Effect of feedback on dynamic response

Feedback changes the dynamic (transient) response of a system: it changes the time constant, bandwidth, damping and stability.

1. Time constant and speed

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):

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

Feedback reduces the time constant by $(1+K)$, so the response is faster and the bandwidth increases from $1/\tau$ to $(1+K)/\tau$.

2. Damping and overshoot (second order)

With $G(s) = \dfrac{K}{s(\tau s+1)}$ and unity feedback:

$$\frac{C(s)}{R(s)} = \frac{K/\tau}{s^2 + s/\tau + K/\tau}, \quad \omega_n = \sqrt{\frac{K}{\tau}}, \quad \zeta = \frac{1}{2\sqrt{K\tau}}$$

Raising the feedback loop gain $K$ increases $\omega_n$ (faster rise) but reduces $\zeta$, so overshoot $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ increases. Adding rate (tachometer) feedback $K_ts$ changes damping to

$$\zeta' = \frac{1 + KK_t}{2\sqrt{K\tau}}$$

which reduces overshoot without changing $\omega_n$.

3. Stability

Feedback moves the poles. It can stabilize an unstable plant (e.g. $1/(s-1)$ with gain $K > 1$ gives the stable pole $s = 1 - K$) or destabilize a stable one when gain is too high (e.g. $K/[s(s+1)(s+2)]$ is unstable for $K > 6$).

4. Reduced effect of disturbances and parameter changes on the transient

Both are reduced by $(1+GH)$, so the transient shape stays nearly the same even when plant parameters drift.