ControllabilityGramian

ControllabilityGramian[ssm]

gives the controllability Gramian of the state-space model ssm.

Details and Options

  • The state-space model ssm can be given as StateSpaceModel[{a,b,}], 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
  • For a StateSpaceModel with a descriptor matrix, ControllabilityGramian returns a pair of matrices {wcs,wcf}, where wcs is associated with the slow subsystem, and wcf is associated with the fast subsystem.
  • The controllability Gramians only exist for descriptor systems in which Det[λ e-a]0 for some λ.

Examples

open allclose all

Basic Examples  (1)

The controllability Gramian of a state-space model:

In[1]:=
Click for copyable input
Out[1]//MatrixForm=

Scope  (4)

Applications  (1)

Properties & Relations  (7)

Possible Issues  (1)

See Also

LyapunovSolve  DiscreteLyapunovSolve  ControllableModelQ  ControllabilityMatrix  ObservabilityGramian

Introduced in 2010
(8.0)
| Updated in 2012
(9.0)