# Control Systems - stability and settling time

## Homework Statement

Given G(s) = 1/[(s^2+s+4)(s+6)] and C(s) = k, find the limit of stability of k. Also, what is the range of k such that the settling time is between 10 and 20 seconds.

Provided above

## The Attempt at a Solution

I have attempted to set this up just as i would for any other control systems problem by finding the transfer function:

T(s) = CG/(1+CG) = k / (s^3 + 7s^2 + 10s + k + 24)

I am currently stuck because I have no idea how to deal with a higher order system. In the past, I have done similar questions only in first or 2nd order where I would change T(s) into the form of characteristic equations covered in class.

Any help will help appreciated! Last edited:

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I think i've figured out the limit of stability portion.

http://imgur.com/MOSEEqG
(please note there are 2 pictures)  Any ideas on how to find the range of k within the settling time?

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rude man
Homework Helper
Gold Member
For most systems (including this one) when an increase in gain leads to less and less stability, operate on the open-loop transfer function C(s)G(s): stable if and only if |C(jw)G(jw)| < 1 for arg[C(jw)G(jw) = pi. This can easily be appreciated if you look at the closed loop T(s) = CG/(1+CG). To go unstable the denominator has to = infinity in both its real and imaginary parts. Note that the "1" in the denominator = 1 +j0.

As for the settling time criterion I know only to apply a step into T(s), time-invert and look at the response. Probably there's a better way.

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Thanks rudeman,

Now that you mention it, i do vaguely remember my professor mentioning this. Any idea where I can find some material to read as a refresher?

rude man
Homework Helper
Gold Member
I'm sure there are lots of posts on the Internet, or any elementary control systems text.

The best all-around stabilkity test is the Nyquist critrerion but it's a bit on the involved side. Nice thing about it is it covers all types of T(s).

Settling time I can't give you any more suggestions I'm afraid. Third-order systems don't have nice formulas like 2nd order ones do, like settling time, overshoot etc etc. At least I've never seen any.

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I've figured it out...

I don't exactly know what the explanation is but I'm dividing the characteristic equation with a 2nd order characteristic equation based off the condition ts = 10,20. After you crunch the numbers, you'll find the range of K from 10 - 20 seconds.

rude man
Homework Helper
Gold Member
I've figured it out...

I don't exactly know what the explanation is but I'm dividing the characteristic equation with a 2nd order characteristic equation based off the condition ts = 10,20. After you crunch the numbers, you'll find the range of K from 10 - 20 seconds.

Sounds interesting. Ya got me!

Here's the best part... i can't seem to verify this on matlab. Either the code is wrong or i'm wrong. I tried using my prof's example on matlab and its still inconclusive...

rude man
Homework Helper
Gold Member
What did you do with matlab? Put a step input into T(s)?