Lead Compensation

$$
D_c(s)=\frac{T_Ds+1}{\alpha T_Ds+1},\ \alpha < 1
$$

The phase contributed by the lead compensator is given by

$$
\phi = \tan^{-1}(T_D \omega) - \tan^{-1}(\alpha T_D \omega)
$$

Maximum phase occurs when the frequency is

$$
\omega_{\mbox{max}} = \frac{1}{T_D\sqrt{\alpha}}
$$

and

$$
\sin \phi_{\mbox{max}}=\frac{1-\alpha}{1+\alpha}\ \mbox{or}\ \alpha = \frac{1-\sin \phi_{\mbox{max}}}{1+\sin \phi_{\mbox{max}}}
$$

Design Procedure

Determine the gain $K$ to satisfy error requirement.
Evaluate the PM of the uncompensated system using the value of $K$ obtained from step 1.
Allow for $5^{\circ}$ to $10^{\circ}$ more margin, and determine the needed phase lead $\phi_{\mbox{max}}$.
Determine $\alpha$ from $\alpha = \frac{1 − \sin \phi_{\mbox{max}}}{1 + \sin \phi_{\mbox{max}}}$.
Find $\omega$ such that $|KD_c(s)G(s)| = 1$, i.e. $|K G(s)|=\sqrt{\alpha}$. The zero is then at $1/T_D = \omega\sqrt{\alpha}$ and the pole is at $1/(\alpha T_D) = \omega /\sqrt{\alpha}$.
Check the resulting PM.
Iterate design by adjusting compensator parameters until all specifications are met. Add additional lead compensator (double-lead compensator) if necessary.

Lag Compensation

$$
D_c(s) = \frac{T_Is+1}{\alpha T_Is+1},\ \alpha > 1
$$

Procedure

Determine the gain $K$ to satisfy error requirement.
Determine the frequency where the PM requirement would be satisfied if the magnitude curve crossed the 0-dB line at this frequency (allow for additional $5^{\circ}$in PM).
Find how much gain reduction is required at that frequency, i.e. compute the gain of the uncompensated system, $|KG(jω)|$, at this frequency, the magnitude of the lag compensator, $|Dc(jω)|$, is the inverse of this gain ($\frac{1}{|KG(j\omega)|}$) such that $|KD_c(j\omega)G(j\omega)| = 1$.
We also know that at high frequencies, $|D_c(j\omega)| \approx \frac{1}{\alpha}$. Find $\alpha$.
Next, place the lag compensator zero, $\frac{1}{T}$ at least $1$ decade below the frequency computed from step 2, i.e. $T = 10/\omega$.
Check the resulting PM.
Iterate design by adjusting compensator parameters until all specifications are met.

总结

Lead compensation, $D_c(s) = \frac{Ts + 1}{\alpha Ts + 1},\ \alpha < 1$, adds phase lead at a frequency band between the two break-points. Lead compensation will increase both the crossover frequency and the speed of the response.

Lag compensation, $D_c(s) = \frac{Ts + 1}{\alpha Ts+1},\ \alpha < 1$, can be used to decrease the frequency response magnitude at frequencies above the two break-points so that $\omega_c$ yields acceptable phase margin. Typically it provides a slower response compare to lead compensation.