the whole speed range of the rotor. Experimental results demonstrate that this method can suppress the unbalance vibration significantly in the operation speeds of MSFW and is suitable for MSFW in that the rotor traverses its critical stable speed frequently.

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1 Abstract-This paper presents a novel method for position sensorless control of high-speed brushless dc (BLDC) motors with low inductance and nonideal back electromotive force (EMF) in order to improve the reliability of the motor system of a magnetically suspended control moment gyro (MSCMG) for space application. The commutation angle error of the traditional line-to-line voltage zero-crossing points (ZCPs) detection method is analyzed. Based on the characteristics measurement of the nonideal back EMF, a two-stage commutation error compensation method is proposed to achieve the high reliable and high accurate commutation in the operating speed region of the proposed sensorless control process. The commutation angle error is compensated by the transformative line voltages, the hysteresis comparators and the appropriate design of the low-pass filters (LPFs) in the low speed and high speed region respectively. High precision commutations are achieved especially in the high speed region to decrease the motor loss in steady state. The simulated and experimental results show that the proposed method can achieve an effective compensation effect in the whole operating speed region.Index Terms-Brushless dc (BLDC) motor, position sensorless control, back electromotive force (EMF) detection.