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. 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.

The second system is slightly more complex, but the Routh array is formed in the same manner.

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 2 sign changes, from +761.7 to -355.5 and from -355.5 to +120. Therefore, there are 2 roots of the characteristic equation with positive real parts and 2 closed-loop poles in the right-half plane. The location of those roots is not available. ...

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.

The second system is slightly more complex, but the Routh array is formed in the same manner.

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 2 sign changes, from +761.7 to -355.5 and from -355.5 to +120. Therefore, there are 2 roots of the characteristic equation with positive real parts and 2 closed-loop poles in the right-half plane. The location of those roots is not available from the Routh array. Since there were no 0s in the first column, there are no poles on the jw axis.

The last system has its gain K left as a variable. We want to determine the upper and lower bounds on K
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