Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011 ### Control Systems and Transfer Functions#### Problem 4: Transfer Function AnalysisGiven:The transfer function for a process is \[ G(s) = \frac{Y'(s)}{U'(s)} = \frac{3}{10s^2 + 2s + 4} \]**Questions:**a) For a step change in the input \( U'(s) = \frac{5}{s} \), sketch the response \( y'(t) \). Show as much detail as possible, including the steady-state value of \( y'(t) \) and whether there is oscillation. *(Note: You do NOT need to solve the differential equation!)*b) What is the decay ratio?**Analysis:**To sketch the response \( y'(t) \) for a step change in the input, you need to consider how the system behaves based on its transfer function. The given transfer function is a second-order system which can potentially include oscillations depending on the damping ratio. The steady-state value can be found by evaluating the final value theorem, and the presence of oscillations can be inferred by analyzing the poles of the transfer function.**Graphical Representation:**If provided, a detailed sketch would include:- The initial transient response.- Any oscillations, if they occur.- The steady-state value which is reached as time goes to infinity.**Decay Ratio:**The decay ratio is a measure of the damping properties of the system, representing how successive peaks of oscillations decrease over time. It can typically be calculated if the damping ratio is known or estimated from the graph.For comprehensive analysis and accurate depiction, detailed calculations or simulations based on the transfer function would be required to achieve precise sketches and numerical values for decay ratios.

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Answered: 4. A process has the transfer function…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

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Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011

### Control Systems and Transfer Functions

#### Problem 4: Transfer Function Analysis

Given:
The transfer function for a process is

\[ G(s) = \frac{Y'(s)}{U'(s)} = \frac{3}{10s^2 + 2s + 4} \]

**Questions:**

a) For a step change in the input \( U'(s) = \frac{5}{s} \), sketch the response \( y'(t) \). Show as much detail as possible, including the steady-state value of \( y'(t) \) and whether there is oscillation.

*(Note: You do NOT need to solve the differential equation!)*

b) What is the decay ratio?

**Analysis:**

To sketch the response \( y'(t) \) for a step change in the input, you need to consider how the system behaves based on its transfer function. The given transfer function is a second-order system which can potentially include oscillations depending on the damping ratio. The steady-state value can be found by evaluating the final value theorem, and the presence of oscillations can be inferred by analyzing the poles of the transfer function.

**Graphical Representation:**

If provided, a detailed sketch would include:
- The initial transient response.
- Any oscillations, if they occur.
- The steady-state value which is reached as time goes to infinity.

**Decay Ratio:**

The decay ratio is a measure of the damping properties of the system, representing how successive peaks of oscillations decrease over time. It can typically be calculated if the damping ratio is known or estimated from the graph.

For comprehensive analysis and accurate depiction, detailed calculations or simulations based on the transfer function would be required to achieve precise sketches and numerical values for decay ratios.

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