Nyquist Plot Steps

7/26/2019 Nyquist Plot Steps

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CONTROL SYSTEM(part 2)

EEE 350

EN. MUHAMMAD NASIRUDDIN

MAHYUDDIN

FREQUENCY DOMAIN ANALYSIS


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Nasiruddin's Horizons 2

n Frequency Response Technique

Continues.

n NYQUIST PLOT


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NYQUIST PLOT

The basis of Nyquist Plot is the polar plot (Plot Kutub).

Polar plot of a transfer function )()( sHsG is a magnitude plot for )()( jHjG

against its phase plot with frequency, , acts as a parameter that changes from

0 to infinity aftersis replaced withj inG(s)H(s).

Mathematically, plotting a polar plot for )()( jHjG is a process of mappingthe positive side of the S-planes imaginary into a )()( jHjG -plane.


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NYQUIST PLOT

Generally a polar plot or nyquist plot of a system is done by the aid of computer.

However, a sketch can be done if the following information:

The behaviour of the magnitude and phase for )()( jHjG at 0 frequency (w=0)

and infinite frequency (w=).

The intersection point between the polar plot and the real, imaginary axis in the

G(jw)H(jw)-plane, and the values of w at the intersection point.


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n Worked Example:

Sketch a polar plot for the following transfer function.

)5)(1(

10

)(

++= ssssG


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Solution:

First, substitute s withjwin the transfer function,

)5(6

)5(6*

)5(6-

10

)55(

10

)5)((

10)5)(1(

10)(

32

32

32

223

2

www

www

www

wwww

www

www

w

---

---

-+=

+---

=

++-=

++

=

j

j

j

jj

jj

jjj

jG


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NYQUIST PLOT

234

22

)5(36

)5(1060)(

www

www

w--

---=

jjG

At frequency 0=w , we only observe the most significant terms that take the effect. For

this case,

000

2510)(

=== ==w

ww

wjj

jG .

Magnitude for G(jw) at frequency 0=w ,

===== ww

ww

wwww

2lim

2lim)(lim)(

0000 j

jGjG


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NYQUIST PLOT

Phase for G(jw) at frequency w=0,

o902

lim)(0

0-==

= ww

ww j

jG

At w , we shall look at the most significant term that takes effect when the frequency

approaches infinity. The term of G(jw) is3)(

10)(

w

ww

jjG = .

For magnitude,

010

lim)(

10lim)(

33===

www

www

jjG


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NYQUIST PLOTFor phase,

( )o270

10lim|)(

3-=

=

w

w

w

wj

jG

The point of intersection of the plot with the real axis,

5

5

0)-10(5

0)5(36

)-(510-

0)(Im

2

2

234

2

=

=

=

=-+

=

w

w

w

www

ww

wjG


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NYQUIST PLOT

The intersection point between the polar plot and the real axis is

when 5=w at,

3

1|)(

5 -==wwjG

The intersection between the polar plot with the imaginary axis

can be obtained by setting the real part of 0)( =wjG .

Re 0)( =wjG


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Therefore,

0)( =jG

=

=

-+

-

w

www

w0

)5(36

60

234

2

Nyquist Diagram0 2


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Real Axis

-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

B

-20 dB

-10 dB-6 dB-4 dB-2 dB

System: Open Loop L

Real: -0.327

Imag: -0.000358

Frequency (rad/sec): -2.27


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Nyquist stability criterion

yquist stability criterion is a graphical method to determine the stability of

a closed-loop system by examining the behaviour of the frequency domain in

response to the open-loop system.

yquist stability criterion determines the stability of the closed-loop system

based on the open-loop transfer function of that system.


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The stability of a closed-loop system can be determined by means of characteristicequation, that is )()(1)( sHsGsF += in the S-plane when s equals to the points on the

Nyquist path. Then, we need to study the behaviour of the plot, comparing with

the origin in the S-plane. This plot is called theNyquist Plotfor 1+G(s)H(s).

However, to simplify things, it is easy to construct a Nyquist plot for G(s)H(s) in

the G(s)H(s)-plane rather than in 1+G(s)H(s)-plane like what we did for Polar plot

(remember?)


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NYQUIST PLOTThere are two types of stability to be examined in any control system:

Open-loop stability

Closed-loop stability

By using the Nyquist criterion,

1. The stability of open loop system can be found by studying the behaviour of the

Nyquist plot forG(s)H(s) in relative to the origin of G(s)H(s)-plane although the

poles ofG(s)H(s)are not known.

