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# sos2ss

Convert digital filter second-order section parameters to state-space form

## Syntax

[A,B,C,D] = sos2ss(sos)
[A,B,C,D] = sos2ss(sos,g)

## Description

sos2ss converts a second-order section representation of a digital filter to an equivalent state-space representation.

[A,B,C,D] = sos2ss(sos) converts sos, a system expressed in second-order section form, to a single-input, single-output state-space representation:

$\begin{array}{ll}x\left(n+1\right)\hfill & =Ax\left(n\right)+Bu\left(n\right),\hfill \\ y\left(n\right)\hfill & =Cx\left(n\right)+Du\left(n\right).\hfill \end{array}$

The discrete transfer function in second-order section form is given by

$H\left(z\right)=\prod _{k=1}^{L}{H}_{k}\left(z\right)=\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}.$

sos is a L × 6 matrix organized as

$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right].$

The entries of sos must be real for proper conversion to state space. The returned matrix A is of size 2L × 2L, B is a 2L × 1 column vector, C is a 1 × 2L row vector, and D is a 1 × 1 scalar.

[A,B,C,D] = sos2ss(sos,g) converts to state space a system sos in second-order section form with gain g:

$H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right).$

## Examples

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### State-Space Representation of a Second-Order Section System

Compute the state-space representation of a simple second-order section system with a gain of 2.

```sos = [1  1  1  1   0  -1 ;
-2  3  1  1  10   1];
[A,B,C,D] = sos2ss(sos,2)
```
```A =

-10     0    10     1
1     0     0     0
0     1     0     0
0     0     1     0

B =

1
0
0
0

C =

42     4   -32    -2

D =

-4

```

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### Algorithms

sos2ss first converts from second-order sections to transfer function using sos2tf, and then from transfer function to state-space using tf2ss.