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2.5 Proof of Nyquist Stability Criterion 29

2.5 Proof of Nyquist Stability Criterion

This section is provided for those who are interested to understand the Nyquist
stability criterion in depth. Those readers who are not interested to know the control
theory in depth can skip this section and those readers who want to know the control
theory in depth can refer to advanced control theory book such as the book “Mod-
ern Control Engineering” published by Prentice-hall international, Inc. The book
is two volumes and more than 1000 pages which is not suitable for engineers who
want to know the basic control theory for real industrial applications. This section is
extracted from the mentioned book and for industrial application the material given
by next section is sufficient. This section has been added because it is interesting
and is not too complicated and some modification has been made those material
extracted from the above book to make it clearer.
Let F( s) be the ratio of two polynomial of function of s. The mapping theorem
is used to prove the Nyquist stability criteria. The polynomials may have roots with
the multiplicity of s = 0. Let the number of poles which can easily be found from
the denominator be n and let the number of zeros which can easily be found from
numerator be m. The zeros and poles lie in a closed contour which does not pass
through any zeros or poles.
This closed contour is then mapped into F( s) plane as a closed curve, as repre-
sentative point s traces out the entire contour in the s plane in the clockwise direc-
tion. Let p = n − m be the difference between zeros and poles. The total number of
p encirclement of the origin of F( s) plane clockwise direction, is p. Note that from
this mapping only the difference of poles and zeros can be found. The proof of the
mapping theorem is beyond the scope of this book.
Note that a positive p shows the excess of poles over zeros and negative p indi-
cates an excess of zeros than poles. In control system with characteristic equation
1 Gs H+ ()⋅ ()s the number of poles and zeros can easily be found from the charac-
teristic function. G( s) is the forward path transfer function, which includes the con-
trol transfer function too and H( s) is the feedback transfer function. In this theorem
the number of poles and zeros is immaterial only the number of encirclements of
the origin is important.
Now the mapping theorem is used to prove the Nyquist stability criteria. For
linear control system, let the closed contour in the s plane be all in the right portion
of s plane with a semicircle with radios infinity which includes the entire imaginary
axis. This is known as Nyquist path. The contour contains the complete iω axis
from ω = − ∞ to ω = + ∞, where i is the complex operator. The direction of the path
is clockwise. The Nyquist path encloses the entire left portion of the s plane. The
Nyquist path also encloses all poles and zeros of the function of 1 Gs H+ ()⋅ ()s that
have positive real parts. It is necessary that the contour does not pass through any
zeros and poles. If there are no zeros of the mentioned function in the right half of s
plane, then there cannot be a closed loop pole in the right half of the s plane then the
system is stable. If there is a pole or poles of G( s)H( s) in the origin the mapping is
indeterminate and this is avoided by including a small detour around it.