Final answer:
To find the system characteristics, we need to express the poles in the standard form. The first pole (p1) = -3 - j7, and the second pole (p2) = -3 + j7. The natural frequency (ωn) is 7 rad/s, the damping ratio (ξ) is 3/7, the time constant (Tp) is 1/7 s, the percent overshoot (%OS) can be calculated as exp(-3/7 * π / sqrt(1 - (3/7)^2)) * 100, and the settling time (Ts) is 4/3 s. The response type can be determined based on the value of the damping ratio (ξ).
Step-by-step explanation:
System Characteristics:
To find the system characteristics, we need to express the poles in the standard form. The first pole (p1) = -3 - j7, and the second pole (p2) = -3 + j7.
- The natural frequency (ωn) can be calculated as the magnitude of the imaginary part of the poles i.e., ωn = |Im(p1)| = |Im(p2)| = 7 rad/s.
- The damping ratio (ξ) can be calculated by dividing the real part of the pole (Re(p1) or Re(p2)) by twice the magnitude of the imaginary part i.e., ξ = |Re(p1)/(2 * Im(p1))| = |Re(p2)/(2 * Im(p2))| = 3/7.
- The time constant (Tp) can be calculated as the reciprocal of the natural frequency i.e., Tp = 1/ωn = 1/7 s.
- The percent overshoot (%OS) can be calculated using the damping ratio (ξ) by the formula %OS = exp(-ξ * π / sqrt(1 - ξ^2)) * 100. In this case, %OS = exp(-3/7 * π / sqrt(1 - (3/7)^2)) * 100.
- The settling time (Ts) can be approximated as 4 / (ξ * ωn) = 4 / (3/7 * 7) = 4/3 s.
- The response type can be determined by the damping ratio (ξ). If ξ < 1, it is an underdamped system. If ξ = 1, it is a critically damped system. If ξ > 1, it is an overdamped system.