Topic: Control Systems
A system described by the transfer function $H(s)=\frac{1}{s^{3}+\alpha s^{2}+ks+3}$ is stable. The constraints on $\alpha$ and k are
- A. $\alpha>0$, $\alpha k<3$
- B. $\alpha>0$, $\alpha k>3$
- C. $\alpha>0$, $\alpha k>0$
- D. $\alpha>0$, $\alpha k<0$
Answer: B. $\alpha>0$, $\alpha k>3$
Explanation: The characteristic equation is $s³ + \alpha s² + ks + 3 = 0$.
Constructing the Routh array:
$s³$: 1, k
$s²$: $\alpha$, 3
$s^1$: $\frac{\alpha k - 3}{\alpha}$, 0
$s^0$: 3
For the system to be stable, all elements in the first column must be strictly positive. Thus, $\alpha > 0$ and $\frac{\alpha k - 3}{\alpha} > 0 \Rightarrow \alpha k > 3$.