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 ...

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