Short note: Nyquist stability criterion

The Nyquist stability criterion is a graphical frequency-domain method that finds the number of closed-loop poles in the right half $s$-plane from the plot of the open-loop transfer function $G(j\omega)H(j\omega)$. It is based on Cauchy's principle of the argument.

Statement

If the Nyquist contour (the whole $j\omega$ axis plus an infinite semicircle enclosing the RHP, traversed clockwise) is mapped through $G(s)H(s)$, then

$$N = Z - P \quad\text{or}\quad Z = P + N$$

• $P$ = number of open-loop poles in the RHP
• $Z$ = number of closed-loop poles (zeros of $1+GH$) in the RHP
• $N$ = number of clockwise encirclements of the critical point $-1+j0$

The closed-loop system is stable only if $Z=0$, i.e. $N=-P$ (the plot encircles $-1$ anticlockwise $P$ times). For an open-loop stable system ($P=0$): stable if the Nyquist plot does not encircle $-1+j0$.

Steps

1. Find $P$ from the open-loop poles.
2. Plot $G(j\omega)H(j\omega)$ for $\omega=0\to\infty$, add its mirror image, and close it (poles at origin give infinite clockwise arcs).
3. Count $N$ around $-1+j0$, compute $Z$.

Merits

• Works with experimental frequency-response data.
• Handles time delay and open-loop unstable systems.
• Gives relative stability (gain and phase margins) as well.

Example: $GH=\dfrac{60}{(s+1)(s+2)(s+5)}$ cuts the real axis at $-0.476$; $P=0$, $N=0$, so $Z=0$: stable.