本論文對於輸入維度大於輸出維度的非方陣嚴格真分連續系統提出了一種新型的零點配置法，用於產生具有相同輸入輸出維度的擴增型系統，擴增型系統除了適用於方陣系統的控制方法外，依照原系統的相位特性(極小/非極小)，擴增型系統也將保有其相位特性(極小/非極小)。基於零點配置的擴增型系統，本論文提出了以軌跡追蹤為導向的軌跡追蹤模擬器，可用於提供額外維度的輸出量測值以建構出相應的狀態觀測器及等效輸入干擾估測器，本文針對具有未知干擾的時延/非時延的連續時間/離散採樣系統分別推導出最佳線性二次類比/數位追蹤器，除此之外，針對於連續時間的非極小相位系統，本論文創新提出頻域極座標的定義，以此做為理論基礎，提出一種創新的控制方法論，此方法包括藉由閉迴路零點配置法以建構通用型比例積分追蹤器以及基於比例與高階積分之狀態觀測器與等效輸入干擾估測器。同時，對於離散採樣的非極小相位系統則提出了基於高階比例積分觀測器以及等效輸入干擾估測器以建構出最佳的軌跡追蹤器。本文另結合系統狀態觀測器、卡爾曼濾波器識別法(OKID)，以針對已知/未知、方陣/非方陣、極小/非極小相位、時延/非時延的受干擾系統，提出一種基於新型的等效輸入干擾估測器所建構的最佳軌跡追蹤器，同時提出一些說明性的例題，以驗證所提出的軌跡追蹤器之有效性。

A new zero-assignment-based squaring-up method for the strictly proper continuous-time system with more inputs than outputs has been appropriately utilized in this thesis to produce the squared-up system with equal inputs and outputs and retain the non-minimum-phase (NMP) property of the given NMP system with unstable zeros. The new zero-assignment method can always place zeros of the squared-up system at desired locations to preserve the original MP/NMP property and insure the resulting squared-up system is MP/NMP. As a result, some benefits of the square-system-based control design methodologies can be applied. Based on the squared-up system, a trajectory-tracking-oriented simulator is newly proposed so that some extra output measurements can be supported by the proposed simulator to construct the corresponding observer-based equivalent-input-disturbance (EID) estimator and tracker. In this thesis, the generalized optimal linear quadratic analog and digital trackers are respectively derived for the time-delay/delay-free continuous-time and sampled-data systems with unknown disturbances. Also, a novel definition for the polar coordinates of the strictly proper square continuous-time system in the frequency domain has been proposed as a theoretical tool to present the new controller design methodology for the NMP system. Therein, the universal proportional plus integral (PI) tracker by the closed-loop zero-assignment approach as well as the proportional plus multi-higher-order integral-based observer and EID estimator for the NM phase system are newly presented in this thesis. Sequentially, the universal state-feedback and output-feedback optimal trackers as well as the multi-higher-order integral-based observer and EID estimator for the NM phase sampled-data system are newly presented in this thesis. Furthermore, integrated with the observer Kalman filter identification (OKID) method, new EID estimator-based customized trackers for the known/unknown MP/NMP square/non-square time-delay/delay-free continuous-time/sampled-data systems with unknown disturbances are proposed in this thesis. Illustrative examples are also given to demonstrate the benefits of the proposed methods.

[1] Anderson, B.D.O. and Moore, J.B., Optimal Control: Linear Quadratic Methods, NJ: Prentice-Hall, 1989.
[2] Astrom, K.J., Hagander, P., and Sternby, J., “Zeros of sampled systems,” Automatica, vol. 20, no. 1, pp. 31-38, 1984.
[3] Chang, J. L., “Applying discrete-time proportional integral observers for state and disturbance estimations,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 814-818, 2006.
[4] Emami-Naeini, A. and Dooren, P.V., “Computation of zeros of linear multivariable systems,” Automatica, vol. 18, no. 4, pp. 425-430, 1982.
[5] Ebrahimzadeh, F., Tsai, J. S. H., Liao, Y. T., Chung, M. C., Guo, S. M., Shieh, L. S., and Wang, L., “A generalized optimal linear quadratic tracker with universal applications－Part 1: Continuous-time systems,” International Journal of Systems Science, vol. 48, no. 2, pp. 376-396, 2017.
[6] Ebrahimzadeh, F., Tsai, J. S. H., Chung, M. C, Liao, Y. T., Guo, S. M., Shieh, L. S., and Wang, L., “A generalized optimal linear quadratic tracker with universal applications－Part 2: Discrete-time systems,” International Journal of Systems Science, vol. 48, no. 2, pp. 397-416, 2017.
[7] Huong, D. C., “A fresh approach to the design of observers for time-delay systems,” Transactions of the Institute of Measurement and Control, vol. 40, no. 2, pp. 477-503, 2018.
[8] Ishitobi, M. and Kunimatsu, S., “Zeros of sampled-data models for time delay MIMO systems,” Proceedings of 2016 IEEE Region 10 Conference (TENCON), pp. 13410-3413, 2016.
