Using Kalman Filter With Acceleration And Position Inputs

I would like to use a Kalman filter to estimate the height and vertical velocity of an object being moved up and down in an unknown way based on a noisy position measurement and a

Solution 1:

Regarding your questions:

1. Regarding Q - The process noise - Since the acceleration changes unpredictably from step to step, Q should represent the standard deviation of the acceleration. I would use something like [0 0 0;0 0 0;0 0 sigma^2]. You should try to use the data you have to estimate how much the acceleration changes from step to step.
2. If you measure the position and acceleration at different times, you should use 2 z vectors, thus 2 different H matrices to update your model. When an acceleration measurement arrives, you update your model with the H matrix that relates to that measurement, and the same for the position measurement. In each step, dt should represent the time since the last update of the model.
3. Your guess for R seems right.