Part 2: Root Locus, Transfer Functions and Unit Step Response?

Question

I'm continuing my question referenced here: Part 1 Question / Problem Description

Say I have a new Root Locus shown below

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, $\square$ denotes the closed loop poles, and $\circ$ the open loop zeros.

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

Question:

I don't fully understand how Matt goes from the closed-loop characteristic equation to solving for the value $K$. Is he comparing coefficients? If so, how can I do that for one real axis pole and two imaginary? I can find the closed-loop transfer function, but it results in a cubic polynomial. Also, his process determines the final settled value, but A) and B) both settle to 1, so how can I differentiate the two? Thanks!

Answer 1

From the root locus plot, the open loop transfer function has a real-valued pole at $s_0=0$ and a complex conjugate pole pair $s_1=-1.5+ 2j$ and $s_1^*$. There's also a zero at $s=-2$. Consequently, $G(s)$ is given by

$$G(s)=\frac{s+2}{s(s-s_1)(s-s_1^*)}$$

The closed-loop transfer function is

$$H(s)=\frac{kG(s)}{1+kG(s)}=\frac{k(s+2)}{s(s-s_1)(s-s_1^*)+k(s+2)}\tag{2}$$

The final value of the step response equals $H(0)=1$, which leaves options $A$ and $B$.

Looking at the root-locus plot, we can see that the real-valued pole is (a bit) closer to the imaginary axis than the complex-conjugate pole pair. So that real-valued pole is (slightly) dominant. Based on this observation one can guess that step response $A$ is correct because it clearly shows the influence of the real-valued pole (i.e., a smooth increase with no overshoot), whereas step response $B$ just looks like a step response of a second-order system with a complex-conjugate pole pair.

In sum, apart from looking at the final value of the step response given by $H(0)$, you can check if there are one or more dominant poles that determine the behavior of the step response. You should be familiar with the step responses of first and second-order systems; this will allow you to make an educated guess also for higher order systems. This website gives a good overview of step responses of first, second, and higher order systems.