# Nyquist criterion

A necessary and sufficient condition for the stability of a linear closed-loop system formulated in terms of properties of the open-loop system.

Consider the linear single input-linear single output system with the following transfer function:

$$W(p)=\frac{M(p)}{N(p)},$$

where it is assumed that the degree of the polynomial $M(z)$ does not exceed that of the polynomial $N(z)$ (i.e. $W(p)$ is a proper rational function). The original Nyquist criterion gives necessary and sufficient conditions for the stability of the closed-loop system with unity feedback $u=y$. This is done in terms of the complex-valued function $z=W(i\omega)$ of the real variable $\omega\in[0,\infty)$ (the amplitude-phase characteristic of the open-loop system) which describes a curve in the complex $z$-plane, known as the Nyquist diagram. Suppose that the characteristic polynomial $N(z)$ of the open-loop system has $k$, $0\leq k\leq n$, roots with positive real part and $n-k$ roots with negative real part. The Nyquist criterion is as follows: The closed-loop system is stable if and only if the Nyquist diagram encircles the point $z=-1$ in the counter-clockwise sense $k/2$ times. (An equivalent formulation is: The vector drawn from $-1$ to the point $W(i\omega)$ describes an angle $\pi k$ in the positive sense as $\omega$ goes from $0$ to $+\infty$.)

This criterion was first proposed by H. Nyquist  for feedback amplifiers; it is one of the frequency criteria for the stability of linear systems (similar, e.g., to the Mikhailov criterion, see , ). It is important to note that if the equations of some of the elements of the systems are unknown, the Nyquist diagram can be constructed experimentally, by feeding a harmonic signal of variable frequency to the input of the open feedback .

Generalizations of this criterion have since been developed for multivariable, infinite-dimensional and sampled-data systems, e.g. , , , .

How to Cite This Entry:
Nyquist criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nyquist_criterion&oldid=32802
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article