臺灣博碩士論文加值系統

在最近這幾年，對於非線性而且致動不足系統的閉迴路穩定性以及性
能設計受到很高的重視。基於這個理由，將對球與弧系統設計平衡控制
器，因為球與弧系統是屬於非線性而且致動不足。球與弧系統是由一顆
球還有一個有弧度的台車所組成，在這裡我們主要的控制目標是要驅動
台車讓球能停在弧的頂端，而且還要達到讓台車回到原點。
在遞迴步階控制器設計的過程中，直接對原始系統設計控制器是相當
困難的。為了暸解跟分析球與弧系統的特性，一開始先介紹並且研究線
性化系統。接下來根據設計線性控制器得到的資訊來設計遞迴步階控制
器，可以成功地讓球停在弧的頂端而且台車可以回到原點。接下來，提
出適應性控制器用來克服球與弧系統有受到干擾的情形。我們設計出來
的適應性控制器可以達到控制目的，而且由模擬驗證得知當系統有未知
的干擾時，有加適應性控制會有比較好的控制效果。最後，所提出的控
制器應用在球與弧系統有很好的模擬結果。

In the recent years, there has been major interest in designing controllers of
nonlinear and underactuated systems for the closed-loop stability and performance.
Motivated by these facts, this thesis is going to design the balancing controller for the ball
and arc system because this system is both nonlinear and underactuated. The system
consists of a ball and a cart with the arc. The main control objective here is not only to
drive the cart to bring the ball to stay on the top of the arc, but also to let the cart return to
its origin.
In the procedure of nonlinear backstepping design, the direct control design of this
nonlinear system is quite difficult. In order to realize and analyze the properties of the ball
and arc system, the linearized system model is introduced and investigated in the beginning.
Then, according to the information we have obtained from designing the linearized system,
the backstepping design scheme can be successfully developed for the nonlinear system to
bring the ball to stay on the top of the arc and let the cart return to the origin. In addition,
an adaptive backstepping design method is proposed to deal with the disturbance for the
control of this ball and arc system. Our proposed nonlinear and adaptive controllers can
achieve the control objectives and the resulting closed-loop system with proposed adaptive
backstepping controller can have better performance than the system without adaptation.
Finally, some simulation results are given to illustrate the excellent performance of the
proposed controllers applied to the ball and arc system.

Abstract i
Contents ii
List of Tables iv
List of Figures v
1 Introduction 1
2 System Model and Dynamics 4
2.1 Ball and Arc System . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Nonlinear Backstepping Design 10
3.1 Linearized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Nonlinear Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Adaptive Control Design 27
4.1 Adaptive Backstepping Control . . . . . . . . . . . . . . . . . . . . . 28
4.2 Comparative Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Conclusions and Future Works 37
Appendix A 41
Bibliography 44

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