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R-H relation between K and T for stability

Short answer

Characteristic equation s^3+10s^2+Ts+K=0. Routh array yields stability condition 0<K<10T with T>0. The system is marginally stable at K=10T, oscillating at ω=√T rad/s.

Using Routh-Hurwitz criterion determine the relation between "K" and "T" so that unity feedback control system whose open loop transfer function given below is stable. G(s) = K/(s[s(s+10) + T])

For unity feedback, the characteristic equation is 1+G(s)=01+G(s)=0. All first-column Routh elements must be positive.

Characteristic equation

s[s(s+10)+T]+K=0s3+10s2+Ts+K=0\begin{aligned} s[s(s+10)+T] + K &= 0 \\ s^3 + 10s^2 + Ts + K &= 0 \end{aligned}

Routh array

RowCol 1Col 2
s3s^31TT
s2s^210KK
s1s^110T−K10\frac{10T-K}{10}0
s0s^0KK

Conditions

  • s0s^0 row: K>0K > 0
  • s1s^1 row: 10T−K>0⇒K<10T10T - K > 0 \Rightarrow K < 10T
  • From these, T>0T > 0 (also needed so that all coefficients are positive).
0<K<10T0 < K < 10T

Interpretation: for a given TT, the gain can be raised up to 10T10T. At K=10TK = 10T the system is marginally stable and oscillates at ω=K/10=T\omega = \sqrt{K/10} = \sqrt{T} rad/s (from 10s2+K=010s^2+K=0). For example, with T=5T = 5 the system is stable for 0<K<500<K<50.

Answer: relation for stability: 0<K<10T0 < K < 10T (with T>0T>0).

✓ Answer checked by Claude Opus 5.5 (high effort)

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