代數鑑別法的優點在於能消除未知初始狀態及負載擾動對程序鑑別的影響，其原理為透過對模型方程式的微分，將這些未知項轉換成狄拉克函數形式，再選擇合適的候選函數來去除所衍生的狄拉克函數，例如指數或多項式候選函數。此外，代數鑑別法適用於常見的輸出入訊號，包括步階、矩形脈波及多重脈波響應。然而，現有的代數鑑別法也存在一些缺點，例如模型階次無法判定，鑑別模型的正確性對雜訊相當敏感，和未知擾動須在鑑別開始前發生。
本論文提出基於多重脈波響應的改良式代數鑑別法，在鑑別計算過程中，將多重脈波響應分解成多個獨立的步階響應，並使用指數候選函數來消除各自的未知初始狀態及擾動項。針對模型階次未知的問題，提取各個步階響應發生於時延後的數據來建立鑑別矩陣，然後利用該矩陣的奇點特性來估測出正確或適合的模型階次。對於雜訊敏感問題，本文發現適當地選擇指數候選函數內的可調參數可有效降低雜訊的影響，因此發展出基於多項式候選函數的鑑別誤差指標，作為選擇該可調參數及鑑別模型時延的標準。當負載擾動發生時間未知時，可將分解的步階響應逐一移除後用於鑑別計算，找出並移除經歷擾動變化的步階響應後即可獲得正確模型。
本鑑別法可提供PID控制器設計所需的模型，本文接著針對各種一階及二階時延模型提出基於內模控制或史密斯預測器架構的雙自由度PID控制器設計法，首先選擇適當的內模控制濾波器來設計具最佳擾動排除性能的PID控制器，然後加入設定點濾波器來確保良好的設定點追蹤性能。模擬結果發現本控制器設計法可處理擾動模型與程序模型不同的情況，並且史密斯預測器架構適合具大時延/時間常數比的程序採用。最後將所提方法實際應用於多變數活性污泥程序的鑑別與控制上，發現不僅可鑑別出極佳的線性模型，而且由此產生的雙環路PID控制器設計，在設定點追蹤和擾動排除性能方面，皆優於現有的PID控制器設計。

The advantage of the algebraic identification method is that it can eliminate the influence of unknown initial states and load disturbances on process identification. However, the available algebraic identification methods also have some shortcomings. For example, the model order cannot be determined, the accuracy of the identified model is quite sensitive to noise, and the unknown disturbance must occur before the identification starts.
In this thesis, the multiple-pulse response is decomposed into multiple independent step responses, and exponential candidate functions are used to eliminate the respective terms for unknown initial states and disturbances. For the problem of unknown model order, the identification matrix is established, and then the correct or appropriate model order is estimated by using the singularity of the matrix. For the problem of noise sensitivity, it is found that proper selection of the adjustable parameter in the exponential candidate function can effectively reduce the effect of noise, so an identification error index is developed as a criterion. Furthermore, when the occurrence time of the load disturbance is unknown, the correct model can be obtained after finding and removing the step response undergoing disturbance changes.
The thesis then proposes a two-degree-of-freedom (2DOF) PID controller design method based on the internal model control (IMC) or Smith predictor structures. First, an appropriate internal model control filter is selected to design a PID controller with the best disturbance rejection performance, and then a set-point filter is added to ensure good set-point tracking performance. Simulation results show that the controller design method can deal with the situation where the disturbance model is different from the process model, and the Smith predictor structure is suited for processes with a large delay/time constant ratio. Finally, the proposed methods are applied to the identification and control of a practical multivariable activated sludge process.

1. Tangirala, A., Principles of System Identification: Theory and Practice. Boca Raton: CRC Press, 2018
2. Keesman, K., System Identification. An Introduction. Springer London, 2011
3. Isermann, R. and M. Münchhof. "Identification of dynamic systems: an introduction with applications". Springer Berlin, Heidelberg. 2010.
4. Allen, G.I., L. Grosenick, and J. Taylor, "A generalized least-square matrix decomposition," Journal of the American Statistical Association, 109(505): p. 145-159, 2014.
5. Baysse, A., F.J. Carrillo, and A. Habbadi, "Least squares and output error identification algorithms for continuous time systems with unknown time delay operating in open or closed loop," IFAC Proceedings Volumes, 45(16): p. 149-154, 2012.
6. Hwang, S.-H. and S.-T. Lai, "Use of two-stage least-squares algorithms for identification of continuous systems with time delay based on pulse responses," Automatica, 40(9): p. 1561-1568, 2004.
7. Liu, M., Q.-G. Wang, B. Huang, and C.C. Hang, "Improved identification of continuous-time delay processes from piecewise step tests," Journal of Process Control, 17(1): p. 51-57, 2007.
8. Belkoura, L., J.-P. Richard, and M. Fliess. "A convolution approach for delay systems identification". in 17th IFAC World Congress. Séoul, South Korea. 2008.
