一般而言，在旋轉機械中轉子會因質量分布不均而有質心偏移的情形發生，這導致轉子運轉時會受到不平衡干擾而產生震動。在控制方面來說，閉迴路控制器能提供系統穩定性及足夠的剛性，而補償的方式最好能在不影系統穩定性的前提下抑制不平衡干擾。故本研究將建立磁浮軸承不平衡干擾補償的前饋控制演算法，並透過模擬及實驗進行驗證。在前饋控制中，將以同步弦波偵測位移訊號的振幅及相對相位，結合反轉移函數矩陣計算前饋補償量及補償相位，並以同步脈衝訊號做為時間基準點進行補償。考慮系統鑑別計算以及訊號感測誤差會使得補償效果受限，因此論文中提出以牛頓法迭代準確的補償量及補償相位，以增進效能。

In a rotating machine, the rotor’s center of gravity usually offsets from the rotational axis due to uneven mass distribution. Such an offset leads to periodical unbalance force and causes rotor to vibrate when it spins. In the magnetically levitated rotor system, while feedback control is used to provide stability and sufficient rigidity to possible disturbances, it is desired that a mechanism to cancel the unbalance force can be added externally without influencing the stability established by the feedback control. Therefore, this thesis devises a feedforward control algorithm for unbalanced compensation for magnetically levitated rotor systems. In the feedforward control algorithm, the amplitude and absolute phase of the displacement signal are firstly detected by using a synchronized sine wave generated from the pulse signal for the speed measurement. The information on amplitude and phase is then combined with the sensitivity transfer function matrix identified from the experiments to calculate the proper feedforward control needed to cancel the unbalance force. In order to account for the errors due to the sensor measurements as well as the inaccuracies in the sensitivity function identification, the thesis proposes to use the Newton method to iteratively compute the feedforward control efforts. The performance of the proposed unbalance compensation algorithm is verified by both simulations and experiments.

摘要 II
ABSTRACT III
致謝 IV
目錄 V
圖目錄 VIII
表目錄 XI
第一章 緒論 12
1.1 研究動機 12
1.2 文獻回顧 13
1.3 論文大綱 15
第二章 磁浮系統與硬體介紹 16
2.1 磁浮系統 16
2.2 磁浮軸承系統線性化 18
2.3 研究設備 19
第三章 靜止轉子系統 22
3.1 系統模型與動態方程式 22
3.2 控制器設計 25
3.2.1 Butterworth filter 26
3.3 平移與角度的解耦控制 27
3.4 轉子懸浮軌跡控制 29
3.5 轉子靜止系統模擬 31
第四章 轉子運轉系統 34
4.1 物理模型與動態方程式 34
4.2 轉子運轉系統模擬 36
第五章 不平衡干擾補償 38
5.1 控制器干擾抑制能力 38
5.2 補償效果與干擾影響分析 39
5.3 牛頓法簡介 41
5.4 牛頓法設計補償電流 41
5.5 牛頓法限制與驗證 42
5.6 振幅及相角偵測 43
5.7 建立同步弦波 44
5.8 SINC CONVOLUTION 46
5.9 不平衡干擾補償模擬 48
第六章 實作成果 53
6.1 建立同步弦波 53
6.2 量測位移訊號振幅及相角 55
6.3 不平衡干擾補償 57
第七章 結論與未來規劃 61
7.1 結論 61
7.2 未來規劃 61
參考文獻 62

