Advantages of closed loop with mathematical support

Two important advantages of a closed-loop (feedback) system over an open-loop system are (1) reduced sensitivity to parameter variations and (2) faster response. (Better disturbance rejection is a third.)

Advantage 1: Reduced sensitivity to parameter changes

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%.)

Advantage 2: Faster response (smaller time constant)

Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

Open loop: for a unit step input

$$c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau$$

Closed loop (unity feedback):

$$\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}$$ $$c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}$$

The time constant becomes $\tau/(1+K)$, so the system responds $(1+K)$ times faster. The loss of gain ($K \to K/(1+K)$) is easily recovered with a preamplifier.

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.