Difference between revisions of "Frequency Domain Analysis"

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In this chapter we study how how stability and robustness of closed loop systems can be determined by investigating how signals propagate around the feedback loop. The Nyquist stability theorem is a key result that provides a way to analyze stability and introduce measures of degrees of stability.
 
In this chapter we study how how stability and robustness of closed loop systems can be determined by investigating how signals propagate around the feedback loop. The Nyquist stability theorem is a key result that provides a way to analyze stability and introduce measures of degrees of stability.
  
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== Textbook Contents ==
 
== Textbook Contents ==
 
{{am05pdf|am06-analysis|16Sep06|Loop Analysis|}}
 
{{am05pdf|am06-analysis|16Sep06|Loop Analysis|}}
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* 3. Stability Margins
 
* 3. Stability Margins
 
* 4. Bode's Relations
 
* 4. Bode's Relations
* 5. The Small Gain Theorem
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* 5. The Notion of Gain
 
* 6. Further Reading
 
* 6. Further Reading
 
* 7. Exercises
 
* 7. Exercises
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== Lecture Materials ==
 
== Lecture Materials ==
 
* [[Lecture: Loop Analysis]]
 
* [[Lecture: Loop Analysis]]
* [[#Homework|Additional Homework Problems]]
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* [[#Exercises|Additional Exercises]]
  
 
== Supplemental Information ==
 
== Supplemental Information ==
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* Wikipedia entries: [http://en.wikipedia.org/wiki/Nyquist_stability_criterion Nyquist stability criterion]
 
* Wikipedia entries: [http://en.wikipedia.org/wiki/Nyquist_stability_criterion Nyquist stability criterion]
 
* [[#Additional Information|Additional Information]]
 
* [[#Additional Information|Additional Information]]
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== Homework ==
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== Chapter Summary ==
<!-- Inline:Category:Chapter 8 HW -->
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This chapter describes the use of the Nyquist criterion for determining the stability of a system:
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<ol>
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<li><p>The ''loop transfer function'' of a feedback system represents the transfer function obtained by breaking the feedback loop and computing the resulting transfer function of the open loop system.  For a simple feedback system
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<center>[[Image:loopanal_fbksys.png]]</center>
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the loop transfer function is given by <math>L = P C</math>
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</p></li>
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<li><p>The ''Nyquist criterion'' provides a way to check the stability of a closed loop system by looking at the properties of the loop transfer function.  For a stable open loop system, the Nyquist criterion states that the system is stable if the contour of the loop transfer function plotted from <math>s = -j\infty</math> to <math>s = j \infty</math> has no net encirclements of the point <math>s=-1</math> when it is plotted on the complex plane.</p></li>
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<li><p>The general Nyquist criterion uses the image of the loop transfer function applied to the ''Nyquist countour''
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<center>[[Image:loopanal_nyqcontour]]</center>
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The number of unstable poles of the closed loop system is given by the number of open loop unstable poles plus the number of clockwise encirclements of the point <math>s = -1</math>.
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<li><p>Stability margins describe</p></li>
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<li><p>''Bode's relations''</p></li>
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<li><p>The ''gain'' of a system</p></li>
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<li><p></p></li>
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<li><p></p></li>
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<li><p></p></li>
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</ol>
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== Exercises ==
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<ncl>Loop Analysis Exercises</ncl>
 
== Frequently Asked Questions ==
 
== Frequently Asked Questions ==
<!-- Inline:Category:Chapter 8 FAQ -->
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<ncl>Loop Analysis FAQ</ncl>
 
== Additional Information ==
 
== Additional Information ==
 
* [http://www.engin.umich.edu/group/ctm/freq/nyq.html Control Tutorials for Matlab, Nyquist Criterion]
 
* [http://www.engin.umich.edu/group/ctm/freq/nyq.html Control Tutorials for Matlab, Nyquist Criterion]

Revision as of 19:21, 4 November 2006

Prev: Transfer Functions Chapter 9 - Loop Analysis Next: PID Control

In this chapter we study how how stability and robustness of closed loop systems can be determined by investigating how signals propagate around the feedback loop. The Nyquist stability theorem is a key result that provides a way to analyze stability and introduce measures of degrees of stability.

Textbook Contents

Loop Analysis (pdf, 16Sep06)

  • 1. Introduction
  • 2. The Nyquist Criterion
  • 3. Stability Margins
  • 4. Bode's Relations
  • 5. The Notion of Gain
  • 6. Further Reading
  • 7. Exercises

Lecture Materials

Supplemental Information

Chapter Summary

This chapter describes the use of the Nyquist criterion for determining the stability of a system:

  1. The loop transfer function of a feedback system represents the transfer function obtained by breaking the feedback loop and computing the resulting transfer function of the open loop system. For a simple feedback system

    Loopanal fbksys.png

    the loop transfer function is given by

  2. The Nyquist criterion provides a way to check the stability of a closed loop system by looking at the properties of the loop transfer function. For a stable open loop system, the Nyquist criterion states that the system is stable if the contour of the loop transfer function plotted from to has no net encirclements of the point when it is plotted on the complex plane.

  3. The general Nyquist criterion uses the image of the loop transfer function applied to the Nyquist countour

    File:Loopanal nyqcontour

    The number of unstable poles of the closed loop system is given by the number of open loop unstable poles plus the number of clockwise encirclements of the point .

  4. <p>Stability margins describe

  5. Bode's relations

  6. The gain of a system

Exercises

Frequently Asked Questions

Additional Information