Final Answer:
The regions in the complex plane where the system poles need to be located to satisfy the specified transient response specifications are as follows: The poles must lie in the left-half plane, and the damping ratio (\(\zeta\)) should range between approximately 0.21 and 1.0 to meet the given requirements. The natural frequency (\(\omega_n\)) must be greater than 0.55 radians/second to ensure the system's behavior aligns with the specified overshoot, peak time, and settling time.
Step-by-step explanation:
To determine the allowable regions for the system poles, we consider the transient response specifications. A maximum overshoot of 316.3% suggests a damping ratio (\(\zeta\)) between 0.21 and 1.0. The peak time (\(t_p\)) and settling time (\(t_s\)) requirements further guide the acceptable pole locations. Given these specifications, the poles must reside in the left-half plane to ensure stability.
The damping ratio (\(\zeta\)) is crucial for understanding the system's response characteristics. For an underdamped response, \(\zeta\) is close to 1, while \(\zeta\) equal to 0 represents a critically damped system. The natural frequency (\(\omega_n\)) influences the speed of response. We determine the range of \(\zeta\) values to satisfy the overshoot, peak time, and settling time requirements.
In conclusion, the system poles should be located in the left-half plane, with the damping ratio (\(\zeta\)) ranging from approximately 0.21 to 1.0. This ensures that the system meets the specified overshoot, peak time, and settling time criteria. The natural frequency (\(\omega_n\)) must be greater than 0.55 radians/second to achieve the desired transient response behavior.