Page 120 - Modern Control of DC-Based Power Systems

84 Modern Control of DC-Based Power Systems

The most widely-used methods to plot the frequency response of a
system are:
• Nyquist or polar plots; and
• Bode or logarithmic diagrams.
Each of these representations can be used to analyze the stability and
design of the control system to obtain the desired system behavior.

3.2 NYQUIST STABILITY CRITERION

Detailed information about how to plot the Nyquist or polar plots
can be found in Ref. [1]. The stability analysis of control systems using
the Nyquist plots is based on the Nyquist stability criterion. This crite-
rion provides a means to determine the stability of the closed-loop trans-
fer function based on the open-loop poles and the open-loop frequency
response of the system.
According to the Nyquist stability criterion, for an LTI system with
the forward transfer functions GsðÞ and feedback transfer function HsðÞ,
the number of the zeros (Z)ofthe 1 1 GsðÞHsðÞ in the right-half s-plane
(which will be equal to the poles of the closed-loop system) can be
given by

Z 5 N 1 P (3.2)

where
• N is the number of the clockwise encirclements of the point 2 1 1 j0
by the locus of GsðÞHsðÞ,as s varies along a closed contour in the s
plane, which encloses the entire right-half s-plane in a clockwise
direction (i.e., Nyquist path).
• P is the number of the poles of GsðÞHsðÞ.
Using Nyquist plots, it is not only possible to determine whether a
system is stable or unstable, but also to infer the degree of stability of a
stable system and information on how to improve the stability, if necessary.

3.3 BODE DIAGRAMS
Bode diagrams or logarithmic plots are composed of two plots, i.e.,
the plot of the magnitude of the open-loop transfer function

(20log GjωðÞHjωðÞ ), and the phase angle plot, both versus the frequency