隨著壓電材料的發現與應用，原子力顯微鏡(AFM)開始利用壓電材料作為平台的驅動器，以壓電平台的方式來做高精密的軌跡掃描定位，利用軌跡定位之結果來建構出一張奈米尺度的三維(3D)影像。然而，壓電平台本身具有高度非線性特性，例如: 磁滯效應、蠕變效應等，會導致AFM掃描成像的失真問題；另外，傳統AFM的掃描路徑為柵欄式掃描軌跡，它是以週期式三角波與連續步階軌跡所建構而成，但三角波軌跡是由無限奇次諧波所組成，所以在追蹤該軌跡時容易引發壓電平台的機械共振問題，導致在高速掃描下會無法準確依照預定軌跡移動。
為了有效地解決上述問題，本論文從軌跡著手以及控制器設計來改善。首先，我們開發出一種新型雙紐線掃描軌跡，它具有平滑軌跡的特性可以減緩壓電平台的機構共振，同時也保留了傳統柵欄式軌跡的直線部分，可以將掃描點的圖像映射失真問題最小化。另外，本論文首次結合模型預測控制(MPC)與長短期記憶(LSTM)來作為壓電平台的控制器，該控制器可以大幅消除磁滯與蠕變等非線性特性，並且提升追蹤軌跡的掃描速度。最後，我們藉由大量模擬證實所設計之控制方法的追蹤誤差，明顯低於比例積分微分控制(PID)及模型預測控制，並且在追蹤雙紐線軌跡的直線部分也有不錯的表現。

With the discovery of piezoelectric material, atomic force microscopy (AFM) uses the piezoelectric material to design the piezo-stage for high precision positioning of nanometer scale. However, the piezo-stage has inherent shortcomings like hysteresis and creep effects that inevitably cause unwanted distortion in the AFM scanned results. In addition, the traditional scan trajectory is the raster scan, which is constructed with a period triangular waveform and continuous steps. However, since the triangular waveform is a signal with infinite odd frequency, it will easily excite the mechanical resonance of the piezo-stage while tracking the trajectory, resulting in inaccurate movements according to the predetermined trajectory at a high-frequency scanning speed requirement.
This thesis focuses on trajectory improvement and controller design to effectively deal with the abovementioned problems. First, we design a novel smooth lemniscate scan trajectory to reduce resonant vibration of the piezo-stage. Furthermore, the proposed trajectory preserves the straight part of the triangular waveform to minimize the mapping distortion of scanning points. Second, this work is the first to combine model predictive control (MPC) and long short-term memory (LSTM) control methods for use in the piezo-stage. The proposed controller can not only mitigate the nonlinear property like hysteresis and creep effect but also increase the scanning rate and the tracking accuracy of the lemniscate scan trajectory. Simulation results show that the tracking error of our proposed controller is smaller than those of the PID and MPC, and it also has an excellent performance in tracking the straight part of the lemniscate scan trajectory.

[1] Á. M. G.-Límaco, J. H. Galeti, E. C. N. Silva, R. T. Higuti, M. J. Connelly, and C. Kitano, “Digital demodulation using i/q signals and optical phase control applied to a vibrometer,” IEEE Sensors Journal, vol. 20, no. 19, pp. 11313-11325, 2020.
[2] J. Deng, S. Liu, Y. Liu, L. Wang, X. Gao, and K. Li, “A 2-dof needle insertion device using inertial piezoelectric actuator,” IEEE Transactions on Industrial Electronics, vol. 69, no. 4, pp. 3918-3927, 2022.
[3] M. Prasad, Aditi, and V. K. Khanna, “Development of mems acoustic sensor with microtunnel for high spl measurement,” IEEE Transactions on Industrial Electronics, vol. 69, no. 3, pp. 3142-3150, 2022.
[4] A. Bazaei, Y. K. Yong, and S. O. R. Moheimani, “Combining spiral scanning and internal model control for sequential afm imaging at video rate,” IEEE/ASME Transactions on Mechatronics, vol. 22, no. 1, pp. 371-380, 2017.
[5] L. Li, J. Huang, S. S. Aphale, and L. Zhu, “A smoothed raster scanning trajectory based on acceleration-continuous b-spline transition for high-speed atomic force microscopy,” IEEE/ASME Transactions on Mechatronics, vol. 26, no. 1, pp. 24-32, 2021.
[6] N. Nikooienejad, A. Alipour, M. Maroufi, and S. O. R. Moheimani, “Video-rate non-raster afm imaging with cycloid trajectory,” IEEE Transactions on Control Systems Technology, vol. 28, no. 2, pp. 436-447, 2020.
