Nyquist criterion using Nyquist plots

To derive this criterion one starts with the rational transfer function of the open loop

and makes the following assumptions:

1. The polynomials and are relatively prime.

2. 
   
   which is always valid for realisable systems, see section 3.3.

The poles of the open loop are the roots of the characteristic equation

For stability analysis just the poles of the closed loop are of interest, i.e. the roots of the characteristic equation, which are determined by setting the denominator of the closed-loop transfer function to zero. From this condition

and

follows. Because of Eq. (5.8) is valid. Thus the function must be investigated in more detail. The zeros of this function match the poles of the closed loop and its poles match the poles of the open loop. Therefore this function can be represented by

where are the poles of the closed loop and the poles of the open loop. With respect to the position of the poles it is assumed according to Figure 5.4 that

Figure 5.4: Poles of the open and closed loop in the plane (multiple poles are counted according to their multiplicity)

a)

from the poles of the closed loop

are lying in the right-half plane,

are on the imaginary axis, and

( ) in the left-half plane.

Accordingly,

b)

from the open-loop poles

are lying in the right-half plane,

on the imaginary axis, and

( ) in the left-half plane.

and are assumed to be known. Then and will be determined from the knowledge about the frequency response locus of . Therefore with the frequency response

is calculated, for which the phase response is given by

When traverses the range of , the change in the phase consists of the parts from the polynomials and and is given by

Each root of the polynomials and , respectively, provides for and , respectively, an amount of , if they lie in the left-half plane, and each root on the right-hand side of the imaginary axis provides an amount of . These phase changes are continuous with respect to .

Each root on the imaginary axis for causes during the traverse of at a stepwise change of in the phase. This discontinuous part of the phase will not be considered in what follows.

Using the terms given above, for the continuous part of the phase change one obtains

If besides and , is also known, then from Eq. (5.14) it can be determined, whether or/and is valid, i.e. whether and how many closed-loop poles are in the right-half plane and on the imaginary axis.

To determine , the locus can be drawn on the Nyquist diagram and the phase angle checked. Expediently one moves this curve by 1 to the left in the plane. Thus for stability analysis of the closed loop the locus of the open loop according to Figure 5.5 has to be drawn.

Figure 5.5: Nyquist diagrams of and

Here is the continuous change in the angle of the vector from the so called critical point (-1,j0) to the moving point on the locus of for . Points where the locus passes through the point (-1,j0) or where it has points at infinity correspond to the zeros and poles of on the imaginary axis, respectively. These discontinuities are not taken into account for the derivation of Eq. (5.14). Figure 5.6 shows an example of a

Figure 5.6: Determination of continuous changes in the angle

where two discontinuous changes of the angle occur. Thereby the continuous change of the angle consists of three parts

The rotation is counter clockwise positive.

As the closed loop is only asymptotically stable for , then from Eq. (5.14) the general case of the Nyquist criterion follows:

The closed loop is asymptotically stable, if and only if the continuous change in the angle of the vector from the critical point (-1,j0) to the moving point of the locus of the open loop is

For the case with a negative gain of the open loop the locus is rotated by 180 relative to the case with a positive . The Nyquist criterion remains valid also in the case of a dead time in the open loop.