Block diagram of feedback control of a dynamical process.
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:
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]
bode, sensitivity, integral, this, article, technical, most, readers, understand, please, help, improve, make, understandable, experts, without, removing, technical, details, june, 2015, learn, when, remove, this, message, discovered, hendrik, wade, bode, form. This article may be too technical for most readers to understand Please help improve it to make it understandable to non experts without removing the technical details June 2015 Learn how and when to remove this message 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 Block diagram of feedback control of a dynamical process 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 0 ln S j w d w 0 ln 1 1 L j w d w p R e p k p 2 lim s s L s 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 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 0 ln S j w d w 0 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 Contents 1 References 2 Further reading 3 External links 4 See alsoReferences edit Megretski The Waterbed Effect MIT OCW 2004Further reading editKarl Johan Astrom 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 Stein G 2003 Respect the unstable IEEE Control Systems Magazine 23 4 12 25 doi 10 1109 MCS 2003 1213600 ISSN 1066 033X Costa Castello Ramon Dormido Sebastian 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 editWaterbedITOOL 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 See also editBode plot Sensitivity control systems Retrieved from https en wikipedia org w index php title Bode 27s sensitivity integral amp oldid 1167108867, wikipedia, wiki, book, books, library,
