Nichols plot

<h2 id="use-in-control-design">Use in control design</h2>
<p>Given a <a href="/facts/Transfer_function/slDsh1lH">transfer function</a>,
</p>

G
        (
        s
        )
        =
        
          
            
              Y
              (
              s
              )
            
            
              X
              (
              s
              )
            
          
        
      
    
    {\displaystyle G(s)={\frac {Y(s)}{X(s)}}}

<p>with the <a href="/facts/Closed-loop_transfer_function/D1A3AA7q">closed-loop transfer function</a> defined as,
</p>

M
        (
        s
        )
        =
        
          
            
              G
              (
              s
              )
            
            
              1
              +
              G
              (
              s
              )
            
          
        
      
    
    {\displaystyle M(s)={\frac {G(s)}{1+G(s)}}}

<p>the Nichols plots displays 
  
    
      
        20
        
          log
          
            10
          
        
        ⁡
        (
        
          |
        
        G
        (
        s
        )
        
          |
        
        )
      
    
    {\displaystyle 20\log _{10}(|G(s)|)}
  
 versus 
  
    
      
        arg
        ⁡
        (
        G
        (
        s
        )
        )
      
    
    {\displaystyle \arg(G(s))}
  
. Loci of constant 
  
    
      
        20
        
          log
          
            10
          
        
        ⁡
        (
        
          |
        
        M
        (
        s
        )
        
          |
        
        )
      
    
    {\displaystyle 20\log _{10}(|M(s)|)}
  
 and 
  
    
      
        arg
        ⁡
        (
        M
        (
        s
        )
        )
      
    
    {\displaystyle \arg(M(s))}
  
 are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function.  Thus, the frequency 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
 is the parameter along the curve.  This plot may be compared to the <a href="/facts/Bode_plot/xcF1ej98">Bode plot</a> in which the two inter-related graphs - 
  
    
      
        20
        
          log
          
            10
          
        
        ⁡
        (
        
          |
        
        G
        (
        s
        )
        
          |
        
        )
      
    
    {\displaystyle 20\log _{10}(|G(s)|)}
  
 versus 
  
    
      
        
          log
          
            10
          
        
        ⁡
        (
        ω
        )
      
    
    {\displaystyle \log _{10}(\omega )}
  
 and 
  
    
      
        arg
        ⁡
        (
        G
        (
        s
        )
        )
      
    
    {\displaystyle \arg(G(s))}
  
 versus 
  
    
      
        
          log
          
            10
          
        
        ⁡
        (
        ω
        )
      
    
    {\displaystyle \log _{10}(\omega )}
  
) - are plotted.
</p><p>In <a href="/facts/Control_system/8CEjIDlS">feedback control design</a>, the plot is useful for assessing the stability and robustness of a linear system.  This application of the Nichols plot is central to the <a href="/facts/Quantitative_feedback_theory/WbhCmuoH">quantitative feedback theory (QFT)</a> of Horowitz and Sidi, which is a well known method for robust control system design.
</p><p>In most cases, 
  
    
      
        arg
        ⁡
        (
        G
        (
        s
        )
        )
      
    
    {\displaystyle \arg(G(s))}
  
 refers to the phase of the system's response. Although similar to a <a href="/facts/Nyquist_plot/J8rxK4yA">Nyquist plot</a>, a Nichols plot is plotted in a <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a> while a Nyquist plot is plotted in a <a href="/facts/Polar_coordinate_system/HBko4YoN">Polar coordinate system</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hall_circles/x4LD31se">Hall circles</a></li>
<li><a href="/facts/Bode_plot/xcF1ej98">Bode plot</a></li>
<li><a href="/facts/Nyquist_plot/J8rxK4yA">Nyquist plot</a></li>
<li><a href="/facts/Transfer_function/slDsh1lH">Transfer function</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://reference.wolfram.com/mathematica/ref/NicholsPlot.html">Mathematica function for creating the Nichols plot</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Isaac M. Howowitz, Synthesis of Feedback Systems, Academic Press, 1963, Lib Congress 63-12033 p. 194-198 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Boris J. Lurie and Paul J. Enright, Classical Feedback Control, Marcel Dekker, 2000, ISBN 0-8247-0370-7 p. 10 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Allen Stubberud, Ivan Williams, and Joseph DeStefano, Shaums Outline Feedback and Control Systems, McGraw-Hill, 1995, ISBN 0-07-017052-5 ch. 17 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Nichols plot open-in-new

Nichols plot