# SparseBalancedTruncation

## Description

The `SparseBalancedTruncation`

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

## Creation

The `reducespec`

function creates a sparse balanced truncation model order reduction object when you use this
syntax.

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

Here, `sys`

is a sparse LTI model (`sparss`

,
`mechss`

). The workflow uses this object to set up MOR tasks and store
results. For the full model order reduction 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

The sparse balanced truncation algorithm performs these steps to reduce the input model
*G* to the desired order *k*.

Find the low-rank approximations

*L*and_{r}*L*of the Gramian factors. This is based on the low-rank alternating directions implicit (LRADI) algorithm, which is an iterative method for solving the Lyapunov equations. For more details, see [1] and [2]._{o}Compute the HSVs

*σ*based on the approximate controllability and observability Gramians._{j}Obtain the reduced order model using the balanced model truncation with absolute error control [3] (see the Algorithms section of

`BalancedTruncation`

).

## References

[1] Benner, Peter, Jing-Rebecca Li,
and Thilo Penzl. “Numerical Solution of Large-Scale Lyapunov Equations, Riccati Equations, and
Linear-Quadratic Optimal Control Problems.” *Numerical Linear Algebra
with Applications* 15, no. 9 (November 2008): 755–77.
https://doi.org/10.1002/nla.622.

[2] Benner, Peter, Martin Köhler, and
Jens Saak. “Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1—Philosophy, Features, and
Application for (Parametric) Model Order Reduction.” In *Model
Reduction of Complex Dynamical Systems*, edited by Peter Benner, Tobias Breiten,
Heike Faßbender, Michael Hinze, Tatjana Stykel, and Ralf Zimmermann, 171:369–92. Cham:
Springer International Publishing, 2021.
https://doi.org/10.1007/978-3-030-72983-7_18.

[3] Varga, A. “Balancing Free
Square-Root Algorithm for Computing Singular Perturbation Approximations.” In *[1991] Proceedings of the 30th IEEE Conference on Decision and
Control*, 1062–65. Brighton, UK: IEEE, 1991.
https://doi.org/10.1109/CDC.1991.261486.

## Version History

**Introduced in R2023b**