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A control system with a PD controller is shown in the figure. If the velocity error constant $K_v=1000$ and the damping ratio $\zeta=0.5$, then the values of $K_P$ and $K_D$ are

1. $K_P=100,\ K_D=0.09$
2. $K_P=100,\ K_D=0.9$
3. $K_P=10,\ K_D=0.09$
4. $K_P=10,\ K_D=0.9$

Correct answer: $K_P=100,\ K_D=0.9$

Solution

For a PD-controlled unity-feedback system, the velocity error constant determines the low-frequency gain, which gives $K_P=100$ from $K_v=1000$ in the given setup. Matching the closed-loop characteristic equation to the standard second-order form with $\zeta=0.5$ gives $K_D=0.9$.

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