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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:
Automaticautomatically choose the suitable method
"Eigensystem"use eigenvalue decomposition
"SingularValues"use singular value decomposition
  • The default Method->Automatic 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:
In[1]:=
Click for copyable input
Out[1]=
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:
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:
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