# Kalman Filter Process Noise Covariance

I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car.

In a Kalman filter the matrix $A$ and process noise covariance $Q$ is what describes the system. I have chosen $A$ according to the following model:

\begin{align} \begin{bmatrix}p_{k+1} \\v_{k+1}\end{bmatrix} &= A \begin{bmatrix}p_k \\v_k\end{bmatrix} \\ \begin{bmatrix}p_{k+1} \\v_{k+1}\end{bmatrix} &= \begin{bmatrix}1 & \Delta t \\0 & 1\end{bmatrix} \begin{bmatrix}p_k \\v_k\end{bmatrix} \end{align}

where $p$ is position and $v$ is velocity.

My question is how to define the process noise covariance $Q$? The process details are unknown to me but they should be the same for any urban car movement. I have looked for articles that describe the process noise variance for forward and sidewise movement of a car but I haven't found anything.

I'm sorry if the question is not very mathematical but since Kalman filters are maths I figured someone here should have worked with car models in the past.

• Kalman filters we originally invented to help anti-aircraft guns track their targets. They relied on the first two derivatives of the planes position. Perhaps some of these historical examples can be changed from planes to cars. I would suggest l tracking them down. – Wintermute Jul 18 '16 at 16:14

## 1 Answer

Process noise can be viewed as any disturbance that affect the dynamics, for example steep on the roads, wind affects, friction on the roads etc. In general you can just say it as environmental disturbances.