Relative stability from Bode plot

Relative stability tells how far a stable system is from becoming unstable. On a Bode plot it is measured by the gain margin (GM) and phase margin (PM), read directly from the magnitude and phase curves of the open-loop transfer function $G(j\omega)H(j\omega)$.

Procedure

1. Draw the magnitude plot (dB) and phase plot of $G(j\omega)H(j\omega)$ on the same frequency axis.
2. Gain crossover frequency $\omega_{gc}$: where the magnitude curve crosses 0 dB.
3. Phase crossover frequency $\omega_{pc}$: where the phase curve crosses $-180^\circ$.
4. Gain margin:

$$GM = -20\log|G(j\omega_{pc})H(j\omega_{pc})|\ \text{dB}$$

i.e. the distance of the magnitude curve below 0 dB at $\omega_{pc}$. 5. Phase margin:

$$PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})$$

i.e. the distance of the phase curve above $-180^\circ$ at $\omega_{gc}$.

 dB  |\
   0 |-\----------+-------- w
     |  \  wgc    |  ^ GM
     |   \        |  v
 deg |            |
-180 |----+-------x-------- w
     |  ^ PM     wpc

Interpretation (minimum-phase systems)

• Larger margins mean more relative stability. GM tells by how much the gain can be increased before instability; PM tells how much extra phase lag (e.g. a time delay) can be added.
• Good design values: GM ≥ 6 dB and PM of about 30°–60°.
• PM is linked to damping: approximately $\zeta \approx PM/100$ for PM below about 70°. Small PM means a large overshoot.

Example: for a system with $\omega_{gc}=1$ rad/s, phase $-150^\circ$ there, and magnitude $-10$ dB at $\omega_{pc}$: PM = 30°, GM = 10 dB, so the system is stable with reasonable margin.