Controllability Gramian

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In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable.

For a time-invariant linear system described by

\dot{x} = A x + B u

the controllability Gramian is given by

W_c (t) = \int\limits_0^t e^{A\tau} B B^T e^{A^T \tau} d\tau

If the matrix Wc is nonsingular for any t > 0, then the pair (A,B) is controllable.

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