# BalancedTruncation

## Description

The `BalancedTruncation`

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

## Creation

The `reducespec`

function creates a balanced truncation MOR object when you use this syntax.

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

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.

## Properties

## Object Functions

`process` | Run model order reduction algorithm |

`view (balanced)` | Plot state contributions when using balanced truncation method |

```
getrom
(balanced)
``` | Obtain reduced-order models when using balanced truncation method |

## Examples

## Algorithms

Balanced truncation first decomposes the system *G* into its stable and
unstable parts:

$$G={G}_{s}+{G}_{u}$$

When you specify `R.Options.Goal`

as `"absolute"`

, the
software uses the balanced truncation method of [1] to reduce
*G _{s}*. This computes the HSVs

*σ*based on the controllability and observability Gramians. For order

_{j}*r*, the absolute error $${\Vert {G}_{s}-{G}_{r}\Vert}_{\infty}$$ is bounded by $$2{\displaystyle \sum _{j=r+1}^{n}{\sigma}_{j}}$$. Here,

*n*is the number of states in

*G*.

_{s}When you specify `R.Options.Goal`

as `"relative"`

, the
software uses the balanced stochastic truncation method of [2] to reduce
*G _{s}*. For square

*G*, this computes the HSVs

_{s}*σ*of the phase matrix $$F={\left(W\text{'}\right)}^{-1}G$$, where

_{j}*W(s)*is a stable, minimum-phase spectral factor of

*GG’*:

$$W\text{'}(s)W(s)=G(s)G\text{'}(s)$$

For order *r*, the relative error $${\Vert {G}_{s}{}^{-1}({G}_{s}-{G}_{r})\Vert}_{\infty}$$ is bounded by

$$\prod _{j=r+1}^{H}\left(\frac{1+{\sigma}_{j}}{1-{\sigma}_{j}}\right)}-1\approx 2{\displaystyle \sum _{j=r+1}^{n}{\sigma}_{j}$$

where $$2{\displaystyle \sum _{j=r+1}^{n}{\sigma}_{j}}\ll 1$$.

## References

[1] Varga, A., "Balancing-Free
Square-Root Algorithm for Computing Singular Perturbation Approximations," *Proc. of
30th IEEE CDC*, Brighton, UK (1991), pp. 1062-1065.

[2] Green, M., "A Relative Error Bound
for Balanced Stochastic Truncation", *IEEE Transactions on Automatic
Control*, Vol. 33, No. 10, 1988

## Version History

**Introduced in R2023b**