THE NYQUIST STABILITY CRITERION

Stan Zak

School of Electrical and Computer EngineeringPurdue University, West Lafayette, IN

November 05, 2018

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Outline

The term “Encircle”

The encirclement property

The Nyquist stability criterion for continuous-time linearfeedback systems

Examples

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Outline

The term “Encircle”

The encirclement property

The Nyquist stability criterion for continuous-time linearfeedback systems

Examples

2 / 20

Outline

The term “Encircle”

The encirclement property

The Nyquist stability criterion for continuous-time linearfeedback systems

Examples

2 / 20

Outline

The term “Encircle”

The encirclement property

The Nyquist stability criterion for continuous-time linearfeedback systems

Examples

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The term “Encircle”

Consider an open-loop transfer function

GH (s) = (s − z1)(s − z2)

where s is a complex variable and z1 and z2 are two real zeros ofGH (s)

Plot GH (s) for values of s along a closed contour in the s-plane

Traverse the contour in the s-plane in a clockwise direction

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The term “Encircle”

Consider an open-loop transfer function

GH (s) = (s − z1)(s − z2)

where s is a complex variable and z1 and z2 are two real zeros ofGH (s)

Plot GH (s) for values of s along a closed contour in the s-plane

Traverse the contour in the s-plane in a clockwise direction

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The term “Encircle”

Consider an open-loop transfer function

GH (s) = (s − z1)(s − z2)

where s is a complex variable and z1 and z2 are two real zeros ofGH (s)

Plot GH (s) for values of s along a closed contour in the s-plane

Traverse the contour in the s-plane in a clockwise direction

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Example 1

The closed curve C in the GH -plane represents a plot of values ofGH (s) for values of s along the circular contour in the s-plane

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Counting the clockwise encirclements of the plotGH (s) of the origin

The zero, z1, of GH (s) is inside the circular contour in the s-planeand the zero, z2 of W (p) is outside this contour

At any point s on the circular contour in the s-plane, the angle ofGH (s) is the sum of the angles of the two vectors

v¯1 = s − z1

andv¯2 = s − z2

to the point s

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Counting the clockwise encirclements of the plotGH (s) of the origin

The zero, z1, of GH (s) is inside the circular contour in the s-planeand the zero, z2 of W (p) is outside this contour

At any point s on the circular contour in the s-plane, the angle ofGH (s) is the sum of the angles of the two vectors

v¯1 = s − z1

andv¯2 = s − z2

to the point s

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Counting the clockwise encirclements of the plotGH (s) of the origin—contd

As we traverse the contour once, the angle φ1 of the vector v¯1

from the zero inside the contour in the s-plane encounters a netchange of−2π radians

The angle φ2 of the vector v¯2 from the zero outsie the contourin

the s-plane encounters no net change

Thus on the plot of GH (s) there is a net change of−2π

Said another way, the plot of GH (s) encircles the origin once inthe clockwise direction

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Counting the clockwise encirclements of the plotGH (s) of the origin—contd

As we traverse the contour once, the angle φ1 of the vector v¯1

from the zero inside the contour in the s-plane encounters a netchange of−2π radians

The angle φ2 of the vector v¯2 from the zero outsie the contourin

the s-plane encounters no net change

Thus on the plot of GH (s) there is a net change of−2π

Said another way, the plot of GH (s) encircles the origin once inthe clockwise direction

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Counting the clockwise encirclements of the plotGH (s) of the origin—contd

As we traverse the contour once, the angle φ1 of the vector v¯1

from the zero inside the contour in the s-plane encounters a netchange of−2π radians

The angle φ2 of the vector v¯2 from the zero outsie the contourin

the s-plane encounters no net change

Thus on the plot of GH (s) there is a net change of−2π

Said another way, the plot of GH (s) encircles the origin once inthe clockwise direction

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Counting the clockwise encirclements of the plotGH (s) of the origin—contd

As we traverse the contour once, the angle φ1 of the vector v¯1

from the zero inside the contour in the s-plane encounters a netchange of−2π radians

The angle φ2 of the vector v¯2 from the zero outsie the contourin

the s-plane encounters no net change

Thus on the plot of GH (s) there is a net change of−2π

Said another way, the plot of GH (s) encircles the origin once inthe clockwise direction

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Counting the clockwise encirclements of the plot ofthe origin for a general rational GH (s)

