NYQUIST PLOT Dr Nasiruddin

8/10/2019 NYQUIST PLOT Dr Nasiruddin

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CONTROL SYSTEM(part 2)

EEE 350

DR. MUHAMMAD NASIRUDDIN

MAHYUDDIN

FREQUENCY DOMAIN ANALYSIS


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Frequency Response Technique

Continues.

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 afters is replaced with j in G(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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NYQUIST PLOT

Worked Example:

Sketch a polar plot for the following transfer function.

)5)(1(

10

)(

ssssG

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NYQUIST PLOT

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

j

j

j

jj

jj

jjj

jG

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NYQUIST PLOT

234

22

)5(36

)5(1060)(

jjG

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

this case,

000

2510)(

jjjG .

Magnitude for G(jw) at frequency 0 ,

2lim

2lim)(lim)(

0000 j

jGjG

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NYQUIST PLOT

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

902

lim)(0

0

j

jG

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

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

10)(

jjG

.

For magnitude,

010

lim)(

10lim)(

33

jjG

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NYQUIST PLOTFor phase,

270

10lim|)(

3

jjG

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

jG

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NYQUIST PLOT

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

when 5 at,

3

1|)(

5 jG

The intersection between the polar plot with the imaginary axis

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

Re 0)( jG

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Therefore,

0)( jG

0)5(36

60

234

2

DR NASIRUDDIN

Nyquist Diagram


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Nyquist Diagram

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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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

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

the origin in the S-plane. This plot is called the Nyquist Plot for 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?)

DR NASIRUDDIN


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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 for G(s)H(s)in relative to the origin of G(s)H(s)-plane although the

poles of G(s)H(s)are not known.

2.

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

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

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yquist Pathwhat is it?

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

the ri ght-hal f S-plane and does not pass thr ough the poles of F (s)=1+G(s)H (s)=0,

located on the imaginary axis(instead, the Nyqui st path encir cles hal f way and

proceed 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 tothe 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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NYQUIST PLOT

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

yquist Criterion:

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

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

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

Nyquist path or on the right half of s-plane.

:0P The number of poles for G(s)H(s)that have been enclosed by the Nyquist

path or on the right half of s-plane.

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NYQUIST PLOT

:1N 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)

:1Z 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.

:1P 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 for G(s)H(s)is the same as poles forF(s)=1+G(s)H(s), then

10 PP

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

000 PZN

with

00P

for closed-loop stability, then,

111 PZN

with

01Z

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NYQUIST PLOT

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 ,hence 00 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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NYQUIST PLOT

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 of G(s)H(s) at first.

ii. Sketch the Nyquist plot for G(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 00P , then the open loop system is stable.

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NYQUIST PLOT

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

vi. Obtain 1Z from 111 PZN .

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

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NYQUIST PLOT

Examples 1

)5()()(

ss

KsHsG

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

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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

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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NYQUIST PLOT

The gain margin can be obtained from the Nyquist plot

by the followings,

In designing a control system, phase margin is chosen

such that it is in range between 30to 60.

XGain

jHjGX

1Margin

)()(

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