Adaptive Simpson's method

2012-08-06T12:21:13Z

217.22.0.226:

The '''Observability Gramian''' is a [[Gramian]] used in [[optimal control]] theory to determine whether or not a linear system is [[Observability|observable]].

For a linear system described by

<math>\dot{x} = A x + B u</math>

<math>y = C x + D u \,</math>

The observability Gramian for a linear time invariant system is given by

<math>W_o (t) = \int\limits_0^t e^{A^T \tau} C^T C e^{A \tau} d\tau</math>

If and only if the matrix <math>W_o</math> is [[nonsingular]] for every <math>t > 0</math>, the pair <math>(A,C)</math> is observable.

==See also==
*[[Controllability Gramian]]
*[[Gramian matrix]]

==References==
{{unreferenced|date=July 2008}}

==External links==
*[http://reference.wolfram.com/mathematica/ref/ObservabilityGramian.html Mathematica function to compute the observability Gramian]

[[Category:Control theory]]

{{applied-math-stub}}

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Adaptive Simpson's method