How to draw the root locus of the open loop transfer function $s e^{as}$

Question

How to draw the root locus of the following open loop transfer function with respect to $a$ \begin{equation} G(s) = s e^{as} \end{equation} where $a \ge 0$ and $a$ is the variable.

Answer 1

I think your $a$ should be negative because only then you will have a positive input output delay. If $a$ is positive this would imply that your output is faster than your input.

If you are using MATLAB you could try to plot approximations to the solutions by using the Padé approximation.

s = tf('s');
G = s*exp(-s);
G_pade = pade(G,3); % 3rd order pade approximation
rlocus(G_pade); 

In general, you should not use the root locus method for transfer functions with delays. As you can see the approximations lead to very complicated plots (check by using higher order Padé approximations), which comes from the fact that you need to use the Lambert function in the complex domain. You should rather use the Nyquist plot, which can handle delays.