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.