Observability Gramian

Observability Gramian: The Observability Gramian is a Gramian used in optimal control theory to determine whether or not a linear system is observable.

For a linear system described by

$\dot{x} = A x + B u$

$y = C x + D u \,$

The observability Gramian is given by

$W_o (t) = \int\limits_0^t e^{A^T \tau} C^T C e^{A \tau} d\tau$

If and only if the matrix $W o$ is nonsingular for every $t > 0$ , the pair $(A, C)$ is observable.

See also

Controllability Gramian

Gramian matrix

References

Categories:
Control theory
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