Girish Mahajan (Editor)

Bode's sensitivity integral

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Bode's sensitivity integral, discovered by Hendrik Wade Bode, is a formula that quantifies some of the limitations in feedback control of linear parameter invariant systems. Let L be the loop transfer function and S be the sensitivity function. Then 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 )

where p k are the poles of L in the right half plane (unstable poles).

If L has at least two more poles than zeros, and has no poles in the right half plane (is stable), the equation simplifies to:

0 ln | S ( j ω ) | d ω = 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."

References

Bode's sensitivity integral Wikipedia