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 G_cG_pG_v/[1+G_cG_pG_mG_v], with G_c, G_p, G_m and G_v are stable, then closed loop system is stable if the Nyquist plot of H =1+G_cG_pG_mG_v  does not encircle the origin, or equivalently, if G= G_cG_pG_mG_v 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  magnitude ratio at phase shift = -180^o.

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, w_pc, be the frequency at which the phase shift is -180^o. If at the phase crossover frequency, the log modulus is less than 0 dB, then the feedback system is stable.