# Pole placement from system requirements

Discussion in 'Math' started by Shagas, May 10, 2015.

1. ### Shagas Thread Starter Active Member

May 13, 2013
802
74
Hello,

Could someone explain how I can get the pole positions from given system requirements? I need to regulate
a fourth order, inherently unstable astatic system given by the identified OL tansfer function:

$G_{OL}(s) = \frac{148}{0.6s^4 + 14.8s^3 + 408s^2 + 642s}$

using a PD or lead regulator (nothing else works). The system requirements are roughly 10% OS and a settling time of less than about 4 or 5 seconds. I know how to find the compensator coefficients once I have the required pole locations but question is how do I find the pole locations? I know how to do it for a second order system , but not for a 4th order system.

2. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
Are you able to determine the location of the open loop poles?
Are there any real poles that you might obviously cancel with a compensator zero?
Is the system unstable in closed loop "as is" without added compensation?

Last edited: May 10, 2015
3. ### Shagas Thread Starter Active Member

May 13, 2013
802
74
I know the open loop poles, they are the poles of the open loop transfer function G(s) mentioned above. There aren't any real poles that i'm itching to cancel, there are two pairs of imaginary poles. Yes the system is unstable
in a closed loop with a compensator of 1, or any other gain value for that matter.

4. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
@Shagas : If the transfer function is actually the one you posted in #1, then I can't agree that it would be unstable under closed loop with negative feedback.
I obtain a gain margin of 40.98 dB - which is a significant safety margin. Perhaps you should double check your transfer function is correctly entered in post #1.
In particular, I note there is no constant term in the denominator, which gives rise to my doubts. For your stated transfer function there would be four poles -

Code (Text):
1.
2. G  =
3.
4.                148
5.     -------------------------
6.                2       3      4
7.     642s + 408s + 14.8s + 0.6s
8.
9. roots(denom(G))
10. ans  =
11.   - 11.499542 + 22.570061i
12.   - 11.499542 - 22.570061i
13.   - 1.6675833
14.     0
Attached I include the root locus for the transfer function...

Last edited: May 11, 2015
5. ### Shagas Thread Starter Active Member

May 13, 2013
802
74
The problem is that the transfer function in post 1 is a linearized identified model and a proportional regulator doesn't work on the real system due to deadzones and other nasties. In any case we spent some time with it in the lab and found a PD
regulator by trial and error which works pretty decently so I guess that it is done. Thanks for the help though.

6. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
Frankly, I doubt I have been of any help.
I find it curious that there is a trend on the forums to ask a question on the first thread post which turns out to provide only part of the story. I'm not sure by what stretch of the imagination (other than by divination) a reader could have discerned the actual nature of the problem(s) you were facing.

7. ### Shagas Thread Starter Active Member

May 13, 2013
802
74
To be honest I only wanted an idea on how to approach finding poles from system requirements, but I agree I should have said more about the situation. In any case I already submitted the report. I appreciate the effort though.