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.  
−  +  {{chaptertable begin}}  
−  +  {{chaptertable left}}  
−  +  
== Textbook Contents ==  == Textbook Contents ==  
{{am05pdfam06analysis16Sep06Loop Analysis}}  {{am05pdfam06analysis16Sep06Loop Analysis}}  
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* 3. Stability Margins  * 3. Stability Margins  
* 4. Bode's Relations  * 4. Bode's Relations  
−  * 5. The  +  * 5. The Notion of Gain 
* 6. Further Reading  * 6. Further Reading  
* 7. Exercises  * 7. Exercises  
−  
+  {{chaptertable right}}  
== Lecture Materials ==  == Lecture Materials ==  
* [[Lecture: Loop Analysis]]  * [[Lecture: Loop Analysis]]  
−  * [[#  +  * [[#ExercisesAdditional 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 InformationAdditional Information]]  * [[#Additional InformationAdditional Information]]  
−  +  {{chaptertable end}}  
−  ==  +  == Chapter Summary == 
−  <  +  
+  This chapter describes the use of the Nyquist criterion for determining the stability of a system:  
+  <ol>  
+  <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  
+  <center>[[Image:loopanal_fbksys.png]]</center>  
+  the loop transfer function is given by <math>L = P C</math>  
+  </p></li>  
+  
+  <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>  
+  
+  <li><p>The general Nyquist criterion uses the image of the loop transfer function applied to the ''Nyquist countour''  
+  <center>[[Image:loopanal_nyqcontour]]</center>  
+  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>.  
+  
+  <li><p>Stability margins describe</p></li>  
+  
+  <li><p>''Bode's relations''</p></li>  
+  
+  <li><p>The ''gain'' of a system</p></li>  
+  
+  <li><p></p></li>  
+  
+  <li><p></p></li>  
+  
+  <li><p></p></li>  
+  
+  </ol>  
+  
+  == Exercises ==  
+  <ncl>Loop Analysis Exercises</ncl>  
== Frequently Asked Questions ==  == Frequently Asked Questions ==  
−  <  +  <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

Lecture MaterialsSupplemental Information

Chapter Summary
This chapter describes the use of the Nyquist criterion for determining the stability of a system:
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
the loop transfer function is given by
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.
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 .
 <p>Stability margins describe
Bode's relations
The gain of a system
Exercises
Frequently Asked Questions