This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

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
  • The observability Gramian:
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:
The observability Gramian of a system:
In[1]:=
Click for copyable input
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:
New in 8