As we traverse the contour once in the clockwise direction, anypoles and zeros of GH (s) outside the countour will contribute nonet change of the angle of GH (s)

Each zero inside the contour in the s-plane will contribute a netchange of−2π

Each pole inside the contour in the s-plane will contribute a netchange of +2π

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Counting the clockwise encirclements of the plot ofthe origin for a general rational GH (s)

As we traverse the contour once in the clockwise direction, anypoles and zeros of GH (s) outside the countour will contribute nonet change of the angle of GH (s)

Each zero inside the contour in the s-plane will contribute a netchange of−2π

Each pole inside the contour in the s-plane will contribute a netchange of +2π

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Counting the clockwise encirclements of the plot ofthe origin for a general rational GH (s)

As we traverse the contour once in the clockwise direction, anypoles and zeros of GH (s) outside the countour will contribute nonet change of the angle of GH (s)

Each zero inside the contour in the s-plane will contribute a netchange of−2π

Each pole inside the contour in the s-plane will contribute a netchange of +2π

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Encirclement Property

As a closed contour in the s-plane is traversed once in the clockwisedirection, the corresponding plot of GH (s) for values of s along thecontour encicles the origin in the GH -plane in the clockwisedirection a net number of times equal to the number of zeros minusthe number of poles contained within the contour in the s-plane

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Example 2

This example comes from A. V. Oppenheim, A. S. Willsky with S. H. Nawab,Signals & Systems, 2nd ed., Prentice Hall, 1997, page 849

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Application of the encirclement property to testingthe stability of a closed-loop system

Consider the closed-loop charactersistic equation in the rationalform

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

or equaivalently the function

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

The closed-loop system is stable there are no zeros of thefunction R(s) in the right half of the s-planeNote that

R(s) = 1 +N (s)

D(s)

=D(s) + N (s)

D(s)

=CLCPOLCP

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Application of the encirclement property to testingthe stability of a closed-loop system

Consider the closed-loop charactersistic equation in the rationalform

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

or equaivalently the function

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

The closed-loop system is stable there are no zeros of thefunction R(s) in the right half of the s-planeNote that

R(s) = 1 +N (s)

D(s)

=D(s) + N (s)

D(s)

=CLCPOLCP

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Application of the encirclement property to testingthe stability of a closed-loop system

Consider the closed-loop charactersistic equation in the rationalform

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

or equaivalently the function

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

The closed-loop system is stable there are no zeros of thefunction R(s) in the right half of the s-planeNote that

R(s) = 1 +N (s)

D(s)

=D(s) + N (s)

D(s)

=CLCPOLCP

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Representing the right-half s-plane

The right-half plane represented as the interior of the semicircularcontour where the radius r tends to∞

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Evaluating R(s) on the contour Γr

Along the semicircular porition of the contour Γr the functionR(s) remains bounded

We have

R(s) =CLCPOLCP

=sn + an−1sn−1 + · · ·+ a1s + a0

sn + cn−1sn−1 + · · ·+ c1s + c0

Thuslim|s|→∞

R(s) = 1

Therefore, as r →∞, the value of R(s) does not change as wetraverse the semicircular part of the contour Γr

The constant value of R(s) along the semicircle is equal to thevalue of R(s) at the end points, that is, the value of R(jω) atω = ±∞

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Evaluating R(s) on the contour Γr

Along the semicircular porition of the contour Γr the functionR(s) remains bounded

We have

R(s) =CLCPOLCP

=sn + an−1sn−1 + · · ·+ a1s + a0

sn + cn−1sn−1 + · · ·+ c1s + c0

Thuslim|s|→∞

R(s) = 1

Therefore, as r →∞, the value of R(s) does not change as wetraverse the semicircular part of the contour Γr

The constant value of R(s) along the semicircle is equal to thevalue of R(s) at the end points, that is, the value of R(jω) atω = ±∞

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Evaluating R(s) on the contour Γr

Along the semicircular porition of the contour Γr the functionR(s) remains bounded

We have

R(s) =CLCPOLCP

=sn + an−1sn−1 + · · ·+ a1s + a0

sn + cn−1sn−1 + · · ·+ c1s + c0

Thuslim|s|→∞

R(s) = 1

Therefore, as r →∞, the value of R(s) does not change as wetraverse the semicircular part of the contour Γr

