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# InternallyBalancedDecomposition

 InternallyBalancedDecomposition[ss] yields the internally balanced decomposition of the StateSpaceModel object ss. The result is a list where s is the similarity transformation matrix and ib is the internally balanced form of ss.
• Possible settings include:
 Automatic automatically choose the suitable method "Eigensystem" use eigenvalue decomposition "SingularValues" use singular value decomposition
• The default Method selects for all exact systems and inexact continuous-time systems, and otherwise.
The internally balanced realization of a state-space model:
The internally balanced realization of a state-space model:
 Out[1]=
 Scope   (2)
The internally balanced realization of a SISO system:
The two realizations are different forms of the same model:
The balanced realization of a MIMO system:
 Applications   (1)
In a balanced realization, each state is just as controllable as it is observable:
Get an approximation to the model by truncating the least controllable and observable mode:
Get an approximation by residualizing the least controllable and observable mode:
The truncated model better approximates the system during the transients, and the residualized model better approximates the system at steady state:
The controllability and observability Gramians of the balanced form are equal. They are diagonal matrices with the Hankel singular values on the leading diagonal:
The original and balanced realizations are related by the similarity transformation:
The state-space model must be a minimal realization, which is both completely controllable and observable:
The state-space model must be asymptotically stable:
It is only marginally stable:
New in 8