Stability Analysis from Bode Plot

 

Stability Analysis from Bode Plot

Bode plots allow us to determine system stability by evaluating Gain Margin (GM) and Phase Margin (PM). These margins measure how close the system is to instability and provide a practical way to design controllers.

Key Concepts

• Gain Crossover Frequency (${\omega }_{gc}$): Frequency where the magnitude plot crosses 0 dB.

• Phase Crossover Frequency (${\omega }_{pc}$): Frequency where the phase plot crosses -180°.

Gain Margin (GM)

• Definition: The amount of gain increase possible before the system becomes unstable.

• Measured at phase crossover frequency (${\omega }_{pc}$).

• Formula:

$$GM=-\text{Magnitude at }{\omega }_{pc}\text{(in dB)}$$

• Interpretation:

  • Positive GM → stable.

  • Negative GM → unstable.

Phase Margin (PM)

• Definition: The additional phase lag required to bring the system to instability.

• Measured at gain crossover frequency (${\omega }_{gc}$).

• Formula:

$$PM={180}^{\circ }+\text{Phase at }{\omega }_{gc}$$

• Interpretation:

  • Larger PM → more stable.

  • Small or negative PM → unstable.

Stability Conditions

• Stable system: GM > 0 dB and PM > 0°.

• Unstable system: GM < 0 dB or PM < 0°.

• Marginally stable: GM ≈ 0 dB or PM ≈ 0°.

Example

For a system with transfer function:

$$G(s)=\frac{10}{s(s+2)}$$

• From Bode plot:

  • Gain crossover frequency: ${\omega }_{gc}=5rad\mathrm{/}s$, Phase ≈ -135°.

  • Phase margin: $PM=180-135={45}^{\circ }$.

  • Phase crossover frequency: ${\omega }_{pc}=10rad\mathrm{/}s$, Magnitude ≈ -6 dB.

  • Gain margin: GM = +6 dB.

Conclusion: Since GM > 0 and PM > 0, the system is stable with good relative stability.