Simplified Nyquist Criterion

160 Controller setting and stability of control loops
5.6.3 Controller design in the Bode diagram
For the practical treatment of linear control systems the representation
of frequency responses in the Bode diagram is preferred above that by
the Nyquist plots because the former method is simpler by far. The
statements on the stability of a closed loop provided by the Nyquist
criterion can be transferred to the representation of the frequency response
of the open control loop in the Bode diagram. This shall only
be dealt with in the following text within the context of the “Simplified
Nyquist Criterion”.
As illustrated in figure 5-26, we can, for stability testing with the “Simplified
Nyquist Criterion” determine the frequency ω π from the phase
response with the definition ϕ 0 (ω π ) =−π and check whether the amplitude
of the frequency response |G 0 (jω π )| is less than one. Alternatively,
with the definition |G 0 (jω d )|=1 we can obtain from the amplitude
response the frequency ω d and check whether the phase angle
ϕ 0 (ω d ) is greater than −π.
We must ensure in every case that the “Simplified Nyquist Criterion” is
applicable.
Summarizing, the “Simplified Nyquist Criterion” for the application in
the Bode diagram gives:
If the open control loop is stable or has integrating behaviour
and the phase response of its frequency response in the Bode
diagram intersects the lines −180 ◦ − n · 360 ◦ with a negative
slope only, then the following applies:
If the amplitude of the frequency response |G 0 (jω π )| is less
than one at frequency values for which the phase response
gives ϕ 0 (ω π ) =−180 ◦ − n · 360 ◦ , then the closed control
loop is stable. Otherwise it is not stable.
For the “Simplified Nyquist Criterion” the point where the phase response
is just tangent to the line −180 ◦ − n · 360 ◦ for infinit values of
ω has to be considered.
Figure 5-26 also shows that it is possible to obtain the gain margin
from the amplitude response |G 0 (jω π )| and the phase margin from
the phase response ϕ 0 (ω d ).