Sorry for what might be a long post, I want to give background.
Initially I had regular Kalman filter, and the state model was defined by Newtonian kinematics, with initial position 0 and speed of 2. I was tracking position (x) and velocity (v), i.e. my state vector is $\begin{bmatrix} x & \dot x \\ \end{bmatrix}^T$:
$$x = x_0 + v_0t$$ $$v = v_0$$
This resulted in a State Transition Matrix: $$ \begin{bmatrix} 1 & \Delta t\\ 0 & 1\\ \end{bmatrix}$$
Now I am trying to implement Extended Kalman Filter. I have given system Acceleration of 2, so that equations go like this together with plugging in initial speed and acceleration:
$$x = x_0 + v_0t + \frac12at^2 \rightarrow x = 2t + t^2$$
$$v = v_0 + at \rightarrow v = 2 + 2t$$
Now I need to find Jacobian Matrix with respect to my state vector and I understand what it is, however, I do not understand, how do I find my State Transition Matrix, if equations that I have are expressed in terms of time and not in terms of the state variables. I assume that first line in State Transition Matrix remains the same, since position changes in same way, i.e: $$position = previous\,position + \Delta time * speed\, over\, that \, time\, period$$ It is the speed that is changing. But I don't know how to define it in my State Transition Matrix. From what I understand, my STM will be different every epoch, am I right? As I said, I know that I need to find Jacobian and know what it is, but I don't know how to find it in this particular case.
Thank you.
