Final answer:
The transfer function for the system with two poles at p1 = -3-j7 and p2 = -3+j7 is (s - p1)(s - p2). By expanding and rearranging the terms, we can determine the damping ratio (ξ), natural frequency (ωn), time constant (Tp), percent overshoot (%OS), settling time (Ts), and the response type of the system. For this system, ξ is approximately -0.21, ωn is 7, Tp is approximately 0.204 seconds, %OS is approximately 48.9%, Ts is approximately 0.816-1.02 seconds, and the response type is overdamped.
Step-by-step explanation:
The system can be represented by the transfer function H(s) = (s - p1)(s - p2), where p1 and p2 are the poles of the system. For the given poles p1 = -3-j7 and p2 = -3+j7, we have H(s) = (s - (-3-j7))(s - (-3+j7)).
By expanding the equation and rearranging the terms, we get H(s) = (s + 3 + j7)(s + 3 - j7).
To find ξ (damping ratio), we compare the poles to the standard form of a second-order system, which is s = -ξωn ± jωn√(1 - ξ²), where ωn is the natural frequency. From the given poles, we can see that ωn = 7.
Since the poles are complex conjugates, ξ can be calculated as ξ = -3 / (2 * ωn) = -3 / (2 * 7). In this case, ξ is approximately -0.21.
The time constant Tp can be calculated using the formula Tp = 1 / (ωn * √(1 - ξ²)), which gives Tp = 1 / (7 * √(1 - (-0.21)²)). The time constant Tp is approximately 0.204 seconds.
The percent overshoot (%OS) can be estimated using the formula %OS = e^(πξ / √(1 - ξ²)) * 100, where e is the mathematical constant. Substituting the value of ξ into the formula, we get %OS = e^(π * -0.21 / √(1 - (-0.21)²)) * 100. The percent overshoot %OS is approximately 48.9%.
The settling time Tₛ is the time it takes for the system to reach and stay within a certain range of the final value. Generally, a settling time is considered to be 4-5 times the time constant Tp. So for this system, Tₛ is approximately 4-5 times 0.204 seconds, which is approximately 0.816-1.02 seconds.
The response type of the system can be determined by the values of ξ. In this case, since ξ is negative (-0.21), the system is classified as overdamped.