I have this dynamic system
$$ J \ddot{\theta} + F\dot{\theta} = u $$
I would like to acquire the state space of the system. This is what I've done $$ x_{1} = \theta, \\ x_{2} = \dot{\theta}, \\ x_{3} = \ddot{\theta} \\ \dot{x_{1}} = x_{2} \\ \dot{x_{2}} = \ddot{\theta} = \frac{1}{J}u - \frac{F}{J} x_{2} \\ \begin{bmatrix} \dot{x_{1}} \\ \dot{x_{2}} \end{bmatrix} = \underbrace{ \begin{bmatrix} 0 & 1 \\ 0 & -\frac{F}{J} \end{bmatrix}}_{A} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} + \underbrace{ \begin{bmatrix} 0 \\ \frac{1}{J} \end{bmatrix}}_{B} u $$
Is it correct? I've tried to double check my work using Laplace transform. $$ G = \frac{Y(s)}{U(s)} = \frac{1}{J s^{2} + Fs} $$
This is the code
>> syms J F
>> b = 1;
>> a = [J, F, 0];
>> [A, B, C, D] = tf2ss(b,a)
A =
[ -F/J, 0]
[ 1, 0]
B =
1
0
C =
[ 0, 1/J]
D =
0
Are the results same? Why B in Matlab has no 1/J?
You have done your calculations correctly and the results are the same. It looks like different because there are infinitely many state space representation of a system, depending on the choice of state variables.
However, your state space representation is incomplete. Because your system does not have an "output" where you can select a linear combination of the states, using a $C$ matrix, i.e. $y=Cx$.
You can put $1/J$ to either $B$ or $C$ matrix. It is the difference between your transfer function and
$$\frac{1/J}{s^2+(F/J)s}$$