I have a transfer function from $x_c$ to $x$
$ \dfrac{x_c}{x} = \dfrac{k}{s + k} $
Now, I want to analyze the stability and find the best possible value for k.
I've tried to convert the closed loop system to an open loop system, but I end up with something that isn't very usable. I end up with a standard feedback system with the constant $1$ and $\frac{1}{k} s$ under the loop. So the open loop transfer function is simply the constant $1$.
Is there any other way to analyze this transfer function?
Well, simply speaking the system will be unstable if the transfer function blows up to infinity. That means that for any infinitely small input one gets infinitely large output. So you just have to find points where $H=\dfrac{x_c}{x} = \dfrac{k}{s + k}=\infty$. Those are the poles of $H$. So if $k\neq\infty$ then $H=\infty$ when $s + k=0$.
Then just use the conditions of stability for closed-loops.