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
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
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
Phase Margin = 180 + f
4. Bode Stability Criterion
where f is the phase shift when
amplitude ratio = 1 or LM=0 dB.
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.