# How to treat noise (in the acceleration) in a Kalman filter when tracking position?

I have an equation of motion for which I am wanting to track the position.

The equation looks like this:

$$\ddot{x}(t) = -a\dot{x}(t) - bx(t) + F(t)$$

where $F(t)$ is an external Stochastic force noise (like brownian motion).

I know if $F = 0$ I can build a Kalman filter like so:

$$\begin{bmatrix} \dot{x} \\ \ddot{x} \\ \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -b & -a \\ \end{bmatrix} . \begin{bmatrix} x \\ \dot{x} \\ \end{bmatrix}$$

And then transform $\begin{bmatrix} 0 & 1 \\ -b & -a \\ \end{bmatrix}$ like so:

\begin{align} A(t) & = \mathscr{L}^{-1}\left\{\left(s\vec{I} - \begin{bmatrix} 0 & 1 \\ -b & -a \\ \end{bmatrix}\right)^{-1}\right\}\\ & = \mathscr{L}^{-1}\left\{\left(\begin{bmatrix} s & 0 \\ 0 & s \\ \end{bmatrix} - \begin{bmatrix} 0 & 1 \\ -b & -a \\ \end{bmatrix}\right)^{-1}\right\}\\ &= \mathscr{L}^{-1}\left\{\left(\begin{bmatrix} s & -1 \\ b & s+a \\ \end{bmatrix}\right)^{-1}\right\} \end{align}

... (at this point I get stuck as I am using Sympy in python to calculate the inverse Laplace transform and it fails for this matrix).

But I want to know how I should treat this Stochastic effect in a Kalman filter.

• Since $F(t)$ is not a deterministic input but a random driving force, I would first try to use a process noise matrix in the computation of the Kalman gains during the Riccati iterations.... Have you tried it? – Fat32 Jan 26 '17 at 22:12
• No, I haven't, I'm only just beginning to work on Kalman filters. I'll have a look into how to implement your suggestion, thanks for your input. – SomeRandomPhysicist Jan 26 '17 at 22:20
• I suggest you the following book for nice introduction to Kalman filtering: Fundamentals of Kalman Filtering_Practical Approach_ZARCHAN the link: amazon.com/… – Fat32 Jan 26 '17 at 22:26