LINEAR CONTROL SYSTEMS

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LINEAR CONTROL SYSTEMS

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

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Lecture 24
Nyquist stability criteria.

• Topics to be covered include
• Nyquist stability criteria.

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Mapping
?????
c'
d
c
4
2
a
a'
b
d'
b'
4
Mapping
?????
s0.5
0.2
1
-2
4
4
5
Mapping
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r0.5
1
-2
5
5
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Mapping
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-2
-5
1
7
6
6
7
Mapping
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r2.5
1
-2
7
8
1kf(s)0
Nyquist fundamental
Nyquist path

• ????? ?????? RHP? 1kf(s)

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Nyquist fundamental
Nyquist path
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Nyquist fundamental
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Example 1 Check the stability of the following
system by Nyquist method.
???? 1 ??????? ????? ?? ???? ??? ????????? ?????
????.
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Example 1 Check the stability of the following
system by Nyquist method.
???? 1 ??????? ????? ?? ???? ??? ????????? ?????
????.
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Nyquist fundamental
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More study
?????? ?????
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Example 2 Check the stability of following
system by Nyquist method.
???? 2 ??????? ????? ?? ???? ??? ????????? ?????
????.
Very important
Very important
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Example 2 Check the stability of following
system by Nyquist method.
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How to calculate?
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Example 2 Check the stability of following
system by Nyquist method.
??????? ????? ?? ???? ??? ????????? ????? ????.
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Example 2 Check the stability of following
system by Nyquist method.
??????? ????? ?? ???? ??? ????????? ????? ????.
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?????? ?????
More study
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?Example 3 Check the stability of following
system from the given Nyquist plot.
???? 3 ??????? ????? ?? ?? ???? ?? ?????
????????? ???? ??? ????? ????.
d(s) is a polynomial.
System is unstable( 1 RHP zero)
System is stable
System is unstable( 2 RHP zero)
System is unstable( 1 RHP zero)
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?Example 3 Check the stability of following
system from the given Nyquist plot.
???? 3 ??????? ????? ?? ?? ???? ?? ?????
????????? ???? ??? ????? ????.
d(s) is a polynomial.
System is unstable( 1 RHP zero)
System is stable
System is unstable( 2 RHP zero)
System is unstable( 1 RHP zero)
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Exercises
???????
1- Assume the following control system
Find the value of k which make the system stable.
Answer kgt1
2- Consider the following control system
Find the value of k which make the system stable.
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Exercises
???????
3- Polar plot of an open loop transfer function
is shown in the following
Discuss about the stability of system.
Answer It is unstable and it has 4 unstable
poles.
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Exercises
???????
4- Discuss the stability of following system by
Nyquist criteria..