Observability Gramian

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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 Wo is nonsingular for any t > 0, the pair (A,C) is observable.

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