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:

where 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:

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]


Further reading[edit]

  • Karl Johan Åström and Richard M. Murray. Feedback Systems: An Introduction for Scientists and Engineers. Chapter 11 - Frequency Domain Design. Princeton University Press, 2008.
  • Stein, G. (2003). "Respect the unstable". IEEE Control Systems Magazine. 23 (4): 12–25. doi:10.1109/MCS.2003.1213600. ISSN 1066-033X.
  • Costa-Castelló, Ramon; Dormido, Sebastián (2015). "An interactive tool to introduce the waterbed effect". IFAC-PapersOnLine. 48 (29): 259–264. doi:10.1016/j.ifacol.2015.11.246. ISSN 2405-8963.

External links[edit]

  • WaterbedITOOL - Interactive software tool to analyze, learn/teach the Waterbed effect in linear control systems.
  • Gunter Stein’s Bode Lecture on fundamental limitations on the achievable sensitivity function expressed by Bode's integral.

See also[edit]