[7] D. Ziegler, T. R. Meyer, A. Amrein, A. L. Bertozzi, and P. D. Ashby, “Ideal scan path for high-speed atomic force microscopy,” IEEE/ASME Transactions on Mechatronics, vol. 22, no. 1, pp. 381-391, 2017.
[8] J.-W. Wu, Y.-T. Lin, Y.-T. Lo, W.-C. Liu, and L.-C. Fu, “Lissajous hierarchical local scanning to increase the speed of atomic force microscopy,” IEEE Transactions on Nanotechnology, vol. 14, no. 5, pp. 810-819, 2015.
[9] Y. Wu, Y. Fang, C. Wang, Z. Fan, and C. Liu, “An optimized scanning-based afm fast imaging method,” IEEE/ASME Transactions on Mechatronics, vol. 25, no. 2, pp. 535-546, 2020.
[10] N. Nikooienejad, M. Maroufi, and S. O. R. Moheimani, “Rosette-scan video-rate atomic force microscopy: trajectory patterning and control design,” Rev. Sci. Instrum., vol. 90, no. 7, 073702, 2019.
[11] Z. Sun, B. Song, N. Xi, R. Yang, L. Hao, Y. Yang, and L. Chen, “Asymmetric hysteresis modeling and compensation approach for nanomanipulation system motion control considering working-range effect,” IEEE Transactions on Industrial Electronics, vol. 64, no. 7, pp. 5513-5523, 2017.
[12] J. Fang, L. Zhang, Z. Long, and M. Y. Wang, “Fuzzy adaptive sliding mode control for the precision position of piezo-actuated nano positioning stage,” International Journal of Precision Engineering and Manufacturing, vol. 19, no. 10, pp. 1447-1456, 2018.
[13] H. Xie, Y. Wen, X. Shen, H. Zhang, and L. Sun, “High-speed afm imaging of nanopositioning stages using H∞ and iterative learning control,” IEEE Transactions on Industrial Electronics, vol. 67, no. 3, pp. 2430-2439, 2020.
[14] Y.-D. Tao, H.-X. Lib and L.-M. Zhu, “Hysteresis modeling with frequency-separation-based gaussian process and its application to sinusoidal scanning for fast imaging of atomic force microscope,” Sensors and Actuators A: Physical, vol. 311, 112070, 2020.
[15] L. Li, S. S. Aphale, and L. Zhu, “Enhanced odd-harmonic repetitive control of nanopositioning stages using spectrum-selection filtering scheme for high-speed raster scanning,” IEEE Transactions on Automation Science and Engineering, vol. 18, no. 3, pp. 1087-1096, 2021.
[16] J. Curie and P. Curie, “Développement, par pression, de l’électricité polaire dans les cristaux hémièdres à faces inclinées,” Comptes Rendus, vol. 91, pp. 294-295, 1880.
[17] M. Yoichi, “Applications of piezoelectric actuator,” NEC Technical Journal, vol. 1, no. 5, pp. 82-86, 2006.
[18] P. J. Chen and S. T. Montgomery, “A macroscopic theory for the existence of the hysteresis and butterfly loops in ferroelectricity,” Ferroelectrics, vol. 23, pp. 199-208, 1980.
[19] P. Chen, X.-X. Bai, L.-J. Qian, and S.-B. Choi, “An approach for hysteresis modeling based on shape function and memory mechanism,” IEEE/ASME Transactions on Mechatronics, vol. 23, no. 3, pp. 1270-1278, 2018.
[20] Y. Liu, D. Du, N. Qi, and J. Zhao, “A distributed parameter maxwell-slip model for the hysteresis in piezoelectric actuators,” IEEE Transactions on Industrial Electronics, vol. 66, no. 9, pp. 7150-7158, 2018.
[21] Z. Li, J. Shan, and U. Gabbert, “A direct inverse model for hysteresis compensation,” IEEE Transactions on Industrial Electronics, vol. 68, no. 5, pp. 4173-4181, 2020.
[22] Y. Fan and U-X. Tan, “Design of a feedforward-feedback controller for a piezoelectric-driven mechanism to achieve high-frequency nonperiodic motion tracking,” IEEE/ASME Transactions on Mechatronics, vol. 24, no. 2, pp. 853-862, 2019.
[23] H. Jung and D.-G. Gweon, “Creep characteristics of piezoelectric actuators,” Review of Scientific Instruments, vol. 71, no. 4, pp. 1895-1900, 1999.
[24] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural Computation, vol. 9, no. 8, pp. 1735-1780, 1997.
