LINEAR CONTROL SYSTEMS

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LINEAR CONTROL SYSTEMS

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

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Lecture 25
Nyquist stability criteria (Continue).

• Topics to be covered include
• Nyquist stability criteria (continue).
• Minimum phase systems.
• Simplified Nyquist stability criterion.

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Nyquist fundamental
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Example 1 Discuss about the RHP roots of
following system.
???? 1 ?? ???? ???? ??? ??? ???? ?????? ????? ??
??? ?????? ????? k ??? ????.
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Example 1 Discuss about the RHP roots of the
following system.
???? 1 ?? ???? ???? ??? ??? ???? ?????? ????? ??
??? ?????? ????? k ??? ????.
Which point is very important?
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Example 1 Discuss about the RHP roots of the
following system for different values of k.
???? 1 ?? ???? ???? ??? ??? ???? ?????? ????? ??
??? ?????? ????? k ??? ????.
Unstable( one RHP root)
Stable
Unstable (two RHP root)
More Study
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Example 2 Discuss about the stability of the
following system for different values of k.
???? 2 ??????? ????? ?? ?? ??? ?????? ????? k
??? ????.
??? ??????????????? ?? ???? ??? ???
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Discuss about the RHP roots of
How?
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Discuss about the RHP roots of
Stable
Unstable 2 RHP roots
Unstable (2 RHP roots)
Unstable (1 RHP roots)
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Minimum Phase systems
????? ????? ???
f(s) is said to be minimum phase if it has no
poles and zeros on the RHP and on the j? axis
(origin is an exception) and there is no delay.
???? f(s) ?? ????? ??? ????? ??? ??? ???? ??? ???
? ???? ?? RHP ? ??? ???? j? ?????? (????
?????? ???) ? ????? ????? ?????.
Important note
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Nyquist fundamental for minimum phase systems
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Check the stability of following system by
Nyquist method.
System is minimum phase
f-1
90Ty
f1
90Ty
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Check the stability of following system by
Nyquist method.
Simplified Nyquist path plot
polar plot
f-1 is the angle of polar plot around -1
f1 is the angle of polar plot around 1
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Simplified Nyquist method
f-1 is the angle of polar plot around -1
f1 is the angle of polar plot around 1
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Example 3 Discuss about the RHP roots of the
following system for different values of k.
???? 3 ?? ???? ???? ??? RHP ????? ??? ?? ???
?????? ????? k ??? ????.
Clearly System is minimum phase so we use
simplified Nyquist method
No RHP root.
One RHP root.
No RHP root.
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Example 4 Discuss about the RHP roots of the
following system for different values of k.
???? 4 ?? ???? ???? ??? RHP ????? ??? ?? ???
?????? ????? k ??? ????.
Clearly System is minimum phase so we use
simplified Nyquist method
1
No RHP root.
One RHP root.
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Example 5 Discuss about the RHP roots of the
following system for different values of k.
???? 5 ?? ???? ???? ??? RHP ????? ??? ?? ???
?????? ????? k ??? ????.
Clearly System is minimum phase so we use
simplified Nyquist method
1
Two RHP roots.
One RHP root.
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Example 6 Discuss about the RHP roots of
following system for different value of k.
???? 6 ?? ???? ???? ??? RHP ????? ??? ?? ???
?????? ????? k ??? ????.
Clearly System is minimum phase so we use
simplified Nyquist method
1
Two RHP roots.
One RHP root.
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Simplified Nyquist method
f-1 is the angle of polar plot around -1
f1 is the angle of polar plot around 1
Important remark If any of f-1 or f1 is greater
than zero the system is unstable but if they were
less than zero one must check it!
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Simplified Nyquist method
Unstable
Stability depends on Ty
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Exercises
???????
1- The open loop transfer function of a
unity-feedback (negative sign) is
Apply the Nyquist criterion to determine the
range of k for stability. Let n1,2,3 and 4
2- The characteristic equation of a linear
control system is
Apply the Nyquist criterion to determine the
range of k for stability.
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Exercises
???????
3- The open loop transfer function of a
unity-feedback (negative sign) with PD controller
is
Select the value of Kp so that the parabolic
error constant be 100. Find the equivalent
open-loop transfer function Geq(s) for stability
analysis with Kd as a gain factor. Sketch the
Nyquist plot and check the stability for
different values of Kd.
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Exercises
???????
4- The polar plot of an open loop transfer
function of a minimum phase system is
Determine the steady state error of the system to
a unit step.
????
5- The open loop transfer function of a
unity-feedback (negative sign) is
Derive an expression that make the system stable.
????