Final answer:
Design the gain K for a unity feedback control system with G(s) to achieve 5% overshoot by finding the corresponding damping ratio and natural frequency. Then calculate the settling time using the established second-order system formulas.
Step-by-step explanation:
The student is seeking to design a gain value, K, for a unity feedback control system to achieve a specific overshoot characteristic, and to determine the system's settling time. The system's transfer function is given as G(s)=K/(s+1)(s+3)(s+5). To design for a 5% overshoot, we need to determine the desired damping ratio (ζ) that corresponds to this overshoot level.
A 5% overshoot corresponds to a damping ratio of approximately 0.7, which can be found using standard overshoot formula or tables found in control systems textbooks. Using this information, we apply the standard second-order system characteristics to find K. Once K is found, the settling time can be determined using the formula 4/(ζωn), where ωn is the natural frequency of the system.
However, to completely answer this question, we would need to perform a root locus or use similar methods to find the appropriate gain K and then determine the natural frequency. Knowing all these parameters allows us to calculate the precise settling time for the system with the given damping ratio and gain value.