[9] Joshi, V. V., Xie, L. B., Park, J. J., Shieh, L. S., Chen, Y. H., Grigoriadisd, K., and Tsai, J. S. H., “Digital modeling and control of multiple time-delayed distributed power grid,” Applied Mathematical Modelling, vol. 36, pp. 4118-4134, 2012.
[10] Juang, J. N., Applied System Identification, Englewood Cliffs, NJ: Prentice-Hall, 1994.
[11] Kim, J., Chang, P.H., and Jin, M., “Fuzzy PID controller design using time-delay estimation,” Transactions of the Institute of Measurement and Control, vol. 39, no. 9, pp. 1329-1338, 2017.
[12] Liu J., Gusrialdi, A., Obradovic, D., and Hirsche, S., “Study on the effect of time delay on the performance of distributed power grids with networked cooperative control,” Proceedings of the 1st IFAC Workshop on Estimation and Control of Networked Systems, Venice, pp. 180-185, 2009.
[13] Lewis, F. L., Applied Optimal Control and Estimation, NJ: Prentice-Hall, 1992.
[14] Lewis, F.L. and Syrmos, V.L., Optimal Control, NJ: John Wiley & Sons, 1995.
[15] Liao, Y. L., An Efficient and Universal EID Estimation Mechanism for Compensation-Improvement of Square/Non-square Systems with/without Disturbances, Department of Electrical Engineering, N.C.K.U., Tainan, Taiwan, R.O.C., Master Thesis, July 2021.
[16] Qu, Z., Wiese, D., Annaswamy, A.M., Lavretsky, E., “Squaring-Up method in the presence of transmission zeros,” IFAC Proceeding, vol. 47 4164-4169, 2014.
[17] Salloum, R., Moaveni, B., and Arvan, R., “Identification and robust controller design for an electromechanical actuator with time delay,” Transactions of the Institute of Measurement and Control, vol. 37, no. 9, pp. 1109-1117, 2015.
[18] Tsai, J. S. H., Shieh, L. S., and Yates, R. E., “Fast and stable algorithms for computing the principal ?th root of a complex matrix and the matrix sector function,” Computers & Mathematics with Applications, vol. 15, no. 11, pp. 903-913, 1988.
[19] Tsai, J. S. H., Ebrahimzadeh, F., Lin, Y. Y., Guo, S. M., Shieh, L. S., Juang, Y. T., “An efficient robust servo design for non-minimum phase discrete-time systems with unknown matched/mismatched input disturbances,” Automation, Control and Intelligent Systems, vol. 5, no. 2, pp. 14-28, 2017.
[20] Tsai, J. S. H., Kuo, L. Y., Guo, S. M., Shieh, L. S., and Canelon, J. I., “Application of the observer/Kalman filter identification method to unknown time-delay disturbed systems and the associated optimal digital tracker design,” International Journal of Systems Science, vol. 49, no. 16, pp. 2935-2961, 2019.
[21] Tsai, T. J., Tsai, J. S H., Guo, S M., & Chen, G., “Observer-based optimal/sub-optimal digital trackers for analog neutral systems with multiple discrete and distributed time delays,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, vol. 13, no. 6, pp. 743-789, 2006.
[22] Tsai, J. S. H., Liao, Y. T., Lin, Z. W., Ebrahimzadeh, F., Guo, S. M., Shieh, L. S., and Canelon, J. I., “A new PI-based optimal linear quadratic state-estimate tracker for discrete-time non-square non-minimum phase systems,” International Journal of Systems Science, vol. 49, no. 9, pp. 1856-1877, 2018.
[23] Tsai, J. S. H., Liao, Y. T., Ebrahimzadeh, F., Lai, S. Y., Su, T. J., Guo, S. M., Shieh, L. S., and Tsai, T. J., “A new PI optimal linear quadratic state-estimate tracker for continuous-time non-square non-minimum phase systems,” International Journal of Systems Science, vol. 48, no. 7, pp. 1438-1459, 2017.
[24] Tsai, J. S.H., Su, T. J., Cheng, J. C., Lin, Y. Y., Giap, V. N., Guo, S.M., Shieh, L.S., “Robust observer-based optimal linear quadratic tracker for five-degree-of-freedom sampled-data active magnetic bearing system,” International Journal of Systems Science, vol. 49, no. 6, pp. 1273-1299, 2018.
[25] Tsai, J. S. H., Wang, H. H., Guo, S. M., Shieh, L. S., and Canelon, J.I., “A case study on the universal compensation-improvement mechanism: A robust PID filter-shaped optimal PI tracker for systems with/without disturbances,” Journal of the Franklin Institute, vol. 355, pp. 3583-3618, 2018.
[26] Wu, C. Y., Tsai, J. S. H., Guo, S. M., Shieh, L. S., Canelon, J. I., Ebrahimzadeh, F., and Wang, L., “A novel on-line observer/Kalman filter identification method and its application to input-constrained active fault-tolerant tracker design for unknown stochastic systems,” Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 352, pp. 1119-1151, 2015.
[27] Wang, J., Zhang, Y., Wang, W., Zhao, J., “Optimal design of PI/PD controller for non-minimum phase system,” Transactions of the Institute of Measurement and Control, vol. 28, no. 1, pp. 27-35, 2006.