[1] Yeh, Ting-Jen. Modeling, analysis and control of magnetically levitated rotating machines. Diss. Massachusetts Institute of Technology, 1996.
[2] Higuchi, T. "Application of Periodic Learning Control with Inverse Transfer Function Compensation in Totally Active Magnetic Bearing." Proceedings of the 2nd international Symposium on Magnetic Bearings. 1990.
[3] De Miras, J., and A. Charara. "Unbalance cancellation with rotating reference control for a horizontal shaft." Proceedings of the Sixth International Symposium on Magnetic Bearings. 1998.
[4] Schweitzer, Gerhard, and Eric H. Maslen. Magnetic bearings: theory, design, and application to rotating machinery. Vol. 2009. Berlin: Springer, 2009.
[5] Jiaxi He, Zhiquan Deng, Cong Peng, Kexiang Li, "Optimal Vibration Control for a Centrifugal Compressor with Magnetic Bearings by a Phase-shift Notch Filter." Proceedings of the Sixteenth International Symposium on Magnetic Bearings. 2018
[6] Kejian, Jiang, Zhu Changsheng, and Chen Liangliang. "Unbalance compensation by recursive seeking unbalance mass position in active magnetic bearing-rotor system." IEEE Transactions on Industrial Electronics 62.9 (2015): 5655-5664.
[7] Kang, Christopher, and Tsu-Chin Tsao. "Control of magnetic bearings for rotor unbalance with plug-in time-varying resonators." Journal of dynamic systems, measurement, and control 138.1 (2016): 011001.
[8] Wei SONG, Chengtao YU, Yuemei SUN, Xiuxiang CHEN, "Radial and Axial Integrated Auxiliary Bearing Device for Active Magnetic Bearing System." Proceedings of the Sixteenth International Symposium on Magnetic Bearings. 2018
[9] Knospe, Carl R., and Samir M. Tamer. "Experiments in robust unbalance response control." Proc. 5th Intl. Symp. on Magnetic Bearings, Kanazawa, Japan. 1996.
[10] Lum, Kai-Yew, Vincent T. Coppola, and Dennis S. Bernstein. "Adaptive autocentering control for an active magnetic bearing supporting a rotor with unknown mass imbalance." IEEE Transactions on Control Systems Technology 4.5 (1996): 587-597.
[11] Herzog, Raoul, et al. "Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings." IEEE Transactions on control systems technology 4.5 (1996): 580-586.
[12] Cui, Peiling, et al. "Harmonic current suppression of an AMB rotor system at variable rotation speed based on multiple phase-shift notch filters." IEEE Transactions on Industrial Electronics 63.11 (2016): 6962-6969.
[13] Setiawan, Joga D., Ranjan Mukherjee, and Eric H. Maslen. "Adaptive compensation of sensor runout for magnetic bearings with uncertain parameters: Theory and experiments." Journal of Dynamic Systems, Measurement, and Control 123.2 (2001): 211-218.
[14] Setiawan, Joga D., Ranjan Mukherjee, and Eric H. Maslen. "Synchronous sensor runout and unbalance compensation in active magnetic bearings using bias current excitation." Journal of dynamic systems, measurement, and control 124.1 (2002): 14-24.
[15] Setiawan, J. D. "Variable magnetic stiffness approach for simultaneous sensor runout and mass unbalance compensation in active magnetic bearings." Proc. of the 7th int. Symp. on Magnetic Bearings, Aug. 2000. 2000.
[16] Knospe, Carl R., Samir M. Tamer, and Stephen J. Fedigan. "Design of robust adaptive unbalance response controllers for rotors with magnetic bearings." (1996).
[17] Lee, J. "Q-parametrization Control of Magnetic Bearing Systems with Imbalance." Proc. of the Fifth Int. Symp. on Magnetic Bearings. 1996.
[18] Knospe, C. R. "Robustness of unbalance response controllers." Proc. of the 3rd Inter. Sympo. on Magnetic Bearings (1992): 580-589.
[19] Zhao, H., L. Zhao, and W. Jiang. "Simulation and experimental research on unbalance vibration control of AMB system." Seventh International Symposium on Magnetic Bearings, Zurich. 2000.
[20] Markert, Richard, Norbert Skricka, and Xiantong Zhang. "Unbalance compensation on flexible rotors by magnetic bearings using transfer functions." Proc. 8th International Symposium on Magnetic Bearings, Mito, Japan. 2002