A second order control system is designed using PD controller having $K_{P}, K_{D}$ as proportional and derivative constant respectively.

$\frac{G (s)}{1 + G (s) H (s)} = \frac{4}{s^{2} + s + 4}$

For maximum overshoot to be zero and $K_{P} = 1$ . Minimum value of $K_{D}$ is

1.164

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0.75

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4.65

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Solution

The correct option is C 0.75
$\frac{G (s)}{1 + G (s) H (s)} = \frac{4}{s^{2} + s + 4}$

$G (s) = \frac{4}{s (s + 1)}$

Now after implementing PD controller CLTF will be

$\frac{G_{1} (s)}{1 + G_{1} (s) H (s)} = \frac{4 (1 + K_{D} s)}{s^{2} + (1 + 4 K_{D}) s + 4}$

Now maximum overshoot to be zero

$ξ \geq 1$

$ω_{n} = 2$

$2 ξ ω_{n} = 1 + 4 K_{D}$

$ξ = \frac{1 + 4 K_{D}}{4}$

So, $\frac{1 + 4 K_{D}}{4} \geq 1$

$1 + 4 K_{D} \geq 4$

$4 K_{D} \geq 3$

$K_{D} \geq 0.75$

So minimum value of $K_{D}$ will be 0.75.

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