The problem is the following. I have to control a spaceship (assuming no gravity, no mass, no friction / no external forces). In other words, the first equation in this question leads to $0 = 0$. My goal is to reach a position for a given acceleration (i.e. throttle).
The movement equation (assuming a constant acceleration) is the following: $x(t) = 0.5 * a * t^2 + v0 * t + x0$ where $v0$ is the initial velocity and $x0$ the initial position ($x0 = 0$). Finally, I get $a = (goal - t*V0) / t^2$ where $goal= target - traveled_distance$
I put this equation in my system. There is a sensor to measure the velocity. Each time the velocity is measure, I can send a new acceleration command.
I got the following plots (the first one is the acceleration, the second one the velocity, and the last one the trajectory)
On the last plot, we can clearly identify the response of a second order system. I also observed that increasing $t$ reduces the overshoot (so I guess that $t$ is acting like a proportional gain).
My question is what would be the corresponding transfer function? I tried to identify it with a second order differential equation but did not get any convincing results.