時間延遲系統在控制領域中為一個常見的問題，傳統對於時間延遲項的分析是以有理數的近似法做逼近，以此方式所得的系統模型在分析以及設計控制器時往往會有問題產生。隨著電腦科技的進步，過往的理論又重新被提及，例如Lambert W function就是一個在最近被提到的方法之一，此方法可以讓帶有延遲時間的系統直接做分析，並以特徵值分佈的方式來進行回授控制器設計。在本篇論文中會比較此兩種方法的優缺點以及使用上的限制，並且以一帶有時間延遲的倒單擺系統來對Lambert W function設計出的控制做驗證，以了解其在現實中的可行性。

Time delay system is a common problem in the control fields. In order to analyze time delay system, rational approximation for time delay term was often used in traditional design methods. However, the approach may cause unexpected errors with the design. With the improvement of the computer technology, theorems developed in the past can be used to deal with new problems and a Lambert W function is one of the methods mentioned in recent years. This method can be used to analyze the delay term in a direct way and design a feedback controller by placement of system’s desirable eigenvalues. In this thesis, the traditional Pade-approximation based delay system control design method is compared with Lambert W function based design method. The designed controller is applied into the control of an inverted pendulum system with time delay term to verify the feasibility of the Lambert W function.

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