Find the range of K within which the following characteristic polynomial is stable ### Equation: Polynomial ExpressionThe polynomial expression shown is:\[ s^4 + 2s^3 + 3s^2 + 4.2s + K \]**Explanation:**This is a fourth-degree polynomial in the variable $ s $. It consists of five terms, each involving powers of $ s $:- **$ s^4 $**: The highest degree term, which indicates this is a quartic (fourth-degree) polynomial.- **$ 2s^3 $**: The cubic term, where the coefficient is 2.- **$ 3s^2 $**: The quadratic term with a coefficient of 3.- **$ 4.2s $**: The linear term with a coefficient of 4.2.- **$ K $**: A constant term, which is typically unknown or provided in specific contexts.Polynomials like this are fundamental in algebra and calculus and are often used in solving equations, modeling real-world situations, or analyzing behaviors of functions.

Given characteristic polynomial.To find:The range of in which the polynomial is stable.Concept…

Answered: Find the range of K within which the…

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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Question

Find the range of K within which the following characteristic polynomial is stable

### Equation: Polynomial Expression

The polynomial expression shown is:

\[ s^4 + 2s^3 + 3s^2 + 4.2s + K \]

**Explanation:**

This is a fourth-degree polynomial in the variable $ s $. It consists of five terms, each involving powers of $ s $:

- **$ s^4 $**: The highest degree term, which indicates this is a quartic (fourth-degree) polynomial.
- **$ 2s^3 $**: The cubic term, where the coefficient is 2.
- **$ 3s^2 $**: The quadratic term with a coefficient of 3.
- **$ 4.2s $**: The linear term with a coefficient of 4.2.
- **$ K $**: A constant term, which is typically unknown or provided in specific contexts.

Polynomials like this are fundamental in algebra and calculus and are often used in solving equations, modeling real-world situations, or analyzing behaviors of functions.

Expert Solution

Step 1: Analysis and Introduction

Given characteristic polynomial.

$s to the power of 4 plus 2 s cubed plus 3 s squared plus 4.2 s plus K$

To find:

The range of $K$ in which the polynomial is stable.

Concept used:

Routh Table:

$s to the power of 4$	$a subscript 4$	$a subscript 2$	$a subscript 0$
$s cubed$	$a subscript 3$	$a subscript 1$	$0$
$s squared$	$negative fraction numerator open vertical bar table row cell a subscript 4 end cell cell a subscript 2 end cell row cell a subscript 3 end cell cell a subscript 1 end cell end table close vertical bar over denominator a subscript 3 end fraction equals b subscript 1$	$negative fraction numerator open vertical bar table row cell a subscript 4 end cell cell a subscript 0 end cell row cell a subscript 3 end cell 0 end table close vertical bar over denominator a subscript 3 end fraction equals b subscript 2$	$negative fraction numerator open vertical bar table row cell a subscript 4 end cell 0 row cell a subscript 3 end cell 0 end table close vertical bar over denominator a subscript 3 end fraction equals 0$
$s to the power of 1$	$negative fraction numerator open vertical bar table row cell a subscript 3 end cell cell a subscript 1 end cell row cell b subscript 1 end cell cell b subscript 2 end cell end table close vertical bar over denominator b subscript 1 end fraction equals c subscript 1$	$negative fraction numerator open vertical bar table row cell a subscript 3 end cell 0 row cell b subscript 1 end cell 0 end table close vertical bar over denominator b subscript 1 end fraction equals 0$	$negative fraction numerator open vertical bar table row cell a subscript 3 end cell 0 row cell b subscript 1 end cell 0 end table close vertical bar over denominator b subscript 1 end fraction equals 0$
$s to the power of 0$	$negative fraction numerator open vertical bar table row cell b subscript 1 end cell cell b subscript 2 end cell row cell c subscript 1 end cell 0 end table close vertical bar over denominator c subscript 1 end fraction equals d subscript 1$	$negative fraction numerator open vertical bar table row cell b subscript 1 end cell 0 row cell c subscript 1 end cell 0 end table close vertical bar over denominator c subscript 1 end fraction equals 0$	$negative fraction numerator open vertical bar table row cell b subscript 1 end cell 0 row cell c subscript 1 end cell 0 end table close vertical bar over denominator c subscript 1 end fraction equals 0$

For the system is to be stable, the following condition must be satisfied:

The necessary condition is that the coefficients of the characteristic polynomial should be positive.

The sufficient condition is that all the elements of the first column of the Routh array should have the same sign. This means that all the elements of the first column of the Routh array should be either positive or negative.