Given the following feedback control systems, apply the Routh-Hurwitz criterion of stability to: a) Find the range of K for stability b) Find the period of oscillation when the system is marginally stable R(s) + E(s) K s+1 1 s3+ 6s2+9s +4 C(s)
Given the following feedback control systems, apply the Routh-Hurwitz criterion of stability to: a) Find the range of K for stability b) Find the period of oscillation when the system is marginally stable R(s) + E(s) K s+1 1 s3+ 6s2+9s +4 C(s)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
Related questions
Question
![### Application of the Routh-Hurwitz Criterion to Feedback Control Systems
**Problem Statement:**
Given the following feedback control system, apply the Routh-Hurwitz criterion of stability to:
a) Find the range of \( K \) for stability.
b) Find the period of oscillation when the system is marginally stable.
#### System Diagram:
\[
\begin{array}{c}
{\large\text{Block Diagram Representation:}}\\
\begin{array}{cccc}
R(s) & \longrightarrow &\oplus & \rightarrow & K/(s+1) & \rightarrow & 1/(s^3 + 6s^2 + 9s + 4) & \rightarrow & C(s) \\
& &\longleftarrow &\circlearrowleft & \text{(Feedback Loop)}
\end{array}
\end{array}
\]
**Explanation of the Block Diagram:**
- **\(R(s)\)**: Input signal to the system.
- **\(K/(s+1)\)**: Transfer function of the controller with gain \(K\).
- **\(1/(s^3 + 6s^2 + 9s + 4)\)**: Transfer function of the plant.
- **\(C(s)\)**: Output signal of the system.
- The feedback loop indicates that the output \(C(s)\) is fed back and subtracted from the input \(R(s)\).
**Tasks to be Performed:**
**a) Finding the Range of \( K \) for Stability:**
Use the Routh-Hurwitz criterion to determine the values of \( K \) for which the system remains stable.
**b) Finding the Period of Oscillation for Marginal Stability:**
Determine the period of oscillation when the system transitions to a marginally stable state, which means the system will exhibit sustained oscillations without growing or decaying in amplitude. This generally occurs when there is a pair of purely imaginary roots in the characteristic equation derived from the system's transfer function.
This exercise will help you understand the practical application of the Routh-Hurwitz stability criterion in analyzing and designing control systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe36fb7f7-6381-4add-b52b-9085dfd27844%2Fb5ffb1dd-159e-4ec0-8349-9a736496d5be%2Fjjltxoh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Application of the Routh-Hurwitz Criterion to Feedback Control Systems
**Problem Statement:**
Given the following feedback control system, apply the Routh-Hurwitz criterion of stability to:
a) Find the range of \( K \) for stability.
b) Find the period of oscillation when the system is marginally stable.
#### System Diagram:
\[
\begin{array}{c}
{\large\text{Block Diagram Representation:}}\\
\begin{array}{cccc}
R(s) & \longrightarrow &\oplus & \rightarrow & K/(s+1) & \rightarrow & 1/(s^3 + 6s^2 + 9s + 4) & \rightarrow & C(s) \\
& &\longleftarrow &\circlearrowleft & \text{(Feedback Loop)}
\end{array}
\end{array}
\]
**Explanation of the Block Diagram:**
- **\(R(s)\)**: Input signal to the system.
- **\(K/(s+1)\)**: Transfer function of the controller with gain \(K\).
- **\(1/(s^3 + 6s^2 + 9s + 4)\)**: Transfer function of the plant.
- **\(C(s)\)**: Output signal of the system.
- The feedback loop indicates that the output \(C(s)\) is fed back and subtracted from the input \(R(s)\).
**Tasks to be Performed:**
**a) Finding the Range of \( K \) for Stability:**
Use the Routh-Hurwitz criterion to determine the values of \( K \) for which the system remains stable.
**b) Finding the Period of Oscillation for Marginal Stability:**
Determine the period of oscillation when the system transitions to a marginally stable state, which means the system will exhibit sustained oscillations without growing or decaying in amplitude. This generally occurs when there is a pair of purely imaginary roots in the characteristic equation derived from the system's transfer function.
This exercise will help you understand the practical application of the Routh-Hurwitz stability criterion in analyzing and designing control systems.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,