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:

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$$. --- Would you like any further details or have any questions? Here are 5 related questions that might interest you: 1. How does the Routh-Hurwitz criterion apply to third-order systems? 2. What are some methods for determining the stability of a system besides the Routh-Hurwitz criterion? 3. How does changing the coefficient of $$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.