Why closed loop rejects disturbances better

Control system

A control system is an arrangement of components that regulates, commands or directs a system so that its output reaches and stays at a desired value, e.g. speed control of a motor or temperature control of a room.

Why closed loop rejects disturbances better

An open-loop system does not measure the output, so any disturbance passes straight to the output. A closed-loop system senses the change in output and generates a corrective control action.

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

With $R = 0$ (superposition):

$$\text{Open loop: } C_D = G_2D, \qquad \text{Closed loop: } C_D = \frac{G_2}{1+G_1G_2H}D$$

The disturbance effect is reduced by the factor $(1+G_1G_2H)$. For large loop gain, $C_D \approx D/(G_1H)$.

Illustration

In a DC motor speed system, let a load torque cause a 100 rpm speed drop in open loop. With tachometer feedback and loop gain $G_1G_2H = 19$, the drop becomes $100/(1+19) = 5$ rpm. The tachometer sees the fall in speed, the error increases, and the amplifier raises the armature voltage, restoring the speed almost fully. With an integral controller the steady-state drop becomes zero.