Introduction:
The Routh-Hurwitz stability criterion is a mathematical method used to determine the stability of a linear time-invariant (LTI) system. It provides a straightforward and efficient way to analyze the stability of a system without explicitly calculating the system's poles.

Routh-Hurwitz Algorithm:
The Routh-Hurwitz algorithm involves constructing a Routh array using the coefficients of the characteristic equation. The algorithm can be summarized in the following steps:

1. Write down the characteristic equation in descending powers of 's'.
2. Create the first two rows of the Routh array using the coefficients of the even and odd powers of 's'.
3. Continue filling the remaining rows of the Routh array using the following formulas:
- For the elements in the first column, use the coefficients of the previous row.
- For the elements in the second column, use the coefficients of the previous two rows.
- Repeat this pattern until all the rows are filled.
4. Analyze the Routh array to determine the system's stability.

Application of Routh-Hurwitz Algorithm:
Let's apply the Routh-Hurwitz algorithm to the given characteristic equation:

Q(s) = S^5 + 2S^4 + 2S^3 + 4S^2 + 11S + 10 = 0

Step 1:
Write down the characteristic equation in descending powers of 's':

Q(s) = 1S^5 + 2S^4 + 2S^3 + 4S^2 + 11S + 10 = 0

Step 2:
Create the first two rows of the Routh array:

R1: 1 2 11
R2: 2 4 0

Step 3:
Calculate the remaining rows of the Routh array:

R3: 2 4 0
R4: 4 0 0
R5: 4 0 0
R6: 0 0 0

Step 4:
Analyze the Routh array to determine the system's stability:

Rule 1: The first column of the Routh array should not contain any zeros. If it does, the system has poles on the right-half plane, indicating instability. In this case, there are no zeros in the first column, so the system is potentially stable.

Rule 2: All the elements in a row must have the same sign. If any row violates this rule, the system has poles on the imaginary axis, indicating marginal stability. In this case, all the elements in the R6 row are zeros, which means the system may have poles on the imaginary axis.

Rule 3: The number of sign changes in a row indicates the number of poles in the right-half plane. In this case, there are no sign changes in any row, indicating that all the poles of the system are in the left-half plane.

Conclusion:
Based on the Routh-Hurwitz analysis, the system described by the given characteristic equation is potentially stable with all its poles in the left-half plane. However, further analysis is required to determine if there are any poles on the imaginary axis