1 Lag -lead Compensator

When a single lead or lag compensator cannot guarantee the specified design criteria, a lag-lead compensator is used.

In lag-lead compensator the lag part precedes the lead part. A continuous time lag-lead compensator is given by

$\displaystyle C(s) = K \; \frac{1+\tau_1s}{1+\alpha_1 \tau_1 s} \; \frac{1+\tau_2s}{1+\alpha_2 \tau_2 s}$ where, $\displaystyle \; \alpha_1>1, \;\; \alpha_2<1$

The corner frequencies are $\dfrac{1}{\alpha_1 \tau_1 }$ , $\dfrac{1}{\tau_1 }$ , $\dfrac{1}{\tau_2 }$ , $\dfrac{1}{\alpha_2 \tau_2 }$ . The frequency response is shown in Figure 1.

$\includegraphics[width=5.0in]{m5l7fig2.eps}$

Figure 1: Frequency response of a lag-lead compensator

In a nutshell,

• If it is not specified which type of compensator has to be designed, one should first check the PM and BW of the uncompensated system with adjustable gain K.

• If the BW is smaller than the acceptable BW one may go for lead compensator. If the BW is large, lead compensator may not be useful since it provides high frequency amplification.

• One may go for a lag compensator when BW is large provided the open loop system is stable.

• If the lag compensator results in a too low BW (slow speed of response), a lag-lead compensator may be used.