# NCFBalancedTruncation

Balanced truncation of normalized coprime factors model order reduction specification

*Since R2023b*

## Description

The `NCFBalancedTruncation`

object stores model order reduction
specifications for the balanced truncation of normalized coprime factors (NCF) of ordinary
(nonsparse) linear time-invariant (LTI) models.

## Creation

The `reducespec`

function creates an NCF balanced truncation model order reduction (MOR) object when you use
this syntax.

`R = reducespec(sys,"ncf")`

Here, `sys`

is any nonsparse LTI model. The workflow uses this object to
set up MOR tasks and store results. For the full workflow, see Task-Based Model Order Reduction Workflow.

This method requires Robust Control Toolbox™ software.

## Properties

## Object Functions

`process` | Run model order reduction algorithm |

`view (ncf)` | Plot state contributions when using balanced truncation of normalized coprime factors method |

`getrom (ncf)` | Obtain reduced-order models when using balanced truncation of normalized coprime factors method |

## Examples

## Tips

You can use this method to reduce the plant

*G*or controller*K*while preserving closed-loop stability of the following SISO or MIMO feedback loop.Stability of this loop is preserved as long as the approximation error of the reduced plant is smaller than the robustness margin for this loop given by

`ncfmargin(G,K)`

.For controllers computed with

`ncfsyn`

(Robust Control Toolbox), reducing the controller*K*that_{s}`ncfsyn`

computes for the shaped controller*G*is preferable. Both_{s}*K*and_{s}*G*are returned by_{s}`ncfsyn`

in the`info`

output argument. You can then compute*K*, the reduced controller for the original plant_{r}*G*, from*K*=_{r}*W*_{1}*K*_{sr}*W*_{2}, where*W*_{1}and*W*_{2}are the shaping weights used with`ncfsyn`

. For an example, see Reduce Controller Order While Preserving Stability and Robustness.For controllers obtained by other techniques, reduction with this method also preserves stability if the error does not exceed the

`ncfmargin`

margin. However, such reduction can partially remove integral action and introduce steady-state tracking errors. Therefore, removing any integrator terms from the controller before reduction and replacing them in the reduced controller is recommended.

## Algorithms

The balanced truncation of normalized coprime factors algorithm performs these steps to
reduce the input model *G* to the desired order *k*.

Find the left normalized coprime factorization [

*M*,_{l}*N*] of_{l}`G`

, where`G`

=*M*\_{l}*N*(see_{l}`lncf`

(Robust Control Toolbox)).Obtain the

*k*th-order approximation [*M*,_{k}*N*] of [_{k}*M*,_{l}*N*], using balanced model truncation with absolute error control (see the Algorithms section of_{l}`BalancedTruncation`

).Set

`Gred`

=*M*\_{k}*N*._{k}

## Version History

**Introduced in R2023b**

## See Also

`process`

| `view (ncf)`

| `getrom (ncf)`

| `lncf`

(Robust Control Toolbox)