Notes on the Nyquist Stability Criterion

David A. Torrey
Clarkson University, Capital Region Campus
80 Nott Terrace
Schenectady, NY 12308
dtorrey@clarkson.edu

1 Overview

These notes were put together to help explain the basis of the Nyquist stability criterion. They should not be viewed as a replacement for the development provided in a control systems text such as [1]. The Nyquist stability criterion is a consequence of the Cauchy integral theorem from the calculus of complex numbers, but we take a more intuitive approach based on some examples coupled with careful observations.

2 Fundamentals

Consider the closed-loop control system shown in Fig. 1. The overall transfer function is


	T(s)=C(s)R(s)=G(s)/(1+G(s)H(s))	(1)

The stability of the closed loop system is dictated by the roots of the characteristic polynomial 1+G(s)H(s).

Figure 1 - The block diagram of a closed-loop control system.

Take the transfer functions G(s) and H(s) to be the ratio of two polynomials in s. That is,


	G(s)=N_G(s)/D_G(s)	(2)
	H(s)=N_H(s)/D_H(s)	(3)

It follows that our characteristic polynomial is


	D_G(s)D_H(s)+N_G(s)N_H(s)	(4)

From Eq. 4 we note that:

The poles of our characteristic polynomial are the poles of our open loop system.
The zeros of our characteristic polynomial are the poles of our closed loop system.

3 Contour Mapping

The process of mapping takes a contour on the s-plane and generates a contour F(s) on the F-plane. Consider Contour A to be defined by a circle in the s-plane such that


	Contour A: s=c+re^{jθ}	(5)

as θ is swept from 2π to 0. That is, Contour A is a circle that is swept in the clockwise direction. The center of Contour A is at point c, and the radius of the contour is r. Figure 2 shows a circular contour consistent with Eq. 5. It also shows three points α, β, and γ relative to Contour A that will be used in the following discussion.

Owing to the structure of Eq. 4, we are interested in mapping Contour A through F(s). We can generate sufficient insight into the mapping process with consideration of five cases:

Mapping function F(s) contains a zero that is outside of Contour A.
Mapping function F(s) contains a pole that is outside of Contour A.
Mapping function F(s) contains a zero within Contour A.
Mapping function F(s) contains a pole within Contour A.
Mapping function F(s) contains a pole and a zero within Contour A.

Figure 2 - Contour A and the points α, β, and γ that will be used to explore mapping of complex functions.

Careful consideration of these five cases indicates that:

A pole within Contour A creates counter-clockwise rotation along the map which is Contour B.
A zero within Contour A creates clockwise rotation along Contour B.
For Contour B to encircle the origin, there must be an unequal number of poles and zeros within Contour A.

We conclude that if P is the number of poles within Contour A, Z is the number of zeros within Contour A, and N is the number of counterclockwise encirclements of the origin in Contour B, then


	Z=P−N	(6)
	N=P−Z	(7)

4 The Nyquist Stability Criterion

Going back to Eq. 4,


	1+G(s)H(s)=D_G(s)D_H(s)+N_G(s)N_H(s)	(8)

If we define our mapping function F(s)=1+G(s)H(s), then:

P is the number of poles within Contour A.
Z is the number of zeros within Contour A.
N is the number of counterclockwise encirclements around the origin in Contour B.

It follows that the number of closed loop poles (zeros within Contour A) within Contour A is


	Z=P−N	(9)

The Nyquist stability criterion is a frequency response method because the important part of Contour A is when s=jω as the contour travels along the imaginary axis. Generally as ω→∞, |G(jω)H(jω)|→0, so Contour B will sit at zero as Contour A swings around from jω→∞ to jω→−∞.

The map of the open right-half plane through F(s)=G(s)H(s) is the Nyquist plot. Analysis of the plot is used to determine N, from which the stability of the closed loop system can be determined through Eq. 9. Since our concern focuses on the number of encirclements of the -1 point, it will be natural to examine the behavior of the Nyquist plot near this point.

5 References

[1] N. S. Nise, Control Systems Engineering, 6th ed., J. Wiley and Sons, 2011.

Notes on the Nyquist Stability Criterion