Gain Margin and Phase Margin for Controls

We consider this just for unity feedback systems. This essentially controls how much variation is allowed before instability kicks in.

Gain Margin and Phase Margin

Gain margin $G_M$ is the magnitude below 0 dB at the frequency where the phase becomes 180°.

The phase margin is the phase angle above -180° at the frequency where the gain becomes 0 dB.

If $G_M$ and ${\Phi }_M$ are positive, the closed-loop system is stable for unity feedback.

Main Idea

How much variation of gain and phase is allowed to the open-loop system while the closed-loop system remains stable? (when the loop is closed by a unity feedback)

The reciprocal of the gain margin is the maximum amount that the gain can be before the system become unstable by only looping at the open-loop frequency response.

• $GM=20{\log }_{10}\left(\frac{1}{|G(j{\omega }_{180})|}\right)$

Ex. For the gain main, $G_M=14dB$ your critical $K=5$ because $G_M=20\log \left(5\right)=-20\log \left(\frac{1}{5}\right)\text{ on the bode plot}$

For the phase margin, this represents how much phase (time) delay you can allow without making the system unstable, when you go below the critical angle of 180°, the closed-loop system becomes unstable

• $PM=180^{\circ }+\angle G(j{\omega }_{gc})$ You can find the time delay like this for a phase margin of 87°…
• $\Delta t=\frac{87^{\circ }}{\omega }=\frac{1.5\text{[rad]}}{1.01\text{[rad/s]}}\approx 1.5\text{ sec.}$

Also note that a Re / Im plot is for the closed loop system but the bode plot is for the open-loop system.

Solving

From the GM and PM equations, we basically plug in to the magnitude and phase equations defined in Frequency Domain Analysis…

• GM: $<G\left(j\omega \right)=-180$ and solve for $\omega$
  • Now find the amplitude ($\left|G\left(j\omega \right)\right|$) at the found $\omega$
  • Use $20{\log }_{10}\left(\frac{1}{K}\right)=-20{\log }_{10}K=GM$
• PM: Now find $\omega$ that satisfies $\left|G\left(j\omega \right)\right|=1$ (probably use wolfram)
  • Find the phase angle $<G\left(j\omega \right)$ at the found $\omega$
  • The phase margin (PM) is now = -180°- $<G\left(j\omega \right)$

The PM is directly related to $\zeta$ as $PM={\tan }^{-1}\left(\frac{2\zeta }{\sqrt{\sqrt{1+4{\zeta }^2}-2{\zeta }^2}}\right)$!
Also note that overshoot has a 1:1 relationship to the damping ratio $\zeta$ which has a 1:1 relationship with the phase margin. To get k, you can use the closed-loop transfer function to solve for ${\omega }_n$, we should now have $\zeta$, and we can use a comaprison of coefficients in the characteristic polynomial to derive a required k!