本論文主要是對於含有狀態時間延遲和輸入時間延遲的資料採樣系統，提出兩種數位重新設計之控制器。一種為以脈幅調變和脈寬調變法則設計之最佳化追蹤器，另一種為以觀察器為基礎之次最佳化追蹤器。我們將先直接對連續系統的最佳控制性能指標做離散化。同時建構一個新的擴增狀態向量，其包含原本系統狀態時延和輸入時延變數，去得到一個不含狀態時延和輸入時延的新擴增系統。然後藉由求解離散化後的二次式最佳控制性能指標，使得數位控制的閉迴路系統其輸出可以密切地吻合參考輸入。在完成上述設計之後，我們再將這個新的數位重新設計之脈幅調變控制器轉換成等效的脈寬調變控制器。第二種方法是當假若系統狀態是無法量得時，我們將利用新建構的擴增系統（其系統矩陣為奇異矩陣），設計一個以觀察器為基礎之次最佳化追蹤器。

The thesis explores two important subjects in control system. One is optimal digital redesigns of PAM and PWM controllers for sampled-data system with state and input delays, the other is a new observer-based suboptimal tracker for the sampled-data time-delay system. We directly convert the continuous-time optimal tracking formulation into the discretized form. And an extended state vector involving all state and input time-delay variables of the original system is constructed to obtain an extended new system without state and input delays. Then, by solving the discretized quadratic optimal tracking formulation makes the output of the digitally controlled closed-loop system to closely match the reference signal. After completing the design, we translate the newly developed digital-redesign pulse-amplitude modulated (PAM) controller into the equivalent pulse-width modulated (PWM) controller. On the other hand, if the states are unmeasured, we utilize the extended system, in which the system matrix is singular, to design an observer-based suboptimal tracker. We also employ applications of the matrix
sign function and model conversion method to solve the difficulty of the observer design.

[1] Anderson, B. D. O. and J. B. Moore, Optimal Control Linear Quadratic Methods, Prentice Hall, Englewood Cliffs, New Jersey, 1989.
[2] Astrom, K. J. and B. Wittenmark, Computer controlled systems, Prentice-Hall, Upper Saddle River, N.J., 1997.
[3] Bateman, H., “The control of an elastic fluid”, Bull. Am. Math. Soc., vol. 51, pp. 601-646, 1964.
[4] Callender, A., D. R. Hartree and A. Porter, “Time lag in a control system”, Transactions R. Soc., Series A, vol. 235, pp. 415-444, 1935 and
Op. Cit., vol. 161, pp. 460-476, 1937.
[5] Chosky, N. H., “Time-lag systems- a bibliography”, IRE. Transactions Automatic Control, vol. AC-5, pp. 66-70, 1960.
[6] Chen, T. and B. A. Francis, Optimal Sampled-Data Control Systems, Spring-Verlag, New York, 1995.
[7] Chu, Jian, “Application of a discrete optimal tracking controller to an industrial electric heater with pure delays”, J. Proc. Cont. vol. 5. no. 1, pp. 3-8, 1995.
[8] Denman, E. D. and A. N. Beavers, “The matrix sign function and computations in systems”, Application Math. Comput., vol. 2, pp. 63-94, 1976.
[9] Fujimoto, H., A. Kawamura and M. Tomizuka, “Generalized digital redesign method for linear feedback system based on n-delay control”, IEEE/ASME Trans. Mechaton,vol. 4, pp. 101-109, June 1999.
[10] Fujimoto, H., J. Hori and A. Kawamura, “Perfect tracking control based on multi-rate feedback control with generalized sampling periods”, IEEE Trans. On Industrial Electronics, vol. 48, no. 3, pp. 636-644, 2001.
[11] Guo, S. M., W. Wang and L. S. Shieh, “Discretisation of two degree-of-freedom controller and system with state, input and output delays”, IEE Proc.-Control Theory and Applications, vol.147, no. 1, pp. 87-96, January
2000.
