# Use MatLab to convert from state space to transfer function (symbolic)

This script requires the MatLab's symbolic toolbox

```% Start by clearing everything
clear all
clc
```

Declare all symbolic variables

```syms a3 a2 a1 s b0 b1
```

Define state space system

```A=[0 1 0; 0 0 1; -a3 -a2 -a1];
B=[0; 0; 1;];
C=[b1 b0 0];
D=0;
```

Find State Transition Matrix transition matrix

```Phi=inv(s*eye(3)-A)
```
```
Phi =

[ (s^2 + a1*s + a2)/(s^3 + a1*s^2 + a2*s + a3),     (a1 + s)/(s^3 + a1*s^2 + a2*s + a3),   1/(s^3 + a1*s^2 + a2*s + a3)]
[               -a3/(s^3 + a1*s^2 + a2*s + a3), (s*(a1 + s))/(s^3 + a1*s^2 + a2*s + a3),   s/(s^3 + a1*s^2 + a2*s + a3)]
[           -(a3*s)/(s^3 + a1*s^2 + a2*s + a3), -(a3 + a2*s)/(s^3 + a1*s^2 + a2*s + a3), s^2/(s^3 + a1*s^2 + a2*s + a3)]

```

Find transfer function

```H=C*Phi*B+D
```
```
H =

b1/(s^3 + a1*s^2 + a2*s + a3) + (b0*s)/(s^3 + a1*s^2 + a2*s + a3)

```
```%Display
pretty(simple(H))
```
```
b1 + b0 s
----------------------
3       2
s  + a1 s  + a2 s + a3
```