Bode's sensitivity integral
Encyclopedia
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:
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."
Hendrik Wade Bode
Hendrik Wade Bode , was an American engineer, researcher, inventor, author and scientist], of Dutch ancestry. As a pioneer of modern control theory and electronic telecommunications he revolutionized both the content and methodology of his chosen fields of research.He made important contributions...
, is a formula that quantifies some of the limitations in feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
control of linear parameter invariant systems. Let L be the loop transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
and S be the sensitivity function. Then 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
Zero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
, 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."
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