ObservabilityGramian

ObservabilityGramian[ss]
gives the observability Gramian of the StateSpaceModel object ss.

The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a 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 observability Gramian is computed as the controllability Gramian of the dual system.

The observability Gramian of a system:

Out[1]=

The observability Gramian of a continuous-time system:

The observability Gramian of a discrete-time system:

The observability Gramian of a symbolic system:

Check if a system is observable:

The observability Gramian has the dimension of the state matrix:

If the observability Gramian has full rank, the system is observable:

The observability Gramian of an observable and asymptotically stable system is symmetric and positive definite:

The observability Gramian of a continuous-time (discrete-time) system satisfies a continuous (discrete) Lyapunov equation:

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

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