Hi so we have a sinus signal that is sent to a feedback loop system and I'm trying to understand signals and stability better. I let $\arg(G_o(iw_o))=0$ and get that $y$ has no bound although $r=0$.
Suppose that we have a stable system by the nyquist criteria. Let r=0, then $-e(t)=y(t)$ but we also have that $y(t)_{\text{new}}=G_oe(t)=-G_oy(t)$. Further let $\arg(G_o(iw_o)=0$ and $|G(iw_o)|=k>1$. Then $G(iw_o)=k$ thus $y(t)_{\text{new}}=-ky(t)$. So $y$ has no bound. The feedback inverts y and going through $G_o$ the absolute value increases with a factor of $k$.
Since the system is stable there must be a damping effect on $y$ I can't see right?
Editlink below the text. you also need to specify all the time instants when the switch is in position 2 and position 1, e.g. "Switch is in position $2$ for $t$ ranging from $-\infty$ to $0^-$. At $t=0$ it switches instantly to position $1$ and remains there from $t=0^+$ onwards." or whatever you want to tell us about what the switch does. Also, $G_0(\omega)$ is a constant $k > 1$? and there is no filtering going on, just gains of $k$ and $-1$? $\endgroup$