Why is it that a second order system with an additional pole or zero can be approximated into the general second order system for analysis?

This was asked during our lecture and I am still looking for the answer.

In my reading I've only learned that when an additional pole or zero is added to the system, the step response of this system starts to resemble the step response of the original system. I am not sure if this is the answer to the question.


1 Answer 1


I think there must be some other restrictions placed on your professor's comment. Adding a pole to a general second order system makes it a third order system and the dynamics can change dramatically.

One example I can think of in which adding a pole or zero does not change the system dynamics dramatically is when the pole or zero is at much higher frequency than the pair of poles of the second order system. In this case, the dynamics only change at very short timescales. The step response, for example, would only change at the very beginning and the damping or oscillatory behavior later in the response would remain essentially the same.

At any rate, I wouldn't fret too much about making approximations of such simple systems. Once you are actually working with them in the field you will likely use numerical software to understand them, and such software has no need to make any approximations for a third order system.

  • $\begingroup$ I agree with the technical part of your answer, but I'm not 100% behind saying 'don't worry about it.' A lot of stuff learned in school will be done by computers in the real world, but there's definitely value to understanding how the computers do what they do, and OP's professor probably has a reason for bringing up that point. $\endgroup$ Mar 27, 2015 at 17:48
  • $\begingroup$ @TrevorArchibald You're right. It is important to understand the fundamentals in order to be able to ensure that the computer isn't telling you lies (which they do quite often in my experience). $\endgroup$ Mar 27, 2015 at 19:05

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