Modern Control System Theory and Design, 2nd Edition

6.5.  NYQUIST STABILITY CRITERION

The Nyquist stability criterion [5] is a very valuable tool that determines the degree of stability, or instability, of a feedback control system. In addition, it is the basis for other methods that are used to improve both the steady-state and the transient response of a feedback control system. Application of the Nyquist stability criterion requires a polar plot of the open-loop transfer function, G(jω)H(jω), which is usually referred to as the Nyquist diagram.

The Nyquist criterion determines the number of roots of the characteristic equation that have positive real parts from a polar plot of the open-loop transfer function, G(jω)H(jω), in the complex plane. Let us consider the characteristic equation

System stability can be determined from Eq. (6.54) by identifying the location of its roots in the complex plane. Assuming that G(s) and H(s), in their general form, are functions of s which are given by

and

then we can say that

Substituting Eq. (6.57) into Eq. (6.54), we obtain the following equivalent expression for F(s):

In terms of factors, we ...