Closed loop reduces internal disturbances

Internal disturbances are changes inside the system itself: variation of plant parameters (ageing, temperature, wear, supply changes) and internal noise. Feedback reduces their effect because the output is measured and the error is corrected.

1. Parameter variation

The sensitivity of the overall gain $T$ to a change in $G$ is

$$S_G^T = \frac{\partial T/T}{\partial G/G} = \frac{\partial T}{\partial G}\cdot\frac{G}{T}$$

• Open loop: $T = G$, so $S_G^T = 1$ (a 10% change in $G$ gives a 10% change in output).
• Closed loop: $T = \dfrac{G}{1+GH}$, so $\dfrac{\partial T}{\partial G} = \dfrac{1}{(1+GH)^2}$ and

$$S_G^T = \frac{1}{(1+GH)^2}\cdot\frac{G(1+GH)}{G} = \frac{1}{1+GH}$$

Since $|1+GH| \gg 1$, the effect of a change in $G$ is reduced by the factor $(1+GH)$. (With $GH = 99$, a 10% change in $G$ changes the output by only 0.1%.)

2. Disturbance generated inside the plant

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

A DC motor's armature resistance rises as it heats. In open loop the speed falls in proportion. With tacho feedback and loop gain $K = 49$, the change in speed is only $1/(1+49) = 2\%$ of the open-loop change.

Hence the closed-loop system is far less affected by internal disturbances, as long as the loop gain is high and the sensor itself is accurate (the sensitivity to $H$ is $S_H^T = -GH/(1+GH) \approx -1$).