# Why is it impossible to create an observer for this not fully observable system?

Consider a 1D point-mass moving along an axis. A force $u$ is applied as control. There is no gravity or other forces involved. The system can be described in state space equations as:

\begin{align} A &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \\ B &= \begin{bmatrix} 0 \\ 0 \\ \dfrac{1}{M} \end{bmatrix} \\ C &= \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \\ D &= [0] \end{align}

The shown system is controllable, but not observable. Not even structurally observable and most certainly not fully observable. Thus, it should be impossible to construct an observer for this system.

However, if I know the initial state of the system, I can compute the full state at every time, i.e. by integrating the system's output. How does this fall in line with the concept of observability? How would I incorporate the initial state into the equations?

I can't find the error in my train of thought, but I am certain there is one. Do I misunderstand observability?