I'm thinking specifically of systems with ideal PID controllers. The transfer function will always have an integrator term, which, if I understand correctly, means it will always have infinite DC gain. In other words, the magnitude bode plot will go to infinity either as frequency goes to infinity or as it goes to negative infinity. Something like this where there's no clear cutoff point on the left side and where MatLab basically says the bandwidth is undefined, as shown in the screenshot below:
and yet, bandwidth is a common target parameter when doing PID tuning (or at least, that's the impression I've gotten from my control systems course).
How does that make sense? What am I missing?