# What does it mean that a zero is slow?

I am studying control systems, and I have encountered the definizion of slow zero. I am searching on internet and on books, but I don't understand the meaning this. I know that if a zero is too slow, it introduces a reasonance peak and an overshoot in the step response, but what is a slow zero?

I thiught it was a zero close to the imaginary axis in the root locus, so a zero at low frequencies, but I am not sure, since I cannot find a definition.

I know that a pole that is close from the imaginary axis settle quicker than a pole far from the imaginary axis, but I did some simulations in Matlab, and using a lead compensator, I have seen that if I decrease frequency of the zero in the lead compensator, the overshoot in the step response increases:

where the lead compensators I have used are:

lead = (1+10*0.05*s)/((1+0.05*s));          %red line
lead_2 = (1+30*0.083*s)/((1+0.083*s));      %blu line
lead_3 = (1+50*0.083*s)/((1+0.083*s));      %green line


can somebody help me?

## 1 Answer

Again I refer you to the Murray's book. Which is free by the way

http://www.cds.caltech.edu/~murray/books/AM08/pdf/fbs-limits_18Aug2019.pdf

From the book "Furthermore a zero is said to be “slow” if its magnitude is smaller than the intended closed loop bandwidth."

• wow. i thought a zero is "slow" for the same reason a pole is "slow". a slow pole is one associated with an impulse response that is a decaying exponential having a long time constant. longer time constant means lower corner frequency of the associated frequency response. so for the corner frequency associated with a zero (that would be a + 6 dB/oct corner), the association between the corner frequency and the speed of the zero is the same relationship as with poles.  it's an interesting article, though, and i found the references. – robert bristow-johnson Feb 12 at 20:33