TABLE OF CONTENTS In this chapter we consider transfer functions that are quotients of polynomials in s rational functions of s with real coefficients. Suppose that one has a feedback system for which G p( s) = P 1( s)/ Q 1( s) and H( s) = P 2( s)/ Q 2( s), with P 1( s), P 2( s), Q 1( s) and Q 2( s) polynomials in s whose coefficients are real. Then the transfer function of the system is:
To determine the stability of the system, it would seem that we need to know the zeros of the polynomial P 1( s) P 2( s) + Q 1( s) Q 2( s) the poles of the transfer function. This would be problematic as it is proved in modern algebra that there can be no general formula for the roots of polynomials of degree five or higher in terms of the coefficients of the polynomial [vdW91].
In fact, we do not need to know the roots of the polynomial P 1( s) P 2( s) + Q 1( s) Q 2( s); we only need to know whether the roots of the polynomial are in the right half-plane. This problem was studied in the late 1800's, and a number of criteria...
