To determine the stability of the system with the given characteristic equation using the Routh-Hurwitz criterion, we need to construct the Routh array. The characteristic equation is:
s^6 + 4s^5 + 3s^4 + 2s^3 + s^2 + 4s + 4 = 0
The Routh array is formed as follows:
Row 1: 1 3 4
Row 2: 4 2 4
Row 3: 2.5 4
Row 4: 4
To apply the Routh-Hurwitz criterion, we need to check the signs of the elements in the first column of the Routh array. If all the elements have the same sign, then the system is stable. If any element has a different sign, the system is unstable.
In this case, we can see that the first column elements are 1, 4, 2.5, and 4. Since these elements do not all have the same sign (1 and 2.5 are positive, while 4 is negative), the system is unstable according to the Routh-Hurwitz criterion.
Therefore, based on the Routh-Hurwitz criterion, the system described by the given characteristic equation is unstable.