The constant value of R(s) along the semicircle is equal to thevalue of R(s) at the end points, that is, the value of R(jω) atω = ±∞

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Evaluating R(s) on the contour Γr

Along the semicircular porition of the contour Γr the functionR(s) remains bounded

We have

R(s) =CLCPOLCP

=sn + an−1sn−1 + · · ·+ a1s + a0

sn + cn−1sn−1 + · · ·+ c1s + c0

Thuslim|s|→∞

R(s) = 1

Therefore, as r →∞, the value of R(s) does not change as wetraverse the semicircular part of the contour Γr

The constant value of R(s) along the semicircle is equal to thevalue of R(s) at the end points, that is, the value of R(jω) atω = ±∞

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Evaluating R(s) on the contour Γr

Along the semicircular porition of the contour Γr the functionR(s) remains bounded

We have

R(s) =CLCPOLCP

=sn + an−1sn−1 + · · ·+ a1s + a0

sn + cn−1sn−1 + · · ·+ c1s + c0

Thuslim|s|→∞

R(s) = 1

Therefore, as r →∞, the value of R(s) does not change as wetraverse the semicircular part of the contour Γr

The constant value of R(s) along the semicircle is equal to thevalue of R(s) at the end points, that is, the value of R(jω) atω = ±∞

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Evaluating R(s) on the contour Γr—contd

The plot of R(s) along the contour Γr can be obtained by plottingR(s) along the part of the contour that coincides with theimaginary axis—that is, the plot of R(jω) as ω varies from−∞ to+∞Note that

R(jω) = 1 + G(jω)H (jω)

Thus R(s) along the contour Γr can be drawn from knowledge ofthe open-loop transfer function G(jω)H (jω)

The plot of G(jω)H (jω) as ω varies from−∞ to +∞ is called theNyquist plot

We have, G(jω)H (jω) = R(jω)− 1

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Evaluating R(s) on the contour Γr—contd

The plot of R(s) along the contour Γr can be obtained by plottingR(s) along the part of the contour that coincides with theimaginary axis—that is, the plot of R(jω) as ω varies from−∞ to+∞Note that

R(jω) = 1 + G(jω)H (jω)

Thus R(s) along the contour Γr can be drawn from knowledge ofthe open-loop transfer function G(jω)H (jω)

The plot of G(jω)H (jω) as ω varies from−∞ to +∞ is called theNyquist plot

We have, G(jω)H (jω) = R(jω)− 1

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Evaluating R(s) on the contour Γr—contd

The plot of R(s) along the contour Γr can be obtained by plottingR(s) along the part of the contour that coincides with theimaginary axis—that is, the plot of R(jω) as ω varies from−∞ to+∞Note that

R(jω) = 1 + G(jω)H (jω)

Thus R(s) along the contour Γr can be drawn from knowledge ofthe open-loop transfer function G(jω)H (jω)

The plot of G(jω)H (jω) as ω varies from−∞ to +∞ is called theNyquist plot

We have, G(jω)H (jω) = R(jω)− 1

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Evaluating R(s) on the contour Γr—contd

The plot of R(s) along the contour Γr can be obtained by plottingR(s) along the part of the contour that coincides with theimaginary axis—that is, the plot of R(jω) as ω varies from−∞ to+∞Note that

R(jω) = 1 + G(jω)H (jω)

Thus R(s) along the contour Γr can be drawn from knowledge ofthe open-loop transfer function G(jω)H (jω)

The plot of G(jω)H (jω) as ω varies from−∞ to +∞ is called theNyquist plot

We have, G(jω)H (jω) = R(jω)− 1

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Evaluating R(s) on the contour Γr—contd

The plot of R(s) along the contour Γr can be obtained by plottingR(s) along the part of the contour that coincides with theimaginary axis—that is, the plot of R(jω) as ω varies from−∞ to+∞Note that

R(jω) = 1 + G(jω)H (jω)

Thus R(s) along the contour Γr can be drawn from knowledge ofthe open-loop transfer function G(jω)H (jω)

The plot of G(jω)H (jω) as ω varies from−∞ to +∞ is called theNyquist plot

We have, G(jω)H (jω) = R(jω)− 1

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Plots of R(jω) = 1 + G(jω)H (jω) in the(1 + GH )-plane and GH -plane

