The controllability Gramian of a state-space model:

The controllability Gramian of a continuous-time system:

The controllability Gramian of a discrete-time system:

The controllability Gramian of a symbolic system:

The controllability Gramian of a descriptor system:

Check if a system is controllable:

The controllability Gramian has the dimension of the state matrix:

If the controllability Gramian has full rank, the system is controllable:

The controllability Gramian of a controllable and asymptotically stable system is symmetric and positive definite:

The controllability Gramians of asymptotically stable systems satisfy the corresponding Lyapunov equations:

The controllability Gramian is the observability Gramian of the dual system:

A descriptor state-space model gives two Gramians:

The systems is completely controllable if and only if the sum is positive definite:

The fast and slow subsystem Gramians are computed from the Kronecker decomposition:

Split the decoupled states:

The slow subsystem yields a Gramian for the slow states and a zero matrix:

The fast subsystem yields a Gramian for the fast states and a zero matrix:

Inversing the Kronecker transformation gives the Gramians for the original system:

This gives the same result as using ControllabilityGramian directly:

The controllability Gramian is not defined for a system that is not asymptotically stable: