線性二次式(LQ)最佳化是一個常運用於控制系統中的方法，線性二次式可以在系統的轉移函數不知道的情況下，使用開路的測試數據進行最佳化設計，使設計更為簡單。另一方面，一個非零的輸入權重，會造成輸出的誤差，為了要達到完美指令跟隨，線性二次式控制設計必須把輸入權重拿掉。然而在拿掉輸入權重後，計算數據化最佳化設計(DBLQ)時，會造成資訊矩陣接近奇異，使運算結果有很大的誤差，對於這個問題，本研究使用超高精確度解決數值困難的問題，超高精確度提供更高的精確度和更大的運算空間。

The linear quadratic (LQ) optimization is a known approach for control system synthesis. In addition, a LQ control design can also be conducted based sorely on the open-loop plant test data, when a plant dynamic model is not explicitly known. On the other hand, the presence of a nonzero penalty on the control input causes an error to appear in the closed-loop output. In order to achieve a perfect command following operation, a LQ control design must be performed without penalizing the control input. However, the removal of the penalty on the control input also brings the information matrix of the data-based LQ (DBLQ) control design close to singular. In this work, the numerical difficulty of such a DBLQ computation is resolved by developing an ultra-precision (UP) arithmetic package and by conducting DBLQ computations using the UP package.

[1] Chan, J. T., 1996,”Optimal Output Feedback Regulator-A Numerical Synthesis Approach for Input-Output Data”, ASME/JDSMC, Vol.118.
[2] Chan, J. T., 1996,”Data-Based Synthesis of Multivariable LQ Regulator”, Automatica, Vol. 32, No. 3.
[3] Chan, J. T.,”Output Feedback Realization of State Systems”, The 39th IEEE Conference on Decision and Control, Sydney, Australia, December, 2000. Proceedings Vol.1, pp.3670-3675
[4] Chan, J. T., Tsair-Ren Tang. , 2002,” Singular LQ singal tracker – a data-based synthesis approach”,The 15th IFAC World Congress, Barcelona, Spain, July.
[5] Kelly, H.J., 1964, ”A transformation Approach to Singular in Optimal Trajectory and Control Problems”, SIAM,J. of Control, vol.2,234-240.
[6] Goh, B.S. ,1967,” Optimal Singular Control for Multi-unput Linear Systems”, J. of Mathematical Analysis and Applications, 20, 534-539
[7] Speyer, J.L., and D.H. Jacobson, 1971,”Necessary and sufficient Conditions of Optimality for Singular Control Problem; A transformation Approach”, J. of Mathematical Analysis and Applications, 33, 163-187.
[8] Jacobson, D.H., 1971,”Totally Singular Quadratic Minimization Problem”, IEEE tran.Auto. Conte., AC-16,Dec..
[9] Moore, J.B, and P.J. Moylan, 1971,” Generalizations of singular Optimal Control Theory”, Automatica, vol. 7,591-598
[10] Gene F. Franklin and J. David Powell,”Digital Control of Dynamic Systems”, Addison-Wesley publishing company, California, 1980.
[11] ANSI/IEEE 754-1985. ”American National Standard- IEEE Standard for Binary Floating-Point Arithmetic”. American National Standards Institute, Inc., New York, 1985.
[12] Dorf, R.c. ,1980, Modern control system, .Edition, Addison Wesley, New York,N.Y..
[13] 唐才仁，基於取樣數據之最佳化追蹤器設計-工程驗證之壹，國立成功大學航空工程研究所碩士論文,1996
[14] 陳政雄，數據化控制設計之效能評估(I):與傳統控制設計之比較，國立成功大學航空工程研究所碩士論文,2004
[15] 魏榮良，基於系統逆動態之數據化最佳控制設計，國立成功大學航空工程研究所碩士論文,2005