# Does the idea of bandwidth make sense for a system with an infinite DC gain?

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?

• In other words, the magnitude bode plot will go to infinity either as frequency goes to infinity or as it goes to negative infinity. <-- it goes to negative infinity in decibels but to zero in real numbers and, that statement is not directly correlated with your previous sentence as having infinite DC gain. Jun 11 at 13:33
• The term "bandwidth" can be applied to several physical phenomena and is not restricted to lowpass or other magnitude functions. It is simply a matter of definition. For example, we even could ask "in which frequency range does the PHASE of the shown circuit reaches ta value of more than 180 deg?". And we are able to find this range (app between 2 and 50 rad/s) and call it "bandwidth".
– LvW
Jun 11 at 14:36
• @LvW, hmm. So what kind of bandwidth would we be talking about in the context of a target parameter for a system with a PID controller? Jun 11 at 23:05