Relative and absolute stability; R-H of s⁵+s⁴+2s³+2s²+3s+5

Absolute and relative stability

• Absolute stability answers yes or no: is the system stable or not? A system is absolutely stable if all closed-loop poles lie in the left half of the s-plane, so every bounded input gives a bounded output and the natural response decays to zero. The Routh–Hurwitz criterion checks absolute stability.
• Relative stability measures how stable a stable system is, i.e. how far its poles are from the $j\omega$ axis (instability boundary). Poles further left decay faster and give less oscillation. It is measured by the real part of the dominant poles (settling time), damping ratio, or in the frequency domain by gain margin and phase margin. In Routh's method, it is found by shifting the axis ($s = z - \sigma$) and testing again.

Stability of $s^5 + s^4 + 2s^3 + 2s^2 + 3s + 5 = 0$

Only the first element of the $s^3$ row is zero (the row is not all zero), so it is replaced by a small positive number $\varepsilon$. Let $\varepsilon \to 0^+$:

There are 2 sign changes ($+\infty \to -2$ and $-2 \to +5$).

Answer: two roots lie in the right half s-plane, so the system is unstable. The other three roots are in the LHP. The actual roots are $\approx 0.72\pm j1.17$, $-0.60\pm j1.34$ and $-1.24$.