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:

$\int_0^\infty \ln |S(i \omega)| d \omega = \int_0^\infty \ln \left| \frac{1}{1+L(i \omega)} \right| d \omega = \pi \sum Re(p_k) - \frac{\pi}{2} \lim_{s\rightarrow\infty} s L(s)$

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:

$\int_0^\infty \ln |S(i \omega)| d \omega = 0$

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."

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