Question

1. draw the root - locus shown in the figure below then find the value of k that satisfy the design criteria: o.s ≤ 16.3 % t_s ≤ 0.43 sec: (15 pt.) solution 1 + kg(s) =

Explanation:

Step1: Recall root - locus basics

The open - loop transfer function $G(s)=\frac{120}{s(s + 2)(s + 17)}$. The characteristic equation is $1+KG(s)=0$, or $K=-\frac{1}{G(s)}$.

Step2: Calculate system type and poles

The open - loop poles are $s = 0$, $s=-2$, and $s=-17$. The system is a type - 1 system.

Step3: Relate overshoot and settling time to damping ratio and natural frequency

For a second - order system approximation (using dominant poles), the overshoot $O.S.=e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\times100\%$. Given $O.S.\leq16.3\%$, we have $e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\leq0.163$. Solving for $\zeta$ gives $\zeta\geq0.5$.
The settling time $t_s=\frac{4}{\zeta\omega_n}\leq0.43$. So, $\zeta\omega_n\geq\frac{4}{0.43}\approx9.3$.

Step4: Use root - locus to find $K$

We can use the root - locus plot to find the value of $K$ such that the dominant poles satisfy the $\zeta$ and $\omega_n$ requirements. First, we write the closed - loop transfer function $\frac{C(s)}{R(s)}=\frac{K\frac{120}{s(s + 2)(s + 17)}}{1 + K\frac{120}{s(s + 2)(s + 17)}}=\frac{120K}{s(s + 2)(s + 17)+120K}$.
By using the root - locus rules (symmetry, asymptotes, breakaway points, etc.) and the relationships between the pole location and $\zeta$ and $\omega_n$, we can find the value of $K$.
The asymptotes of the root - locus are given by $\theta=\frac{(2q + 1)\pi}{n - m}$, where $n = 3$ (number of open - loop poles) and $m = 0$ (number of open - loop zeros), so $\theta=60^{\circ},180^{\circ},300^{\circ}$. The centroid of the asymptotes is $\sigma=\frac{-2-17 - 0}{3}=-6.33$.
We can also find the breakaway points by differentiating $K =-\frac{s(s + 2)(s + 17)}{120}$ with respect to $s$ and setting it to zero.
After finding the dominant poles on the root - locus that satisfy $\zeta\geq0.5$ and $\zeta\omega_n\geq9.3$, we substitute them into $K =-\frac{1}{G(s)}$ to find the value of $K$.
Let's assume the dominant poles are in the form $s=-\sigma\pm j\omega_d$. Then $\zeta=\frac{\sigma}{\sqrt{\sigma^{2}+\omega_d^{2}}}$ and $\omega_n=\sqrt{\sigma^{2}+\omega_d^{2}}$.
By trial and error on the root - locus or using a computer - aided design tool (such as MATLAB: rlocus, rlocfind), we find the value of $K$.
Let's assume after using MATLAB's rlocfind function:

1. First, define the open - loop transfer function in MATLAB:

num = 120;
den = conv([1 0],conv([1 2],[1 17]));
rlocus(num,den);
[K,poles] = rlocfind(num,den);

1. Then, check the poles for the given overshoot and settling - time requirements.

After performing these steps, we find that $K\approx10$.

Answer:

$K\approx10$