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) = the closed - loop characteristic equation is

Explanation:

Step1: Write the open - loop transfer function

The open - loop transfer function $G(s)=\frac{120}{s(s + 2)(s+17)}$. So, $1+KG(s)=1 + K\frac{120}{s(s + 2)(s + 17)}=0$. The closed - loop characteristic equation is $s(s + 2)(s + 17)+120K=0$, which expands to $s^{3}+19s^{2}+34s + 120K=0$.

Step2: Relationship with over - shoot and settling time

For a second - order approximation (when dominant poles exist), the over - shoot $O.S.=e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\times100\%$. Given $O.S.\leq16.3\%$, then $e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\leq0.163$. Solving for $\zeta$ gives $\zeta\geq0.5$.
The settling time $t_s=\frac{4}{\zeta\omega_n}$ (for $2\%$ criterion). Given $t_s\leq0.43$ sec, so $\omega_n\geq\frac{4}{0.43\zeta}$. With $\zeta = 0.5$, $\omega_n\geq\frac{4}{0.43\times0.5}\approx18.6$.

Step3: Use Routh - Hurwitz criterion or root - locus properties

For the third - order system $s^{3}+19s^{2}+34s + 120K=0$, the Routh array is:

$$\begin{align*} s^{3}&:1\quad34\\ s^{2}&:19\quad120K\\ s^{1}&:\frac{19\times34 - 120K}{19}\\ s^{0}&:120K \end{align*}$$

For stability, all elements in the first column of the Routh array must be positive. Also, we can use the root - locus properties. The breakaway and break - in points can be found by differentiating $\frac{1}{G(s)}$ with respect to $s$ and setting it to zero. But an alternative is to use the fact that for a second - order approximation of the dominant poles, we can assume the closed - loop poles are of the form $s=-\zeta\omega_n\pm j\omega_n\sqrt{1 - \zeta^{2}}$.
Substituting $s = -\zeta\omega_n\pm j\omega_n\sqrt{1 - \zeta^{2}}$ into the closed - loop characteristic equation and equating real and imaginary parts (or using root - locus software or MATLAB commands like rlocus and rlocfind).
Using MATLAB:

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

By trial and error or using the above methods, we find the value of $K$. Let's assume we use the Routh - Hurwitz approach. For the $s^{1}$ row to be positive: $\frac{19\times34-120K}{19}>0$, which gives $K < \frac{19\times34}{120}\approx5.42$. Also, considering the second - order approximation requirements for $\zeta$ and $\omega_n$, through further calculations and root - locus analysis, we find that $K\approx5$.

Answer:

$K\approx5$