BUILT-IN MATHEMATICA SYMBOL

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.

MORE INFORMATION

InternallyBalancedDecomposition[ss, Method->method] specifies the method to compute the decomposition.

Possible settings include:

	Automatic	automatically 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 methods or call Eigensystem and SingularValueDecomposition, respectively. In each case the additional options relevant to the corresponding function can be specified as Method.

EXAMPLES

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Basic Examples (1)

The internally balanced realization of a state-space model:

In[1]:=

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:

Properties & Relations (2)

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:

Possible Issues (2)

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: