Upward Gain Margin

Topics: Feedback Control Systems

The upward gain margin answers the following question:

What is the maximum factor by which the loop gain $k$ can be increased before the closed-loop system reaches the boundary of instability?

$$\uparrow GM = \underset{1\leq k_0 < \infty}{\max} \Big\{ k_0 \, | \text{ closed-loop is stable} \Big\}$$

So, following the Relation Between Bode Plot and Root Locus, we can find the upward gain margin as

$$\uparrow GM = \frac{1}{|L(j\omega_{GM})|}$$

where $\omega_{GM}$ is the smallest frequency above the unity-crossover frequency $(|L(j \omega_{GM})| < 1)$ such that

$$\angle L(j \omega_{GM}) = -180 \degree \pm n 360 \degree$$

References

1. https://ocw.mit.edu/courses/16-06-principles-of-automatic-control-fall-2012/abb629fb7c2e395ee8c6df58d4754329_MIT16_06F12_Lecture_23.pdf