Problem

How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system

$$ \dot{x}=ax\tag{1}\label{1}$$

which has a state space representation given by

$${\bf\dot{x}}={\bf A}{\bf x} \tag{2}\label{2}$$

one can introduce the integrated state by augmenting the ${\bf A}$ matrix above by writing

$$\frac{d}{dt}\int xdt=x \tag{3}\label{3}$$

and taking $\int xdt$ to be the augmented state. This gives the following augmented state space model:

$$\frac{d}{dt}\left[\matrix{x \cr \int xdt} \right]=\left[\matrix{a & 0\cr 1 & 0}\right]\left[\matrix{x \cr \int xdt} \right] \tag{4}\label{4}$$

Solution Attempt

I can take the derivative of $\ref{1}$ to get

$$\ddot{x}=a\dot{x} \tag{5}\label{5}$$

This gives the following augmented state space model:

$$\frac{d}{dt}\left[\matrix{\dot{x} \cr x} \right]=\left[\matrix{a & 0\cr 0 & a}\right]\left[\matrix{\dot{x} \cr x} \right] \tag{6}\label{6}$$

But this seems wrong because now I have essentially the same equation twice... So I tried substituting $\ref{1}$ into $\ref{5}$ yielding

$$ \ddot{x}=a^2x \tag{7}\label{7}$$

This gives the following augmented state space model:

$$\frac{d}{dt}\left[\matrix{\dot{x} \cr x} \right]=\left[\matrix{0 & a^2\cr 0 & a}\right]\left[\matrix{\dot{x} \cr x} \right] \tag{8}\label{8}$$

I think somehow what I am doing above is incorrect, because if the initial state of $x$ is zero, the system remains static.

Any help greatly appreciated and thanks in advance!!