In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system
the controllability Gramian is given by
If and only if the matrix is nonsingular, i.e. has full rank, for any, then the pair is controllable.
Linear time-variant state space models of form
- ,
are controllable in an interval if and only if the rows of the matrix product where is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of . To have linear independency Gramian matrix have to be nonsingular, i.e., invertible.