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R-H range of K and oscillation for K(s+13)/[s(s+3)(s+7)]

Short answer

Characteristic equation s^3+10s^2+(21+K)s+13K=0. Routh array gives stability for 0<K<70. At K=70 the s^1 row vanishes, giving marginal stability with sustained oscillation frequency ω=√91≈9.54 rad/s.

A unity feedback system has the feedforward transfer function G(s) = K(s+13)/[s(s+3)(s+7)]. Using R-H Criterion, find the range of K for the system to be stable. When does the system just oscillate and what would be that frequency during sustained oscillation?

The closed-loop characteristic equation 1+G(s)=01+G(s)=0 is tested with the Routh array. The system just oscillates (is marginally stable) when the s1s^1 row becomes zero.

Characteristic equation

s(s+3)(s+7)+K(s+13)=0s3+10s2+(21+K)s+13K=0\begin{aligned} s(s+3)(s+7) + K(s+13) &= 0 \\ s^3 + 10s^2 + (21+K)s + 13K &= 0 \end{aligned}

Routh array

RowCol 1Col 2
s3s^3121+K21+K
s2s^21013K13K
s1s^110(21+K)−13K10=210−3K10\frac{10(21+K)-13K}{10}=\frac{210-3K}{10}0
s0s^013K13K

Range of K

  • s1s^1 row: 210−3K>0⇒K<70210-3K>0 \Rightarrow K<70
  • s0s^0 row: 13K>0⇒K>013K>0 \Rightarrow K>0
0<K<700 < K < 70

Sustained oscillation

At K=70K = 70 the s1s^1 row is zero, so the system just oscillates. Auxiliary equation from the s2s^2 row:

10s2+13K=010s2+910=0s2=−91⇒s=±j9.54\begin{aligned} 10s^2 + 13K &= 0 \\ 10s^2 + 910 &= 0 \\ s^2 &= -91 \Rightarrow s = \pm j9.54 \end{aligned}

Answer: stable for 0<K<700<K<70; it just oscillates at K=70K=70, with frequency ω=91=9.54\omega=\sqrt{91}=9.54 rad/s.

✓ Answer checked by Claude Opus 5.5 (high effort)

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