本論文呈現以平方和方法為基礎改善幾種非線性控制系統之設計。首先應用平方和方法改善顯式模式追隨控制設計，使得非線性系統可直接融入擴增系統中，避免因線性化而喪失系統特性。其次，應用李亞普諾夫第二方法取代舊有線性調變方法設計控制器並確保其穩定性。第二、應用平方和方法改善T-S模糊模型之建模與控制器設計。多數自然系統無法經由線性化模型恰當地被描述，應用多項式來描述T-S模糊模型會比傳統以線性子系統為主之T-S模糊模型更具彈性且精確。因此，本論文基於平方和方法提出一演算法來確保多項式T-S模糊系統之穩定性。此外，一以非線性子控制器為基礎之切換結構用來取代慣用之平行分佈補償方法，其優點在於構造簡單並且易於實現。第三，本論文發展一套系統化設計流程來找尋恰當之李亞普諾夫函數使每一個遞迴步階迴路均穩定。相較於傳統遞迴步階設計中使用之嘗試錯誤法，平方和方法更有系統地與有效率地找尋李亞普諾夫函數。此外，演算法亦考量控制器輸入之飽和條件，使得遞迴步階控制系統同時滿足穩定性、性能需求以及控制輸入限制等。最後，相關的電腦模擬結果均驗證所提出之演算法可有效地達成上述個別之改善。

This dissertation presents several improvements of nonlinear control systems design based on sum of squares (SOS) techniques. Firstly, the SOS technique is employed to improve the explicit model following control design so that the nonlinear model is merged directly in the augmented system to avoid the loss of characteristics of system due to the linearization. Then, the Lyapunov’s second method is applied to substitute the linear quadratic regulator method for controller design and ensure the stability of control system. Secondary, the SOS technique is employed to improve the modeling and controller design of T-S fuzzy model. As the formulation of T-S fuzzy model via polynomials is more flexible and accurate than the conventional linear subsystem based T-S fuzzy model. The dissertation proposes an algorithm to ensure the stability of polynomial T-S fuzzy system based on SOS techniques. Moreover, a nonlinear sub-controller based switching mechanism is proposed to instead the conventional PDC control scheme, which advantage is the simplified architecture and is easy to be carried out. Thirdly, this dissertation develops a systematical design process for the search of Lyapunov function candidate such that each backstepping recursive channel is stable. Compare to the trial–and–error method in conventional backstepping approach, SOS technique is more efficiently and systematically. Furthermore, the input saturation conditions are also addressed in the proposed algorithm. Finally, computer simulation results illustrate that the proposed algorithms successfully validate the desired performances for each improvement.

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