# InternallyBalancedDecomposition

yields the internally balanced decomposition of the state-space model ssm.

# Details and Options

• For a standard StateSpaceModel the result is a list {p,bssm}, where p is the similarity transformation matrix and bssm is the internally balanced form of ssm.
• For a descriptor StateSpaceModel the result is a list {{p, q},bssm}, where p and q are a pair of transformation matrices.
• InternallyBalancedDecomposition accepts a Method option with the following settings:
•  Automatic automatically choose method "Eigensystem" use eigenvalue decomposition "SingularValues" use singular value decomposition
• The methods "Eigensystem" and "SingularValues" call Eigensystem and SingularValueDecomposition, respectively. In each case, the additional options relevant to the corresponding function can be specified as Method->{"name",opt1-> val1,opt2-> val2,}.

# Examples

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## Basic Examples(1)

The internally balanced realization of a state-space model:

## Scope(3)

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:

The balanced realization of a descriptor 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 submatrices of the balanced model:

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 ControllabilityGramian and ObservabilityGramian are equal for a balanced system:

The diagonal entries are given by Hankel singular values for the original system:

The original and balanced realizations are related by a similarity transformation:

Use StateSpaceTransform to transform the original system:

The system matrices from the balanced and transformed system are identical:

## Possible Issues(2)

The state-space model must be both completely controllable and observable:

The state-space model must be asymptotically stable:

It is only marginally stable:

Wolfram Research (2010), InternallyBalancedDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html (updated 2012).

#### Text

Wolfram Research (2010), InternallyBalancedDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html (updated 2012).

#### CMS

Wolfram Language. 2010. "InternallyBalancedDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html.

#### APA

Wolfram Language. (2010). InternallyBalancedDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html

#### BibTeX

@misc{reference.wolfram_2024_internallybalanceddecomposition, author="Wolfram Research", title="{InternallyBalancedDecomposition}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html}", note=[Accessed: 22-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_internallybalanceddecomposition, organization={Wolfram Research}, title={InternallyBalancedDecomposition}, year={2012}, url={https://reference.wolfram.com/language/ref/InternallyBalancedDecomposition.html}, note=[Accessed: 22-May-2024 ]}