# Homework Help: Determine values of K>0 such that the poles are in left-hand plane?

1. Apr 9, 2014

### annas425

Given the transfer function G(s) = 1 / (s(s+1)(s^2 + 4s + 13)), how would I determine the range of the values of K>0 such that the closed-loop poles are in the left-hand plane?

Picture of block diagram with transfer function:

Not sure if this is right at all, but I know that a system is stable when K>0, so if the poles were to be in the left-hand plane (i.e., be stable), would it just be for all K>0? I am assuming it's more involved than just that, so I would really appreciate some help please! :)

2. Apr 9, 2014

### maajdl

Start by solving question 3).

Note also that in question 2 you are given the information that for too large K, the system become unstable.
Therefore you hypothesis that K>0 for stability must be wrong. (there was no reason for that)

3. Apr 9, 2014

### annas425

Well, doesn't the prompt just say that K must be greater than 0?? I am confused...

4. Apr 9, 2014

### rude man

To give you a feel for how k has to range as a function of ω, form an expression for the phase shift of G(s) as a function of ω. This is not hard. The phase shift will be a transcendental equation in ω. The phase shift is independent of k. Then use Excel to run ω over a range of values starting at zero with constant small increments, and ask it to compute the phase shift for each ω. When the phase shift gets to π, put that value of ω into G(s) and solve for the range of k for which |G(s)| < 1. That range of k gives a stable closed loop.

There may be more than one value of ω for which the phase shift is π, so don't quit when you hit pay dirt the first time.

Routh-Hurwitz is seldom used in 'real life' because usually the transfer function is known as gain and phase plots but not in closed form.

I never learned how to do Root-Locus, don't regret it, so can't help you there either. Sorry about both.