I'm trying to derive the state space model for the following system: $L\ddot{\theta}-g\sin(\theta)-\ddot{z}\sin(\theta) = 0$
I'm allowed to use that $\sin(\theta) \approx \theta$. And I must set $x_1 = \theta, x_2 = \dot{\theta}$, input $u = \ddot{z}$, output $y = \theta$.
This is what I've come up with: \begin{equation*} \dot{x} = \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = Ax + Bu = \begin{bmatrix} 0 & 1 \\ \frac{g}{L} & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ \frac{\theta}{L} \end{bmatrix} u \end{equation*} \begin{equation*} y = Cx + Du = \begin{bmatrix} 1 & 0 \end{bmatrix}x \end{equation*}
But I can't have the $\theta$ in $B$, since that is equal to $x_1 = y = \theta$, right? How should I solve it?
Thanks in advance!