Summary Week 9
1. Complex Mapping Theorem.
For a given transfer function, G(s), and a given simple closed
contour G, let Z be the number of zeros of G(s)
inside G and let P be the number of poles of
G(s) inside G. Then as s traverses G in the clockwise manner, the mapping
G(s) will encircle the origin N=Z-P times in the clockwise manner.
2. Nyquist Stability Criterion.
Suppose the the closed loop transfer function is given
by GcGpGv/[1+GcGpGmGv],
with Gc, Gp, Gm and Gv are
stable, then closed loop system is stable if the Nyquist plot of H =1+GcGpGmGv
does not encircle the origin, or equivalently, if G= GcGpGmGv
does
not encircle the point (-1,0), i.e. -1+0i.
3. Stability Margins
Motivation: to account for the uncertainties due to
modeling errors/ imprecision.
Gain Margin = 1/x
where x is the value of amplitude ratio at phase shift
= -180o.
Phase Margin = 180 + f
where f is the phase shift when
amplitude ratio = 1 or LM=0 dB.
4. Bode Stability Criterion
Let the phase crossover frequency, wpc,
be the frequency at which the phase shift is -180o. If at the
phase crossover frequency, the log modulus is less than 0 dB, then the
feedback system is stable.