本論文藉由以PI濾波器為基底之頻率整形的方法提出一個適用於非極小相位多變數系統的最佳比例加積分狀態回授之線性二次類比追蹤器且其能夠達成類似極小相位的追蹤性能。此外本論文所提出的追蹤器對於任意有著劇烈變化的參考軌跡仍能有著良好的追蹤性能。其它相關的應用也會介紹在本論文中，例如 (1)適用於非極小相位系統的一個基於PID濾波器頻率整形之比例加積分狀態回授最佳線性二次類比估測器。(2)適用於非極小相位系統的一個以最佳比例加積分狀態回授觀測器基底之線性二次類比追蹤器。關於本論文所提出之追蹤器與其應用，將以一些說明性的例題來驗證所提出方法的有效性。

This thesis proposes a new optimal proportional-plus-integral (PI) filter-based frequency shaping PI state-feedback linear quadratic analog tracker (LQAT) for non-minimum phase multivariable systems to achieve a minimum-phase-like tracking performance. In addition, the proposed tracker works well for the arbitrary reference trajectory with drastic variations. Some other applications are also presented in this thesis, such as (i) An optimal PID filter-based frequency shaping PI state-feedback linear quadratic analog estimators for non-minimum phase systems. (ii) An optimal PI state-feedback observer-based linear quadratic analog tracker for non-minimum phase systems. Some illustrative examples show the effectiveness of the proposed method.

[1] Anderson, B. D. O. and Mingori, D. L., “Use of frequency dependence in linear quadratic problems to frequency shape robustness,” Journal of Guidance and Control, vol. 8, no. 3, pp. 397-401, 1985.
[2] Anderson, B. D. O. and Moore, J. B., Optimal Control: Linear Quadratic Method, Englewood Cliffs: NJ, Prentice-Hall, 1989.
[3] Doyle, J. C., Francis, B. A., and Tannenbaum, A. R., Feedback Control Theory, NY: Macmillan, 1990.
[4] Ebrahimzadeh, F., Tsai, J. S. H., Liao, Y. T., Chung, M. C., Guo, S. M., Shieh, L. S., and Wang, L., “A new generalized optimal linear quadratic tracker with universal applications-Part 1: Continuous-time systems,” International Journal of Systems Science, 2015 (under review process).
[5] Emami-Naeini, A. and Dooren, P. V., “Computation of zeros of linear multivariable systems,” International Federation of Automatic Control, vol. 18, no. 4, pp. 425-430, 1982.
[6] Ferreira, P. G., “On degenerate systems,” International Journal of Control, vol. 24, no. 4, pp. 585-588, 1976.
[7] Gangsaas, D., Bruce, K. R., Blight, D. J., and Ly, U. L., “Application of modern synthesis to aircraft control: Three case studies,” IEEE Transactions on Automatic Control, vol. 31, no. 11, pp. 995-1014, 1986.
[8] Gupta, N. K., “Frequency-shaped cost functionals: Extension of linear quadratic Gaussian design methods,” Journal of Guidance and Control, vol. 3, no. 6, pp. 529-535, 1980.
[9] Huang, Y. J. and Wang, Y. J., “Robust PID controller design for non-minimum phase time delay systems,” ISA Transactions, vol. 40, no. 1, pp 31-39, 2001.
[10] Latawiec, K., Banka S., and Tokarzewski, J., “Control zeros and non-minimum phase LTI MIMO systems,” Annual reviews in Control, vol. 24, no. 1, pp. 105-112, 2000.
[11] Lewis, F. L., Applied Optimal Control and Estimation, Englewood Cliffs: NJ, Prentice-Hall, 1992.
[12] Lewis, F. L., Syrmos, V. L., Optimal Control. NJ, John Wiley & Sons, 1995.
[13] Madsen, J. M., Shieh, L. S. and Guo, S. M., “State-space digital PID controller design for multivariable analog systems with multiple time delays,” Asia Journal of Control, vol. 8, no. 2, pp. 161-173, 2006.
[14] Moore, J. B., Glover, K., and Telford, A., “All stabilizing controllers as frequency shaped state estimate feedback,” IEEE Transactions on Automatic Control, vol. 35, no. 2, pp. 203-208, 1990.
[15] Skogestad, S. and Postlethwaite, I., Multivariable Feedback Control: Analysis and Design, NY, John Wiley & Sons, 2005.
[16] Tanaka, R., Shibasaki, H., Ogawa, H., Murakami, T., and Ishida, Y., “Controller design approach based on linear programming,” ISA Transactions, vol. 52, no. 6, pp. 744-751, 2013.
[17] Thompson, C. M., Coleman, E. E., and Blight, J. D., “Integral LQG controller design for a fighter aircraft,” A.I.A.A. Navigation and Control Conference, pp. 866-895, 1987.
[18] Tsai, J. S. H., Lin, J. Y., Shieh, L. S., Chandra, J., and Guo, S.M., “Self-tuning fault-tolerant digital PID controller for MIMO analog systems with partial actuator and system component failures,” IMA Journal of Mathematical Control and Information, vol. 25, no. 2, pp. 221-238, 2008.
[19] Tsai, J. S. H., Du, Y. Y., Zhuang, W. Z., Guo, S. M., Chen, C. W., and Shieh, L. S., “Optimal anti-windup digital redesign of MIMO control systems under input constraints,” IET Control Theory & Applications, vol. 5, no.3, pp. 447-464, 2011.
[20] Tsai, J. S. H., Hsu, W. T., Lin, L. G., Guo, S. M., and Tan, J. W., “A modified NARMAX model-based self-tuner with fault tolerance for unknown nonlinear stochastic hybrid systems with an input-output direct feed-through term,” ISA Transactions, vol. 53, no. 1, pp. 56-75, 2014.
[21] Tsai, J. S. H., Ebrahimzadeh, F., Lai, S. Y., Guo, S. M., Shieh, L. S., and Wang L., “A new proportional-plus-integral optimal linear quadratic state-estimate tracker-Part 1: Continuous-time non-minimum phase systems,” (To be submitted) ,
[22] Tsai, J. S. H., Hsu, W. T., Tsai, T. J., Lin, K., Guo, S. M., and Shieh, L. S., “Realization of causal current output-based optimal full/reduced-order observer and tracker for the linear sampled-data system with a direct transmission term,” Optimal Control Application and Methods; Online publication, Jul. 2014: http://onlinelibrary. wiley.com/doi/10.1002/oca.2125/abstract
[23] Wang, Y. J., “Determination of all feasible robust PID controllers for open-loop unstable plus time delay processes with gain margin and phase margin specifications,” ISA Transactions, vol. 53, no.2, pp. 628-646, 2014.
[24] Zhang, Y., Shieh, L. S., Liu, C. R., and Guo, S. M., “Digital PID controller design for multivariable analogue systems with computational input-delay,” IMA Journal of Mathematical Control and Information, vol. 21, no. 4, pp. 433-456, 2004.
[25] Zhou, H. Q., Shieh, L. S., Liu, C. R., and Wang, Q. G., “State-space digital PI controller design for linear stochastic multivariable systems with input delays,” The Canadian Journal of Chemical Engineering, vol. 84, no. 2, pp. 230-238, 2006.