Find the percent overshoot, settling time, rise time, and peak time for T(s) 14.145 (s2+0.824s+2.892)(s+5)
The Correct Answer and Explanation is :
To determine the percent overshoot, settling time, rise time, and peak time for the transfer function ( T(s) = \frac{14.145}{(s^2 + 0.824s + 2.892)(s + 5)} ), we need to analyze the system’s behavior in the time domain. Here’s a step-by-step process to derive these characteristics.
Step 1: Identify the system’s poles
We begin by simplifying the transfer function into a standard second-order system form and finding the poles of the system. The given transfer function has two terms in the denominator:
- ( (s^2 + 0.824s + 2.892) ), which corresponds to the second-order part, and
- ( (s + 5) ), a first-order pole at ( s = -5 ).
First, we can solve for the poles of the second-order system by using the quadratic formula on ( s^2 + 0.824s + 2.892 = 0 ):
[
s = \frac{-0.824 \pm \sqrt{(0.824)^2 – 4(1)(2.892)}}{2(1)}
]
Solving this:
[
s = \frac{-0.824 \pm \sqrt{0.6781 – 11.568}}{2} = \frac{-0.824 \pm \sqrt{-10.8899}}{2}
]
This gives complex conjugate poles, indicating underdamped behavior. We can express these poles as ( s = \sigma \pm j \omega_n ), where ( \sigma ) is the real part, and ( \omega_n ) is the imaginary part.
Step 2: Determine the damping ratio and natural frequency
From the quadratic equation, we can find the damping ratio ( \zeta ) and the natural frequency ( \omega_n ). For a second-order system ( s^2 + 2\zeta \omega_n s + \omega_n^2 ), the values are:
- ( \zeta = \frac{0.824}{2\sqrt{2.892}} \approx 0.423 ),
- ( \omega_n = \sqrt{2.892} \approx 1.7 ).
Step 3: Time domain specifications
Using these values, we can calculate the time-domain specifications.
- Percent Overshoot (PO):
The formula for percent overshoot is: [
PO = 100e^{\frac{-\zeta \pi}{\sqrt{1 – \zeta^2}}}
] Substituting ( \zeta = 0.423 ): [
PO = 100e^{\frac{-0.423 \pi}{\sqrt{1 – 0.423^2}}} \approx 16.75\%
] - Settling Time (Ts):
Settling time for a second-order system is approximately: [
Ts = \frac{4}{\zeta \omega_n} = \frac{4}{0.423 \times 1.7} \approx 5.8 \, \text{seconds}
] - Rise Time (Tr):
The rise time is approximated for a second-order system by: [
Tr \approx \frac{\pi – \theta}{\omega_n \sqrt{1 – \zeta^2}}
] where ( \theta = \arccos(\zeta) ). Using ( \zeta = 0.423 ): [
Tr \approx \frac{\pi – \arccos(0.423)}{1.7} \approx 3.1 \, \text{seconds}
] - Peak Time (Tp):
The peak time for a second-order system is: [
Tp = \frac{\pi}{\omega_n \sqrt{1 – \zeta^2}} \approx \frac{\pi}{1.7 \times \sqrt{1 – 0.423^2}} \approx 3.1 \, \text{seconds}
]
Conclusion
The calculated time-domain specifications for the system are:
- Percent Overshoot (PO): 16.75%
- Settling Time (Ts): 5.8 seconds
- Rise Time (Tr): 3.1 seconds
- Peak Time (Tp): 3.1 seconds
These parameters provide insights into the transient response of the system.