BUILT-IN MATHEMATICA SYMBOL

ControllabilityGramian

ControllabilityGramian[ss]
gives the controllability Gramian of the StateSpaceModel object ss.

The state-space model ss can be given as StateSpaceModel, where a and b represent the state and input matrices in either the continuous-time system or the discrete-time system:

		continuous-time system
		discrete-time system

		continuous-time system
		discrete-time system

For asymptotically stable systems, the Gramian can be computed as the solution of a Lyapunov equation:

		continuous-time system
		discrete-time system

The controllability Gramian of a state-space model:

Out[1]//MatrixForm=

The controllability Gramian of a continuous-time system:

The controllability Gramian of a discrete-time system:

The controllability Gramian of a symbolic 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:

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