Kalman filter - question about matrices

Question

In order to familiarize myself with Kalman filter I decided to see how it work step by step. I would gladly use your help. Let's say I have a falling object with air resistance proportional to its velocity. Since my measurements are not perfect let's say that position measurement is gaussian with standard deviation of $15m$, so is velocity with deviation $5 \frac{m}{s}$. Equation for this system is $\ddot{x} = g - \frac{b}{m}\dot{x}$, where $b,m$ are constant. Now how do I find matrices $A$ and $B$. Do I simply convert to space-state form? Also, how do I determine the processing noise $\mathbf{v}$? And finally, how do I find covariance matrices of of processing($\mathbf{Q}$) and measurement ($\mathbf{R}$)? If you are kind enough to help me I would gladly ask you some more questions tomorrow.

Answer 1

According to the scenario you formulated, you only have measurement noise and no input noise. You are correct that you might need a state space model. For example,

$$ \begin{bmatrix}\dot{x} \\ \ddot{x}\end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & -\frac{b}{m}\end{bmatrix} \begin{bmatrix}x \\ \dot{x}\end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u + \begin{bmatrix} 0 \\ 1 \end{bmatrix} v, $$

$$ \begin{bmatrix}y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix}x \\ \dot{x}\end{bmatrix} + \begin{bmatrix}w_1 \\ w_2\end{bmatrix}, $$

with $u$ always equal to $g$, $v$ the input noise (which assumed to be zero), $y_1$ the measured position, $y_2$ the measured velocity and $w_1$ and $w_2$ zero mean Gaussian noise.

Since we did not take into account input noise, therefore you can also view it as a zero mean Gaussian noise with also zero (standard) deviation. When calculating the covariance matrices you can take the variance (standard deviation squared) on the diagonal, if all random variables are independent (which I assume $w_1$ and $w_2$ are). This would yield,

$$ Q = \mathbb{E}(v^2) = \begin{bmatrix}0 \end{bmatrix}, $$

$$ R = \mathbb{E}\left([w_1\ w_2]^T [w_1\ w_2]\right)= \begin{bmatrix}225 & 0 \\ 0 & 25 \end{bmatrix}. $$

But depending on whether you want to implement a Kalman filter in continues or discrete time, you might first have to discretize the state space model.