Find the range of K for stability of the system with characteristic equation  s^4 + 3s^3 +3s^2 + 2s +k

To find the range of \( K \) for the stability of the system, we need to ensure that all the roots of the characteristic equation

\[
s^4 + 3s^3 + 3s^2 + 2s + K = 0
\]

have negative real parts. This can be determined using the **Routh-Hurwitz criterion**.

### Step 1: Form the Routh array
The characteristic equation is \( s^4 + 3s^3 + 3s^2 + 2s + K = 0 \).

The Routh array is constructed as follows:

\[
\begin{array}{c|cc}
s^4 & 1 & 3 \\
s^3 & 3 & 2 \\
s^2 & \frac{9 - 2K}{3} & K \\
s^1 & \frac{6K - 9}{9 - 2K} & 0 \\
s^0 & K & \\
\end{array}
\]

### Step 2: Determine conditions for stability
For the system to be stable, all elements in the first column of the Routh array must be positive.

1. **Row \( s^2 \):** \( \frac{9 - 2K}{3} > 0 \)
   - \( 9 - 2K > 0 \)
   - \( K < 4.5 \)

2. **Row \( s^1 \):** \( \frac{6K - 9}{9 - 2K} > 0 \)
   - For the numerator \( 6K - 9 > 0 \)
   - \( K > 1.5 \)
   - For the denominator \( 9 - 2K > 0 \)
   - \( K < 4.5 \) (already derived)

3. **Row \( s^0 \):** \( K > 0 \) (since it must be positive).

### Step 3: Combine the conditions
Combining these inequalities:

\[
1.5 < K < 4.5
\]

### Conclusion
The system is stable for \( 1.5 < K < 4.5 \).

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Would you like any further details or have any questions?

Here are 5 related questions that might interest you:

1. How does the Routh-Hurwitz criterion apply to third-order systems?
2. What are some methods for determining the stability of a system besides the Routh-Hurwitz criterion?
3. How does changing the coefficient of \( s^3 \) in the characteristic equation affect the stability range of \( K \)?
4. What happens to the system stability if \( K = 4.5 \)?
5. Can you explain how the Routh array is derived for higher-order systems?

**Tip:** Always check the boundary conditions separately when determining the stability range to ensure all possible cases are covered.

Explore how to find the range of K for stability of a system using the Routh-Hurwitz criterion. Understand the conditions under which a characteristic equation s^4 + 3s^3 + 3s^2 + 2s + K remains stable, ensuring all roots have negative real parts. Suitable for advanced undergraduate and graduate students.

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