2. The stability of closed loop system can be found by studying the behaviour of

Nyquist plot forG(s)H(s)in relative to the (-1,j0) point.


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Nyquist Path what is it?

-a path that goes in counterclockwise direction (arah lawan jam) that encloses

the right-half S-plane and does not pass through the poles of F(s)=1+G(s)H(s)=0,

located on the imaginary axis(instead, the Nyquist path encircles half way and

roceed downwards)


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The Nyquist stability criterion methods can be summarized as follows:

1. The Nyquist path is determined in S-plane.

2. Nyquist plot for G(s)H(s) is sketched in the G(s)H(s)-plane with s value equals to

the points value along the Nyquist-path.

3. The open-loop and closed-loop stability for a system can be determined by

observing the behaviour of the Nyquist plot for G(s)H(s) relative to the origin

(0,j0) and point (-1,j0).


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The followings are the symbols used to determine the system stability by using

Nyquist Criterion:

0N : The number of encirclement around the origin (0,j0) by theNyquist plotfor

G(s)H(s)(positive if the encirclement(kepungan) is counterclockwise direction.

:0Z The number of zeros forG(s)H(s)that have been enclosed (dikepung)by the

Nyquist pathor on the right half of s-plane.

:0P The number of poles forG(s)H(s)that have been enclosed by theNyquist

path or on the right half of s-plane.


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:1-N The number of encirclement around the point (-1,j0) by the Nyquist plot forG(s)H(s)(positive if the encirclement is in counterclockwise direction)

:1-Z The number of zeros forF(s)=1 + G(s)H(s)that have been enclosed by theNyquist path or on the right half of S-plane.

:1- The number of poles forF(s)=1+G(s)H(s)that have been enclosed by theNyquist path or on the right half of s-plane.

Since poles forG(s)H(s)is the same as poles forF(s)=1+G(s)H(s), then

10 -=


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By Nyquist Criterion, for open-loop system stability, the following should be adhered,

000 PZN -=

with

00=P

for closed-loop stability, then,

111 --- -= PZN

with

01=-Z


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Nyquist Stability Criterion can be stated as follow:

i. For open-loop system to be stable, the Nyquist plot for G(s)H(s) must encloses or

encircles(mengepung) origin (0,j0) as many as the number of zeros of G(s)H(s) that

situates on the right half of S-plane. The encirclement must be in counterclockwise

direction ,hence00

ZN = .

ii. For closed-loop system to be stable, the Nyquist plot for G(s)H(s)must encircles the

point (-1,j0) in clockwise direction with number of encirclements as many as the

number of poles of G(s)H(s) that located on the right-half of S-plane, hence

011 PPN -=-= -- .


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Steps in determining the stability using Nyquist Stability Criterion:

i. From the characteristic equation, F(s)=1+G(s)H(s)=0, the Nyquist path on the S-

plane is constructed from the behaviour of zero-pole ofG(s)H(s)at first.

ii. Sketch the Nyquist plot forG(s)(s) on the G(s)H(s) plane.

iii. Determine the value of 10 -NandN from the behaviour of Nyquist plot for G(s)H(s)

with respect to origin point (0,j0) and point (-1,j0).

iv. Obtain the value of 0P (if not known) from

000 PZN -= ( 0Z is known)

If 00=P , then the open loop system is stable.


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v. Then, after 0P is known, obtain the value of 1-P by 0P= 1-P .

vi. Obtain 1-Z from 111 --- -= PZN .

If 1-Z =0, then, the closed-loop system is stable.


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Examples 1

)5()()(

+=

ss

KsHsG

Determine the system stability when K changes from 0 to infiniti.


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n Gain margin and phase margin from Nyquist plot.

Gain cross-over frequency is the frequency at which the

point on the Nyquist Plot for G(s)H(s) has magnitude equals

to 1.

1)()(1

==ww

sHsG

Phase cross-over frequency is the frequency at which thepoint on the Nyquist plot for G(s)H(s) has phase difference

of 180


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n The gain margin can be obtained from the Nyquist plot

by the followings,

n In designing a control system, phase margin is chosen

such that it is in range between 30 to 60.

XGain

jHjGX

1Margin

)()(

=

= ww


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Nyquist Plot Steps

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