# Root Locus, Transfer Functions and Unit Step Response?

Consider the generic feedback loop, and the transfer function $$G(s)$$ shown by the following root locus plot.

Where $$\mathbf{x}$$ denotes the open-loop poles and $$\square$$ denotes the closed loop poles.

I want to determine if this root-locus produces any of the following output responses to a unit step reference signal:

Attempt:

I attempted to build the open loop transfer function:

$$G_{OL} = \frac{K}{s^2+2s+2}$$ Find the value $$K$$ using the FVT for the step response and graph a) $$\lim_{s\to 0} = s\cdot\frac{1}{s}\frac{K}{s^2+2s+2} = 1$$ which gives $$K = 2$$, plug this back in and see if the step response matches any of the figures. It is possible none of the figures match, but I think I am doing something incorrect.

Root locus shows all the possibilities of K, but I am also given the closed loop poles. Can I use those to get K? Is the unit step response in relation to the Closed Loop Transfer Function? or the open loop? I can see that building the connection between the graphs is important, I seem to be missing something though. Thanks!

Edit/Update

I believe the corresponding Step Response Graph is C) but I'm still uncertain on if I followed the right process: Finding the closed-loop transfer function I get $$G_{CL} = \frac{K}{s^2+2s + 2 + K}$$ Applying FVT to graph C I get $$K = 3$$, which makes $$G_{CL} = \frac{3}{s^2+2s + 5}$$ and plotting the unit step response with Matlab gives C). What if I didn't have access to Matlab? Is there an easier way? Is it sufficient to match the closed loop poles after I determine $$K$$?

• Note that the denominator of $G(s)$ should be $s^2+2s+2$ instead of $s^2-2s+2$. Jan 19, 2021 at 20:26
• @MattL. You're right, thanks for catching that. I updated the question Jan 19, 2021 at 20:36

The closed loop poles are the roots of the polynomial

$$D(s)=s^2+2s+2+K\tag{1}$$

and, according to the root locus plot, they are $$s_{1,2}=-1\pm 2j$$. Consequently, we get

$$2+K=|1+2j|^2=5\quad\Longrightarrow\quad K=3\tag{2}$$

With $$K=3$$ we obtain $$H(0)=\frac{K}{2+K}=\frac35\tag{3}$$

which leaves step response $$C$$ as the only option.

• How did you know step 2? are you simply factoring $D(s)$? Then what is $H(0)$? is that simply the application of FVT to the closed-loop transfer function. Physical meaning, I suppose. I get you're just plugging in $0$ Jan 19, 2021 at 22:16
• @Clark: Eq. $(2)$ is just a consequence of the quadratic polynomial $(1)$ having two complex conjugate zeros. So the constant part $2+K$ must equal the squared magnitude of these zeros. And $H(0)$ is the DC response of the filter, which is the value the step response converges to. Jan 20, 2021 at 8:09