# Kalman Filter application to non-linear system.

I want to use the Kalman filter to have a better estimate of the state of a system which I know its equations of motion:

$\ddot{\theta}=\frac{-Mr\dot{\theta}^2sin(\theta)cos(\theta)-(m+M)gsin(\theta)}{R(M+msin(\theta)^2}$

Well as you see the system is far from linear. I could make an approximation of the trigonometric functions to a linear function when $\theta$ is close to 0 but I want to use the whole domain from $0$ to $2\pi$.

Can I use the Kalman filter in this situation? Should I use a different Kalman filter for different ranges of $\theta$ having each filter a different linear approximation of the trigonometric functions? What should be my A matrix?

$\mathbf{x}^{n|n}=\mathbf{A}\mathbf{x}^{n|n-1}+\mathbf{\epsilon_x}$

with $\mathbf{x}=\begin{bmatrix} \theta\\ \dot{\theta}\\ \ddot{\theta} \end{bmatrix}$

• You may want to check out the Extended Kalman Filter, which is a generalization of the Kalman Filter for nonlinear systems. Commented Mar 16, 2014 at 8:32
• However, if you want the filter to work in the whole range of the nonlinearity, chances are that the extended Kalman filter will not give very good results, because it is based on linearizations. There exist alternative techniques such as particle filters and things called unscented filters but the who area of nonlinear estimation is still a wide.open research problem.
– Pait
Commented Mar 16, 2014 at 18:24