a. To determine the number of sign changes in the first column of the Routh array, we need to write the coefficients of the characteristic equation in a specific pattern. The characteristic equation in this case is Q(s) = s¹ + 2s³ + 3s² + 4s + 3.
We arrange the coefficients in the Routh array as follows:
Row 1: 1, 3
Row 2: 2, 4
Row 3: (calculation based on the previous rows)
To count the number of sign changes in the first column, we observe that there are no sign changes between 1 and 3. Therefore, the number of sign changes in the first column is 0.
b. The number of roots in the right half of the s-plane can be determined by counting the number of sign changes in the first column of the Routh array. Since there are no sign changes in the first column (as determined in part a), there are no roots in the right half of the s-plane.
c. The Routh-Hurwitz criterion helps determine the stability of a system based on the coefficients of the characteristic equation. According to the criterion, for a system to be stable, all the coefficients in the first column of the Routh array must be positive.
In this case, since there are no sign changes in the first column (as determined in part a), all the coefficients are positive. Therefore, we can conclude that the system is stable based on the Routh-Hurwitz criterion.