9. Ibn Taarit, K., L. Belkoura, M. Ksouri, and J.-P. Richard, "A fast identification algorithm for systems with delayed inputs," International Journal of Systems Science, 42(3): p. 449-456, 2011.
10. Belkoura, L. and Y. Tian, "A convolution approach for parameters and delay systems identification," Journal of Control Engineering and Applied Informatics, 2016.
11. Mhamdi, A., K. Ibn Taarit, and M. Ksouri, "Online algebraic identification approach of continuous linear time delay systems," Asian Journal of Control, 20(1): p. 343-355, 2018.
12. 萬廸翔, "基於常見響應和多項式候選函數之連續時延系統代數鑑別法," 碩士論文, 化學工程學系, 國立成功大學, 2018.
13. 簡昀平, "基於開環或閉環多重脈波響應之代數鑑別法," 碩士論文, 化學工程學系, 國立成功大學, 2020.
14. Ziegler, J.G. and N.B. Nichols, "Optimum settings for automatic controllers," Trans. ASME, 64(11): p. 220-222, 759-768, 1942.
15. Francis, B. and W. Wonham, "The internal model principle of control theory," Automatica, 12: p. 457-465, 1976.
16. Smith, O.J.M., Feedback Control Systems. McGraw-Hill, 1958
17. Lee, Y., S. Park, M. Lee, and C. Brosilow, "PID controller tuning for desired closed-loop responses for SI/SO systems," AIChE Journal, 44(1): p. 106-115, 1998.
18. Shamsuzzoha, M. and M. Lee, "IMC−PID controller design for improved disturbance rejection of time-delayed processes," Industrial & Engineering Chemistry Research, 46(7): p. 2077-2091, 2007.
19. Zhang, Y., A. Kumar, and M.-S. Chiu, "VRFT-based predictor design for processes with inverse response," Journal of the Taiwan Institute of Chemical Engineers, 130: p. 104-113, 2022.
20. Vilanova, R. and R. Katebi, "2-DoF decoupling controller formulation for set-point following on decentraliced PI/PID MIMO systems," IFAC Proceedings Volumes, 45(3): p. 235-240, 2012.
21. Vilanova, R., R. katebi, and V. Alfaro. "Multi-loop PI-based control strategies for the activated sludge process". in 2009 IEEE Conference on Emerging Technologies & Factory Automation. 2009.
22. Dirac, P.A.M., The Principles of Quantum Mechanics. Vol. 4: Clarendon Press, 1981
23. Schwartz, L., Théorie des Distributions. Hermann, 1966
24. Lee, Y., J. Lee, and S. Park, "PID controller tuning for integrating and unstable processes with time delay," Chemical Engineering Science, 55(17): p. 3481-3493, 2000.
25. Wang, Q.G., T.H. Lee, K.K. Tan, and Q. Bi. "A new approach to analysis and design of Smith-predictor controllers". in Proceedings of 1995 American Control Conference - ACC'95. 1995.
26. Horn, I.G., J.R. Arulandu, C.J. Gombas, J.G. VanAntwerp, and R.D. Braatz, "Improved filter design in internal model control," Industrial & Engineering Chemistry Research, 35(10): p. 3437-3441, 1996.
27. Chen, D. and D.E. Seborg, "PI/PID controller design based on direct synthesis and disturbance rejection," Industrial & Engineering Chemistry Research, 41(19): p. 4807-4822, 2002.
28. Morari, M. and E. Zafiriou, Robust Process Control. Prentice Hall, 1989
29. Kaya, I. "Controller design for integrating processes using user-specified gain and phase margin specifications and two degree-of-freedom IMC structure". in Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003. 2003.
30. Kaya, I., "Controller design for integrating processes with inverse response and dead time based on standard forms," Electrical Engineering, 100(3): p. 2011-2022, 2018.
31. Rao, A.S. and M. Chidambaram, "Analytical design of modified Smith predictor in a two-degrees-of-freedom control scheme for second order unstable processes with time delay," ISA Transactions, 47(4): p. 407-419, 2008.
32. KJ, Å. and T. Hägglund, "PID controllers: theory, design and tuning," Research Triangle Park, Instrument Society of America, 1995.
33. Alcantara, S., C. Pedret, R. Vilanova, P. Balaguer, and A. Ibeas. "Control configuration for inverse response processes". in 2008 16th Mediterranean Conference on Control and Automation. IEEE. 2008.
34. Chien, I.L., Y.C. Chung, B.S. Chen, and C.Y. Chuang, "Simple PID controller tuning method for processes with inverse response plus dead time or large overshoot response plus dead time," Industrial and Engineering Chemistry Research, 42(20): p. 4461-4477, 2003.
35. Alfaro, V., R. Vilanova, and O. Arrieta, Analytical Robust Tuning of PI Controllers for First-Order-Plus-Dead-Time Processes. 2008 International Conference on Emerging Technologies and Factory Automation: IEEE. 273-280, 2008