# Documentation

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

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

• Additive error bound:

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

Here, σi are denoted the ith Hankel singular value of the original system G.

• Multiplicative (relative) error bound:

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

Here, σi are denoted the ith 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

Main interface to model approximation algorithms

Normalized Coprime Balanced Model Reduction Command

Method

Description

Normalized coprime balanced truncation

Additive Error Model Reduction Commands

Method

Description

Square-root balanced model truncation

Schur balanced model truncation

Hankel minimum degree approximation

Multiplicative Error Model Reduction Command

Method

Description

Balanced stochastic truncation

Additional Model Reduction Tools

Method

Description

Modal realization and truncation

Slow and fast state decomposition

Stable and antistable state projection