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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 stability of the system, we need to analyze the roots of the characteristic equation, which are the values of s that satisfy the equation.

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

Combining like terms, we have 3s^4 + 3s^3 + 3s^2 + K = 0.

To make the analysis easier, let's rearrange the equation in descending powers of s:

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

Now, let's apply the Routh-Hurwitz stability criterion. According to this criterion, for the system to be stable, all the coefficients of the characteristic equation must be positive.

For the coefficients of s^4 and s^2, we have 3 and 3, respectively, which are positive.

For the coefficients of s^3 and s^1, we have 3 and 0, respectively, which are also positive.

However, we have the parameter K, which can be positive, zero, or negative. To satisfy the Routh-Hurwitz criterion, K must be positive.

Therefore, the correct answer is ii. 14 > K > 0.

In this range, the system will be stable.