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R-H stability of 2/(2s⁵+3s⁴+2s³+s²+2s)

Short answer

Characteristic equation 2s^5+3s^4+2s^3+s^2+2s+2=0. Routh array first column: 2,3,1.33,−0.5,6,2. Two sign changes indicate 2 poles in RHP; the closed-loop system is unstable.

Check stability for the system with open loop transfer function G(s)H(s) = 2/(2s⁵ + 3s⁴ + 2s³ + s² + 2s) using R-H criterion.

Result: 2 sign changes, so the closed-loop system has 2 roots in the RHP and is unstable.

The closed-loop stability is decided by the characteristic equation 1+G(s)H(s)=01+G(s)H(s)=0.

2s5+3s4+2s3+s2+2s+2=0\begin{aligned} 2s^5+3s^4+2s^3+s^2+2s + 2 &= 0 \end{aligned}

Routh array

RowCol 1Col 2Col 3
s5s^5222
s4s^4312
s3s^33(2)−2(1)3=43\frac{3(2)-2(1)}{3}=\frac{4}{3}3(2)−2(2)3=23\frac{3(2)-2(2)}{3}=\frac{2}{3}0
s2s^243(1)−3⋅2343=−0.5\frac{\frac43(1)-3\cdot\frac23}{\frac43}=-0.543(2)−043=2\frac{\frac43(2)-0}{\frac43}=2
s1s^1−0.5⋅23−43(2)−0.5=6\frac{-0.5\cdot\frac23-\frac43(2)}{-0.5}=60
s0s^02

First column: 2, 3, 1.33, −0.5, 6, 22,\ 3,\ 1.33,\ -0.5,\ 6,\ 2.

There are 2 sign changes (1.33→−0.51.33 \to -0.5 and −0.5→6-0.5 \to 6), so 2 closed-loop poles lie in the right half s-plane.

Answer: the closed-loop system is unstable, with 2 poles in the RHP and 3 in the LHP. The roots are approximately 0.51±j0.750.51\pm j0.75, −0.76±j0.80-0.76\pm j0.80 and −1-1.

✓ Answer checked by Claude Opus 5.5 (high effort)

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