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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 Previous: Nyquist criterion using Nyquist   Contents
Christian Schmid 2005-05-09