# Find transfer function from root locus and step response diagram?

I have the following 2 diagrams:

The step response is of magnitude of 3.

I'm a beginner at this so I've done something stupid probably but I have trouble finding answers regarding control engineering on the internet. This is what I tried doing: I found the poles and the zeros from the root locus. z=-5,+4 p=-6,-10,-3 I think my transfer function is given from this formula but I'm not sure if we have an H(s) in the feedback and it is not stated : $$T(s)= \frac{KG(s)}{1+KG(s)}$$ From the poles and the zeros my open-loop transfer function G(s) is : $$G(s)= \frac{(s+5)(s-4)}{(s+10)(s+6)(s+3)}$$ Doing the calculations I find : $$T(s)= \frac{Ks^2+Ks-20K}{s^3+(K+19)s^2+(108+K)+180-20K}$$ From the step response(final value is 4) and the final value theorem I find $\frac{-20K}{180-20K}=-4/3=>K=5.14$ I divided 4 by 3 because the diagram above is for a step of magnitude of 3. Now testing the step response of the function I found I get this:

It's similar to the first response but not identical.

What am I missing here?

• I use step(T) to generate the last diagram. The others are given by the exercise. – John Katsantas May 7 '17 at 12:44

$$G(s) = \frac{K\, (s+5)\, (s-4)\, }{(s+10)\, (s+6)\, (s+3)},$$
with $3\, G(0) = -4$. Substituting in the equation for $G$ and solving for $K$ yields $K = 12$.