Derivative feedback controller: block diagram and effect

Derivative (rate or tachometer) feedback control feeds back a signal proportional to the rate of change of the output, $K_t\,dc/dt$, through an inner (minor) loop, in addition to the usual unity feedback. In a position servo this signal comes from a tachogenerator on the motor shaft.

Block diagram

R(s)    E(s)        +-------------------+     C(s)
--->(+)---->(+)---->| wn^2/(s(s+2z wn)) |--+---->
     ^-      ^-     +-------------------+  |
     |       |                             |
     |       +-------[ Kt s ]<-------------+
     |                                     |
     +-------------------------------------+

Transfer function

Inner loop (forward $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$, feedback $K_ts$):

$$\begin{aligned} G_1(s)&=\frac{G}{1+GK_ts}=\frac{\omega_n^2}{s(s+2\zeta\omega_n)+K_t\omega_n^2s}=\frac{\omega_n^2}{s\left(s+2\zeta\omega_n+K_t\omega_n^2\right)} \end{aligned}$$

Closing the unity outer loop:

$$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+\left(2\zeta\omega_n+K_t\omega_n^2\right)s+\omega_n^2}$$

Comparing with $s^2+2\zeta'\omega_ns+\omega_n^2$:

$$\zeta'=\zeta+\frac{K_t\omega_n}{2},\qquad \omega_n'=\omega_n$$

Effect on transient performance

• Damping increases ($\zeta'>\zeta$) while $\omega_n$ is unchanged.
• Peak overshoot reduces, since $M_p=e^{-\pi\zeta'/\sqrt{1-\zeta'^2}}$ falls as $\zeta'$ rises.
• Settling time reduces: $t_s=4/(\zeta'\omega_n)$.
• Rise time increases slightly (slower initial rise).
• Unlike derivative error (PD) control, it adds no zero to the closed loop, so it does not cause the extra overshoot a zero can produce.

Drawback: steady-state error

$$K_v=\lim_{s\to0}sG_1(s)=\frac{\omega_n}{2\zeta+K_t\omega_n},\qquad e_{ss}(\text{ramp})=\frac{2\zeta+K_t\omega_n}{\omega_n}$$

So the ramp error increases; this is usually offset by raising the forward gain.

Example: $\omega_n=4$ rad/s, $\zeta=0.25$ ($M_p=44\%$). With $K_t=0.15$: $\zeta'=0.25+0.15\times4/2=0.55$, and $M_p$ falls to about 12.6%.