** Nyquist stability criterion** is based on the principle of argument which states that if there are

*P*poles and

*Z*zeros of the transfer function enclosed in s-plane contour, then the corresponding

*GH*plane contour will encircle the origin

*Z*-times in the clockwise direction and

*P*-times in anticlockwise direction, i.e. (

*P*−

*Z*) times in anticlockwise direction.

The Routh–Hurwitz criterion and root locus methods are used to determine the stability of the linear single-input single-output (SISO) system by finding the location of the roots of the characteristic equation in s-plane. The Nyquist stability criterion derived by ** H. Nyquist** is a semi-graphical method that helps in determining the absolute stability of the closed-loop system graphically from frequency response of loop transfer function (Nyquist plot) without determining the closed-loop poles. The Nyquist plot of a loop transfer function

*G(s)H(s)*is a graphical representation of the frequency response analysis when the frequency ω is varied from -∞ to ∞. Since most of the linear control systems are analyzed by using their frequency responses, Nyquist plot will be convenient in determining the stability of the system

Table of Contents

## Concept of Nyquist stability criterion

To illustrate this let us consider a contour in the *s*-plane which encloses one zero of *G*(*s*) *H*(*s*), say *s* = *α*_{1.} The remaining poles and zeros are placed in the *s*-plane outside the contour as shown in given figure below.

For any non-singular point in the *s*-plane contour, there is a corresponding point *G*(*s*) *H*(*s*) in the *GH*-plane contour. Let *s* follows a prescribed path as shown in the clockwise direction making one circle and returning to its original position. The phasor (*s* − *α*_{1}) will cover an angle of −2*π* while the net angle covered by the other two phasors, namely (*s* − *β*_{1}) and (*s* − *α*_{2}) is zero. The phasor representing the magnitude of *G*(*s*) *H*(*s*) in the *GH-*plane will undergo a net phase change of −2*π*.

That is, this phasor will encircle the origin once in the clockwise direction as shown in figure (b) below. If the contour in the *s*-plane encloses two zeros, *α*_{1} and *α*_{2} in the *s*-plane, the contour in the GH-plane will encircle the origin two-times. Generalizing this, it can be stated that for each zero of G(s) H(s) in the s-plane enclosed, there is the corresponding encirclement of origin by the contour in the *GH*-plane once in the clockwise direction.

When the contour in the *s*-plane encloses a pole and a zero say at *α* and *β* respectively, both the phasors (*s* − *α*) and (*s* − *β*) will generate an angle of 2*π* as s traverses the prescribed path. However since (*s* − *β*) will be in the denominator of G(s) *H*(*s*), the contour in the *GH-*plane will experience one clockwise and one counter clockwise encirclement of the origin.

If we have *p* number of poles and *z* number of zeros enclosed by the contour in the *s*-plane, the: contour in the *GH-*plane will encircle the origin by (*p − z*) times in counterclockwise direction [since (*s* − *p*) is in the denominator of the open loop transfer function, *G*(*s*) *H*(*s*)].

For a system to be stable, there should not be any zeros of the closed loop transfer function in the right half of *s*-plane. Therefore, *a closed loop system is considered stable, if the number of encirclements of the contour in GH-plane around the origin is equal to the open-loop pole of the transfer function*, *G*(*s*) *H*(*s*).

We can express

*G*(*s*) *H*(*s*) = [1 + *G*(*s*) *H*(*s*)] − 1

From this, it can be said that the contour *G*(*s*) *H*(*s*) in *GH-*plane corresponding to Nyquist contour in the *s*-plane is the same as contour of [1 + *G*(*s*) *H*(*s*)] drawn from the point (−1 + *j*0).

Thus it can be stated that if the open-loop transfer function *G*(*s*) *H*(*s*) corresponding to Nyquist contour in the *s*-plane encircles the point at (−1 + *j*0) in the counterclockwise direction the number of times equal to the right half *s*-plane poles of *G*(*s*) *H*(*s*), it can then be said that the closed-loop control system is stable.

## Nyquist Path or Nyquist Contour

The transfer function of a feedback control system is expressed as

\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}

The denominator is equated to 0 to represent the characteristic equation as

1+G(s)H(s)=0

The study of the stability of the closed-loop system is done by determining whether the characteristic equation has any root in the right half of the s-plane. Or, in other words, we have to determine whether C(s)/R(s) has any pole located in the right half of the s-plane. We use a contour in the s-plane which encloses the whole of the right-hand half of the s-plane. The contour will encircle in the clockwise direction having a radius of infinity. If there is any pole on the jω-axis, these are bypasses with small semicircles taking around them. If the system does not have any pole or zero at the origin or in the jω-axis, the contour is drawn as shown in the below figure (b). A few examples of the application of the Nyquist criterion for stability study will be taken up now.

Advantages of Nyquist Plot

The advantages of Nyquist plot are:

- The Nyquist plot helps in determining the relative stability of the system in addition to the absolute stability of the system.
- It determines the stability of the closed-loop system from the open-loop transfer function without calculating the roots of the characteristic equation.
- The Nyquist plot gives the degree of instability of an unstable system and indicates the ways in which the stability of the system can be improved.
- It gives information related to frequency domain characteristics such as Mr, ωr, BW etc..
- It can easily be applied to systems with a pure time delay that cannot be analyzed using the root locus method or Routh–Hurwitz criterion.

#### Example:

The open-loop transfer function of a unity feedback control system is given as

G(s)H(s)=\frac{K}{s(s^{2}+s+4)}Determine the value of *K* for which the system is stable by applying Nyquist criterion.

#### Solve:

We have,

G(s)H(s)=\frac{K}{s(s^{2}+s+4)}Putting s=jω, than we get using rationalization,

G(s)H(s)=\frac{-K[\omega +j(4-\omega ^{2})]}{j\omega [(4-\omega ^{2})^{2}+\omega ^{2}]}The crossing of Nyquist plot at the real axis can be found out by equating the imaginary part of the above to 0.

Thus, 4 – ω^{2} = 0

or, ω = ± 2

The magnitude |G(jω) H(jω)| at ω = ± 2, i.e. at ω^{2} = 4 is found as

The Nyquist plot for *G*(*jω*) *H*(*jω*) has been drawn as in above figure. From this figure we notice that the intercept on the real axis at *ω* = ± 2 is -\frac{K}{4}, which is greater than −1.

The value of *K* is found as

According to Nyquist stability criterion, if there are *P* poles and *Z* zeros in the transfer function enclosed by the *s*-plane contour, the corresponding contour in the *GH-*plane must encircle origin *Z* times in the clockwise direction and *P* times in the anticlockwise direction resulting encircling (*P* − *Z*) times in the anticlockwise direction. Again *G*(*s*) *H*(*s*) contour around the origin is the same as contour of 1 + *G*(*s*) *H*(*s*) drawn from (−1 + *j*0) point on the real axis.

Here the encirclement of −1 + *j*0 has been in the clockwise direction two times, hence *N* = −2. There is no pole on the right-hand side of the s-plane and hence *P* = 0

Putting, P – Z = N,

0 – Z = – 2 or Z = 2

Thus, for *K* > 4, the system is unstable. The system will be stable if *K* < 4.

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