Higher-order singular value decomposition: Difference between revisions

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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:
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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>
 
where <math>p_k</math> are the [[Pole (complex analysis)|poles]] of ''L'' in the right half plane (unstable poles).
 
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:
 
<math>\int_0^\infty \ln |S(i \omega)| d \omega = 0</math>
 
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>
 
==References==
{{reflist}}
 
==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
 
{{DEFAULTSORT:Bode's Sensitivity Integral}}
[[Category:Control theory]]
 
 
{{science-stub}}

Latest revision as of 10:09, 16 December 2014

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