Downward Gain Margin

Topics: Feedback Control Systems

The concept is very similar to the Upward Gain Margin but in the opposite direction. The downward gain margin answers the following question:

How much can the system's gain be decreased before the closed-loop system reaches the boundary of instability?

$$\downarrow GM = \underset{0\leq k_0 < 1}{\min} \Big\{ k_0 \, | \text{ closed-loop is stable} \Big\}$$

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

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

where $\omega_{GM}$ is the largest frequency below 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