GATE 2024 EC – Question 38
A satellite attitude control system, as shown below, has a plant with transfer function $G(s)=\dfrac1{s^2}$ cascaded with a compensator $C(s)=\dfrac{K(s+\alpha)}{s+4}$, where $K$ and $\alpha$ are positive real constants.
In order for the closed-loop system to have poles at $-1\pm j\sqrt3$, the value of $\alpha$ must be ________.
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Correct answer: (B) 1
Explanation
The characteristic equation is $s^2(s+4)+K(s+\alpha)=s^3+4s^2+Ks+K\alpha=0$. With poles at $-1\pm j\sqrt3$ and a third real pole $p$: the sum of roots is $-4$, so $-2+p=-4$ and $p=-2$. The pair contributes $(s^2+2s+4)$, so the polynomial is $(s^2+2s+4)(s+2)=s^3+4s^2+8s+8$. Hence $K=8$ and $K\alpha=8$, giving $\alpha=1$.