Question

The open loop transfer function of a unity feedback system is

\(G(S) = \frac{K}{{S\left( {{S^2} + S + 2} \right)\left( {S + 3} \right)}}\)

The range of K for which the system is stable is

1. \(\frac{{21}}{{4}} > K > 0\)
2. 13 > K > 0
3. \(\frac{{21}}{{44}} < K < \infty\)
4. -6 < K < ∞

Answer 1

Correct Answer - Option 1 : \(\frac{{21}}{{4}} > K > 0\)

\(G\left( s \right) = \frac{k}{{s\left( {{s^2}\; + \;s\; + \;2} \right)\left( {s\; + \;3} \right)}}\)

C.E = 1 + G(s) H(s) = 0

s(s² + s + 2) (s + 3) + K = 0

s(s³ + 3s² + s² + 3s + 2s + 6 + K) = 0

s⁴ + 3s³ + s³ + 3s² + 2s² + 6s + K = 0

s⁴ + 4s³ + 5s² + 6s + K = 0

S4	1	5	K
S3	4	6	0
S2	\(\frac{{20 - 6}}{4}\)	\(\frac{{4K}}{4}\)	0
S1	\(\frac{{3.5\; \times \;6 - 4K}}{{3.5}}\)	0
S0	K

 

3.5 × 6 - 4K > 0

21 > 4K

\(K < \frac{{21}}{4}\) and K > 0