Finding the maximum gain limit is very easy. Assume that we have a transfer function.
$$G(s) = \frac {num (s)}{den (s)} $$
If we want to compute the maximum gain limit when system $G (s) $ is implemented into a feedback system. We find the roots from.
$$0=Knum (s) + den (s) $$
Where $K $ is the gain limit.
But how would I do if my transfer function looked like this:
$$G(s) = \frac {num (s)}{den (s)}e^{-Ls} $$
Then I would have a feedback system and then to compute the maximum gain limit:
$$0=Knum (s)e^{-Ls}+ den (s) $$
The only problem is that $s $ is a complex number. How can I find the roots from that equation?
I have solver a nondelayed problem by using this code
https://github.com/DanielMartensson/matavecontrol/blob/master/sourcecode/findmaxgain.m
Can I solve this with Padé approximation or will this result a bad solution?
When you have time-delays and work in continuous time, you cannot analyze poles in the classical meaning. Instead, you can for instance do standard design and then analyze stability using phase margins, or you do design where you explicitly consider the delay using for instance a Smith predictor. Or as you indicate, use an approximation of the time-delay. Or go to discrete-time and do the design there.
Concering your code for analyzing stability when tweaking a gain, it looks like you should take a look at the very standard theory of root-loci analysis (rotort in Swedish). It can be found in essentially any introductory course/text on control. The command is rlocus in MATLAB