How feedback rejects disturbance input

A feedback system senses the effect of the disturbance on the output and produces a correcting control action, so the disturbance is largely cancelled.

Disturbance rejection. Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

• Open loop: $C_D(s) = G_2(s)D(s)$.
• Closed loop (feedback $H$), with $R = 0$ (superposition):

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

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

Example: $G_1G_2H = 99$ means only 1% of the disturbance effect remains at the output, compared with 100% in open loop. In a motor speed system, a sudden load torque lowers the speed; the tachometer detects this, the error rises, and the amplifier increases armature voltage to restore the speed.