## Model Reduction Techniques

Robust Control Toolbox™ software offers several algorithms
for model approximation and order reduction. These algorithms let
you control the absolute or relative approximation error, and are
all based on the Hankel singular values of the system.

Robust control theory quantifies a system uncertainty as either *additive* or *multiplicative *types.
These model
reduction routines are also categorized into two groups: *additive
error* and *multiplicative error* types.
In other words, some model reduction routines produce a reduced-order
model `Gred`

of the original model `G`

with
a bound on the error $${\Vert G-Gred\Vert}_{\infty}$$,
the peak gain across frequency. Others produce a reduced-order model
with a bound on the relative error $${\Vert {G}^{-1}\left(G-Gred\right)\Vert}_{\infty}$$.

These theoretical bounds are based on the "tails"
of the Hankel singular values of the model, which are given as follows.

Additive error bound:

$${\Vert G-Gred\Vert}_{\infty}\le 2{\displaystyle \sum _{k+1}^{n}{\sigma}_{i}}$$

Here, *σ*_{i} are
denoted the* i*th Hankel singular value of the
original system `G`

.

Multiplicative (relative) error bound:

$${\Vert {G}^{-1}\left(G-Gred\right)\Vert}_{\infty}\le {\displaystyle \prod _{k+1}^{n}\left(1+2{\sigma}_{i}\left(\sqrt{1+{\sigma}_{i}^{2}}+{\sigma}_{i}\right)\right)-1}$$

Here, *σ*_{i} are
denoted the* i*th Hankel singular value of the
phase matrix of the model `G`

(see the `bstmr`

reference page).

### Commands for Model Reduction

**Top-Level Model Reduction Command**

Method | Description |

`reduce`
| Main interface to model approximation algorithms |

**Normalized Coprime Balanced Model Reduction
Command**

Method | Description |

`ncfmr`
| Normalized coprime balanced truncation |

**Additive Error Model
Reduction Commands **

Method | Description |

`balancmr`
| Square-root balanced model truncation |

`schurmr`
| Schur balanced model truncation |

`hankelmr`
| Hankel minimum degree approximation |

**Multiplicative Error Model Reduction Command**

Method | Description |

`bstmr`
| Balanced stochastic truncation |

**Additional Model Reduction Tools**

Method | Description |

`modreal`
| Modal realization and truncation |

`slowfast`
| Slow and fast state decomposition |

`stabsep`
| Stable and antistable state projection |