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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
  • The controllability 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 controllability Gramian of a state-space model:
The controllability Gramian of a state-space model:
In[1]:=
Click for copyable input
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:
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