Bode's sensitivity integral

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:

${\displaystyle \int _{0}^{\infty }\ln |S(j\omega )|d\omega =\int _{0}^{\infty }\ln \left|{\frac {1}{1+L(j\omega )}}\right|d\omega =\pi \sum Re(p_{k})-{\frac {\pi }{2}}\lim _{s\rightarrow \infty }sL(s)}$

where $p_{k}$ 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:

${\displaystyle \int _{0}^{\infty }\ln |S(j\omega )|d\omega =0}$

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."^[1]

References

1. Megretski: The Waterbed Effect. MIT OCW, 2004

Further reading

• Karl Johan Åström and Richard M. Murray. Feedback Systems: An Introduction for Scientists and Engineers. Chapter 11 - Frequency Domain Design. Princeton University Press, 2008. http://www.cds.caltech.edu/~murray/amwiki/Frequency_Domain_Design