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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: | | Hello, I'm Pansy, a 21 year old from Adare, Australia.<br>My hobbies include (but are not limited to) Vintage clothing, Painting and watching Arrested Development.<br><br>Here is my website :: [http://ahairlossblog.Wordpress.com/ vivicomb ebay] |
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| :<math>\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)</math>
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| where <math>p_k</math> are the [[Pole (complex analysis)|poles]] of ''L'' in the right half plane (unstable poles).
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| If ''L'' has at least two more poles than [[Zero (complex analysis)|zeros]], and has no poles in the right half plane (is stable), the equation simplifies to:
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| <math>\int_0^\infty \ln |S(i \omega)| d \omega = 0</math> | |
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| 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."<ref>[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-245-multivariable-control-systems-spring-2004/lecture-notes/lec5_6245_2004.pdf Megretski: The Waterbed Effect. MIT OCW, 2004]</ref>
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * 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
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| {{DEFAULTSORT:Bode's Sensitivity Integral}}
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| [[Category:Control theory]]
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| {{science-stub}}
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Revision as of 21:21, 23 February 2014
Hello, I'm Pansy, a 21 year old from Adare, Australia.
My hobbies include (but are not limited to) Vintage clothing, Painting and watching Arrested Development.
Here is my website :: vivicomb ebay