Topic: Control Systems
The characteristic equation of a system is given by: $s^{4}+2s^{3}+3s^{2}+4s+K=0$. Determine the range of values of K for which the system is stable according to the Routh-Hurwitz criterion.
- A. $0<K<6$
- B. $0<K<8$
- C. $0<K<10$
- D. $0<K<2$
Answer: D. $0<K<2$
Explanation: Constructing the Routh array for the polynomial $s^{4}+2s^{3}+3s^{2}+4s+K=0$:
$s^{4}$: 1, 3, K
$s^{3}$: 2, 4, 0
$s^{2}$: $\frac{2(3) - 1(4)}{2} = 1$, $\frac{2(K) - 1(0)}{2} = K$
$s^{1}$: $\frac{1(4) - 2(K)}{1} = 4 - 2K$
$s^{0}$: K
For the system to be strictly stable, all elements in the first column must be strictly positive. Thus, we require:
$4 - 2K > 0 \implies K < 2$
$K > 0$
Therefore, the range for stability is $0 < K < 2$.