Can bandwidth of the closed loop be bigger than bandwidth of individual elements in it? This is a question from control theory but I hope you can help me with it. I was pointed to this community. 

So in my control loop I have elements that have cut-off frequency of 3-5kHz and I designed a PID controller for the inner loop and those elements and suddenly I get a closed loop bandwidth of 30-50kHz cut-off frequency.

So the loop is in the figure bellow.

So the PID controller, or voltage controller I designed is a PID with a derivative filter. Current sources are modeled as 1st order filters with 3-5kHz cutoff frequency. How is it possible to get such a high bandwidth of the closed loop if the inner elements have a lower cut-off frequency?

Thanks
Yes, that's perfectly possible and to be expected.

Below is a toy example that starts with a Butterworth low pass filter and puts it in the forward path of a feedback loop. The feedback path has a gain of two.

This makes the whole loop unstable, but it also makes the bandwidth very high.

The original filter's frequency response is below.



And this is the closed loop response.



R Code Below
    #30035

library("signal")

bf &lt;- butter(5, 0.1)
freqz(bf)
dev.copy(png, 'Q30035/Q30035-Butterworth.png')
dev.off()

k &lt;- 2
bf2 &lt;- bf
bf2$a &lt;- bf$a + k*bf$b
bf2$b &lt;- bf$a*k
freqz(bf2)
dev.copy(png, 'Q30035/Q30035-Feedback.png')
dev.off()