**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 G_{c}G_{p}G_{v}/[1+G_{c}G_{p}G_{m}G_{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_{c}G_{p}G_{m}G_{v}
does not encircle the origin, or equivalently, if G= G_{c}G_{p}G_{m}G_{v
}does
not encircle the point (-1,0), i.e. -1+0*i*.

**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
= -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.