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##

Simplified forms of the Nyquist criterion

It follows from Eq. (5.15) that for an open-loop stable
system, that is and , then
. Therefore the Nyquist criterion can be
reformulated as follows:

If the open loop is asymptotically stable, then the closed loop is
only asymptotically stable, if the frequency response locus of the open loop does
neither revolve around or pass through the critical point (-1,j0).

Another form of the simplified Nyquist criterion for with
poles at is the so called 'left-hand
rule':

The open loop has only poles in the left-half plane with the
exception of a single or double pole at (P, I or
behaviour). In this case the closed loop is only stable,
if the critical point (-1,j0) is on the *left* hand-side of
the locus
in the direction of increasing values of
.

This form of the Nyquist criterion is sufficient for most cases. The
part of the locus that is significant is that closest to the
critical point. For very complicated curves one should go back to
the general case. The left-hand rule can be graphically derived from
the generalised locus according to
section A.2. The orthogonal
(
)-net is observed and asymptotic stability of the
closed loop is given, if a curve with passes through the
critical point (-1,j0). Such a curve is always on the left-hand side
of
.

** Next:** The Nyquist criterion using
** Up:** Algebraic stability criteria
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Christian Schmid 2005-05-09