Some results of finding the roots of time-delay systems by Lambert W function

A unified formula is used to find the roots of the transcendental characteristic equations for both an integer and fractional -order time delay systems by using Lambert W function solutions. However, the unified formula is applicable only to the special case of time delay systems with repeating poles alone. In the basis, an iteration approach is presented to find the roots of the transcendental characteristic equations in general. It removes the restriction on the repeating poles and time-delay system without zeros. It enables us to apply the presented approach to solve more generalized delay differential equations. Some examples are given to illustrate the proposed method and show that it is effective. It is shown that a certain iteration equation can only find the partial roots of all roots.