R(s) 1 C(s) Ke s+1 s² + 2s + 1 a) Calculate the equivalent transfer function? b) Check the stability of the system using Routh's stability criterion. What will be the range of values of k, for a stable system?

R(s) 1 C(s) Ke s+1 s² + 2s + 1 a) Calculate the equivalent transfer function? b) Check the stability of the system using Routh's stability criterion. What will be the range of values of k, for a stable system?

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

Q1. For the following system:
R(s)
1
1
C(s)
s+1
s2 + 2s + 1
a) Calculate the equivalent transfer function?
b) Check the stability of the system using Routh's stability criterion. What will be the range
of values of k, for a stable system?
c) Plot Root Locus for the above transfer function using MATLAB. (Hints: Assume ke =1,
define the transfer function first)
d) If the input is a unit step, use ilaplace function in MATLAB to find the response c(t).
(Assume k = 1)
e) Plot the step response of the system using the calculated step response in d).
f) Plot the step response using the step function in MATLAB on the same figure.

g) When the system is stable, which value of k, will provide the best performance for step
response? Why? Consider performance indicators like: peak time, rise time, ... etc.
Hint: Plot the responses using different k, values on the same figure. (Only evaluate the
odd positive integers in the range of values of k. you found in part b) for a stable system)
Q2. Repeat a) from Q1, after adding proportional integral to the system.
ka
(*. 1
b) Check the stability of the system using Routh's stability criterion. What will be the range
of values of ka for a stable system?
c) Plot Root Locus for the above transfer function using MATLAB. (Hints: Assume ka = 1,
define the transfer function first)
d) If the input is a unit step, use ilaplace function in MATLAB to find the response c(t).
(Assume ka = 1)
e) Plot the step response of the system using the calculated step response in d).
f) Plot the step response using the step function in MATLAB on the same figure.
g) When the system is stable, which value of ka will provide the best performance for step
response? Why? Consider performance indicators like: peak time, rise time, ... etc.
Hint: Plot the responses using different ka on the same figure. (Only evaluate these values
of ka = 0.5, 0.75, 1, 1.25)
Q3. Replot the step responses with the best performance from Q1 and Q2 on the same figure
using hold on. You should only have the plot with the best performance from each question.
(use legend, title, and axes labels)
Compare the best response from Q1 with the best one from Q2 using the performance
indicators (peak time, settling time, rise time, percentage overshoot). Which one looks better?
Why?

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