Sensitivity (control systems)

<h2 id="sensitivity-function">Sensitivity function</h2>
<p>Let 
  
    
      
        G
        (
        s
        )
      
    
    {\displaystyle G(s)}
  
 and 
  
    
      
        C
        (
        s
        )
      
    
    {\displaystyle C(s)}
  
 denote the plant and controller's transfer function in a basic closed loop control system written in the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace</a> domain using unity <a href="/facts/Negative_feedback/G9byVgpN">negative feedback</a>.
</p><h3>Sensitivity function as a measure of robustness to parameter variation</h3>
<p>The closed-loop transfer function is given by
</p><p>
  
    
      
        T
        (
        s
        )
        =
        
          
            
              G
              (
              s
              )
              C
              (
              s
              )
            
            
              1
              +
              G
              (
              s
              )
              C
              (
              s
              )
            
          
        
        .
      
    
    {\displaystyle T(s)={\frac {G(s)C(s)}{1+G(s)C(s)}}.}

</p><p>Differentiating 
  
    
      
        T
      
    
    {\displaystyle T}
  
 with respect to 
  
    
      
        G
      
    
    {\displaystyle G}
  
 yields 
</p><p>
  
    
      
        
          
            
              d
              T
            
            
              d
              G
            
          
        
        =
        
          
            d
            
              d
              G
            
          
        
        
          [
          
            
              
                G
                C
              
              
                1
                +
                G
                C
              
            
          
          ]
        
        =
        
          
            C
            
              (
              1
              +
              C
              G
              
                )
                
                  2
                
              
            
          
        
        =
        S
        
          
            T
            G
          
        
        ,
      
    
    {\displaystyle {\frac {dT}{dG}}={\frac {d}{dG}}\left[{\frac {GC}{1+GC}}\right]={\frac {C}{(1+CG)^{2}}}=S{\frac {T}{G}},}

</p><p>where 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is defined as the function 
</p><p>
  
    
      
        S
        (
        s
        )
        =
        
          
            1
            
              1
              +
              G
              (
              s
              )
              C
              (
              s
              )
            
          
        
      
    
    {\displaystyle S(s)={\frac {1}{1+G(s)C(s)}}}
  
 
</p><p>and is known as the sensitivity function. Lower values of 
  
    
      
        
          |
        
        S
        
          |
        
      
    
    {\displaystyle |S|}
  
 implies that relative errors in the plant parameters has less effects in the relative error of the closed-loop transfer function.  
</p>
<h3>Sensitivity function as a measure of disturbance attenuation</h3>
<p>The sensitivity function also describes the transfer function from external disturbance to process output. In fact, assuming an additive disturbance <i>n</i> after the output  
</p><p>of the plant, the transfer functions of the closed loop system are given by
</p><p>
  
    
      
        Y
        (
        s
        )
        =
        
          
            
              C
              (
              s
              )
              G
              (
              s
              )
            
            
              1
              +
              C
              (
              s
              )
              G
              (
              s
              )
            
          
        
        R
        (
        s
        )
        +
        
          
            1
            
              1
              +
              C
              (
              s
              )
              G
              (
              s
              )
            
          
        
        N
        (
        s
        )
        .
      
    
    {\displaystyle Y(s)={\frac {C(s)G(s)}{1+C(s)G(s)}}R(s)+{\frac {1}{1+C(s)G(s)}}N(s).}

</p><p>Hence, lower values of 
  
    
      
        
          |
        
        S
        
          |
        
      
    
    {\displaystyle |S|}
  
 suggest further attenuation of the external disturbance. The sensitivity function tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that 
  
    
      
        
          |
        
        S
        (
        j
        ω
        )
        
          |
        
      
    
    {\displaystyle |S(j\omega )|}
  
 is less than one are reduced by an amount equal to the distance to the critical point 
  
    
      
        −
        1
      
    
    {\displaystyle -1}
  
 and disturbances with frequencies such that 
  
    
      
        
          |
        
        S
        (
        j
        ω
        )
        
          |
        
      
    
    {\displaystyle |S(j\omega )|}
  
 is larger than one are amplified by the feedback.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="sensitivity-peak-and-sensitivity-circle">Sensitivity peak and sensitivity circle</h2>
<h3>Sensitivity peak</h3>
<p>It is important that the largest value of the sensitivity function be limited for a control system. The nominal sensitivity peak 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 is defined as<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>
  
    
      
        
          M
          
            s
          
        
        =
        
          max
          
            0
            ≤
            ω
            <
            ∞
          
        
        
          |
          
            S
            (
            j
            ω
            )
          
          |
        
        =
        
          max
          
            0
            ≤
            ω
            <
            ∞
          
        
        
          |
          
            
              1
              
                1
                +
                G
                (
                j
                ω
                )
                C
                (
                j
                ω
                )
              
            
          
          |
        
      
    
    {\displaystyle M_{s}=\max _{0\leq \omega <\infty }\left|S(j\omega )\right|=\max _{0\leq \omega <\infty }\left|{\frac {1}{1+G(j\omega )C(j\omega )}}\right|}

</p><p>and it is common to require that the maximum value of the sensitivity function, 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
, be in a range of 1.3 to 2.
</p>
<h3>Sensitivity circle</h3>
<p>The quantity 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 is the inverse of the shortest distance from the <a href="/facts/Nyquist_plot/J8rxK4yA">Nyquist curve</a> of the loop transfer function to the critical point 
  
    
      
        −
        1
      
    
    {\displaystyle -1}
  
. A sensitivity 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 guarantees that the distance from the critical point to the Nyquist curve is always greater than 
  
    
      
        
          
            1
            
              M
              
                s
              
            
          
        
      
    
    {\displaystyle {\frac {1}{M_{s}}}}
  
 and the Nyquist curve of the loop transfer function is always outside a circle around the critical point 
  
    
      
        −
        1
        +
        0
        j
      
    
    {\displaystyle -1+0j}
  
 with the radius 
  
    
      
        
          
            1
            
              M
              
                s
              
            
          
        
      
    
    {\displaystyle {\frac {1}{M_{s}}}}
  
, known as the sensitivity circle. 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 defines the maximum value of the sensitivity function and the inverse of 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 gives you the shortest distance from the open-loop transfer function 
  
    
      
        L
        (
        j
        ω
        )
      
    
    {\displaystyle L(j\omega )}
  
 to the critical point 
  
    
      
        −
        1
        +
        0
        j
      
    
    {\displaystyle -1+0j}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Robust_control/SDh0dFfp">Robust control</a></li>
<li><a href="/facts/PID_controller/7XcYk0CH">PID controller</a></li>
<li><a href="/facts/Bode%2527s_sensitivity_integral/eFJCwr0d">Bode's sensitivity integral</a></li></ul>

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

<ol>
<li id="fn:1"><p>K.J. Astrom, "Model uncertainty and robust control," in Lecture Notes on Iterative Identification and Control Design. Lund, Sweden: Lund Institute of Technology, Jan. 2000, pp. 63–100. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>K.J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning, 2nd ed. Research Triangle Park, NC 27709, USA: ISA - The Instrumentation, Systems, and Automation Society, 1995. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>A. G. Yepes, et al., "Analysis and design of resonant current controllers for voltage-source converters by means of Nyquist diagrams and sensitivity function" in IEEE Trans. on Industrial Electronics, vol. 58, No. 11, Nov. 2011, pp. 5231–5250. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Karl Johan Åström and Richard M. Murray. Feedback systems : an introduction for scientists and engineers. Princeton University Press, Princeton, NJ, 2008. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Sensitivity (control systems) open-in-new

Sensitivity (control systems)