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Absolute Stability tells you if a system is stable or not.
For example, if you find that the real part of one of your poles is
positive (right-half plane), then your system is unstable.
Relative Stability tells you how stable the system is.
Consider the use of the phase margin in evaluating closed loop
stabliity. Phase margin is the phase difference from -180 degrees
for the loop transfer function. As long as the phase margin is positive,
the system is stable. However, the further away you are from -180 degrees
the system is considered to be better in terms of stability. A phase margin
of 60 is better than a phase margin of 40 degrees. This is because if
there are any changes in component values, the phase curve might be
shifted toward the -180 level and a phase margin of 60 provides much
more room for these.