Higher-order singular value decomposition: Difference between revisions

Revision as of 17:49, 27 January 2014

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:

\int_{0}^{\infty} \ln | S (i ω) | d ω = \int_{0}^{\infty} \ln | \frac{1}{1 + L (i ω)} | d ω = π \sum R e (p_{k}) - \frac{π}{2} \lim_{s \to \infty} s L (s)

where $p_{k}$ 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:

$\int_{0}^{\infty} \ln | S (i ω) | d ω = 0$

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]

References

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↑ Megretski: The Waterbed Effect. MIT OCW, 2004

[1] Megretski: The Waterbed Effect. MIT OCW, 2004

[1]

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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:
+:<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}}

Higher-order singular value decomposition: Difference between revisions

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