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

Question

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?

Answer 1

So what kind of bandwidth would we be talking about in the context of a target parameter for a system with a PID controller?

• The term "bandwidth" is a parameter which specifies some circuit properties in the frequency domain (e.g. filters, amplification range for small-band or broadband amplifiers,....). Hence, the "bandwidth" is a parameter which tells us which frequency range should pass the circuit (low-/high-/band-pass) or must be attenuated (band-stop filter).

• Such specifications do not apply for a PID controller. In contrast, a PID controller mainly has to fulfill some specifications in the time domain . Therefore, it makes not much sense and it is not necessary to specify resp. define a parameter like "bandwidth" for a PID controller.