Difference between revisions of "Frequency Domain Analysis"
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−  {{chheader  +  {{chheaderTransfer FunctionsFrequency Domain AnalysisPID 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.  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 ==  
−  {{am05pdf  +  {{am05pdfam08analysis28Sep12Frequency Domain Analysis}} 
* 1. Introduction  * 1. Introduction  
* 2. The Nyquist Criterion  * 2. The Nyquist Criterion  
* 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 ==  
* [[#Frequently Asked QuestionsFrequently Asked Questions]]  * [[#Frequently Asked QuestionsFrequently Asked Questions]]  
−  * Wikipedia entries:  +  * [[#ErrataErrata]] 
+  * 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.png]]</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>.</p></li>  
+  
+  <li><p>Stability margins describe the robustness of a system to perturbations in the dynamics. We define the ''phase crossover frequency'', <amsmath>\omega_180</amsmath> as the smallest frequency where the phase of the loop transfer function is <math>180^\circ</math> and the ''gain crossover frequency'', <math>\omega_{gc}</math> as the small frequency where the loop transfer function has unit magnitude. The ''gain margin'' and ''phase margin'' are given by  
+  <center><amsmath>  
+  g_m=\frac{1}{L(j\omega_{180})} \qquad \varphi_m=\pi+\arg{L(j\omega_{gc})}  
+  </amsmath></center>  
+  These margins describe the the maximum variation in gain and phase in the loop transfer function under which the system remains stable. Two other margins are the ''stability margin'', which is the shortest distance frmo the Nyquist curve to the critical point <math>s=1</math>, and the ''delay margin'', which is the smallest time delay required to make the system unstable.  
+  </p></li>  
+  
+  <li><p>''Bode's relations'' relate the gain and phase of a transfer function with no poles or zeros in the right half plane. They show that  
+  <center><amsmath>  
+  \arg{G(j\omega_0)} \approx  
+  \frac{\pi}{2} \frac{d\log{G(j\omega)}} {d\log{\omega}}.  
+  </amsmath></center>  
+  A ''nonminimum phase'' sytem is one for which there is more phase lag than the amount given by Bode's relations. Systems with right have plane poles or zeros are nonminimum phase.  
+  </p></li>  
+  
+  <li><p>The ''gain'' of an input/output system is defined as  
+  <center><amsmath>  
+  \gamma=\sup_{u\in\mathcal U}\frac{\y\}{\u\},  
+  </amsmath></center>  
+  where sup is the supremum. The ''small gain theorem'' states that if two systems with gains <math>\gamma_1</math> and <math>\gamma_2</math> are connected in a feedback loop, then the closed loop system is stable if <math>\gamma_1\gamma_2</math>.  
+  </p></li>  
+  
+  </ol>  
+  
+  {{chaptertable begin}}  
+  {{chaptertable left}}  
+  
+  <!  
+  == Additional Exercises ==  
+  The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text.  
+  <ncl>Frequency Domain Analysis Exercises</ncl>  
+  >  
== Frequently Asked Questions ==  == Frequently Asked Questions ==  
−  <  +  <ncl>Frequency Domain Analysis FAQ</ncl> 
+  == Errata ==  
+  <ncl>Frequency Domain Analysis errata v2.11b</ncl>  
+  * [[:Category:Frequency Domain Analysis errataFull list of errata starting from first printing]]  
+  * {{submitbug}}  
+  {{chaptertable right}}  
+  == MATLAB code ==  
+  The following MATLAB scripts are available for producing figures that appear in this chapter.  
+  * Figure 9.4: {{matlabfileloopanalnyquist_threepole.m}}  
+  * Figure 9.5: {{matlabfileloopanalsketch.m}}  
+  * Figure 9.6: {{matlabfileloopanalcongctrl_nyquist.m}}, {{matlabfile.congctrl_params.m}}, {{matlabfile.congctrl_equil.m}}  
+  * Figure 9.7: {{matlabfileloopanalnyqcondstab.m}}  
+  * Figure 9.8: {{matlabfileloopanalinvpend_nyquist.m}}  
+  * Figure 9.10: {{matlabfileloopanalsimple.m}}  
+  * Figure 9.11: {{matlabfileloopanalniklas.m}}  
+  * Figure 9.12: {{matlabfileloopanalosc_integral.m}}  
+  * Figure 9.14: {{matlabfileloopanalsteering_fwdrev.m}}  
+  * Figure 9.17: {{matlabfileloopanaldesfcn_relay.m}}, {{matlabfileloopanaldesfcn_hystsim.m}}, {{matlabfileloopanalrelaysine.mdl}}  
+  See the [[softwaresoftware page]] for more information on how to run these scripts.  
+  
== Additional Information ==  == Additional Information ==  
+  * [http://www.engin.umich.edu/group/ctm/freq/nyq.html Control Tutorials for Matlab, Nyquist Criterion]  
+  {{chaptertable end}} 
Latest revision as of 23:03, 11 November 2012
Prev: Transfer Functions  Chapter 9  Frequency Domain 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 ContentsFrequency Domain Analysis (pdf, 28Sep12)

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
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 . Stability margins describe the robustness of a system to perturbations in the dynamics. We define the phase crossover frequency, as the smallest frequency where the phase of the loop transfer function is and the gain crossover frequency, as the small frequency where the loop transfer function has unit magnitude. The gain margin and phase margin are given by
These margins describe the the maximum variation in gain and phase in the loop transfer function under which the system remains stable. Two other margins are the stability margin, which is the shortest distance frmo the Nyquist curve to the critical point , and the delay margin, which is the smallest time delay required to make the system unstable.
Bode's relations relate the gain and phase of a transfer function with no poles or zeros in the right half plane. They show that
A nonminimum phase sytem is one for which there is more phase lag than the amount given by Bode's relations. Systems with right have plane poles or zeros are nonminimum phase.
The gain of an input/output system is defined as
where sup is the supremum. The small gain theorem states that if two systems with gains and are connected in a feedback loop, then the closed loop system is stable if .
Frequently Asked Questions
Errata

MATLAB codeThe following MATLAB scripts are available for producing figures that appear in this chapter.
See the software page for more information on how to run these scripts. Additional Information 