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R-H range of K for K/[s(s²+s+1)(s+2)]

Short answer

Characteristic equation: s⁴ + 3s³ + 3s² + 2s + K =0. Routh array requires K>0 and 2−9K/7>0, so K<14/9≈1.556. The stable range is 0 < K < 14/9.

Find the range of 'K' for stable operation using R-H criteria. [Figure: R(s) → summing point (+, −) → K → 1/[s(s²+s+1)(s+2)] → C(s); unity negative feedback]

For unity feedback, the characteristic equation is 1+G(s)=01+G(s)=0, with G(s)=Ks(s2+s+1)(s+2)G(s)=\frac{K}{s(s^2+s+1)(s+2)}.

Characteristic equation

s(s2+s+1)(s+2)+K=0s4+3s3+3s2+2s+K=0\begin{aligned} s(s^2+s+1)(s+2) + K &= 0 \\ s^4 + 3s^3 + 3s^2 + 2s + K &= 0 \end{aligned}

Routh array

RowCol 1Col 2Col 3
s4s^413KK
s3s^3320
s2s^23(3)−1(2)3=73\frac{3(3)-1(2)}{3}=\frac{7}{3}KK
s1s^173(2)−3K73=2−9K7\frac{\frac73(2)-3K}{\frac73}=2-\frac{9K}{7}0
s0s^0KK

Conditions

  • s1s^1 row: 2−9K7>0⇒K<149=1.5562 - \frac{9K}{7} > 0 \Rightarrow K < \frac{14}{9} = 1.556
  • s0s^0 row: K>0K > 0
0<K<1.5560 < K < 1.556

At K=14/9K = 14/9 the system is marginally stable. The auxiliary equation 73s2+149=0\frac73 s^2 + \frac{14}{9} = 0 gives ω=2/3=0.816\omega = \sqrt{2/3} = 0.816 rad/s.

Answer: stable for 0<K<1490 < K < \frac{14}{9} (= 1.556).

✓ Answer checked by Claude Opus 5.5 (high effort)

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