What does absolute stability mean?

Whether the system is stable or not, i.e. all closed-loop poles are in the left half plane

The gain margin of a stable system

Which description fits relative stability?

It measures how far the poles are from the jω axis, e.g. through GM, PM or damping ratio

It is a yes/no test using Routh

It applies only to unstable systems

The s^3 row of s^6+2s^5+8s^4+12s^3+20s^2+16s+16 is all zero. What does that indicate?

Roots symmetric about the origin, found from the s^4 auxiliary equation

The polynomial has a negative coefficient

What is the verdict for the system?

Marginally stable, with simple jω roots and no RHP roots

Asked 2/27
2078 Bhadra · 2+6 marks

Relative and absolute stability; R-H of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16

Short answer

Absolute stability means all poles in the left half-plane; relative stability quantifies distance from jω-axis. Four simple poles lie on the jω-axis and two in the left half-plane, so the system is marginally stable.

What is relative and absolute stability? Check the stability of system with characteristic equation: s⁶ + 2s⁵ + 8s⁴ + 12s³ + 20s² + 16s + 16 = 0 using R-H criteria.

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.

Absolute stability	Relative stability
Stable or unstable (qualitative)	Degree of stability (quantitative)
Checks sign of real parts of poles	Checks distance of poles from $j\omega$ axis
Routh–Hurwitz, pole location	Shifted Routh, GM, PM, damping ratio

Stability of $s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0$

Routh array

Row	Col 1	Col 2	Col 3	Col 4
$s^6$	1	8	20	16
$s^5$	2	12	16	0
$s^4$	$\frac{2(8)-1(12)}{2}=2$	$\frac{2(20)-1(16)}{2}=12$	16
$s^3$	$\frac{2(12)-2(12)}{2}=0$	$\frac{2(16)-2(16)}{2}=0$

The $s^3$ row is all zeros, so there are roots placed symmetrically about the origin. Auxiliary equation from the $s^4$ row:

A(s) = 2s^4 + 12s^2 + 16, \qquad \frac{dA}{ds} = 8s^3 + 24s

Row	Col 1	Col 2
$s^3$	8	24
$s^2$	$\frac{8(12)-2(24)}{8}=6$	$\frac{8(16)-0}{8}=16$
$s^1$	$\frac{6(24)-8(16)}{6}=2.67$	0
$s^0$	16

First column: $1,\ 2,\ 2,\ 8,\ 6,\ 2.67,\ 16$ . No sign change, so no root is in the RHP.

Roots of the auxiliary equation

\begin{aligned} 2s^4+12s^2+16 &= 2(s^2+2)(s^2+4) = 0 \\ s &= \pm j1.414,\ \pm j2 \end{aligned}

These four roots lie on the $j\omega$ axis and are not repeated. The other factor is

\frac{s^6+2s^5+8s^4+12s^3+20s^2+16s+16}{(s^2+2)(s^2+4)} = s^2+2s+2 \;\Rightarrow\; s = -1\pm j1

Region	Number	Roots
Right half plane	0	none
$j\omega$ axis	4	$\pm j1.414$ , $\pm j2$
Left half plane	2	$-1\pm j1$

Conclusion

There is no root in the RHP, and the four roots on the $j\omega$ axis are simple (non-repeated). The system is marginally stable (not absolutely stable): its response contains sustained oscillations at $\omega = 1.414$ rad/s and $\omega = 2$ rad/s.

✓ Answer checked by Claude Opus 5.5 (high effort)

Relative and absolute stability; R-H of s⁵+s⁴+2s³+2s²+3s+5 · Control Systems, chapter 4 · asked 2×
R-H root distribution of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16 · Control Systems, chapter 4 · asked 4×
R-H stability of s⁴+8s³+18s²+16s+50 · Control Systems, chapter 4

All Control Systems chapter 4 questions →