Given a unity feedback system with the transfer function $G(s) = \frac{16}{s(s+5)}$, determine the damping ratio, overshoot, and settling time for a step input.

1. Problem Description

Given a unity feedback system with the transfer function $G(s) = \frac{16}{s(s+5)}$, determine the damping ratio, overshoot, and settling time for a step input.

2. Solution Steps

First, determine the closed-loop transfer function, $T(s)$, for a unity feedback system:

$T(s) = \frac{G(s)}{1 + G(s)}$

Substitute the given $G(s)$:

$T(s) = \frac{\frac{16}{s(s+5)}}{1 + \frac{16}{s(s+5)}} = \frac{16}{s(s+5) + 16} = \frac{16}{s^2 + 5s + 16}$

The general form of a second-order system is:

$T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$

Comparing the given $T(s)$ with the general form, we can identify $\omega_n^2 = 16$, so:

$\omega_n = \sqrt{16} = 4$

Also, $2\zeta\omega_n = 5$. Substitute $\omega_n = 4$:

$2\zeta(4) = 5$

$\zeta = \frac{5}{8} = 0.625$

(i) Damping Ratio:

The damping ratio $\zeta$ is 0.

6

2

5.

(ii) Overshoot:

The percentage overshoot (OS) is given by:

$OS = e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}} \times 100\%$

Substitute $\zeta = 0.625$:

$OS = e^{-\frac{\pi(0.625)}{\sqrt{1-(0.625)^2}}} \times 100\% = e^{-\frac{1.9635}{\sqrt{1-0.390625}}} \times 100\% = e^{-\frac{1.9635}{\sqrt{0.609375}}} \times 100\% = e^{-\frac{1.9635}{0.7806}} \times 100\% = e^{-2.5154} \times 100\% = 0.08076 \times 100\% = 8.076\%$

(iii) Settling Time:

For a 2% settling time ($t_s$), the formula is:

$t_s = \frac{4}{\zeta\omega_n}$

Substitute $\zeta = 0.625$ and $\omega_n = 4$:

$t_s = \frac{4}{0.625 \times 4} = \frac{4}{2.5} = 1.6$

3. Final Answer

(i) Damping Ratio: $\zeta = 0.625$

(ii) Overshoot: $OS = 8.076\%$

(iii) Settling Time: $t_s = 1.6$ seconds