The Routh array is a tabular procedure for determining how many roots of a polynomial are in the right-half of the s-plane. We can also determine if there are any roots on the jw axis and their locations. An important use of the Routh array is to determine upper and lower limits on the value of some parameter, such as gain, so that all roots of the closed-loop characteristic equation are in the left-half plane.
This is the polynomial that the Routh array uses.
The number of right-half plane roots of the characteristic equation (closed-loop poles) is given by the number of sign changes in the first column of the array. By inspection, there are no sign changes. Therefore, there are no roots with positive real parts. Since the array was constructed without a 0 appearing anywhere in the first column, there are no roots on the jw axis.
Note that the number of terms in each row decreases by 1 at each odd-powered row, and that the last element in each even-powered row is the constant coefficient from the characteristic equation. Since there are no sign changes, there are no roots in the right-half plane. There are no roots on the jw axis since there were no 0s in the first column.
The third system is the same as the second system except that the gain has been increased by a factor of 10. Note that several of the coefficients in the characteristic equation have changed. Also note that there is a negative coefficient in the polynomial. That guarantees that there is at least one unstable root. Since the constant coefficient is positive, there is an even number of unstable roots.
Examination of the first column of the array shows that there are ...