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The term encircle one more time

Clckwise encirclements of the point−1 + j0

A closed contour C is said to make N clockwise encirclements of thepoint−1 + j0 if a radial line drawn from the point−1 + j0 to a pointon C rotates in a clckwise direction through 2πN radians in goingcompletely round C

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Example to illustrate the term encircleDetermine the number of clockwise encirclements by the contour Cof the points

(a) the origin;

(b) the point−1 + j0

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Example to illustrate the term encircleDetermine the number of clockwise encirclements by the contour Cof the points

(a) the origin;

(b) the point−1 + j0

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The Nyquist criterion

Let Z be number of closed loop poles in the right half s-plane,that is, the number of poles of

G(s)

1 + G(s)H (s)

in the right half plane

Let P be the number of the open-loop poles in the right halfplane, that is, the number of poles of G(s)H (s) in the right halfplane

Let N be the number of clockwise encirclememnts of the point−1 + j0 by the Nyquist plot of G(jω)H (jω)

The criterion

Z = N + P

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The Nyquist criterion

Let Z be number of closed loop poles in the right half s-plane,that is, the number of poles of

G(s)

1 + G(s)H (s)

in the right half plane

Let P be the number of the open-loop poles in the right halfplane, that is, the number of poles of G(s)H (s) in the right halfplane

Let N be the number of clockwise encirclememnts of the point−1 + j0 by the Nyquist plot of G(jω)H (jω)

The criterion

Z = N + P

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The Nyquist criterion

Let Z be number of closed loop poles in the right half s-plane,that is, the number of poles of

G(s)

1 + G(s)H (s)

in the right half plane

Let P be the number of the open-loop poles in the right halfplane, that is, the number of poles of G(s)H (s) in the right halfplane

Let N be the number of clockwise encirclememnts of the point−1 + j0 by the Nyquist plot of G(jω)H (jω)

The criterion

Z = N + P

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The Nyquist criterion

Let Z be number of closed loop poles in the right half s-plane,that is, the number of poles of

G(s)

1 + G(s)H (s)

in the right half plane

Let P be the number of the open-loop poles in the right halfplane, that is, the number of poles of G(s)H (s) in the right halfplane

Let N be the number of clockwise encirclememnts of the point−1 + j0 by the Nyquist plot of G(jω)H (jω)

The criterion

Z = N + P

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The Nyquist stability criterion

Since G(jω)H (jω) = R(jω)− 1, the plot G(jω)H (jω) encircles thepoint−1 + j0 exactly as many times as

R(jω) = 1 + G(jω)H (jω)

encirles the origin

The criterionFor the closed-loop system to be stable, the net number of clockwiseencirclements of the point−1 + j0 by the Nyquist plot of theopen-loop transfer function G(jω)H (jω) must equal the number ofright-half poles of G(s)H (s), that is,

N = −P ≤ 0

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The Nyquist stability criterion

Since G(jω)H (jω) = R(jω)− 1, the plot G(jω)H (jω) encircles thepoint−1 + j0 exactly as many times as

R(jω) = 1 + G(jω)H (jω)

encirles the origin

The criterionFor the closed-loop system to be stable, the net number of clockwiseencirclements of the point−1 + j0 by the Nyquist plot of theopen-loop transfer function G(jω)H (jω) must equal the number ofright-half poles of G(s)H (s), that is,

N = −P ≤ 0

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Example 3—Nyquist plot of GH (s) = s+1(s−1)( 1

2 s+1)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Nyquist Diagram

Real Axis

Ima

gin

ary

Axis

System: sys

Real: -0.554

Imag: 0.752

Frequency (rad/s): -0.761

System: sys

Real: -0.201

Imag: -0.775

Frequency (rad/s): 1.49

System: sys

Real: -0.00845

Imag: -0.412

Frequency (rad/s): 4.43

System: sys

Real: -0.883

Imag: -0.44

Frequency (rad/s): 0.321

System: sys

Real: -1

Imag: 1.22e-16

Frequency (rad/s): -0

System: sys

Real: 1.22e-33

Imag: 2e-17

Frequency (rad/s): -1e+17

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Example 4—Nyquist plot of GH (s) = 1s(s−1)

-infinity -8 -6 -4 -2 0

Real Axis

-15

-10

-5

0

5

10

15

Ima

g A

xis

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