二軸球平台系統軌跡追蹤之積分式順滑模態控制器設計

Abstract

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。

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.