GATE 2018 EC – Question 51
For a unity feedback control system with the forward path transfer function
$$G(s)=\frac{K}{s(s+2)}$$
The peak resonant magnitude $M_r$ of the closed-loop frequency response is 2. The corresponding value of the gain $K$ (correct to two decimal places) is ________.
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Correct answer: 14.86 to 15.00
Explanation
The closed loop is a standard second-order system with $\omega_n^2=K$ and $2\zeta\omega_n=2$. The peak magnitude is $M_r=\frac{1}{2\zeta\sqrt{1-\zeta^2}}=2$, so $\zeta^2(1-\zeta^2)=\frac1{16}$ and $\zeta^2=\frac{1-\sqrt{0.75}}{2}=0.067$, i.e. $\zeta=0.2588$. Then $\omega_n=\frac{1}{\zeta}=3.864$ and $K=\omega_n^2=14.93$.