# Documentation

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# slowfast

Slow and fast modes decomposition

## Syntax

```[G1,G2] = slowfast(G,ns)
```

## Description

`slowfast` computes the slow and fast modes decompositions of a system G(s) such that

G(s) = [G1(s)] + [G2(s)]

`G(s)` contains the `N` slowest modes (modes with the smallest absolute value) of `G`.

$\left[{G}_{1}\left(s\right)\right]:=\left({\stackrel{^}{A}}_{11},{\stackrel{^}{B}}_{1},{\stackrel{^}{C}}_{1},{\stackrel{^}{D}}_{1}\right)$ denotes the slow part of G(s). The slow poles have low frequency and magnitude values.

$\left[{G}_{2}\left(s\right)\right]:=\left({\stackrel{^}{A}}_{22},{\stackrel{^}{B}}_{2},{\stackrel{^}{C}}_{2},{\stackrel{^}{D}}_{2}\right)$ denotes the fast part. The fast poles have high frequency and magnitude values.

The variable `ns` denotes the index where the modes will be split.

Use `freqsep` to separate slow and fast modes at a specified cutoff frequency instead of a specified number of modes.

## References

M.G. Safonov, E.A. Jonckheere, M. Verma and D.J.N. Limebeer, “Synthesis of Positive Real Multivariable Feedback Systems”, Int. J. Control, vol. 45, no. 3, pp. 817-842, 1987.