Small damping (zeta = 0.1, Wn = 4pi/5T)
Medium damping (zeta = 0.4, Wn = 11pi/20T)
Large damping (zeta = 0.8, Wn = pi/4T)

In this page, we are going to show you more analysis on the pole locations and the corresponding time response.

Small damping (zeta = 0.1, Wn = 4pi/5T)

First let's consider the following discrete transfer function.

The following commands will show you the poles of this transfer function. Enter the following commands to an new m-file and run it in the Matlab command window. You should see the pole map shown below.

numDz=[1];
denDz=[1 1.2 0.57];
[poles,zeros] = pzmap (numDz,denDz)
pzmap (numDz,denDz)

axis([-1 1 -1 1])
zgrid

From this pole map, we see that poles are located at the natural frequency of 4pi/5T rad/sample (T is the sampling time in sample/sec) and the damping ratio of 0.1. Assuming that we have the sampling time of 1/20 sec/sample and by using three equations shown below,

we can determine that this system should have the rise time of 0.03 sec, the settling time of 0.9 sec, and the maximum overshoot of 70% (0.7 more than the steady-state value). Let's confirm this by obtaining the step response. Add the following commands to the above m-file and rerun it. You should see the following step response.

[x] = dstep (numDz,denDz,51);
t = 0:0.05:2.5;
stairs (t,x)

The plot shows all of the rise time, the settling time, and the overshoot as what we expected.

Medium damping (zeta = 0.4, Wn = 11pi/20T)

Now consider the next discrete transfer function.

Let's follow the same steps as what we did above. Create an new m-file and enter the following commands. Running this m-file in the command window gives you the following pole map.

numDz=[1];
denDz=[1 0 0.25];
[poles,zeros]=pzmap (numDz,denDz)
pzmap (numDz,denDz)

axis([-1 1 -1 1])
zgrid

From this pole map, we see that poles are located at the natural frequency of 11pi/20T rad/sample and the damping ratio of 0.4. Assuming the sampling time of 1/20 sec as before and by using the above three equations, we can determine that the step response should have the rise time of 0.05 sec, settling time of 0.3 sec, and the maximum overshoot of 25%. Let's obtain the step response and see if these are correct. Add the following commands to the above m-file and rerun it. You should see the following step response.

[x] = dstep (numDz,denDz,51);
t = 0:0.05:2.5;
stairs (t,x)

Once again, this step response shows all of the rise time, the settling time and the overshoot as what we expected.

Large damping (zeta = 0.8, Wn = pi/4T)

Just as before, enter the following commands to an new m-file and run it in the Matlab command window. You should see the following pole map.

numDz=[1];
denDz=[1 -0.98 0.3];
[poles, zeros]=pzmap (numDz,denDz)
pzmap (numDz,denDz)

axis([-1 1 -1 1])
zgrid

From this plot, we see that the poles are located at the natural frequency of pi/4T rad/sample and the damping ratio of 0.8. Once again assuming the sampling time of 1/20 sec, we can determine that this system should have the rise time of 0.1 sec, the settling time of 0.36 sec and the overshoot of 1%. Let's confirm this by adding the following commands to the above m-file and rerunning it. You should see the following step response.

[x] = dstep (numDz,denDz,51);
t = 0:0.05:2.5;
stairs (t,x)

This time both the rise time and the settling time came out a little slower than what we expected; however, the overshoot still came out as what we expected.

Using these three examples, we proved that the pole locations can be used to roughly estimate the transient response of a system. This analysis is particularly useful in the root-locus design method where you are asked to pick poles (by choosing a gain using rlocfind) to obtain a satisfactory response.

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