Bode's sensitivity integral

Bode's sensitivity integral, discovered by <a href="/facts/Hendrik_Wade_Bode/aGVSy4aJ">Hendrik Wade Bode</a>, is a formula that quantifies some of the limitations in <a href="/facts/Feedback/XnuWH3zp">feedback</a> control of linear parameter invariant systems. Let L be the loop <a href="/facts/Transfer_function/slDsh1lH">transfer function</a> and S be the <a href="/facts/Sensitivity_(control_systems)/0VrLab58">sensitivity function</a>. 
In the diagram, P is a dynamical process that has a transfer function P(s). The controller, C, has the transfer function C(s). The controller attempts to cause the process output, y, to track the reference input, r. Disturbances, d, and measurement noise, n, may cause undesired deviations of the output. Loop gain is defined by L(s) = P(s)C(s).
The following holds:

∫
          
            0
          
          
            ∞
          
        
        ln
        ⁡
        
          |
        
        S
        (
        j
        ω
        )
        
          |
        
        d
        ω
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        ln
        ⁡
        
          |
          
            
              1
              
                1
                +
                L
                (
                j
                ω
                )
              
            
          
          |
        
        d
        ω
        =
        π
        ∑
        R
        e
        (
        
          p
          
            k
          
        
        )
        −
        
          
            π
            2
          
        
        
          lim
          
            s
            →
            ∞
          
        
        s
        L
        (
        s
        )
      
    
    {\displaystyle \int _{0}^{\infty }\ln |S(j\omega )|d\omega =\int _{0}^{\infty }\ln \left|{\frac {1}{1+L(j\omega )}}\right|d\omega =\pi \sum Re(p_{k})-{\frac {\pi }{2}}\lim _{s\rightarrow \infty }sL(s)}

where 
 
 
 
 
 p
 
 k
 
 
 
 
 {\displaystyle p_{k}}
 
 are the <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> of L in the right half plane (unstable poles).
If L has at least two more poles than <a href="/facts/Zero_(complex_analysis)/gnA3U8VP">zeros</a>, and has no poles in the right half plane (is stable), the equation simplifies to:

∫
          
            0
          
          
            ∞
          
        
        ln
        ⁡
        
          |
        
        S
        (
        j
        ω
        )
        
          |
        
        d
        ω
        =
        0
      
    
    {\displaystyle \int _{0}^{\infty }\ln |S(j\omega )|d\omega =0}

This equality shows that if sensitivity to disturbance is suppressed at some frequency range, it is necessarily increased at some other range. This has been called the "waterbed effect."

Bode's sensitivity integral open-in-new

Bode's sensitivity integral