Q.2 Check whether the following system given by its characteristic equation is stable or not and shows the location of roots on the s-plane q(s) = s5 + 10s4 + 45s³ +90s² + 164s + 200
Q.2 Check whether the following system given by its characteristic equation is stable or not and shows the location of roots on the s-plane q(s) = s5 + 10s4 + 45s³ +90s² + 164s + 200
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q.2 Check whether the following system given by its characteristic equation is stable or not and shows the location of roots on the s-plane
![Here's the transcription for the educational website:
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**Question 2: Stability and Root Location Analysis**
**Objective:**
Determine whether the given system, described by its characteristic equation, is stable. Additionally, identify the location of the roots on the s-plane.
**Characteristic Equation:**
\[ q(s) = s^5 + 10s^4 + 45s^3 + 90s^2 + 164s + 200 \]
**Tasks:**
1. Analyze the stability of the system using appropriate methods.
2. Plot the roots on the s-plane to visualize their locations.
---
**Explanation:**
In this problem, we are given a characteristic polynomial and asked to determine the stability of the system. The polynomial is a fifth-degree polynomial, and stability can be checked using several methods:
1. **Routh-Hurwitz Criterion:**
This method involves constructing the Routh array and analyzing the sign changes in the first column.
2. **Root Locus:**
Alternatively, the roots of the polynomial can be found, and their locations on the complex s-plane can be analyzed. For stability, all roots must lie in the left-half of the s-plane.
There are no graphs or diagrams provided in the prompt. However, the subsequent steps will typically involve plotting the given polynomial's roots on the complex plane for visual confirmation of stability.
---
Ensure to follow the appropriate steps to construct the Routh array or compute the roots using numerical methods or software tools for complex polynomials. The roots’ positions will visually correspond to specific locations on the s-plane, assisting in concluding the system's stability.
**Note:** The text covered by the scribbles contains further elaborations or possibly other parts of a different question which are not needed for this particular task.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facf42901-461c-4318-866a-72a73aa9949c%2F657fc88b-ffc7-483e-93f4-8c72748cd049%2Fqko88g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here's the transcription for the educational website:
---
**Question 2: Stability and Root Location Analysis**
**Objective:**
Determine whether the given system, described by its characteristic equation, is stable. Additionally, identify the location of the roots on the s-plane.
**Characteristic Equation:**
\[ q(s) = s^5 + 10s^4 + 45s^3 + 90s^2 + 164s + 200 \]
**Tasks:**
1. Analyze the stability of the system using appropriate methods.
2. Plot the roots on the s-plane to visualize their locations.
---
**Explanation:**
In this problem, we are given a characteristic polynomial and asked to determine the stability of the system. The polynomial is a fifth-degree polynomial, and stability can be checked using several methods:
1. **Routh-Hurwitz Criterion:**
This method involves constructing the Routh array and analyzing the sign changes in the first column.
2. **Root Locus:**
Alternatively, the roots of the polynomial can be found, and their locations on the complex s-plane can be analyzed. For stability, all roots must lie in the left-half of the s-plane.
There are no graphs or diagrams provided in the prompt. However, the subsequent steps will typically involve plotting the given polynomial's roots on the complex plane for visual confirmation of stability.
---
Ensure to follow the appropriate steps to construct the Routh array or compute the roots using numerical methods or software tools for complex polynomials. The roots’ positions will visually correspond to specific locations on the s-plane, assisting in concluding the system's stability.
**Note:** The text covered by the scribbles contains further elaborations or possibly other parts of a different question which are not needed for this particular task.
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