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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.

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

For multi-input, multi-output (MIMO) systems, if the loop gain L(s) has entries with pole excess of at least two, the theorem generalizes to $${\displaystyle \int _{0}^{\infty }\ln |\det S(j\omega )|\,d\omega =\pi \sum \operatorname {Re} (p_{k}),}$$ where ${\displaystyle p_{k}}$ are the unstable poles of L(s).^[2]

• 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. hdl:2117/79317. ISSN 2405-8963.

• 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.
• Use of Bode's Integral Theorem (circa 1945) – NASA publication.

• Bode plot
• Law of conservation of complexity
• Sensitivity (control systems)
• Waterbed theory