Characteristic Equation and Stability

The characteristic equation of a feedback system is an important factor in determining the stability of the system. In this question, we are given the characteristic equation s^3 + 4s^2 + 10s + 11 = 0, and we need to determine if it results in stable operation of the feedback system.

Stability Criteria

To analyze the stability of a system, we commonly use the Routh-Hurwitz stability criterion. According to this criterion, for a system to be stable, all the coefficients of the characteristic equation must have the same sign.

Calculating Coefficients

Let's calculate the coefficients of the given characteristic equation:

Coefficient of s^3: 1
Coefficient of s^2: 4
Coefficient of s: 10
Constant term: 11

Applying Routh-Hurwitz Criterion

To apply the Routh-Hurwitz criterion, we need to create a Routh array using the coefficients of the characteristic equation:

| 1 | 10 |
| 4 | 11 |
| 4.4 | 0 |

In the first row of the Routh array, we have the coefficients of the odd powers of s, and in the second row, we have the coefficients of the even powers of s.

Checking Sign of Coefficients

Now, let's check the sign of the coefficients in the first column of the Routh array:

1 > 0 (positive)
4 > 0 (positive)
4.4 > 0 (positive)

Since all the coefficients in the first column have the same positive sign, we can conclude that the given characteristic equation results in stable operation of the feedback system.

Conclusion

The given characteristic equation s^3 + 4s^2 + 10s + 11 = 0 results in stable operation of the feedback system. This conclusion is based on the Routh-Hurwitz stability criterion, which states that for a system to be stable, all the coefficients of the characteristic equation must have the same sign. In this case, all the coefficients have a positive sign, indicating stability.