[25] F. A. Gers, N. N. Schraudolph, and J. Schmidhuber, “Learning precise timing with lstm recurrent networks,” Journal of Machine Learning Research, vol. 3, pp. 115-143, 2002.
[26] P. Mahalakshmi and N. S. Fatima, “Summarization of text and image captioning in information retrieval using deep learning techniques,” IEEE Access, vol. 10, pp. 18289-18297, 2022.
[27] J.-H. Hsu, M.-H. Su, C.-H. Wu, and Y.-H. Chen, “Attention-based convolution skip bidirectional long short-term memory network for speech emotion recognition,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 29, pp. 1675-1686, 2021.
[28] X. Hou, K. Wang, C. Zhong, and Z. Wei, “ST-trader: a spatial-temporal deep neural network for modeling stock market movement,” IEEE/CAA Journal of Automatica Sinica, vol. 8, no. 5, pp. 1015-1024, 2021.
[29] G. Gelly and J.-L. Gauvain, “Optimization of rnn-based speech activity detection,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 26, no. 3, pp. 646-656, 2018.
[30] C. F. Lui, Y. Liu, and M. Xie, “A supervised bidirectional long short-term memory network for data-driven dynamic soft sensor modeling” IEEE Transactions on Instrumentation and Measurement, vol. 71, 2504713, 2022.
[31] Y. Qin, D. Chen, S. Xiang, and C. Zhu, “Gated dual attention unit neural networks for remaining useful life prediction of rolling bearings,” IEEE Transactions on Industrial Informatics, vol. 17, no. 9, pp. 6438-6447, 2021.
[32] Y. Yu, L. Quan, Z. Mi, J. Lu, S. Chang, and Y. Yuan, “Improved model predictive control with prescribed performance for aggregated thermostatically controlled loads,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 2, pp. 430-439, 2022.
[33] C. Alfaro, R. Guzman, L. G. d. Vicuña, J. Miret, and M. Castilla, “Dual-loop continuous control set model-predictive control for a three-phase unity power factor rectifier,” IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1447-1460, 2022.
[34] J. Yan, L. Jin, Z. Yuan, and Z. Liu, “RNN for receding horizon control of redundant robot manipulators,” IEEE Transactions on Industrial Electronics, vol. 69, no. 2, pp. 1608-1619, 2022.
[35] N. Qi, Y. Fang, X. Ren, and Y. Wu, “Varying-gain modeling and advanced dmpc control of an afm system,” IEEE Transactions on Nanotechnology, vol. 14, no. 1, pp. 82-92, 2015.
[36] M. S. Rana, H. R. Pota, and I. R. Petersen, “Nonlinearity effects reduction of an afm piezoelectric tube scanner using mimo mpc,” IEEE/ASME Transactions on Mechatronics, vol. 20, no. 3, pp. 1458-1469, 2015.
[37] M. A. Janaideh, M. Rakotondrabe, and O. Aljanaideh, “Further results on hysteresis compensation of smart micropositioning systems with the inverse prandtl–ishlinskii compensator,” IEEE Transactions on Control Systems Technology, vol. 24, no. 2, pp. 428-439, 2016.
[38] G.-Y. Gu, C.-X. Li, L.-M. Zhu, and C.-Y. Su, “Modeling and identification of piezoelectric-actuated stages cascading hysteresis nonlinearity with linear dynamics,” IEEE/ASME Transactions on Mechatronics, vol. 21, no. 3, pp. 1792-1797, 2016.
[39] J. Vörös, “Modeling and identification of systems with backlash,” Automatica, vol. 46, no. 2, pp. 369-374, 2010.
[40] J. M. Maciejowski, “Predictive control: with constraints,” 2002.
[41] H. K. Khalil, “Nonlinear systems,” 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
[42] E. Peci, “Robustness and stability of long short-term memory recurrent neural networks,” Cybernetics and Robotics, 2021.
[43] Z.-P. Jiang and Y. Wang, “Input-to-state stability for discrete-time nonlinear systems,” Automatica, vol. 37, no. 6, pp. 857-869, 2001.
[44] G. H. Golub and C. F. van Loan, “Matrix Computations”. JHU Press, fourth ed., 2013.
[45] D. O. Cajueiroa and E. M. Hemerly, “Direct adaptive control using feedforward neural networks,” Sba Controle & Automação, vol. 14, no. 4, pp. 348-358, 2003.
[46] T. Yabuta and T. Yamada, “Learning control using neural networks,” Proc. of IEEE International Conference on Robotics and Automation, Sacramento, USA, Apr. 9-11, 1991.