車與桿倒單擺系統(car-pole inverted pendulum system)是一個不穩定的系統，同時具有非極小相(non-minimum phase)之特性，其機械結構並不複雜，故適用於驗證各種控制理論。
本論文中首先介紹如何建立車與桿倒單擺系統。本系統主要包括了電樞控制直流馬達(armature controlled DC motor)、車子、單擺、感測器使用光學式旋轉型編碼器。然後推導出車與桿倒單擺系統機械部份數學模型，並且建立直流馬達模型，利用識別方法求出馬達參數。在控制器部份，吾人使用了極點配置法與LQR控制法則模擬，比較其結果，LQR控制法則較極點配置法之性能為佳。故採用LQR控制器，利用運動控制軸卡來實現LQR控制器，控制介面部份是利用Visual C++撰寫，利於控制器參數調設。另外討論車與桿倒單擺系統在斜坡上之平衡情形，平衡控制器亦以LQR控制器實現。

The car-pole inverted pendulum system is an unstable and non-minimum phase system. The mechanism of this system is not complicated so that this system provides a platform for verifying the effectiveness of different control schemes.
In this thesis, it is first shown how to build up a car-pole inverted pendulum system. This system includes an armature controlled DC motor, a car, a pendulum, and an optical rotary encoder. Then the mathematical model of the car-pole inverted pendulum system is derived and also the DC motor model is built. The system identification method is then used to determine DC motor parameters. For the control schemes, the pole-placement method and the LQR control law are considered to design the state feedback controller. By comparing the simulation results, performance of the LQR control law is better than that of the pole-placement method. Therefore, the LQR control law is used in the balance control of this system. The control law is implemented through a motion control card. A control panel interface is coded in Visual C++ for controller parameter setting. Moreover balance control of the car-pole inverted system on an incline is studied. The balance controller also uses the LQR control law.

[1] C. E. Lin and Y. R. Sheu, “A Hybrid-Control Approach for Pendulum-Car Control,” IEEE Trans. on Industrial Electronics, Vol. 39, No. 3, pp. 208-214, 1992.
[2] I. I. Kim and J. H. Lee, “A New Approach to Adaptive Membership Function for Fuzzy Interface System,” Knowledge-Based Intelligent Information Engineering Systems, Third International Conference, pp. 112-116, 1999.
[3] K. Furuta, “Design of Variable Structure Controllers,” IEEE Proc. of the 39th Conf. on Decision and Control, pp. 1685-1690, 2000.
[4] M. Widjaja and S. Yurkovich, “Intelligent Control for Swing Up and Balancing of an Inverted Pendulum System,” Proc. of the 4th IEEE Conf. on Control Appl., pp. 534-542, 1995.
[5] S. Kawaji and K. Ogasawara, “Swing Up Control of a Pendulum using Genetic Algorithms,” Proc. of the 33rd IEEE Conf. on Decision and Control, pp. 3530-3532, 1994.
[6] S. U. Cheang and W. J. Chen, “Stabilizing Control of an Inverted Pendulum System Based on Loop Shaping Design Procedure,” IEEE APEC, pp. 272-280, 1997.
[7] S. J. Huang and C. L. Huang, “Control of an Inverted Pendulum Using Gray Prediction Model,” IEEE Trans. on Industry Applications, Vol. 36, No. 2, pp. 452-458, 2000.
[8] H. Osinga and J. Hauser, “On Geometry of Optimal Control: the Inverted Pendulum Example,” Proc. of American Control Conference, pp. 25-27, 2001.
[9] J. L. Lin and W. J. Yang, “Modified IVSC Stabilization and Swing Up Control of an Inverted Pendulum,” Automatic Control Conference of Taiwan R.O.C., pp.150-153, 1997.
[10] 辛俊光，「永磁式直流有刷馬達之參數自動鑑別系統」，國立成功大學航空太空工程學系碩士論文，民國八十六年六月。
[11] B. C. Kuo, Automatic Control Systems, Prentice-Hall International Editions, Upper Saddle River, N.J., 1997.