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[Kyle0819]

Consider a system whose characteristic equation is s^4+3s^3 +3s^2 + 2s^4 + K = 0.
determine limits on design parameter, K, for stability,
i. 14/9 > K > 0.
ii. 14 > K > 0.
iii. 9/14 > K > 0.
iv. 9 > K > 0.

 

[🤖]

To determine the limits on the design parameter K for stability, we need to analyze the roots of the characteristic equation.

The characteristic equation is given as:
s^4 + 3s^3 + 3s^2 + 2s + K = 0

For stability, all the roots of the characteristic equation must have negative real parts. This is known as the Routh-Hurwitz stability criterion.

To apply the Routh-Hurwitz criterion, we need to create the Routh array. Let's do that:

Coefficients of the characteristic equation:
a = 1, b = 3, c = 3, d = 2, e = K

Routh array:

Row 1: 1 3
Row 2: 3 2
Row 3: (6K - 9)/3 K

Now, let's analyze the signs of the elements in the first column of the Routh array.

For stability, all the elements in the first column must be positive.

i. 14/9 > K > 0:
Substituting K = 14/9, we get (6(14/9) - 9)/3 = (84/9 - 9)/3 = (84 - 81)/3 = 3/3 = 1, which is positive.
Substituting K = 0, we get (6(0) - 9)/3 = -9/3 = -3, which is negative.

Therefore, for 14/9 > K > 0, the first column of the Routh array has elements with alternating signs, which indicates the presence of poles with non-negative real parts. Hence, this range of K does not guarantee stability.

ii. 14 > K > 0:
Substituting K = 14, we get (6(14) - 9)/3 = (84 - 9)/3 = 75/3 = 25, which is positive.
Substituting K = 0, we get (6(0) - 9)/3 = -9/3 = -3, which is negative.

Therefore, for 14 > K > 0, the first column of the Routh array has elements with alternating signs, which again indicates the presence of poles with non-negative real parts. Hence, this range of K also does not guarantee stability.

iii. 9/14 > K > 0:
Substituting K = 9/14, we get (6(9/14) - 9)/3 = (54/14 - 9)/3 = (27/7 - 9)/3 = (27 - 63)/21 = -36/21 = -12/7, which is negative.
Substituting K = 0, we get (6(0) - 9)/3 = -9/3 = -3, which is negative.

Therefore, for 9/14 > K > 0, the first column of the Routh array has all negative elements. This indicates that all the roots have negative real parts, ensuring stability.

iv. 9 > K > 0:
Substituting K = 9, we get (6(9) - 9)/3 = (54 - 9)/3 = 45/3 = 15, which is positive.
Substituting K = 0, we get (6(0) - 9)/3 = -9/3 = -3, which is negative.

Therefore, for 9 > K > 0, the first column of the Routh array has all positive elements. This indicates that all the roots have negative real parts, ensuring stability.

Based on the analysis, the correct answer is iv. 9 > K > 0.

 

[Kyle0819]

wrong equation!!! s^4+3s^3 +3s^2 + 2s^4 + K = 0.

 

[Cee Jay]

The characteristic equation of a system describes the roots of the system, which in turn determine the stability of the system. For a system to be stable, all roots of the characteristic equation must have negative real parts.

Let's first simplify the characteristic equation by combining the terms and we get:

s^4 + 2s^4 + 3s^3 + 3s^2 + K = 0
3s^4 + 3s^3 + 3s^2 + K = 0

Now, your characteristic equation becomes:

3s^4 + 3s^3 + 3s^2 + K = 0

The Hurwitz Criterion is a method to analyze the stability of the system by using the coefficients of the characteristic equation. These coefficients should create a Hurwitz matrix that is a strictly diagonally dominant matrix for the system to be stable.

The Hurwitz matrix from your coefficient will be a 4x4 matrix:

| 3 K 0 0 |
| 3 3 K 0 |
| 0 3 3 K |
| 0 0 3 3 |

We can see that for the principle minors of the Hurwitz matrix to be positive, K has to satisfy the following conditions:

I) 3K > 0 => K > 0

II) 3K*3 - 3*3 > 0 => K > 1

III)3K*3*3 - 3*3*3 > 0 => K > 1

By following the first three conditions, we find that K>1. However, we need to check the fourth condition,

IV) 3*3*3*3 - 3*3*3*3 + 3*3*K > 0 => K > 9/14

Therefore, the only condition that has to hold true is the most restrictive of the above four conditions.

Hence the range for K for the system to be stable is 9/14 < K < ∞.

So the correct answer is
iv. 9/14 > K > 0.