Question

Problem 7a (FOR 5 BONUS POINTS) Does the presence of an entire row of zeros always mean that the system has jo poles? If so how is it determined? If not what does it signify?

Answer 1

Routh-Hurwitz Criteria Special case:

Row of All Zeros: (Row of zeros in Routh table indicates the existance of Poles on the jw axis and those are determined as follows)

If we have a row of all zeros, the row directly above it is known as the Auxiliary Polynomial, and can be very helpful.

The roots of the auxiliary polynomial give us the precise locations of complex conjugate roots that lie on the jω axis.

However, one important point to notice is that if there are repeated roots on the jω axis, the system is actually unstable.

Therefore, we must use the auxiliary polynomial to determine whether the roots are repeated or not.

The auxiliary equation is to be differentiated with respect to s and the coefficients of this equation replaces the all zero row.Routh array can be further calculated using these new values.