Final answer:
The student seeks help with designing a lead controller to reduce the system's settling time by half, while maintaining the percent overshoot by placing the lead zero at s=-8 and determining the lead pole location.
Step-by-step explanation:
The student is asking for help in designing a lead controller for a system described in Ex840, with the goal of halving the settling time while maintaining the same percent overshoot. The specific requirements include placing the lead zero at s=-8 and finding the location of the controller-lead pole.
The settling time is directly related to the poles of the closed-loop transfer function of a system. To achieve a twofold reduction in settling time, the real parts of the dominant poles must be moved further to the left in the s-plane. This is accomplished by adding a lead compensator, which includes a zero (which speeds up the response) and a pole (which ensures the stability of the system).
To keep the same percent overshoot, the damping ratio (ΞΆ) of the system should remain constant. Percent overshoot is related to the damping ratio, and for less-than-critical damping (underdamped systems), the system oscillates before settling. The position of the lead pole should be chosen to ensure that the modified system has the same damping ratio as the original system.