Stability Margins
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Most control systems behave in a pattern such that
they become unstable if the gain increases past a certain value. For these systems, in addition to
determining the absolute stability of a system, the Nyquist diagram provides
qualitative information as to the degree of stability. The (-1,0) point plays the same role in the Nyquist diagram
as the imaginary axis does in the in the root locus diagram. Consider the root locus of Figure 1(a);
the crossing of the imaginary axis tells that for K<K*, the system is
unstable and the corresponding Nyquist curve in Figure 1(b) does not encircle
the (-1,0) point. When K=K*, the
system is marginally stable and the Nyquist curve in Figure 1(c), passes through the (-1,0) point. Finally, when K>K*, the system is
unstable and the Nyquist curve in Figure 1(d) encircles the (-1,0)
point. (Click here to see a larger version of Figure 1). For a stable system, the closer the Nyquist curve
approaches the (-1,0) point, the less stable the system is. Two quantities that measure the stability margin of a
system are directly related to the proximity of the frequency response drawn on
a polar plot to the (-1 ,0) point, are called gain margin and phase margin.
Figure 1. Relationship between imaginary axis in root locus diagram and (-1,0) point in Nyquist diagram |
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Gain Margin |
The factor by which the gain can be increased
before the system becomes unstable is called gain margin, gm. The gain margin of a system can be
determined from its open loop frequency response. It is the reciprocal of the open loop gain at the frequency
where phase is –180 (known as the phase crossover frequency). Looking at a typical Nyquist diagram such
as the one shown in Figure 2(a) one
can see that
Normally gain margins are expressed in dBs,
ie., -20log10(OA)
dBs. For a stable system OA is less than 1, hence the
gain margin of a stable system is positive when expressed in dBs. This figure can be read straight off the
open loop Bode plot as shown in Figure 2(b).
Figure 2 A good gain margin is desirable from the point of
view of robustness. However, one can
not select an arbitrary large gain margin because the dynamic characteristics
of the system are correlated with it and the response may become unacceptably
overdamped and sluggish. |
Phase Margin |
The gain margin alone is not enough to express the
relative stability of a system.
Consider the system shown in Figure 3(a). This system has a very good gain margin, but it only requires a
small additional phase lag to render the system unstable. Hence a further criterion is essential to
cater for the systems of this type.
The criterion is terms the phase margin, pm, and is defined as the
additional phase lag that will make the system marginally stable. The phase margin can also be determined easily
from the open loop frequency response.
It is the clockwise angle through which the unit vector must be
rotated before it lies on the negative real axis. The frequency at which the open loop system gain is unity is
temed the gain crossover frequency, wgc. Hence the phase margin is the difference of the
phase shift of the system and –180o at the gain crossover
frequency. Phase margin can be read straight off the open loop Bode plot as
shown in Figure 2(b). Situations may also arise where the system
exhibits a good phase margin, but has a poor gain margin (Figure 3(b)). That is why both criteria must be
satisfied adequately to provide good overall stability. A good rule of thumb for adequate gain and
phase margins are 12 dBs and 45 o to 60 o.
Figure 3 |
