大部分的非線性系統，含有參數不確定與干擾問題，此易造成系統輸出惡化、性能變差，也可能造成系統不穩定。為解決此問題，本文提出了H_∞模糊滑動的綜合控制法。
在本文之綜合控制器中的模糊控制項，主要由本文以滑動面及滑動面的導數為輸入之模糊規則表中所得。而滑動控制項，依照Lyapunov穩定理論來確保系統的漸近穩定，並配合前述之模糊控制項來滿足系統預設規格。不用傳統的試誤法，在本文中的滑動控制參數，係由將系統的誤差方程式轉化為H_∞標準控制問題，而由求解黎卡迪方程式(Riccati equation)而得。因此，本文控制器之控制增益可將外界輸入(例如系統的干擾或不確定性)到系統的受控輸出(追蹤誤差、控制能量)之轉移函數的H_∞-norm壓縮到最低。進而為移除滑動控制項的Chattering現象，本文將控制器中sgn函數轉換成飽和函數(sat)。此後，再利用本文所提出之不等式來確保含有未模式化不確定系統的穩定性。
最後本文使用含有不確定項跟干擾的機械手臂來做電腦模擬，並追蹤預設之路徑，電腦模擬結果顯示，此控制器已被最佳化處理，而且系統的預設性能皆能圓滿達成。

Most of nonlinear systems encounter the problems of system parameter variation, plant uncertainties and disturbances, which are likely to cause deterioration of the output response and degrade system performance as well as even result in system instability. For solving this problem, the H_∞ fuzzy sliding-mode controller is proposed in this research. The Fuzzy control term in the proposed composite controller is mainly determined by the sliding surface and its derivative by following the fuzzy control rules listed in the attached table. The sliding-mode control term is constructed in the proposed controller to ensure the system asymptotically stability in the sense of Lyapunov while keeping the system response be well shaped to match the desired specifications by the previous fuzzy control term. Instead of the traditional trial and error process, the control parameters of the designed sliding surfaces in this paper are optimally obtained by solving Algebraic Riccati Equation in formulating the system error equation into the standard H_∞-optimal control problem. Thus, the control gains in this paper are optimally selected to minimize the ill-effect of the exogenous inputs (e.g. disturbance and plant uncertainties) on the controlled output (e.g. tracking error and energy cost). Moreover, to remove the undesirable chattering in the sliding-mode component, the sign function is replaced by the saturation function. Moreover, the un-modeled dynamics remained in the closed-loop system is then checked by the inequality proposed in this paper to guarantee the system absolute stability. Finally this paper uses a submarine with plant uncertainties which is controlled and simulated by the proposed composite controller to track a pre-specified desired path.

[1] Babaoglu, O.K. “Designing an automatic control system for a submarine,” M.S. thesis, Naval Postgraduate School, Monterey, California, 1988.

[2] C.N. Hwang, “Tracking of controllers for robot manipulators,” Master Dissertation, Michigan state University, 1986.

[3] C.N. Hwang, “Design of Robust Controllers for Manipulators,” Journal of the National Cheng-Kung University, vol.26. , Sci. Eng. & Med. Section, pp. 213-234, 1991.

[4] C.N. Hwang, “Synthesis Procedure for Nonlinear Systems,” Proc. Natl. Sci. Counc. ROC(A) Vol. 17 ,No. 4,. pp. 279-294 , 1993.

[5] C.N. Hwang, “Formulation of H2 and H∞ Optimal Control Problems – A Variational Approach,” Journal of the Chinese Institute of Engineering’s, Vol. 16, No. 6, pp.853-866, 1993.

[6] C.N. Hwang, “A Variational Approach to H_2 and H_∞ Optimal Control Problems for Linear Nonautonomous System,” Proc. Natl. Sci. Counc. ROC(A) vol. 17, NO. 6, pp. 853-866, 1993.

[7] Chung-Chun Kung and Chia-Chang Liao “Fuzzy-Sliding Mode Controller Design For Tracking Control Of Non-Linear System,” Proceadings of the American Control Conference Baltimore, Maryland, vol. 1, pp. 180-184, July 1994.

[8] C.S. Shieh and Ming-Rong Lee, “A study of a robust fuzzy controller for perturbed nonlinear systems,” Circuit System Signal Processing, Vol. 25, No. 6,,pp. 715-727, 2006.

[9] Doyle, J.C., Glover, K. Khargonekar, P.P. and Francis, B.A., “State-Space Solutions to Standard H2 and H∞ Control Problems,” IEEE Transactions on Automatic Control, Vol. 34, No.8, pp.831-847, 1989.

[10] D. Dumlu and Y. Istefanopulos, “Design of an adaptive controller for submersibles via multimodel gain scheduling,” Ocean Engng, vol. 22, No. 6, pp. 593-614, 1995.

[11] Jun Yang ; Shihua Li ; Xinghuo Yu, “Sliding-mode control for systems with mismatched uncertainties via a disturbance observer,” IEEE Transactions on electronics, Vol. 60, NO. 1, pp. 160-169, January 2013.

[12] J. Shi, H. Liu and N. Bajcinca, “Robust control of robotic manipulators based on integral sliding mode,” International Journal of Control
Vol. 81, No. 10, pp. 1537–1548, October 2008.

[13] Mamdani, E.H., “Application of fuzzy algorithms for control of
simple dynamic plant ,” PROC. IEE, Vol. 121, No. 12, pp. 1585-1588, 1974.

[14] R. J. Richards, and D. P. Stoten, “Depth Control of A Submersible Vehicle,” Int. Shipbuilding Progress, Vol. 28, pp. 30-39, Feb. 1981.

[15] Utkin, V.I., “Sliding Modes in Control and Optimization,” London, UK: Springer-Verlag. (1992)

[16] Utkin, V.I., Jingxin Shi. “Integral Sliding Mode in Systems Operating under Uncertainty Conditions,” Proceedings of the 35th IEEE Conference on Decision and Control, vol. 4, pp. 4591-4596, 1996.

[17] Utkin, V.I. and Hao-Chi Chang “Sliding Mode Control on Electro-Mechanical Systems,” Mathematical Problems in Engineering, Vol. 8(4-5), pp. 451-473, 2002.

[18] Zames, G., Francis, B.A., “A new approach to classical frequency methods : Feedback and minimax sensitivity,” 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, pp. 867-874, 1981.

[19] Zadeh, L. A.,“Fuzzy sets,” Information Contr., vol. 8, pp. 338-353, 1965.
[20] 楊憲東、葉芳柏，“線性與非線性H_∞控制理論”，全華科技圖書股份有限公司，1997。

[21] 楊憲東，“非線性控制講義”，國立成功大學航太系，2011。

[22] 李祖聖，“高等模糊控制講義”， 國立成功大學電機系。

[23] 李允中、王小璠、蘇木春，“模糊理論及其應用”，全華科技圖書股份有限公司，2003。

[24] 孫宗瀛、楊英魁，“Fuzzy 控制：理論、實作與應用”，全華科技圖書股份有限公司，2005。

[25] 蘇義閎、黃正能，“H_∞-模糊滑動複合控制設計與其在水下載具上之應用”，國立成功大學系統及船舶機電工程研究所碩士論文，2010。

[26] 彭宇煊、黃正能，“機械臂之H_∞-滑動控制設計”， 國立成功大學系統及船舶機電工程研究所碩士論文，2013。