[12] Guo, S. M., L. S. Shieh, G. Chen and C. F. Lin, “Effective chaotic orbit tracker: a prediction-based digital redesign approach”, IEEE Transaction on Circuits and Systems – I, Fundamental Theory and Applications, vol. 47,
no. 11, pp. 1557-1570, Nov. 2000.
[13] Ieko, T., Y. Ochi and K. Kanai, “Digital redesign of linear state-feedback law via principle of equivalent areas”, Journal of Guidance, Control, and Dynamics, vol. 24, pp. 857-859, 2001.
[14] Kuo, B. C., Digital control systems, Holt, Rinehart and Winston, New York, 1980.
[15] Lewis, F. L. and V. L. Syrmos, Optimal control, John Wiley, NY, 1986.
[16] Malek-Zavarei, M. and M. Jamshidi, Time-delay systems, North-Holland, Amsterdam, 1987.
[17] Ogata, Katsuhiko, Discrete-time control systems, Prentice-Hall, Englewood Cliffs, N.J., 1987.
[18] Rafee, N., T. Chen and O. P. Malik, “A technique for optimal digital redesign of analog controllers”, IEEE Trans. Control Systems Technology, vol. 5, no. 1, pp. 89-99, 1997.
[19] Roberts, J. D., “Linear model reduction and solution of the algebraic Riccati equation by use of the sign function”, Int. J. Control, vol. 32,
pp. 677-687, 1980.
[20] Shieh, L. S., W. M. Wang and M. K. Appu Panicker, “Design of PAM and PWM digital controllers for cascaded analog systems”, ISA Transactions, vol. 37, Issue: 3, pp. 201-213, July 1998.
[21] Shieh, L. S., I. C. Lin and J. S. H. Tsai, “Design of PWM controller for sampled-data system using digitally redesigned PAM controller”, IEE Proc.-Control Theory and Applications, vol.142, no. 6, pp. 87-96, November 1995.
[22] Shieh, L. S., G. Chen and J. S. H. Tsai, “Hybrid suboptimal control of multi-rate multi-loop sampled-data systems”, Int. J. Systems and Science, vol. 23, pp. 839-854,1992.
[23] Shieh, L. S., H. M. Did and S. Ganesan, “Linear quadratic regulators with eigenvalues placement in a specific region”, Automatica, vol. 24, pp. 819-823, 1988.
[24] Shieh, L. S., X. M. Zhao and J. L. Zhang, “Locally optimal digital redesign of continuous-time systems”, IEEE Transactions Industrial
Electronics, vol. 36, pp. 511, 1989.
[25] Shieh, L. S., Y. T. Tsay and R. E. Yates, “Some properties of matrix sign functions derived from continued fractions”, Proc. IEE-D, vol. 130, pp. 111-118, 1983.
[26] Shieh, L. S., Y. T. Tsay and C. T. Wang, “Matrix sector functions and their applications to system theory”, Proc. IEE-D, vol. 131, pp. 171-181, 1984.
[27] Shieh, L. S., J. S. H. Tsai and R. E. Yates, “The generalized matrix sector function and the separation of matrix eigenvalues”, IMA Jl math.
Control Inf., vol. 2, pp. 251-258, 1985.
[28] Shieh, L. S., X. M. Zhao and J. W. Sunkel, “Hybrid state-space self-tuning control using dual-rate sampling”, Proc. IEE-D , Control Theory Application, vol. 138, pp.50-58, 1991.
[29] Tsai, J. S. H., L. S. Shieh and R. E. Yates, “Fast and stable algorithms for computingthe principal nth root of a complex matrix and the matrix sector function”, Comput. Math. Application, vol. 15, No. 11, pp. 903-913, 1988.
[30] Tsay, Y. T., L. S. Shieh and J. S. H. Tsai, “A fast method for computing the principal nth root of complex matrices”, Linear Algebra Application, vol.76, pp. 205-221, 1986.
[31] Weiss, R., “Transporting lag- an annotated bibliography”, IRE. Transactions Automatic Control, vol. 4, pp. 56-68, 1959.