My professor gave the class an extra credit problem to try and make up for the low test average (less than 50%) from our first exam.

The problem is a root locus design. The Gp(s) = 1/s(s^2+4s+5). He gave an example using a PI compensator, in which he used a K value of 3, resulting in:

Overshoot:38.36%
Tp=4.55
Ts= 9.04

He wants us to get a settling time, Ts, less than 5 seconds. However, this seems to be impossible. The only thing that I thought might work is using PID instead of PI. However, we were not taught this in class, nor r we going to be tested on it. I have tryed but failed, getting my best result of Ts=5.37 seconds using a k value of 4.9.

Please Help!

 

Extra credit usually implies something above and beyond that which you are expected to know - I'd say go for the PID...

 

So you know that one of the roots is at the origin and the other two roots of the original transfer function are a complex conjugate pair or they lie on the real axis. To get the entire system to behave the way you want you must either move the poles, by varying a parameter, or add additional features (poles or zeros).

A couple of questions:
1. What does the contour of 5 seconds settling time look like?
2. What region has settling time less than 5 seconds?
3. What region has settling time greater than 5 seconds?
4. What is the minimum number of features you need to add?
5. What one thing can you add near the pole at the origin to improve the settling time?

While you're working on all of this remember the following:
The positive real part of any pole must disappear or the system WILL!