Because of the simplicity of the graphical construction of the frequency response characteristics of a given transfer function the application of the Nyquist criterion is often more simple using Bode plots. The continuous change of the angle of the vector from the critical point (-1,j0) to the locus of must be expressed by the amplitude and phase response of . From Figure 5.7
Regarding the intersections of the locus of with the real axis in the range , the transfer from the upper to the lower half plane in the direction of increasing values are treated as positive intersections while the reverse transfer are negative intersections (Figure 5.7). The change of the angle is zero if the count of positive intersections is equal to the count of negative intersections . The change of the angle depends also on the number of positive and negative intersections and if the open loop does not have poles on the imaginary axis, the change of the angle is
The open loop with the transfer function has poles in the left-half plane and possibly a single () or double pole () at . If the locus of has positive and negative intersections with the real axis to the left of the critical point, then the closed loop is only asymptotically stable, if
is valid. For the special case, that the open loop is stable (, ), the number of positive and negative intersections must be equal.
From this it follows that the difference of the number of positive and negative intersections in the case of is an integer and for not an integer. From this follows immediately, that for the number is even, for the number is uneven and therefore in all cases is an even number, such that the closed loop is asymptotically stable. This is only valid if .
The Nyquist criterion can now be transferred directly into the representation using frequency response characteristics. The magnitude response , which corresponds to the locus , is always positive at the intersections of the locus with the real axis in the range of . These points of intersection correspond to the crossings of the phase response with lines , etc., i.e. a uneven multiple of 180. In the case of a positive intersection of the locus, the phase response at the lines crosses from below to top and reverse from top to below on a negative intersection as shown in Figure 5.9. In the following these crossings
The open loop with the transfer function has poles in the right-half plane, and possibly a single or double pole at . are the number of positive and of negative crossings of the phase response over the lines in the frequency range where is valid. The closed loop is only asymptotically stable, if
is valid. For the special case of an open-loop stable system (, )
must be valid.
Table 5.1 shows some examples of the Nyquist criterion
in the representation using frequency response characteristics.
Finally the 'left-hand rule' will be given using Bode diagrams, because this version is for the most cases sufficient and simple to apply.
The open loop has only poles in the left-half plane with the exception of possibly one single or one multiple pole at (P, I or behaviour). In this case the closed loop is only asymptotically stable, if has a phase of for the crossover frequency at .
This stability criterion offers the possibility of a practical assessment of the 'quality of stability' of a control loop. The larger the distance of the locus from the critical point the farther is the closed loop from the stability margin. As a measure of this distance the terms gain margin and phase margin are introduced according to Figure 5.10.
A well damped control system should yield the following characteristics:
The crossover frequency is a measure of the dynamical quality of the control loop. The higher the higher the bandwidth of the closed loop, and the faster the reaction on command inputs or disturbances. As the bandwidth that frequency is understood, at which the magnitude of the closed-loop frequency response has fallen off approximately to zero.