How to get second derivative of state as a output of state-space model?

Question

Let's say modeling the free-falling object on state-space. Obviously, the equation of motion is a second order differential equation of the vertical position of the object. And one would normally set vertical position "x" and its derivative "x dot" as element of state to configure a state-space model. However, the output is only a linear combination of states(and input). In this case what would be the best option to get "acceleration" or second derivative "x double dot" as an output?

This is quite a basic question, but can't find a good example.

Edit)
Consider a falling ball with vertical force F and positional sensor y. How can I represent an additional acceleration sensor "y_2" on the model below?

$$\begin{matrix} \dot{x} \\ \ddot{x} \end{matrix} = \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} *\begin{matrix} x \\ \dot{x} \end{matrix} + \begin{matrix} 0 & 0 \\ 1/m & -1 \end{matrix}*\begin{matrix} F \\ g\end{matrix}$$ $$\begin{matrix} y \end{matrix} = \begin{matrix} 1& 0 \end{matrix}*\begin{matrix} x \\\dot{x} \end{matrix} + 0 *\begin{matrix} F \\ g\end{matrix}$$

Answer 1

An ideal acceleration sensor can be modeled using the original equation itself.

$$ a = Cx+ Du = \begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} x\\ \dot{x} \end{pmatrix} + \begin{pmatrix} 1/m & -1 \end{pmatrix} \begin{pmatrix} F\\g \end{pmatrix} $$

Attach the above C and D matrices as the second row of the corresponding matrices in your original output equation. The original output equation represented an ideal position sensor. Now the augmented output equation represents a position sensor (Ist row) and an acceleration sensor (IInd row).

$$ y_1 = \begin{pmatrix} y\\ a\end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix} \begin{pmatrix} x\\ \dot{x} \end{pmatrix} + \begin{pmatrix} 0 & 0\\ 1/m & -1 \end{pmatrix} \begin{pmatrix} F\\g \end{pmatrix} $$

The above is for an ideal sensor. If you want to estimate the acceleration, you may need an estimator which uses the measured states and the inputs and outputs an estimate. If you want to model a non ideal sensor, you will need the above equations as well as additional states and inputs to model the bandwidth, bias, noise etc. of the non ideal sensor.