本文旨在探討原子力顯微鏡(Atomic force microscopy)探針之物理模型，建構其運動數學模式與設計控制器。首先利用漢米頓(Hamilton’s principle)原理得到探針系統的偏微分運動方程式和邊界條件，以分離變數法(The separation of variables method)求出其頻率方程式與特徵函數，分析探針系統之自然頻率，再經由拉普拉式轉換(Laplace transform)求出系統開迴路轉移函數，並利用Matlab套裝軟體求得系統極點與零點分佈情形。最後由共點控制(Collocated)來設計致動器與感測器，從根軌跡圖可以看出極點與零點分佈交錯於虛軸上，具極小相位系統之特性，可作為稍後在控制系統設計之考量依據。
根據根軌跡法(Root locus method)分析閉迴路系統的軌跡變化，避免極零點對消情形，而影響系統穩定性，接著依據根軌跡設計比例微分控制器與相位領先補償器，最後利用基因演算法(Genetic Algorithms, GA)，設定收尋之目標函數，對控制器參數做最佳參數之收尋，可減少計算時間與增加精確性。結果顯示，設計之控制器能使系統無窮多個振動模態穩定且有效抑制，以不產生超越量與增加響應速度為原則，本論文所設計之控制器可以符合期望目標，達到精密控制之實用性。

The objective of thesis is to derive the mathematical model and design the controller of the micro-cantilever probe measurement system of atomic force microscopy. However, most of the distributed parameter control system for atomic force microscopy are based on reduced-order models, but the problems of result are often in “ computation errors” or “control and observation spillover”. Therefore, developing some realizable controllers which cannot only stabilize all the vibration modes but also make such system efficient for good tracking is crucial. First, the governing equations of motion and associated boundary conditions are derived from Hamilton’s principle. Using the separation of variables method, the dynamic characteristics such as natural frequencies and mode shapes function of the system can be analyzed. And then, the corresponding open loop transfer function will be obtained from Laplace transform. Finally, the actuator and sensor will be designed from collocated control. The results show that the poles and zeros patterns will interlace along the imaginary axis in the root locus and have characteristic of minimum phase property. It is the essential characteristics for designing the controller for such system.
According to root locus method the tracking property in the closed loop system can be analyzed. Avoid all the poles and zeros cancellation with each other to influence the system to be steady, then the augmented root locus method is used to design the controllers such as Proportional-Derivative controller and Phase Lead compensator. It can be shown that system cannot only eliminate infinite dimensional vibration modes but also make system stable. Finally, the objective function of Genetic Algorithms can be designed for to find out the best parameter of controller in order to reduce computational process and improve accuracy. It can be seen from computer simulation, the designed control system for atomic force microscopy can accord with performance index to reach precision control.

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