Lec25 Control

LINEAR CONTROL SYSTEMSAli KarimpourAssistant ProfessorFerdowsi University of Mashhad

Lecture 25 Ali Karimpour Apr 2009

Lecture 25Topics to be covered include:

Nyquist stability criteria (continue).Minimum phase systems. Simplified Nyquist stability criterion. Nyquist stability criteria (Continue).


*Nyquist fundamental


1: k .Example 1: Discuss about the RHP roots of following system.


1: k .Example 1: Discuss about the RHP roots of the following system.Which point is very important?


1: k .Example 1: Discuss about the RHP roots of the following system for different values of k.Unstable( one RHP root)StableUnstable(two RHP root)More Study:


* 2: k .Example 2: Discuss about the stability of the following system for different values of k. :


Discuss about the RHP roots of:How?


Discuss about the RHP roots of:StableUnstable2 RHP rootsUnstable (2 RHP roots)Unstable (1 RHP roots)


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:


*Nyquist fundamental for minimum phase systems


*Check the stability of following system by Nyquist method.System is minimum phase90Ty-190Ty1


Check the stability of following system by Nyquist method.-1 is the angle of polar plot around -11 is the angle of polar plot around 1polar plotSimplified Nyquist path plot


Simplified Nyquist method-1 is the angle of polar plot around -11 is the angle of polar plot around 1


Clearly System is minimum phase so we use simplified Nyquist method 3: RHP k .Example 3: Discuss about the RHP roots of the following system for different values of k.No RHP root.One RHP root.No RHP root.


Clearly System is minimum phase so we use simplified Nyquist method 4: RHP k .Example 4: Discuss about the RHP roots of the following system for different values of k.1No RHP root.One RHP root.


Clearly System is minimum phase so we use simplified Nyquist method 5: RHP k .Example 5: Discuss about the RHP roots of the following system for different values of k.1Two RHP roots.One RHP root.


Clearly System is minimum phase so we use simplified Nyquist method 6: RHP k .Example 6: Discuss about the RHP roots of following system for different value of k.1Two RHP roots.One RHP root.


Simplified Nyquist method-1 is the angle of polar plot around -11 is the angle of polar plot around 1Important remark: If any of -1 or 1 is greater than zero the system is unstable but if they were less than zero one must check it!


Simplified Nyquist methodUnstableStability depends on Ty


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 n=1,2,3 and 42- The characteristic equation of a linear control system is:Apply the Nyquist criterion to determine the range of k for stability.


Exercises3- 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.


Exercises4- 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.:

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Lec25 Control