Question

Choose the transfer function that best fits the output response to a unit step input change shown in the figure:

 

• A. \(\frac{(as + 1)e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)^2}\)
• B. \(\frac{(as + 1)e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)}\)
• C. \(\frac{(as + 1)}{(\tau_1 s + 1)(\tau_2 s + 1)^2}\)
• D. \(\frac{(as + 1)^2 e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)^2}\)

Hint:

When fitting transfer functions to observed data, consider how each component—zeros, poles, and delays—affects the system's dynamic response. This holistic approach aids in more accurately capturing the system's characteristics.

Answer 1

The Correct Option is A

Step 1: Analyze the Transfer Function Configuration.
The presence of a zero and a time delay along with the squared second pole in this configuration provides a dynamic response that begins with a delayed start, quickly rises, overshoots, and then settles, matching the behavior observed in the response curve. 
Step 2: Explain the Fitting Criteria.
The time delay (\(e^{-\theta s}\)) accounts for the initial pause before the response begins.
The zero (\(as + 1\)) enhances the system's response speed post-delay.
The squared pole term \((\tau_2 s + 1)^2\) introduces the necessary damping to control the overshoot and allows the system to settle at a new steady state efficiently.