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 倒單擺系統是個典型的控制試驗裝置，從機器人領域的廣泛應用到太空火箭的導引系統，都是為了驗證和實踐各種控制技術。我們介紹的倒單擺類型是能在水平和垂直平面移動的系統，此系統被廣泛的運用在火箭的控制與抗震建築物。此典型的控制試驗裝置，提供許多具有挑戰性的控制設計問題。我們將以非線性李亞普諾夫穩定準則為基礎的遞迴步階控制器設計應用於二維倒單擺系統。主要控制目標是不僅要保持單擺在直立位置，還要使台車達到我們所需的參考坐標。為了瞭解和分析二維倒單擺系統的特性，首先我們先把二維倒單擺系統線性化後再去設計其控制器。根據線性化系統控制的概念去設計這個非線性系統，並且成功的達成我們的控制目的。此外，適應性遞迴步階控制器可以對抗未知的外力干擾以及實現平衡及軌跡追蹤的目的。最後，為了驗證以上理論的推演，會有一些模擬結果來說明控制器在二維倒單擺系統上有很好的表現結果。
 Inverted pendulum systems are classical control test rigs for verification and practice of various control techniques with wide areas of applications from robotics to space rocket guidance systems. In this thesis, the type of inverted pendulum systems moving in the horizontal and vertical plane is introduced and investigated. This system is highly motivated by applications such as the control of rockets and the antiseismic control of buildings. It is well established benchmark problem that provides many challenging issues to control design. The nonlinear Lyapunov-based controllers with backstepping design schemes are developed for the manipulation of the 2-dimensional inverted pendulum system. The main control objective is not only to maintain the pendulum at the upright position, but also to drive the cart to reach the desired reference coordinate.In order to analyze and realize the system properties of the 2-dimensionalinverted pendulum system, in the beginning, the linear controllers are designed for the balancing control of the linearized 2-dimensional inverted pendulum system. According to the information from the linearized case, the nonlinear backstepping controllers can be successfully developed to achieve our control objectives for the original nonlinear system. In addition, an adaptive backstepping design scheme is proposed to cope with unknown disturbance for attaining both balancing and trajectory tracking purpose. Finally, some simulation results are given to illustrate the excellent performance of the proposed controllers applied to a 2-dimensional inverted pendulum system.
 Abstract iContents iiList of Tables iiiList of Figures iv1 Introduction 12 System Model and Dynamics 52.1 2-dimensional Inverted Pendulum System . . . . . . . . . . . . . . . . 52.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Nonlinear Backstepping Design 103.1 Linearized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Nonlinear Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Adaptive Backstepping Control 314.1 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Conclusions and Future Works 43Appendix A 46Bibliography 49