弦系統基本特性與適應性邊界控制

The Fundamental Characteristics and Adaptive Boundary Control of the String System

呂柏逸(Pai-Yat Lu)

指導教授 ： 黃健生

中原大學/工學院/機械工程研究所/博士(2002年)

https://doi.org/10.6840/cycu200200170

摘要

本論文將注意力集中於兩方面，一者為弦系統之週期性能量變化，另一者則是分佈參數（distributed parameter system, DPS）弦系統之振動控制。於振動控制方面，提出最佳之回授增益與適應性計算力矩（Adaptive Computed-torque）。
對於沒有施加邊界控制之弦系統，可使用數學的方式加以證明出系統之所有特徵值均為有界（bounded），且證明出系統之總能量的確呈現週期性變化。另外為達成系統的減振設計了一邊界阻尼控制器，並針對所提出之邊界阻尼控制器來計算所需之最佳回授增益，發現所計算出之最佳回授增益，能將系統之橫向振動量以近似指數性（exponentially）收斂的型態減振至零，同時使用半差分方程（semi-difference scheme）將系統轉換成集中參數系統（lumped parameter system, LPS）並證明此弦系統呈指數穩定。
此弦系統於右邊界由一質量、阻尼及彈簧（mass-damper-spring, MDS）所組成的控制機構，全系統則考慮弦線與控制機構耦合（coupling）之狀態。文中移動弦與邊界控制機構的統御方程式（governing equations）及邊界條件（boundary conditions）乃利用漢米爾頓原理（Hamilton’s principle）推導而得。將適應性控制法則所設計出之控制律應用於控制機構上，來探討線性之二維與三維弦系統的振動控制。並藉由所設計之控制法則，來達成消耗系統能量並抑制振動為目的。
文中主要是將原廣泛應用於集中參數系統之機械手臂控制的適應性計算力矩（Adaptive Computed-torque）方法延伸應用於分佈參數（distributed parameter system, DPS）弦系統之邊界控制上，而且系統中所存在的所有未知參數可利用適應性法則加以線上（On-line）估測。並且藉由此延伸之方法發覺控制力與適應調變率（adaptation law）僅與又邊界弦之位置、速度及斜率有關。李亞波諾夫穩定法（Lyapunov stability）證實系統之追蹤誤差最終會收斂至零。
此外，當弦之張力、質量與流速滿足一特定之關係時，軸向之振動量可以證明出呈現指數性收斂至零。最後，所提出之控制器效能可藉由數值模擬得到應証。

關鍵字

適應性控制 ； 弦系統

並列摘要

This thesis studies the fundamental characteristics of the periodicity of the energy transfer and two control techniques for a class of distributed parameter systems of the string system. For the case that moving string system does not have boundary control, we prove that every solution of the system is bounded and the total mechanical energy is changing periodically. We obtain the best strength for the gain for the boundary control of moving string system with which the transverse vibration decays with most negative exponent to zero exponentially. Also we use semi-difference scheme to convert the system to a lumped system and prove that the later system is exponentially stable. The boundary adaptive computed-torque algorithm was applied to suppress the longitudinal and transverse vibration of the linear 2-dimensional and 3-dimensional moving string system coupled a mass-damper-spring (MDS) mechanism controller at right-hand-side (RHS) boundary. All unknown parameters appearing in the system equation are assumed to be constants and they are estimated on-line by using an adaptation law. The adaptive computed-torque control algorithm applied to robot manipulators of lumped system is extended to design the adaptive boundary controller for the hybrid system. It is found that the control force and update laws depend only on displacement, velocity and the slope of the string at the RHS. Laypunov stability guarantees the convergence of the tracking error to zero. Furthermore, when the tension, mass and the speed of the moving string satisfy a special relation, the longitudinal vibration can also be proved to decay to zero exponentially. Numerical simulations are given to demonstrate the performance of the proposed controller.

並列關鍵字

Adaptive control ； String system

參考文獻

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