Closed loop disturbance rejection and command tracking

A closed-loop system measures the output and corrects any error, so it both rejects disturbances and tracks the command input better than an open-loop system.

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 $G_1$.

In open loop the whole disturbance $G_2D$ reaches the output, because nothing senses it.

Command input tracking

Tracking error for unity feedback: $E(s) = \dfrac{R(s)}{1+G(s)}$, and $e_{ss} = \lim_{s\to0}sE(s)$.

• Step input: $e_{ss} = \dfrac{1}{1+K_p}$; with an integrator in $G$ (type 1), $e_{ss} = 0$.
• In open loop, the output equals the reference only if the plant gain is exactly calibrated; any change in the plant gives an error, since the sensitivity is 1. In closed loop the sensitivity is $1/(1+GH)$.

The closed-loop time constant $\tau/(1+K)$ is also smaller, so the output follows a changing reference more quickly.

Example: in a motor speed system with loop gain 99, a 10% change in motor gain changes the speed by only about 0.1% in closed loop, compared with 10% in open loop, and a load disturbance has only 1% of its open-loop effect.