Final answer:
To meet both the settling time and percent overshoot specifications for the unity negative feedback system, the gain K must be set such that the closed loop poles align with the desired damping ratio and settling time. A root locus analysis is needed to identify the specific range of K that satisfies both criteria, but without this analysis, the exact value of K cannot be determined.
Step-by-step explanation:
The student has asked whether both settling time (Ts) and percent overshoot (P.O.) specifications can be met for a unity negative feedback system with an open loop transfer function of G(s) = K/s(s+5). To determine this, we analyze the system's performance in response to changes in gain K.
Settling Time (Ts)
For a second-order system, the settling time is related to the system's natural frequency and damping ratio. The desired settling time of 1 second or less implies that the poles of the closed loop transfer function must have a real part less than or equal to -4 (since Ts is approximately 4/(zeta*omega_n), where zeta is the damping ratio and omega_n is the natural frequency).
Percent Overshoot (P.O.)
The percent overshoot is related to the damping ratio, with 5% overshoot indicating a damping ratio zeta of approximately 0.7. This ensures the system is underdamped and will have some oscillations but should not exceed 5% of the final value (assuming a second-order approximation).
To meet both specifications, the gain K must be chosen such that the closed loop poles satisfy both the overshoot and settling time requirements. This typically involves creating a root locus and finding the range of K that maintains the desired damping ratio and location of the poles.
Without a complete root locus analysis, it cannot be determined if both specifications can be met simultaneously. However, if it is possible, the gain K would be in the range that satisfies the damping ratio for the desired percent overshoot and places the poles in a position that achieves the required settling time.