二軸球平台系統軌跡追蹤之積分式順滑模態控制器設計

Integral Sliding Mode Controller Design for Trajectory Tracking of Two-axis Ball-on-Plate System

陳銘甫(Chen, Ming Fu)

指導教授 ： 陳永平

國立交通大學/電機學院/電機學院電機與控制學程/碩士(2015年)

https://doi.org/10.6842/NCTU.2015.00056

摘要

本論文研製應用於二軸球平台之控制器設計，首先以拉格朗日方程式推導出系統之非線性動態方程式，包括兩條內部方程式及兩條外部方程式，接著針對內部方程式於平衡點附近作線性化處理，再使用李亞普諾夫函式的穩定理論設計積分型順滑模態法則，對含有扭力輸入的外部方程式進行線性化，得出非時變之線性狀態方程式，再利用極點配置法完成軌跡追蹤之控制器。為了探討對球質量變動之強健性，本論文除了利用積分型順滑模態法則之外，也設計傳統線性控制器，並以三種數值模擬情況來說明積分型順滑模態法則之可行性。前兩種分別對傳統線性控制法則及積分型順滑模態法則進行模擬，由結果可知只有積分型順滑模態法則具有對球質量變動之強健性，但存在高頻的切跳現象，在第三種模擬情況中，選定不同的順滑層厚度進行模擬，可知增加厚度可有效消除切跳現象。

關鍵字

積分型順滑模態控制 ； 極點配置法 ； 李亞普諾夫函式穩定理論

並列摘要

This thesis proposes the controller design of Two-axis Ball-on-Plate System whose dynamic equations derived by the Lagrange’s method are nonlinear and consist of two internal and two external equations. The internal equations are linearized around equilibrium point, while the external equations are linearized by the integral sliding-mode algorithm based on the stability theory of Lyapunov function. The total system is finally changed into a linear time-invariant system with an input related to the integrand of the integral sliding function. Further, design the trajectory tracking control via the pole-placement method and illustrate the feasibility of the proposed controller by numerical simulation on three cases. The first two cases present the performance obtained by the traditional linear state-feedback control and the proposed controller under the mass uncertainties. It is easy to check that only the proposed control using integral sliding-mode algorithm is robust to the uncertainties; however, there exists high-frequency chatting in control torque. To improve the performance, different sliding layer thicknesses are selected in the third case which demonstrate that the chattering phenomenon is indeed ameliorated as the sliding layer thickness increases.

並列關鍵字

Integral sliding mode algorithm ； pole placement method ； Lyapunov stability theory

參考文獻

[2] Zeeshan, A., Nauman, N., Khan, M.J., “Design, Control and Implementation of a Ball on Plate Balancing System,” Applied Sciences and Technology (IBCAST), 2012 9th International Bhurban Conference, pp. 22-26, 2012.

[3] Huida duan, Yantao Tian , Guangbin Wang, “Trajectory tracking Controller design of high-order Ball and Plate System based on Improved Auto-disturbance Rejection Controller,” Systems and Control in Aeronautics and Astronautics (ISSCAA), 2010 3rd International Symposium, pp. 80-85, 2010.

[4] Moarref, M., Saadat, M., Vossoughi, G. “Mechatronic design and position control of a novel ball and plate system,” Control and Automation, 2008 16th Mediterranean Conference, pp. 1071-1076, 2008.

[5] Wang Hongrui, Tian Yantao, Fu Siyan, Sui Zhen, “Nonlinear Control for Output Regulation of Ball and Plate System,” Control Conference, 2008. CCC 2008. 27th, pp. 382-387, Chinese, 2008.

[6] Yuan, D., Zhang, Z., “Modeling and control scheme of the ball plate trajectory-tracking pneumatic system with a touch screen and a rotary cylinder,” Control Theory & Applications, IET, pp. 573-589, 2010.