The correct answer is C. Both (A) and (B).
A phase-lead compensator increases the phase margin of a system, which makes it less susceptible to overshoot and ringing. A phase-lag compensator increases the gain margin of a system, which makes it less susceptible to instability. Both of these effects can be used to increase the damping of a badly underdamped system.
A phase-lead compensator is a type of control system compensator that increases the phase margin of a system. This is done by adding a lead network to the system, which has a positive phase angle. The lead network effectively delays the phase of the output signal relative to the input signal, which increases the phase margin of the system.
A phase-lag compensator is a type of control system compensator that increases the gain margin of a system. This is done by adding a lag network to the system, which has a negative phase angle. The lag network effectively attenuates the high frequencies of the output signal, which increases the gain margin of the system.
Both phase-lead and phase-lag compensators can be used to increase the damping of a badly underdamped system. However, the choice of which type of compensator to use depends on the specific characteristics of the system. In general, phase-lead compensators are used for systems that have a low phase margin, while phase-lag compensators are used for systems that have a low gain margin.
In the case of a badly underdamped system, both the phase margin and the gain margin are low. Therefore, both a phase-lead compensator and a phase-lag compensator can be used to increase the damping of the system.