Control system and why feedback makes response faster

Control system

A control system is an arrangement of components connected to command, direct or regulate a system so that the output reaches the desired value. When the output is measured and fed back to correct the input, it is a feedback (closed-loop) system.

Feedback makes the response faster

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):

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

Comparing:

The time constant is reduced by $(1+K)$, so the output reaches its final value $(1+K)$ times sooner.

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.

 c(t)
  |        .----------- closed loop (fast)
  |      /    ..------- open loop (slow)
  |     / .'
  |    /.'
  |   /'
  +-------------------------> t

(The curves are scaled to the same final value; in practice the reduced closed-loop gain is made up by a preamplifier.)

For a second-order system $K/[s(s+a)]$ with unity feedback, $\omega_n = \sqrt{K}$ also rises with gain, so rise time and peak time fall. Hence adding feedback makes the response faster.