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R-H root distribution for 200/[s(s³+6s²+11s+6)]

Short answer

Characteristic equation s^4+6s^3+11s^2+6s+200=0. Routh array shows two sign changes, so 2 roots lie in RHP, 2 in LHP, and none on the jω axis. The system is unstable.

Use RH criterion to determine number of roots on right side, left side and on the imaginary axis itself of s-plane for the system below. [Figure: unity negative feedback system with forward path G(s) = 200/[s(s³ + 6s² + 11s + 6)]]

Result: RHP = 2, LHP = 2, jωj\omega axis = 0. The system is unstable.

For unity negative feedback the characteristic equation is 1+G(s)=01+G(s)=0. The number of sign changes in the first column of the Routh array gives the number of roots in the right half plane.

Characteristic equation

s(s3+6s2+11s+6)+200=0s4+6s3+11s2+6s+200=0\begin{aligned} s(s^3+6s^2+11s+6) + 200 &= 0 \\ s^4 + 6s^3 + 11s^2 + 6s + 200 &= 0 \end{aligned}

Routh array

RowCol 1Col 2Col 3
s4s^4111200
s3s^3660
s2s^26(11)−1(6)6=10\frac{6(11)-1(6)}{6}=106(200)−06=200\frac{6(200)-0}{6}=200
s1s^110(6)−6(200)10=−114\frac{10(6)-6(200)}{10}=-1140
s0s^0200

First column: 1, 6, 10, −114, 2001,\ 6,\ 10,\ -114,\ 200.

Sign changes: +10→−114+10 \to -114 (one) and −114→+200-114 \to +200 (two), so there are 2 sign changes.

Root location

RegionNumber of roots
Right half s-plane2
Left half s-plane4−2=24-2 = 2
On jωj\omega axis0 (no zero row)

The numerical roots (s≈1.28±j2.54s \approx 1.28 \pm j2.54 and s≈−4.28±j2.54s \approx -4.28 \pm j2.54) agree with this.

Answer: 2 roots in the RHP, 2 in the LHP, none on the imaginary axis. The closed-loop system is unstable; the gain of 200 is too high.

✓ Answer checked by Claude Opus 5.5 (high effort)

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