Question

Cansider the following system

R₍ₛ₎ → O →G ₍ₛ₎ → s+1/s²+s-2 → C

Design Lead compensator, such that
- the damping ratio is 0.707
- the undamped natural frequency is 7.07rad/s ₍ₛ₎

Answer 1

To design a lead compensator for the given system, calculate the desired values for the dominant poles and the zero of the compensator using the provided damping ratio and undamped natural frequency, and then construct the transfer function for the compensator.

Step-by-step explanation:

To design a lead compensator, we need to first determine its transfer function. The transfer function of the system is given as G(s) = s+1/s²+s-2. To calculate the parameters of the lead compensator, we can use the desired damping ratio (0.707) and undamped natural frequency (7.07 rad/s).

1. First, we calculate the desired values for the dominant poles of the compensated system using the provided damping ratio and undamped natural frequency.
2. Next, we calculate the value of the zero of the lead compensator using the desired dominant pole.
3. Finally, we construct the transfer function of the lead compensator by combining the pole and zero values calculated in the previous steps.

By following these steps, we can design a lead compensator for the given system.

Answer 2

A lead compensator is designed to achieve a damping ratio of 0.707 and an undamped natural frequency of 7.07 rad/s by adjusting the phase and gain of the system, correlating to a critically damped RLC circuit response.

Step-by-step explanation:

To design a lead compensator that achieves a damping ratio of 0.707 and an undamped natural frequency of 7.07 rad/s, we must establish a compensator that adjusts the phase and gain characteristics of the given system. The desired damping ratio (ζ) and undamped natural frequency (ωn) indicate a critically damped system that exhibits an optimal balance between overshoot and settling time within the transient response.

In the context of an RLC series circuit, this would correlate to a system where the resistance, inductance, and capacitance values determine an oscillation pattern that is neither too sluggish (overdamped) nor too oscillatory (underdamped). The lead compensator introduces additional phase lead (which can be helpful for stability and speed of response) and attenuates the gain at high frequencies to meet these specifications.