# Why does my closed loop pole-zero map appear to indicate a critically damped system but a step input results in an under-damped response?

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**EDIT:forgot to show plant

As the question says my pole-zero map for a closed loop system with negative unity feedback appears to indicate that I should expect an critically damped response but when I subject my system to a unit step it is oscillatory. Can anyone offer an explanation or point me in the direction of releveant literature? Thanks

I have a model plant, G, with a time delay (dead time)** of -1 seconds which is defined as:

G = (2*exp(-1*s))/(3*s+1);

**Just as a side note, taking away the dead time has the same final result.

Plotting plant:

plot(t,y), grid;

I have the following Ziegler-nichols tuning parameters:

Kp = 1.8;

Ti = 3;

I have made the following PI controller for the plant:

controller = pidstd(Kp,Ti)

I have calculated the open and closed loop system as follows:

gol = controller*G;

gcl = feedback(gol,1);

The PZ map for the closed loop system is:

pzmap(gcl), grid

Based on the PZ map where there is a pole-zero cancellation at -0.333, leaving a single real pole at -1.2 and a damping of 1, I am expecting a critically damped reponse with no oscillations.

However when I plot a step response I get the following:

step(gcl,40),grid

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### Accepted Answer

Jon
on 17 Dec 2021

Edited: Jon
on 17 Dec 2021

If you remove the time delay the response has no oscillations as you expected. I know you said "**Just as a side note, taking away the dead time has the same final result.", but I think perhaps you didn't evaluate this correctly. Here I show the system without time delay and no oscillation:

G = 2/(3*s+1);

Kp = 1.8

Ti = 3

controller = pidstd(Kp,Ti)

gol = controller*G

gcl = feedback(gol,1)

step(gcl,40),grid

##### 4 Comments

Jon
on 17 Dec 2021

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