Lambert W 函數不論在純粹數學、應用數學以及物理、工程等領域都有諸多的應用。特別是在近代的控制理論中，以常微分方程表示的時滯系統常被使用作為討論系統穩定性分析以控制律合成的例子。本文的主要目標是以探討時滯系統的穩定性作為基礎，進行極點配置問題的研究。首先討論單一時滯系統如何透過 Lambert W 函數求解特徵方程，並且進一步延伸討論雙時滯線性微分方程系統。因為特徵值的位置影響穩定性，我們考慮了單一或兩個時滯的時滯系統, 經由特徵值分配，決定系統的響應符合想要配置系統極點的要求。 The Lambert W function has many applications in the fields of pure and applied mathematics as well as physics and engineering. In particular, differential equations which represent time delay systems are employed to stability analysis and controller synthesis in the modern control theory. The main target of this study is to probe the stability of time delay systems and then to place the system's poles to desire locations. Firstly, we discuss how to solute the characteristic equation generated from a single delay system via Lambert W function, and expand further to two-lag linear delay differential equations.Since the positions of eigenvalues influence stability, the problem of delay systems with single or two delays via eigenvalue assignment are then considered. Finally, the pole placement problem is then solved with considerable controller to drive the delay system to have desire response implied by the location of system poles.
