Nyquist

Stability Theorem

This is a visual way to check stability

Theorem

The Nyquist stability theorem states $N=Z-P$

• $Z=$ # of stable closed-loop poles
• $P$ = # of stable open-loop poles
• $N$ is the number of encirclements of -1 by $L[\Gamma ]$

$N=\text{\# of stable open-loop poles}+\text{\# of unstable open-loop poles}-\text{\# stable open-loop poles}$ $=\text{\# Unstable open-loop poles}$

Corollary

The feedback system is closed-loop stable iff $N=\text{\# of unstable open-loop poles}$

Doing this in practice?

Formally…
Use nyquist(L) to generate the Nyquist plot. For the different regions, build a table that counts the encirclements as follows…

Region	N
A	0
B	1
C	-1
D	0
Where we select region B to determine the region of $k$ because it has $1$ encirclements (where 1 = # unstable poles)?

Note

We get closed-loop stability when $N=\#\text{ of usntable open-loop poles}$

Then for this region, determine its domain (i.e for region B, in the Nyquist plot, its domain is $(-4,-3)$) and $k\in (-4,-3)$. We can solve the equality to determine the range of $k$ for which we have closed-loop stability.

Practically…

Easier way to do this

Basically we get stability when $-\frac{1}{k}$ falls in the region which matches the required number of encirclements $N$ (the number of unstable poles form the Nyquist stability theorem).

Therefore, we want to find the domain where $N=\text{number of unstable poles}$ and set $\frac{-1}{k}$ to be an element of that region. Then solve for $K$.