Suppressing disturbance by increasing G(s) or H(s)

The statement is justified, provided that the gain increased is the one in the loop before the point where the disturbance enters, and that the system stays stable.

Let the disturbance $D(s)$ enter between $G_1(s)$ (controller) and $G_2(s)$ (plant), with feedback $H(s)$.

 R  +      +----+  +     +----+  C
 -->(Σ)--->| G1 |-->(Σ)->| G2 |--+-->
     ^ -   +----+    ^ D +----+  |
     |       +----+              |
     +-------| H  |<-------------+
             +----+

With $R = 0$ (superposition), the output due to the disturbance is

$$C_D(s) = \frac{G_2(s)}{1+G_1(s)G_2(s)H(s)}\,D(s)$$

For large loop gain, $|G_1G_2H| \gg 1$:

$$C_D(s) \approx \frac{G_2}{G_1G_2H}D(s) = \frac{D(s)}{G_1(s)H(s)}$$

So increasing $G_1$ and/or $H$ makes $C_D$ smaller, and the disturbance is suppressed. In open loop, $C_D = G_2D$, with no reduction at all.

Output due to the reference: $C_R = \dfrac{G_1G_2}{1+G_1G_2H}R \approx \dfrac{R}{H}$, which is almost independent of the gains, so tracking is not harmed.

Limits

• Raising $G_2$ alone does not help much: the result tends to $D/(G_1H)$, which does not depend on $G_2$.
• Raising $H$ also lowers the closed-loop gain ($\approx 1/H$), so it must be chosen with the required output level in mind.
• Very high gain can reduce damping or cause instability in higher-order systems, and it amplifies sensor noise.